ELECTRICITY,* SOUND, AND LIGHT MILLIKAN AND MILLS - tr* GIFT OF 't.P. Lewis A SHORT UNIVERSITY COURSE IN ELECTRICITY, SOUND, AND LIGHT BY EGBERT ANDREWS MILLIKAN, PH.D. ASSOCIATE PROFESSOR OF PHYSICS IN THE UNIVERSITY OF CHICAGO JOHN MILLS, A.M. INSTRUCTOR IN PHYSICS IN THE WESTERN RESERVE UNIVERSITY GINN & COMPANY BOSTON NEW YORK CHICAGO LONDON $ G u. ^ M r-y PHYSICS DEPt. PHYSICS DBPT. ENTERED AT STATIONERS' HALL COPYRIGHT, 1908, BY ROBERT A. MILLIKAN AND JOHN MILLS ALL RIGHTS RESERVED 78.9 GINN & COMPANY PRO- PRIETORS BOSTON U.S.A. PEBFACE This book represents primarily an attempt to secure a satisfac- tory articulation of the laboratory and class-room phases of instruc- tion in physics. It is an outgrowth of the conviction that in courses of intermediate grade in colleges, universities, and engineering schools a real insight into the methods of physics, and a thorough grasp of its foundation principles are not readily gained unless theory is presented in immediate connection with such concrete laboratory problems as are calculated to give the student a sound basis for intelligent theoretical work. Nevertheless the book is intended to be much more than a lab- oratory manual. It represents an attempt to present a complete logical development, from the standpoint of theory as well as ex- periment, of the subjects indicated in the title. It is designed to occupy a half year of daily work, two hours per day, in either the freshman, sophomore, or junior years of the college or technical- school course. In the University of Chicago about one half of this time is devoted to class discussions, lecture-table demonstrations, quizzes, and problems, and the remainder to laboratory work. The course is preferably preceded by a similar course in mechanics, molecular physics, and heat, the two courses together constituting a year's work in college physics. The method of treatment is throughout analytical rather than descriptive, although no mathematics beyond trigonometry is presupposed. It is assumed that the student has already had a beginning course in descriptive physics in the high school or elsewhere. Most of the apparatus required is of the stock sort found in all moderately well-equipped college laboratories. A few special pieces have been designed (Figs. 60, 99, etc.) where for one reason or another existing forms seemed ill adapted to the needs of the iii 65423 iv ELECTRICITY, SOUND, AND LIGHT course. In the University of Chicago the apparatus is not, in gen- eral, used in duplicate. Nine or ten experiments are commonly kept going at once, two pupils working together. The classes are limited to twenty-five. The authors' thanks are due to Leeds & Northrup for the cut of their post-office-box bridge (Fig. 51 &), to Queen & Co. for the origi- nal of their tangent galvanometer (Fig. 22), to the American In- strument Company for the original of their voltmeter (Fig. 40), and to William Gaertner & Co. for the originals of the magnetometer (Fig. 14), the voltameter (Fig. 27), the electric calorimeter (Fig. 39), the ballistic galvanometer (Fig. 60), the earth inductor (Fig. 107), the ideal dynamo (Fig. 99), and the spectrometer (Fig. 194). E. A. M. UNIVERSITY OF CHICAGO J- M. CONTENTS CHAPTER PAGE I. MAGNETIC AND ELECTRIC FIELDS OF FORCE .... 1 II. DETERMINATION OF THE STRENGTHS OF MAGNETIC FIELDS AND MAGNETIC POLES 17 III. MEASUREMENT OF ELECTRIC CURRENTS 28 IV. MEASUREMENT OF POTENTIAL DIFFERENCE .... 46 V. MEASUREMENT OF RESISTANCE 58 VI. TEMPERATURE COEFFICIENT OF RESISTANCE ; SPECIFIC RESISTANCE 67 VII. GALVANOMETER CONSTANT OF A MOVING-COIL GALVA- NOMETER 77 VIII. ABSOLUTE MEASUREMENT OF CAPACITY 90 IX. COMPARISON OF CAPACITIES, DETERMINATION OF DIE- LECTRIC CONSTANTS, AND THE RATIO OF THE ELEC- TROSTATIC AND ELECTRO-MAGNETIC UNITS . . . 100 X. ELECTROMOTIVE FORCE AND INTERNAL RESISTANCE . 112 XL COMPARISON OF ELECTROMOTIVE FORCES 121 XII. ELECTRO-MAGNETIC INDUCTION 126 XIII. CONSTANTS OF THE EARTH'S MAGNETIC FIELD . . . 141 XIV. SELF-INDUCTION 146 XV. MAGNETIC INDUCTION IN IRON 158 XVI. ELECTROLYTIC CONDUCTION 176 XVII. VELOCITY OF SOUND IN AIR 187 XVIII. MUSICAL PROPERTIES OF AIR CHAMBERS 201 XIX. LONGITUDINAL VIBRATIONS OF RODS 213 XX. WAVES IN STRINGS 222 XXI. DIFFRACTION OF SOUND AND LIGHT WAVES . . . . 238 XXII. DIFFRACTION GRATING 258 XXIIL REFRACTION OF LIGHT 270 XXIV. TOTAL REFLECTION 285 XXV. PHOTOMETRY 292 XXVI. DISPERSION AND SPECTRA 302 XXVII. POLARIZED LIGHT 316 XXVIII. RADIO-ACTIVITY 333 PROBLEMS 354 APPENDIX OF TABLES 367 INDEX 385 v ELECTRICITY, SOUND, AKD LIGHT CHAPTEE I MAGNETIC AND ELECTRIC FIELDS OF FORCE 1. Quantity of magnetism. It is well known that under cer- tain circumstances bars of iron and of some other metals, when suspended so as to be free to rotate about a vertical axis, turn so as to point north and south. A body which possesses the property of so doing is called a magnet. If the two north-seeking ends of two such magnets are brought near together, they are found to repel each other. The same is true of the two south-seeking ends. But a north-seeking and a south-seeking end are found to attract each other. On account of these opposite characteristics the north- seeking ends of magnets are called N poles, the south-seeking, 8 poles. The above facts may then be stated in the general law : Magnetic poles of like kind repel one another, of unlike kind attract one another. Different magnets of the same length placed at a given distance from a suspended magnet are observed to exert different forces upon it. The quantities of magnetism in the poles, or the pole strengths, are then arbitrarily taken as proportional to the forces exerted. That is, the force which a given pole M exerts, at a given distance, upon some standard pole is taken as the meas- ure of the number of units of magnetism in M. It may then be proved experimentally, by the method used in Chapter II, that the force / exerted between any two poles M and m is directly proportional to the product mM and inversely proportional 2 : ELiGTMCLTY, SOUND, AND LIGHT to :the i ^sqiiar^of tihei distance r between them. The algebraic statement of this experimental relation is (1) in which A; is a factor of proportionality depending upon the choice of the unit of magnetism and upon the medium through which the force acts. It has been decided to choose this unit so that k equals unity for air ; that is, the equation / = mM/r 2 , applied to air, contains the definition of unit magnetic pole. Thus if two magnets are chosen for which m = M , and if r is taken equal to 1 cm., then m is by definition unity if / is found to be equal to 1 dyne. In other words, a unit magnetic pole is a pole of such strength that when placed at a distance of 1 cm. from an equal pole it repels or attracts it with a force of 1 dyne. 2. Quantity of electricity. It is also equally well known that when a glass rod has been rubbed with silk it attracts a pith ball, but after contact with the pith ball, repels it. Similarly, when ebonite has been rubbed with cat's fur it attracts a second pith ball, but after contact with the pith ball, repels it. Furthermore, the pith ball which has touched the rubbed glass and is repelled by it is attracted by the ebonite, while the ball which is repelled by the ebonite is attracted by the glass. On account of this behavior the pith balls are said to have been electrified, or to have received charges of electricity ; and on account of the opposite characteristics of these charges the one is called positive and the other negative. These charges of electricity can be produced in other ways, but in every case it has been decided to call a charge positive when it is repelled by a glass rod which has been rubbed with silk, negative when it is repelled by an ebonite rod which has been rubbed with cat's fur. The above facts may then be stated in the general law : Electrical charges of like sign repel one another, of unlike sign attract one another. It need scarcely be said that in adopting these conventions and in setting up this law no assumption whatever has been made regarding the nature of electricity. It has merely been agreed to call a body electrified, or charged MAGNETIC AND ELECTRIC FIELDS OF FORCE 3 with electricity, which behaves toward pith balls or other objects as does the rubbed glass or the rubbed ebonite. Definitions precisely similar to those used in the quantitative study of magnetism are adopted also in the quantitative study of electricity. Thus if q and Q are two electric charges the magni- tudes of which are measured by the forces which they exert upon a third charge at a given distance from it, then it can be proved experimentally that As in magnetism, so in electricity, the unit of quantity is chosen so that for action between charges separated by air /=le which in the old one-fluid theory was assigned to ordinary matter. 8. Fields of force. Afield of force is simply a region in which force exists. It may be a magnetic, electrical, or gravitational field which is under consideration. The strength of field at any point in such a region is the number of units of force which unit quan- tity (be it mass, pole strength, or charge) experiences at the point considered. Thus the strength of field at a point 1 cm. distant from a unit pole (conceived as concentrated at a point) is unity. Unit field is, then, a field in which unit quantity experiences 1 dyne of force. For example, the strength of magnetic field at a given point in space is ten units if unit pole experiences 10 dynes of force when placed at this point. The direction of a gravitational field at any point is defined as the direction in which a small quantity of matter would tend to move if placed in the field at the point considered. The direction of a magnetic field is defined as the direction in which an isolated N pole would move. The direction of an electric field is defined as the direction in which an isolated positive charge of electricity would move. A line of force in any one of these fields is the direction in which a free mass, a free N pole, or a free positive charge would move if it had no inertia. It is convenient and customary, how- ever, to conceive of as many lines drawn across any square centi- meter taken at right angles to the direction of the force as the field possesses units of strength at the point considered. The line of force then becomes the unit of field intensity or strength ; that is, in a gravitational field a line means a field strength such that MAGNETIC AND ELECTRIC FIELDS OF FORCE 9 a force of 1 dyne acts on every gram placed in it. Thus, since the earth's field strength at the surface is 980 dynes, it is cus- tomary to consider 980 lines of gravitational force as piercing each square centimeter of the earth's surface. In magnetic fields, in connection with which the convention is most commonly used, a line means a field strength such that a force of 1 dyne acts on each unit pole. It has received the special name of a gauss. In electrical fields the more usual term is tube of force, and the con- ception is that as many tubes of force cross any square centimeter at right angles to the direction of the field as there are units in the field strength at that point. With these conventions it is evident that a uniform field is represented by a system of parallel lines of force, and conversely, that where the lines of force are parallel, the field has everywhere the same strength. A convergent system of lines represents a field of increasing strength; a divergent system of lines repre- sents a field of decreasing strength. It need scarcely be said that a line of force has no objective reality. The representation of fields of force by lines is a matter of convenience only. 9. Gravitational potential. The term "potential" was first used in connection with gravitational forces. Potential is a char- acteristic of a point . in space, not, in general, of . tf a body. It may be looked upon merely as an abbreviation for the expression "the potential energy of unit mass at the point considered." Thus at a point a above the surface of the earth (Fig. 3) a gram of mass possesses a certain po- tential energy with reference to a point 6 on the surface of the earth. This potential energy is the amount of work that the gram of mass can do in f ailing to the earth. Since the term " potential " is merely an abbreviation for the potential energy of unit mass, ic is evident that potential is measured in energy or work units. The potential of a point is unity when it requires one erg of work to bring unit mass from the point which is taken as the zero of potential up to the point considered. The potential energy of unit mass at the point a (see Fig. 3) is greater if the point c, below the 10 ELECTRICITY, SOUND, AND LIGHT surface of the earth, is taken as the point of reference instead of the point b. Thus it is evident that some point must be chosen arbitrarily from which to reckon the potential of other points. A point which is infinitely distant from all attracting or repelling bodies is taken as the point from which absolute potential is reckoned, i.e. such a point is taken as the absolute zero of potential. The abso- lute potential, then, of any point in the universe is the number of ergs of work which must be done (by some outside agent) to bring unit mass from this absolute zero up to the point considered. In general, we are more concerned with the difference in potential (usually written P.D.) between two points than with the absolute potentials of the points. The P.D. between two points is then defined as the amount of work which the external agent must do in order to carry the unit mass from the point of the lower to that of the higher potential ; or, what amounts to the same thing, the P.D, is the amount of work which the acting force does in carrying unit mass from the point of higher to that of lower potential. Now the force of gravitation is always an attractive, never a repellent, force. For all points which are at a finite distance from any astronomical body the work which the external agent must do in bringing unit mass from an infinite distance to any point is therefore less than nothing. That is, the work is done not by, but against, the action of the external agent. From the definition of absolute potential, as given above, it is evident that the gravitational potential of all points within a finite distance of any astronomical body must be negative. 10. Magnetic potential. The above definitions hold almost without change for magnetic forces, save that it is necessary to specify whether the unit quantity is an N pole or an 8 pole. The magnetic potential of a point is defined as the amount of work which an external agent must do against the existing magnetic field in order to bring a unit N pole from infinity up to the point considered. Thus the potential in the immediate neighborhood of an N pole is evidently positive, because the unit N pole is repelled, and hence the external agent must do work in order to bring up the pole from infinity. It is equally evident that the potential in the immediate neighborhood of an 8 pole is negative. MAGNETIC AND ELECTRIC FIELDS OF FORCE 11 11. Electrical potential. Similarly, the electrical potential of a point is the amount of work required to bring unit positive charge from infinity up to the point considered. Points in the neighbor- hood of a positive charge have therefore a positive potential ; those in the neighborhood of a negative charge have a negative potential. These definitions apply as completely in the study of so-called current electricity as in that of static electricity. Under all circumstances the term P.D. means the amount of work required to carry unit positive charge between the two points considered. 12. Equipotential surfaces. An equipotential surface is the locus of a system of points all of which have the same potential. Thus the gravitational equipotential surfaces about the earth are approximately spherical surfaces concentric with it, because the amount of work required to bring unit mass to within a certain distance of the center of the earth is the same, no matter from what side the earth is approached. Now it can be shown that the direction of the field of force at any point is perpendicular to the equipotential surface passing through that point. In order to prove this statement it is first necessary to show that the work done hi carrying a body between any two points is independent of the path chosen. If a body is carried from b to a over the path Ida (Fig. 4) a certain amount of work w is done upon it. Now suppose the work of the path Ida is less than that of the path lea. Then when the body returns to b over the path acb it will give up a certain amount of energy w' , i.e. it will do a certain amount of work w'. As a net result of the operation we have expended an amount of work w and have received back a larger amount of work w', and yet have brought everything back to the initial condition. We have therefore created an amount of energy w 1 w. But according to the doctrine of the conservation of energy this is impossible. Hence w' cannot be greater than w. By reversing the operation it can be proved that w cannot be greater than w'. That is, the work of the path Ida is equal to that of any other path lea. 12 ELECTRICITY, SOUND, AND LIGHT Now let nop (Fig. 5) represent any equipotential surface. By the definition of such a surface the work required to move a unit body from any arbitrary zero, say m, over the distance mn is the same as that required to move it over the dis- tance mo. But by the proposition just proved the work of the path mn is equal to the work of the path mo plus on. Hence the work corre- FIG 5 spending to the path on must be zero. If it requires no work to move a body over an equi- potential surface, the existing force can have no component along that surface, i.e. the force must be everywhere normal to an equipotential surface. 13. Potential of a conductor in electrical equilibrium. That the electrical potentials of all points on or within a conductor in the static condition must be the same follows at once from the fact of conductivity. For as soon as a conductor ab (Fig. 6) is brought into the field of a positively charged body c the negative electricity within ab at once moves toward b and a positive charge appears at a until further movement of negative toward b is checked by the action of the negative accumulated at b and the positive . accumulated at a. It is obvious, then, that there can be electrical \ equilibrium within a conductor \only when all electrical forces _ C _ _j_ j within the conductor have been /"*/ ^- reduced to zero. No electrical force, then, can exist within a conductor in the static con- , , FIG 6 ditioii; hence no work can be required to move unit charge from one point to another within the conductor. It follows that all points within or upon a conductor in the static condition must have the same potential. The above reasoning holds as well for hollow as for solid conductors, provided no insulated charged bodies exist within the hollow portion. The experimental verification of these conclusions was first made by Faraday, who covered a large box with tin foil and went inside with very delicate electroscopes. These remained wholly unaffected even when powerful electrical disturbances took place MAGNETIC AND ELECTRIC FIELDS OF FORCE 13 just outside the box. In laboratory practice it is now customary to screen delicate electrostatic instruments from external disturb- ances by surrounding them with sheet metal or wire gauze. Since, then, all points on a conductor have the same potential, it has become customary to speak of the potential of the con- ductor rather than of the potential of points on the conductor, in spite of the fact that the term "potential" is one which in strictness characterizes points in space rather than bodies. Further, since the surface of a conductor in electrical equilibrium is always an equipotential surface, it follows that electrical lines of force always enter or leave a conductor normally to the surface. 14. Mapping equipotential surf aces. It is evident from section 12 that if it is possible to map the direction of the lines of force in any field, it must also be possible to map out the equipotential surfaces in that field. Thus, suppose it to be required to find the intersections with the plane of the paper of the equipotential sur- faces about two isolated spheres whose centers are in the plane of FIG. 7 the paper. A little consideration of the field of force about two such spheres will show that the general shape of the lines of force is that represented by the full lines of the sketch (Fig. 7). By drawing lines everywhere perpendicular to these force lines the equipotential lines represented by the dotted lines of the figure are obtained. 14 ELECTRICITY, SOUND, AND LIGHT EXPERIMENT 1 (A) Object. To map completely a somewhat complicated magnetic field, and to draw the equipotential lines. Directions. The field here plotted will be the combination of the field of the earth and the fields of two magnets. Two straight bar magnets (about one foot long) are to be placed vertically in a box the top of which (about three feet square) is shown in Figure 8. The magnets pass through the holes a and b and their upper ends are flush with the surface of the box. First see that the line connecting a and b is in the magnetic meridian ; then, for the first exercise, place at a a north pole and at & a south pole. With brass pins attach to the box a sheet of paper large enough to cover the entire top. The direction in which the north pole of a short compass needle points when the compass is placed in a magnetic field represents the direction of the magnetic force at the center of the compass needle ; for the two poles are always under the' action of forces which tend to move them in opposite directions along the line connecting them, and which would so move them if they could be de- tached from one another. Hence the following method of plotting the direction of the magnetic lines at once suggests itself. Set the needle at some point in the field and with a sharp pencil place two dots, e.g. 1 and 2 (Fig. 8) directly under the extremities of the needle. Move the needle along until the pole which stood over 1 now stands directly over 2, and make a new dot S to indicate the position of the other pole. Proceed thus until the line either runs into a pole or leaves the paper. In this way plot enough lines to show a complete outline of the field. Plot with especial care the lines which pass near the singular points, i.e. points in which the needle will assume no definite position. As a second exercise plot the field produced when both a and b are north poles. When the outline of these fields has been obtained estimate the FIG. 8 MAGNETIC AND ELECTRIC FIELDS OF FOECE 15 directions of the intermediate lines wherever it is perfectly evident what their general course must be, and thus fill in the field thickly with force lines. Then upon each of the two fields draw, in red ink, a. second system of lines which is everywhere roughly perpendicular to the first system, -.i.e. draw the equipotential lines. (B) Object. To map the equipotential surfaces of an electrical field, and to draw the force lines. Directions. In Figure 9 is shown a tray on the glass bottom of which is pasted a sheet of coordinate paper. The tray is filled to a depth of three or four millimeters with a solution of ammonium chloride or of any con- venient salt. A and B are the two points which are to be maintained at different potentials and about which it is desired to find the equipotential lines which lie in the plane of the liquid. If both of the terminals of an FIG. 9 instrument for detecting an electrical current lie on the same equipotential surface, no current passes through the instrument. If, however, the two terminals lie on surfaces of different potentials, a charge is urged through the instrument from the point of higher to that of lower potential, and the instrument indicates the fact. If one terminal be fixed and the other moved to various points of the field, the equipotential surface on which the first terminal lies will pass through all the positions of the second for which no current flows through the instrument. The telephone T is used for detecting a difference of potential between its terminals C and D. 16 ELECTRICITY, SOUND, AND LIGHT Since the telephone receiver will respond by buzzing only as the result of an intermittent or alternating difference of potential, it is necessary that A and B be maintained at a constantly varying P.D. But whatever the P.D. between A and B the equipotential lines will preserve the same con- figuration, although, of course, assuming different absolute values. This varying P.D. is obtained by connecting A and B to the secondary S of a small induction coil, to the primary P of which is connected a storage battery or dry cell E. Assume two lines on the coordinate paper as axes and locate points from these. Connect A and B as described and set them at two points about 20 cm. apart. Set the electrode C about 1 cm. from A, toward B. With the electrode D locate enough points for which there is no buzzing in the receiver to plot the equipotential line through the point at which C is placed. Move C to a position about one sixth of the remaining dis- tance toward B and repeat. Continue, moving C another sixth, until the whole field has been explored. Plot on coordinate paper to one half scale the lines so found. The lines of force will form a system everywhere at right angles to these equipotential lines. Draw this system in red ink. CHAPTEK II THE DETERMINATION OF THE STRENGTHS OF MAGNETIC FIELDS AND OF MAGNETIC POLES 15. Every magnet possesses equal amounts of N and S magnetism. The preceding chapter dealt only qualitatively with magnetic and electrical phenomena. The present chapter has to do with the exact measurement of magnetic quantities. One of the most important of these quantities is the so-called magnetic moment of a magnet. In order to gain a clear conception of the meaning of this term it is first desirable to consider why we believe that the N and S poles of any particular magnet are of exactly equal strength. The earth behaves like a huge magnet, one pole of which is situated near the north geographical pole, the other near the south geographical pole. According to our convention the former of these is an $ pole, the latter an N pole. Since both of the poles of the earth are very remote in comparison with the length of any ordinary magnet, it follows that if the north and south poles of a suspended magnet are of equal strength, then the attraction which either of the earth's poles exerts upon one of the ends of the suspended magnet cannot be sensibly different from the repul- sion which this same pole exerts upon the other end of the magnet. Conversely, if the attraction which the northern pole of the earth exerts upon the N pole of a magnet is found to be equal to the repulsion which it exerts upon the S pole, then these two poles must be of equal strength. A similar method of reasoning may be applied to the action of the southern pole of the earth. Now experiment shows that a floating magnet experiences no motion of translation toward north or south, but only a motion of rotation about a vertical axis. Since this experiment may be performed with any magnet, it follows that the N and S poles of a magnet are always of equal strength. 17 18 ELECTRICITY, SOUND, AND LIGHT m' 16. Magnetic moment. The moment of force tending to pro- duce rotation in a magnet placed at right angles to the lines of a uniform field, such as that due to the horizontal component of the earth's magnetic field, is the prod- uct of the strength of the field and the quantity known as the magnetic moment of the magnet. Magnetic mo- ment is defined as the product of the pole strength m and the distance I between the poles of the magnet. The application of the term "magnetic mo- ment" to this quantity is appropriate, because if the magnet were placed in a field of unit strength, in which case the force acting on each pole would be m dynes, and were placed at right angles to the direction of the field, as in Figure 10, the moment of force M acting to turn it into the direction of the lines of force would be given by the equation M=ml (1) 17. Determination of the strength of a magnetic field. The particular field for which we shall now outline a method of deter- mining the strength will be the horizontal component of the earth's magnetic field. The measurement of this field strength is intimately connected with the determination of the magnetic moment of a magnet. Neither the field strength H nor the mag- netic moment M of the magnet used can be determined directly by a single experiment ; but two experiments can be performed, each of which gives a relation between M and H. From these two relations both M and H can be found. One experiment consists in suspending a large magnet mm 1 of unknown moment M in the field of which the strength H is sought, and observing the period of its oscillation. This gives an expression for MH, as will be shown in section 20. The other experiment consists in placing the same magnet mm' of magnetic moment M in a position due east or west, as in STRENGTHS OF MAGNETIC FIELDS AND POLES 19 Figure 11 (p. 20), from a very small suspended magnet be w r hich hangs in a horizontal position in the earth's field. The suspended magnet be is then acted upon simultaneously by two fields' which are at right angles to each other : first, the earth's field H which exerts a moment, or couple, tending to swing the needle into a north-and-south position ; and second, the magnetic field F due to the magnet mm' winch exerts a couple tending to swing the needle into an east-and-west position. From the position of equi- librium assumed by the small magnet under the action of these two fields, an expression for M/H is Obtained, as will be shown in section 19. But in order to obtain this expression, the strength of the field F must first be found in terms of the magnetic moment M of the large magnet. 18. Strength of field due to a magnet. The strength of the field produced at o (Fig. 11) by the magnet mm 1 is by definition the force which this field exerts at that point upon unit magnet pole. If, then, r is the distance from the middle of the magnet mm' to the suspended needle, the length of which is very minute in comparison with the distance r, then by the law of force given in section 1 the force which the. pole m exerts upon unit pole at the given distance, namely (r l/2), is /79~V Tne otner Pl e m ' exerts an equal and opposite force of - dynes. Since m = m 1 , the resultant force, that is the field strength F, is given by the equation F = H)' H) 2mlr 2 / l\ 2 ('-5) 20 ELECTRICITY, SOUND, AND LIGHT It will be seen from this equation that if r is taken ten times as large as /, an error of one half of one per cent is introduced by neglecting / 2 /4 in comparison with V. If r is twenty times as large as /, this error is only one eighth of one per cent. Thus it is evident that if r is taken sufficiently large, no observable error is introduced into F by writing simply This gives the field strength F due to the magnet mm f of moment M at the position at which the small magnet is suspended. 19. Determination of M/H. Returning now to the considera- tion of the deflection which is produced in the needle be (Fig. 11) by the presence of the magnet mm', it is evident that when the needle takes up its position of rest it is in equilibrium under the I I FIG. 11 action of two moments of force. If n is the pole strength of the suspended magnet be, then, considering both poles, the moment of force due to H is 2 Hn X ab. The moment of force due to F is 2 Fn X ao. But ab = bo sin and ao = bo cos 0. Hence 2 Hn (bo) sin = 2 j^w (&o) cos 0, (3) or Htsmct)=F' (4) M /c or from (2) _ = ___. (5) This gives the first relation between M and H in terms of the measurable quantities and r. STKENGTHS OF MAGNETIC FIELDS AND POLES 21 20. Determination of ME. The second relation between M and H is obtained from an observation of the period of vibration of the magnet mm' when it is suspended at the point for which H is to be determined, namely the position occupied above by the small magnet be. In order to find this period it is first necessary to find what relation exists between the moment of force Fh which acts to restore the magnet to its north-and-south position at any instant at which, in the course of its vibration, it is displaced from this position by an angle 6. It is evident at once from Figure 12 that this restoring moment is given by ~Fh = 1 Hm X de = Hm X df = Hml sin 6 = MH sin 6. (6) This equation shows that in general the restoring moment is not proportional to the angle of displacement 6, but rather to sin 6. If, however, we keep the angle of swing very small, sin 6 will not differ sensibly from 0, and we may .write Hm Wi=MH0, or ^ = MH. (7) Now a simple harmonic vibration is defined as one in which the restoring moment of force acting upon the vibrating system is always pro- portional to the angle of displacement of the system from its position of rest. We see, there- fore, that so long as the amplitude of vibration of the magnet is small, its motion is a case of simple harmonic motion ; and we may therefore apply to it the general formula for the period of any simple harmonic motion. If t represents the period of a half vibration, /the moment of inertia of the vibrating system, and Fh/6 the force constant of the system (i.e. the constant ratio of the moment of force and the displacement), then the general equation for any simple harmonic motion of rotation is - * See " Mechanics, Molecular Physics, and Heat," pp. 87-91 ; also pp. 74 and 75. 22 ELECTRICITY, SOUND, AND LIGHT Since by equation (7) the force constant of the vibration under consideration is MH, we have at once as the expression for the period ^ of a half swing of the magnet or MH=. (9) C l This equation gives us the second relation between M and If in terms of the measurable quantities t l and /. 21. Expression for H. From the two equations (5) and (9) either M or H can easily be obtained. Thus the division of (9) by (5) gives 2 22. Moment of inertia of the magnet. In order to determine experimentally /, the moment of inertia of the large magnet (mm' of Fig. 11), it is most -convenient to lay upon it a brass ring of known mass W and known radius R (Fig. 14, p. 26) ; to adjust it until its center coincides with the axis of suspension ; and then to take the period t 2 (for small vibrations) of the new system. The moment of inertia of this new system is now /+ J , J being the known moment of inertia of the ring. Hence the formula for the new period is The solution of (9) and (11) for /gives The moment of inertia of the ring is simply WR 2 , where W is the mass in grams, and R the mean radius in centimeters of the ring. For it is obvious that the general formula for moment of inertia, namely 2wr 2 , in which w represents an element of mass and r its distance from the axis, reduces in the case of a ring, for which all the elements are at the same distance from the center, to WR Z . * 2 \ * The rigorous formula for the moment of inertia of a ring is WA--s ~) in which RI and R% are the inner and outer diameters. STRENGTHS OF MAGNETIC FIELDS AND POLES 23 As a check upon the experimental determination of / it is well W'il 2 + 5 2 ) * to take also the theoretical value, namely ^ -> where W \.Zi is the mass of the magnet, I its length, and b its width, f 23. Comparison of magnetic fields. From equation (9) it is evident that if ^ and t 2 represent the half periods of the same * See "Mechanics, Molecular Physics, and Heat," pp. 78-81. t Correction of the formulas for torsion of the suspending fibers. Equation (3) was deduced upon the assumption that no forces other than magnetic ones were concerned in the equilibrium of the magnet when its deflection was represented by the angle (Fig. 11). But evidently the torsion of the suspending fiber, as well as the field Zf, opposes the deflection due to the field F. This torsional element cannot always be neglected. In order to take it into account we have only to consider that by Hooke's law there is a constant ratio between the restoring torsional moment Fh' and the angle of displacement. It is customary to define this ratio as the moment of torsion of the suspension and to denote it by TQ. Thus, by definition, =r , or Fh 7 =T Q . (13) Thus when the magnetic needle be (Fig. 11) was deflected through an angle the restoring torsional couple was To0. Hence the rigorous equation of equi- librium was not (3), but rather 2 Hn (bo) sin + T (f> = 2Fn (bo) cos 0. (14) Now TO can be obtained by a second experiment as follows: The pin to which the upper end of the supporting fiber is attached is always arranged in magnetometers so that it can be rotated through any desired angle about a vertical axis. If, then, this so-called torsion head (see t, Fig. 13), be rotated through, say, 180 (TT radians), the needle, which was before in the magnetic meridian, will now be deflected by the torsion of the fiber alone through an angle of, say, a radians. Since the actual twist in the fiber is then (TT a] radians, the equation of equilibrium under these conditions is evidently 2 ffn (60) sin =7-,, (*-), or T, = Substituting this value for T in (14) and dropping the common terms, we get IT sin + Hsinal - ) = Fcos0, \ (1 ir sin, i + Ei. _-)=* cos*. Now since the experiment should always be arranged so that both a and are small (if a is more than 5 the torsion head may be twisted through 90 24 ELECTRICITY, SOUND, AND LIGHT magnet in two different fields of horizontal intensities H^ and H z respectively, then u ^ EXPERIMENT 2 (A) Object. To determine the ratio M/H. Directions. The apparatus used, commonly known as a magnetometer, is shown in Figure 13. I. Set the suspended magnetic needle nn' in the magnetic meridian. To do this set up a reading telescope and scale in front of the small mirror m and two or three meters distant from it. The mirror and instead of 180), no appreciable error is introduced by replacing sin a-/ sin by a/0. Equation (1C) then reduces to F. (17) Substitution in equation (5) gives M _ r* tan TT ff~ T"i^' The final equation for II, namely (10), then becomes . (10) Rigorously, t in equations (10) and (19) also needs correction for the torsion of the fiber which supports the magnet when its period is determined. But since the magnet mm' is large, if the fiber is properly chosen this correction is unnec- essary. In case it should be needed the method by which it is obtained and applied may be seen from the following. The correct equation for the half period is or t* (Ma +T$ = **!, (20) where TO is the moment of torsion of the thread which holds the large magnet. If twisting the torsion head of this magnet support through an angle of TT radians causes the magnet to turn through [3 radians, we have, since /3 is very small, TO(TT /3) = MHp. Substitution of this value of TO in the above equation (20) The combination of this with the equation (18) found above gives as the final corrected form for the value of H ft STRENGTHS OF MAGNETIC FIELDS AND POLES 25 magnet are rigidly connected, the latter usually consisting of a bit o^ magnetized needle a few millimeters long attached by shellac directly to the back of the mirror. Focus* the telescope upon the reflected image of the scale. Then so adjust the torsion head t that equal deflec- tions of the head to the right and to the left produce equal deflections upon the scale. II. Place the magnet mm' upon the graduated crossbar of the magnet- ometer at such a distance east of the suspended magnet nn' that r/l = 10 (see p. 20) ; then read with the telescope and scale the deflection produced by turning mm' end for end. Repeat this operation several times, and thus determine the order of accuracy of the observation. C FIG. 13 III. Make the same observation when mm' is at an equal distance to the west of the suspended magnet. From the mean of these two deflec- tions determine tan <. This is found with sufficient accuracy by dividing the deflection, measured in centimeters, by four times the distance L from the scale to the mirror of the suspended magnet. The reason for this will be clear when it is remembered that the beam of light turns through twice as large an angle as the mirror, and that turning the magnet mm' end for end produces a deflection of 2 in the mirror. * See directions for focusing the telescope, as given on page 64. 26 ELECTRICITY, SOUND, AND LIGHT IV. Remove mm" and determine a (if necessary) as outlined in the theory of footnote f, page 23. The angle a is found in radians by divid- ing the scale deflection by 2 L. Take the mean of the two results obtained by twisting the torsion head equal distances in opposite directions. V. Obtain a value for tan< for another position of mm' such that r/l = 14. Calculate the value of M/H for each deflection. (B) Object. To determine the value of MH with the same magnet that was used in (A), and for the same locality, and by combination of the results with those of (A) to find H. Directions. I. Eliminate tor- sion from the suspending fiber of mm' by turning the torsion head (Fig. 14) until the stir- rup comes to rest approxi- mately in the meridian. Then suspend mm' from the fiber by placing it in this stirrup so that it rests accurately horizontal. Take the pefiod of the magnet with the stop watch, using a telescope and scale. Make three observa- tions of the time of thirty half oscillations. The ampli- tude should not exceed 3, and the instants of passage through the mid-points, not through the end points, of the swing should be observed. II. From the dimensions of the magnet mm', and from its mass as found by a rough balance, determine /. If time permit check this by the experi- mental method of section 22. III. Calculate H from each of the mean values of tan and the corresponding value of r. (C) Object. To find the value of H in a different locality by comparison with the value found in (B). Directions. Set up the magnet and suspension of (B) at a point to be indicated by the instructor (the proposed location of the tangent galvanom- eter of Experiment 3). Make three observations of the time of thirty half oscillations. From equation (23) and the value of H l and t l obtained in (B) calculate 7/ 2 for this position. FIG. 14 STRENGTHS OF MAGNETIC FIELDS AND POLES 27 EXAMPLE (A) The distance in room No. 9 of mm' west of the small magnet mounted on the mirror of the magnetometer was 100 cm. The distance of the telescope and scale from the mirror was 4.13 cm. The first reversal of the magnet changed the scale reading from 20.25 cm. to 42.55 cm., the second from 42.70 to 20.40, and the third from 20.40 to 42.50, thus making the mean deflection at 100 cm., 22.20 cm., and the mean value of tan0 22.20 -H (4 x 413). Similarly the mean value of tan0 at a distance of 140 cm. was 8.12 -r- (4 x 413). Twisting the torsion head through 180 produced a change in the scale reading of 3.1 cm., thus making the value of a 3.1 -f- (2 x 413) radians. (B) Three observations on the time of thirty half swings of mm' when placed in the position occupied by nn f above gave 148.4 sec., 148.6 sec., and 148.4 sec. Hence t l = 4.947. The mass of the magnet was 61.29 g., its length 10 cm., and its breadth 1.3 cm. The value of 7 computed from these dimensions was 518.6. When a 40 g. ring of 4 cm. mean radius was added the half period was 7.40 sec. The value of / computed from these observations was 517.8. H computed from the observations at r=100 was .1764; from the observations at r = 140, .1757, thus giving a mean value in room No. 9 of .1760. The value of M was 1185, and the number of units of magnetism in each pole was 150, approximately. (C) The half period of the magnet in room No. 19 was 4.818 sec. Hence the mean value of H in room No. 19 was .1856 gausses. CHAPTER III MEASUREMENT OF ELECTRIC CURRENTS 24. Definition of the electric current. The charges (or quan- tities) of electricity described and denned on page 2, section 2, by means of certain attractive and repellent properties which they possess, are found to exhibit new properties as soon as they are set into motion. An electric charge in motion is called an electric current. There are, however, two ways in which the charge may move. Suppose, for instance, that the given charge is contained upon the small round body A (Fig. 15), and that the body with its charge b is rapidly carried to B, drawn perhaps by an opposite charge upon the rod ab. Then, according to the. above definition, an electric current has passed from A to B. FlG 15 "" Very good reasons exist (see sect. 31, p. 38) for supposing that electric currents in liquids (electrolytes) are of this nature. The current probably consists of a swarm of charged particles of matter (ions) actually moving through the liquid under the influence of an electric field, just as the charged pith ball moves through the air. But we have seen in section 3 that there is another way in which the charge upon A may arrive at B. Thus when an un- charged body is placed at B and then connected by a wire with the charged body A (Fig. 16), the instant the connection is made B is found to (~~\ /"""^ possess the same sort of attractive prop- p IG . 10 erties which were before possessed by A, while A is found to have lost some of its attractive power. The electric charge upon A (or part of it) has moved along the wire to B. According to the electron theory, in this case, also, a 28 MEASUREMENT OF ELECTRIC CURRENTS 29 swarm of very minute charged particles has actually streamed through the metal from A toward B, or vice versa; but since no direct proof of the correctness of this view has yet been found, it must be regarded at present as merely an hypothesis. Nevertheless it is certain that a part of the charge which was at first upon A has actually passed in some way along the wire to B. Whether, then, a charge passes from A to B by the first method, in which matter is observed to move with the charge, or by the second method, in which minute particles are only assumed as the carriers of the charge, it is the fact of its passage between A and B which ive shall consider as consti- tuting an electric current. That the properties possessed by a static charge which is made to move rapidly through space by mechanical means are in fact identical in all respects with the properties possessed by a charge which is moving along a wire (see sect. 25) was first proved in an elaborate investigation made in 1876 by the American physicist Henry A. Rowland. The correctness of this conclusion has been confirmed by many subsequent investigations. 25. Magnetic effect of an electric current. The new property possessed by a charge in motion, but not possessed by a charge at rest, is the property of exerting magnetic influences. That a charge at rest does not produce any magnetic effect may be clearly demonstrated as follows. If a charged body is brought east or west of, and near to, a magnet supported upon a point, the needle will indeed at first swing about toward the charged body as though the latter exerted a magnetic effect upon it. That this is in reality, however, an effect due merely to elec- trostatic induction may be convincingly shown by inserting a sheet of copper, zinc, or aluminum between the magnet and the charge, when the former will be found to swing back at once to its north-and-south position. For the sheet cuts off all electrostatic influences in accordance with the principle of electric screening (sect. 13) ; but it has no influence at all upon magnetic forces, as may be shown by inserting it between a magnet and the suspended needle. 30 ELECTRICITY, SOUND, AND LIGHT Now if a strong static charge, for example that upon a Leyden jar, is passed through a wire which is wound one or two hun- dred times around a small glass tube containing an unmagiietized knitting needle, the needle will be found to have been quite strongly magnetized. If the sign of the charge is reversed and the knob of the jar touched to the same end of the wire as at first, a second needle will in general be found to have been oppo- sitely magnetized. This certainly shows that a charge in motion produces some sort of a magnetic effect. Again, if the positive terminal of a static machine is connected to one terminal of a coil of wire of ' many turns in the middle of which hangs a magnetic needle, the needle will be deflected when the machine is in operation ; and if the terminals of the machine are reversed, the direction of deflection of the needle will be reversed also. sr\ s^\ A more convenient way, however, of showing that a charge in motion pro- duces a magnetic effect is to make use of a galvanic cell. Such a cell consists essentially of two 2n dissimilar conducting solids immersed in any conducting liquid (the liquid can- not be a molten metal). It may be looked upon as a self-acting static machine. For if the bodies A and B be connected to the two plates of such a cell (see Fig. 17), they are found to be statically charged, just like the poles of a static machine, the one connected to the copper or carbon being positive, and the one connected to the zinc, negative. (It requires, however, a delicate electroscope to prove the existence of these static charges upon the terminals of a galvanic cell.) If now A and B are connected by a wire, the positive charge upon A dis- charges to B (or if it is preferred so to consider it, B loses its negative charge to A)- but the chemical action which is set up in the cell recharges A (or B) as fast as it is discharged. Hence the galvanic cell, like the continuously turned static machine, produces in the wire a continuous current when its terminals are connected by a conductor. Cu FIG. 17 MEASUKEMENT OF ELECTRIC CURRENTS 31 If the wire connecting the terminals of such a cell is held over and parallel to a magnetized needle, the latter will be strongly deflected (Fig. 18). If the terminals of the cell are interchanged the deflection will be reversed. This experiment was first per- formed by the Danish physicist Oersted in 1819. It constitutes one of the most important discoveries which has ever been made in the history of science, for it established for the first time a connection of some sort between electricity and magnetism and paved the way for the marvelous electrical developments of the nineteenth century. The preceding experiments have shown that reversing the direction in which the charge (or charges) moves past the needle reverses the magnetic effect observable near the path of the charge. According to the two-fluid theory the current consists in the mo- tion of a positive charge in one H direction and a negative charge in the other direction. According to Franklin's one-fluid theory it consists of a motion only of posi- FlG 18 tive electricity in one direction. According to the electron theory it consists of a motion only of negative electrons in the opposite direction. It is wholly immaterial which of these theories is correct so long as we understand the conventions which are in common use regarding direction of cur- rent. According to universal convention the direction of a current is the direction from positive toward negative, i.e. in the case of a galvanic cell from copper, or carbon (in the external circuit), toward zinc. Thus, according to this convention, when B was negatively charged and A uncharged, joining them with a wire caused a momentary electric current to flow from A to B, not from B to A. This definition makes it unnecessary to introduce the terms "positive current" and "negative current," although these terms are sometimes employed. 26. Form of magnetic field about a conductor. The form and direction of the magnetic field which surrounds a current may be obtained by mapping out the field with a compass needle in the 32 ELECTRICITY, SOUJSD, AND LIGHT FIG. 19 manner described in Experiment 1, (A). Thus if the black center of Figure 19 represents the cross section of a conductor which passes vertically through the paper and through which the cur- rent flows from the reader toward the plane of the paper, as indicated by the cross in the middle of the conductor,* then the map of the mag- netic field in the plane of the paper is that shown in the figure. The lines of force are circles about the current as a center, and the direc- tion of these circles, i.e. the direc- tion in which an isolated N pole would rotate about the current, is from left to right, as the observer looks in the direction of the current. This rule is often known as the right-handed-screw rule, for the relation between the direction of rotation of the lines and the direction of motion of the current is the same as that existing between the rotary and forward motions in a right-handed screw. The analogy stops here, however, for the lines of force are not spirals about the current. The plane of their direction is always perpendicular to that of the current. It is to be expected from the above rule that an isolated pole would con- tinue to move indefinitely in a circu- lar path around a conductor carrying a current. That this is indeed the case may be strikingly shown by the following experiment. A vertical X FlG 2 o * In general a dot in the middle of the cross section of a conductor is used to represent the head of an approaching arrow, i.e. a current flowing toward the reader. A cross in the conductor represents the tail of a retreating arrow, i.e. a current flowing away from the reader. MEASUREMENT OF ELECTRIC CURRENTS 33 conductor ac (Fig. 20), free to turn about its own axis, carries a bar magnet NS. A circular mercury cup bb', shown in cross section, admits of a continuous connection (over the path be) of the ter- minals a and c with a battery B. A current is now allowed to flow from a to c. This current produces no field at the point S and therefore may be considered as acting 011 an isolated pole N. Under its action the pole N will actually be found to rotate indefinitely in the direction determined by the right-handed-screw rule. If the direction of the current is reversed, the direction of the rotation is found to be reversed also. As is evident from the figure, connection with the battery may be made at b and d and the experiment performed with the S instead of the N pole. 27. The unit of current strength. Since an electric current is defined as an electric charge in motion, it is natural to measure the strength, or intensity, of current flowing through a conductor by the number of units of charge which pass through any cross section of the conductor per second. This is indeed the definition of current strength in the so-called electrostatic system. Thus if Q represents the number of units of charge (electrostatically measured, sect. 2, p. 2) which pass through the conductor in t seconds, then the mean current intensity / is given by the equation z-f- (D The current flowing at any instant is the rate of passage of quan- tity, i.e. if we let dQ represent the quantity of electricity which passes a given cross section in the short element of time dt, then the rigorous definition of current is given by '-% A wire then carries a unit of current when a charge passes through it at the rate of unit quantity per second. But Oersted was experimenting with a galvanic cell when he discovered the magnetic effect of a current, and neither he nor his contemporaries realized that a current was simply a charge in motion, and that the magnetic effect could be produced just as well by means of the electricity developed by means of a static 34 ELECTRICITY, SOUND, AND LIGHT machine. Hence another unit of current strength was chosen, and one which had no apparent connection with the . unit of electro- static quantity defined on page 3. The size of the magnetic effect was arbitrarily taken as the measure of current strength. But this effect was found to vary both with the length of the conductor and with the distance from the conductor to the point at which it was measured. Hence it was necessary to fix upon a wire of specified length and to measure the magnetic effect at a specified distance from it. A wire 1 cm. long was chosen, and in order to arrange to have all parts of this wire at the same distance from some point at which the magnetic effect was to be measured, it was decided to bend it into an arc of 1 cm. radius and to meas- ure the magnetic effect at the center of this arc. Thus a current of unit strength was said to be flowing i cm J^ in the wire when a length of this wire equal to 1 cm. and bent into an arc of 1 cm. radius created at the center of this s / arc a magnetic field of unit strength. \ / Otherwise stated, unit current is that / current unit length of which placed everywhere at unit distance from unit magnetic pole acts upon it with a force of 1 dyne. Thus if the arc AB is 1 cm. and if r is 1 cm. (see Fig. 21), then / is one unit when the magnetic field strength at is 1 dyne. Similarly / is 10 when the field strength at is 10 dynes, etc. Thus, in this so-called electro-magnetic system, it is the unit of current strength which is first defined and which is therefore the fundamental unit. The unit of quantity in this system is defined as the quantity conveyed per second through every cross section of a conductor which carries a current of unit strength. Thus, just as 7 Q/t was the equation which defined current in the electro- static system, Q having been first defined, so the same equation written in the form It Q defines quantity in the electro-magnetic system, / having been first defined. Thus 100 units of quantity pass in 10 seconds through a conductor which carries a constant current whose strength is 10 units. MEASUREMENT OF ELECTKIC CURRENTS 35 28. Ratio of the electrostatic and the electro-magnetic units of quantity, or of current. It is of interest to inquire which of these two units, the electrostatic or the electro-magnetic, is the larger. In order to answer this question experimentally, it is evident that it would only be necessary to collect a quantity of electricity which had been measured in electrostatic units by an observation of its attraction upon some known charge, and then to discharge this quantity in a known time through the arc of Figure 21, and measure the magnetic effect thus produced. Such a measurement shows that the electro-magnetic unit is enormously larger than the electrostatic unit, and further reveals the surprising fact that the ratio of the electro-magnetic to the electrostatic unit is equal to the velocity of light expressed in centimeters per second, namely 3 X 10 10 . This fact played an important role in the discovery of the intimate relation which exists between electricity and light. 29. Practical units. No name lias been given to the absolute units of current and quantity defined above. For purposes of convenience it has been decided to take as the commercial units one tenth of the absolute units of both current and quantity. Thus the commercial unit of current is 10" 1 absolute electro-magnetic units of current. It is named an ampere in honor of the French physicist Andre Marie Ampere (1775-1836). Similarly the com- mercial unit of quantity is 10" 1 absolute electro-magnetic units of quantity. It is named a coulomb hi honor of the French physicist Charles Augustin Coulomb (1736-1806). 30. Tangent galvanometer. It is evident from the definition of current strength that the direct method of measuring / must consist in meas- uring the strength of magnetic field produced by a known length of the current at a known distance from it. This is most easily accomplished by passing the current through a circular loop of wire of known radius and measuring the field strength at the center of the loop. If the current were of unit 36 ELECTKICITY, SOUND, AND LIGHT strength, then from the definition of current strength it would follow that every centimeter of length of the conductor carrying the current would produce at a point which is one centimeter distant from all parts of that centimeter of length a field strength of one dyne. Since all forces which emanate from point sources vary inversely as the squares of the distances from the sources, it is evident that the forces emanating from points on the wire vary inversely as the squares of the distances from those points. Hence at the center of the loop, a point which is r centimeters distant from all points on the wire, the field strength due to each centimeter of length must be l/r 2 and the field strength due to 2 irr the 2 TTT centimeters of length must therefore be - If the r current has J units of strength, and if there are N loops instead of one, the field strength F at the center must be, therefore, N . Hn 2 7TNI F = - gausses. (3) In order to measure this field the plane of the coil (see Fig. 22) is set in the earth's magnetic meridian and a small suspended magnetic needle is placed at the center. In this position the field F due to the current tends to turn the needle into the east-and-west direction, while the earth's magnetic field tends to keep it in the north-and-south direction. The equation of equilibrium is, therefore (see Fig. 23 and sect. 19), Hnl sin 6 = Fnl cos 0, (4) in which I is the length of the small magnet. Hence from (3) or /=J^Ltanft (5) Thus if the galvanometer is set up at some point for which H is known, the current strength / is determined in absolute units from a measurement of r, N, and tan#. MEASUREMENT OF ELECTRIC CURRENTS 37 31. Electrolysis. It is found that all of the liquids (save molten metals) which are conductors of electricity are solutions of chemical compounds, salts, acids, or bases, and that the passage of an electric current through such liquids is uniformly accompanied by the passage out of the solution of the compo- nents of the dissolved substance. Thus when a current is passed through a solution of copper sulphate (CuS0 4 ), metallic copper (Cu) is deposited upon the plate by which the current leaves the solution (the negative electrode or cathode}, while S0 4 radicals collect about the positive electrode or anode, where their pres- ence may be detected by the acid character which they impart to this portion of the solution. The systematic study of this phe- nomenon was first made by Faraday in 1832. He called the operation of separating the constituents of a compound in solu- tion by means of a current electrolysis. Any liquid which was capable of being decomposed in this way he called an electrolyte. The constituents of the dissolved substance which appeared at the electrodes he called ions. The results of his investigation may be summarized thus. (1) So long as the deposit of but one particular kind of ion, for example silver, was being studied, it was found that the amount of the deposit by weight was proportional solely to the quantity of electricity which had passed through the solution. It was independent of the nature of the solvent, of the nature of the silver compound in the solution (AgCl, AgX0 3 , Agl, etc.), of the concentration of the solution, and of the strength of the current, save as this depended upon quantity through the relation Q = It. The obvious interpretation of this result is that a given atom, or radical, in whatever compound it is found, is always associated with a certain definite quantity of electricity. In other words, electricity, like matter, consists of discrete parts. The charge asso- ciated with an atom of hydrogen was first called by Helmholtz an atom of electricity. (2) When the investigation was extended to different ions of the same valency (combining power with respect to hydrogen), such, for example, as hydrogen, silver, iodine, or bromine, it was 38 ELECTEICITY, SOUKD, AND LIGHT found that the amounts, ~by weight, deposited (or liberated in the case of gases) ly the passage of equal quantities of electricity were exactly proportional to the atomic (or ionic) weights of the ions. Thus an atom of silver weighs about 108 times as much as an atom of hydrogen, and the passage of one coulomb of electricity through a solution containing a silver salt was found to deposit 108 times as many grams of silver as there were grams of hydrogen liberated by the passage of one coulomb of electricity through a solution of hydrochloric acid (HC1) in water. This evidently means that an atom of silver carries exactly the same charge as an atom of hydrogen, or, in general, that all atoms having the same valency carry the same charges of electricity. (3) When substances of different valencies were compared, as for example hydrogen and copper, it was found that the deposits ly weight produced ly the passage of a given quantity of electricity were proportional to the atomic weights of the ions divided ly their valencies. Thus the copper atom has a valency 2 and a weight of 63.2 as compared with the atom of hydrogen. The deposit of copper due to the passage of a given quantity of electricity was / q o not 63.2 times the deposit of hydrogen, but instead exactly - > 2i or 36.6 times this deposit. This signifies that a copper atom car- ries twice as large a charge as a hydrogen atom, i.e. that it carries two of Helmholtz's atoms of electricity, or, in general, that the chemical valencies 1, 2, 3, 4> 5 correspond to electrical charges upon the ions in the ratios 1, 2, 3, 4-> &- These discoveries certainly indicate that chemical attractions are of electrical origin. They constitute the chief ground for the statement made in section 24, page 28, that the passage of a cur- rent through a liquid consists in the movement through the liquid of swarms of electrically charged particles (the ions). The weight of a substance in grams deposited by the passage of one absolute electro-magnetic unit of quantity of electricity is called the electro-chemical equivalent of that substance. Thus the electro-chemical equivalent of silver (atomic weight 107.7, valency 1) is.01118,that of hydrogen (atomic weight 1, valency l)is. 0001038, that of copper (atomic weight 63.2, valency 2) is .00328. MEASUREMENT OF ELECTRIC CURRENTS 39 32. Voltameter. This .phenomenon of electrolysis is applied to the measurement of currents in an instrument known as the voltameter. The name " voltameter " is applied to any arrangement of apparatus whereby the weight or the volume of the ions caused to pass out of an electrolytic solution is used to determine the total quantity of electricity which has passed through the circuit in which the instrument is placed. Because of the accuracy with which weighings can be made, and because of the fact that, by taking the time long enough, a very small current can be made to deposit a considerable weight of metal, the voltameter, rather than the tangent galvanometer, is the instrument which is almost universally used for standardizing current-measuring instruments. However, it is to be remembered that the electro-chemical equiva- lents had first to be determined by reference to the tangent gal- vanometer, or some similar instrument which measures directly the strength of the magnetic field at a known distance from the current, and hence that the fundamental current-measuring instru- ment is not the voltameter, but the galvanometer. 33. Ammeter. Neither the tangent galvanometer nor the voltameter are convenient for ordinary commercial or laboratory work. Instruments called ammeters (see Fig. 24) are universally used for measuring current strength. These instruments are merely galva- nometers, of a type to be described later, which have been empirically calibrated by passing a given current simultaneously through the ammeter and through an electrolytic solution (voltameter). EXPERIMENT 3 Object. To measure a current by means of (1) two copper voltameters, (2) a silver voltameter, (3) a hydrogen voltameter, (4) a tangent galvanom- eter, and (5) an ammeter ; otherwise stated, to test Faraday's laws and to calibrate an ammeter. Directions. I. The solutions. Make up two solutions of pure copper sul- phate, one containing, say, 22 g., and the other 32 g. of crystals to 100 g. of water. Add to each solution about 1 per cent of strong sulphuric acid. 40 ELECTRICITY, SOUND, AND LIGHT Make up for the silver voltameter a neutral silver nitrate (AgNO 3 ) solu- tion by dissolving about 18 g. of crystals in 100 g. of water. Also make up for the hydrogen voltameter a solution of 50 or 60 g. of sulphuric acid to a liter of water. II. Connections and adjustments. Set the coil of the tangent galvanom- eter in the magnetic meridian, i.e. parallel to the needle of a good compass. Then by means of the leveling screws adjust the instrument until the suspending fiber hangs over the center of the circular scale. Finally by means of the torsion head bring the suspended needle into the magnetic meridian. This should be accomplished when each end of the aluminum index, which is attached rigidly to the needle at right angles to its length, is exactly above one of the two zeros of the scale. Connect as in the diagram (Fig. 25), taking especial care to avoid loose contacts. The battery B may consist of three or four storage cells or six or eight fresh dry cells. The order in which various instruments are placed in the circuit is wholly unimportant, but the tangent galvanometer G should be not less than ten feet away from the milliammeter A in order that the magnet in the latter may not influence the needle of the former. Furthermore, since the deposit of a metal or of hydrogen is always upon MEASUREMENT OF ELECTRIC CURRENTS 41 FIG. 26 the plate toward which the current flows in the solution (the cathode), care in getting the direction of the current through the voltameters as indicated by the arrows is of the utmost importance. The light lines in B represent the + terminals, i.e. the carbons of dry cells. The 4- terminals- of storage cells are usually marked. If they are not marked they can easily be determined by connecting the two terminals with a few feet of about No. 30 German silver wire and noting the direction in which a compass needle held near the wire is deflected (see right-handed-screw 7 rule, p. 32). R is a cheap, variable German silver resistance provided with a strong spring clamp for a sliding contact (Fig. 38, p. 53). J/ is a mercury commutator (Fig. 26), so made that connecting the mercury contacts e/and gh causes the current to traverse the tangent galvanometer G in one direction, while connecting the mercury cups eg and fh causes the current to pass through the galvanometer in the opposite direction. The negative electrodes a and b are pure copper plates of at least 50 sq. cm. area per ampere of current and well rounded at the corners ; the positive electrodes are either bent cop- per plates of slightly greater width, as in Figure 25, or else two separate plates elec- trically connected as shown in Figure 27. The silver voltameter has similar electrodes of pure silver, the gain plate having an area of about 300 sq. cm. per ampere. In the instrument shown in Figure 27 the plates slip conveniently into spring clips attached to the lower side of the cover. The elec- trodes entering the graduated tubes and //consist of pieces of platinum foil attached to platinum wires. Each wire may be sealed into the end of a glass tube bent into a U -shape, so that the platinum foil may be well within, the graduated tubes ; or the instrument may be constructed in the manner shown in Figure 28. If the plates have been in recent use proceed as follows.* Place the plates in their respective clips and lower them into the solutions. Fill the FIG. 27 * If the gain plates are not already clean they should be thoroughly polished with glass or sandpaper (not with emery). 42 ELECTEICITY, SOUND, AND LIGHT tubes and H with the sulphuric acid solution. Give the resistance R its maximum value. Close the circuit and then adjust R until the deflec- tion of the tangent galvanometer needle is about 45. (The time occupied by the experiment will be of convenient length if the galvanometer is so wound as to make this correspond to a current of from 200 to 250 milli- amperes.) Allow the current to flow for four or five minutes so as to form a fresh deposit upon the plates. Then remove the gain plates, rinse first in distilled water, then in alcohol, and finally dry over a radiator or, if time is limited, by igniting, in an alcohol or Bunsen flame, the alcohol which cannot be shaken off. Do not touch the deposit at any time with the fingers. ' III. Weighings and readings. AVeigh the plates as follows. By means of a wire hook hang the plate to be weighed from one balance arm and add weights to the other pan until the resting point of the pointer, as determined by the method of oscillations,* is within one or two divisions of the middle of the scale over which the end of the pointer moves. Record this resting point, count- ing the middle division on the scale as 10, and record also the weight which you have placed on the other pan. Do this with each of the three cathodes, then replace them in their respective solutions, f Read the levels of the liquid in the tubes and //. Then, at some accurately ob- served time, turn on the current. Allow it to run until 40 or 50 cc. of hydrogen have been formed, continually adjusting R, if necessary, so that the ammeter reading is kept very con- stant. While the current is flowing read both ends of the tangent galvanometer index, taking great pains in so doing to set the eye directly over the center of the scale. At any convenient time during the run reverse the current in the FIG. 28 tangent galvanometer and again read both ends of the index. In reversing leave the current open long enough to allow the needle to make one half swing. Other- wise the needle may swing completely around. This interval of open circuit, which may well be measured with a stop watch, is to be deducted * See "Mechanics, Molecular Physics, and Heat," Experiment 15. t The plates should not be placed in the solutions more than a minute or two before the deposition is begun, for the copper plates tend to dissolve slowly in the solutions. MEASUKEMENT OF ELECTRIC CURRENTS 43 from the total observed time in order to obtain the actual time of dura- tion of the experiment. At an accurately observed time break the circuit ; remove the plates, rinse all three carefully in distilled water, then in alcohol, being exceedingly careful not to shake off any of the loosely adhering silver granules ; dry over a radiator or by igniting the film of alcohol which clings to the surface, and weigh again as follows. Placing each plate on the same balance pan as before, record the weight which will bring the pointer to within two or three divisions of its first resting point. Record also the new resting point. Add 2 mg. and record the resting point for this added weight. From these last two resting points determine the sensibility of the balance, i.e. the number of scale divi- sions corresponding to an addition of one mg. to the pan. Using this sen- sibility determine the weight, to one tenth mg., which w r ould bring the resting point of the second weighing into coincidence with that found in the first weighing. Thus suppose that, before the deposition, with a weight on the right pan of 35.965 g. the resting point was 9.5, and that after the deposition a weight of 36.195 g. produced a resting point of 8.6, while the addition of 2 mg. to the left side changed the resting point i n T ft f\ to 10.7. The sensibility is then =1.05, and the number of milligrams corresponding to 'the difference in resting points between 9.5 and 8.6 is - = .9 mg. Hence the w r eight, after the deposit, J. . (Jo which would have been necessary to produce a resting point of 9.5 was 36.195 - .0009 = 36.1941. Hence the weight of the deposit was 36.1941 - 35.965 = .2291 g. IV. Calculation of current. The calculation of the quantity of electricity which has accompanied the deposits of silver and copper is easily made from these deposits and the electro-chemical equivalents given on page 38. The current is then obtained from equation (1), page 33. The calculation of the current from the amount of hydrogen collected can be made as follows. Let V represent the observed volume of hydrogen, let VQ represent this volume reduced to 0C. and 76cm. pressure. The quantity of electricity which has passed (measured in absolute units) is equal to ^-, since 1.156 cc. is the volume at 0C. and 760 mm. pressure occupied by .0001038 g. (the electro-chemical equivalent) of hydrogen. V is obtained from V by the equation representing the combination of the laws of Boyle and Charles, namely V P T 7 <> - , C6) V 760 T in which T is the absolute temperature of the room and P the pressure which the hydrogen in the tube exerts. If B represents the existing barom- eter height in millimeters, h the height, also in millimeters, of the final 44 ELECTRICITY, SOUND, AND LIGHT level of the water in the tube above that in the outer vessel (in the instru- ment of Fig. 28 this will of course be negative), and p the pressure of saturated water vapor (expressed in millimeters of mercury at the tempera- ture of the room),* then evidently the number 12 being taken as the approximate ratio between the density of mercury and that of the II 2 SO 4 solution. In the calculation of the current from the readings of the tangent gal- vanometer a sufficiently accurate correction for the torsion of the fiber may be made without determining T . Thus, while no current is flowing through the galvanometer, twist the torsion head through the same angle 6 through which the needle was deflected when the current was flowing. Add to 6 the angle through which the needle is now deflected in order to obtain a corrected value of for substitution in equation (5). The num- ber of turns of the galvanometer will be given by the instructor. The value of H has .been determined in Experiment 2, (C). Measure the mean diameter of the coil. Calculate the current from the data at hand. EXAMPLE After the current had been allowed to flow for several minutes the plates were rinsed, dried, and weighed. The current was then allowed to flow for 24 minutes and 30 seconds, less 7 seconds required for the reversal through the tangent galvanometer, i.e. the current flowed for 1463 seconds. The record of the weighing was as follows. With the first copper plate the weight used was 20.190 g. and the resting point 11.75. After the deposit 20.290 g. gave a resting point of 11.53. Adding 2 mg. gave a new rest- ing point of 9.00. Hence the sensibility was 1.26 and the correct second weight 20.2898 g. Hence the gain was .0998 g. and the current .2070 ampere. With the second copper plate the first weight was 19.695 g. at 12.75 and the second 19.795 at 11.75. As the mass was practically that of the first plate weighed, it was unnecessary to redetermine the sensibil- ity; hence the second weight was 19. 7942 g., the gain .0992g., and the current .2058 ampere. Similarly with the silver plate the first weight was 20. 540 g. at 11.56 and the second 20.878 g. at 11.30. Hence the final weight was 20.8778 g., the gain .3378 g., and the current .206"3 ampere. The initial volume of the hydrogen was 7.24 cc. and the final 47.04cc. Hence the gain was 39.8 cc. The barometer reading was 744 mm. at * See table No. 1 in the Appendix. MEASUREMENT OF ELECTRIC CURRENTS 45 22 C. This reduced to 0C. gave B 741.4; // was 25 mm. ; p from the table was 19.6 mm. Hence P 741.4 2.1 19.6 = 719.7. Hence ^ 273 .39.8 = 34.8, 700 273 + 22 and the value of the current as found from this was .2062 ampere. The readings on the tangent galvanometer were, for the east end, 47. G, for one direction of the current, and 47.7 for the reversed direction. For the west end these readings were 47.5 and 47.6, respectively, giving a mean of 47.6. Twisting the torsion head through 48 caused a deflec- tion of the needle of .5. Hence 6 48.1. The average radius of the coil was 15 cm., and there were 24 turns, //"from Experiment 2 was .1856. Hence the current was .2053 ampere. The ammeter reading was kept constant throughout the experiment at .207 ampere. The average of all the determinations of current was .2059 ampere, and the greatest deviation of any single determination was .3 per cent. The error in the ammeter at this point was approximately 1 milliampere. CHAPTEE IV THE MEASUREMENT OF POTENTIAL DIFFERENCE 34. Potential difference between points in the neighborhood of charged bodies. The gravitational P.D. between two points has been denned as the amount of work required to carry unit mass between the two points against the force of the existing gravita- tional field. An exactly analogous definition was given for the electrical P.D. between two points, viz. the number of ergs of work required to carry unit positive [ i ] A B charge between the two points \ / FlG 29 against the force of the existing electrical field. This definition holds under all circumstances, whether static or current phe- nomena are under consideration. Thus the P.D. between the points A and B (Fig. 29) near a charge P is simply the number of ergs of work required to carry unit 4- charge against the re- f , ^ pulsion of P from B to A ; or, \^^J FJG 30 x^x inversely, the* work which the electric field does in carrying unit -f charge from A to B. Or again, if the points A and B represent the localities of + and charges respectively (see Fig. 30), the P.D. between A and B is the number of ergs of work required to x A B carry unit -f charge from B to A. If the points A and B in Fig- ure 29 are connected by a con- ductor of any sort, as in Figure 31, we have seen (sect. 13) that a current at once flows through the conductor, and for convenience we have taken the direction of this current the same as the direc- tion of the electric force, i.e. from A, the point of higher potential, to B, the point of lower potential. We have seen further that in this case the current can be only momentary, for since AB is an 46 MEASUREMENT OF POTENTIAL DIFFERENCE 47 insulated conductor, the passage of a current from A to B means the appearance of a + charge at B, and of a charge at A, since + and charges always appear simultaneously and in equal amount. But since the accumulation of a + charge at B and a charge at A means the creation of a field of force between A and B whose direction is opposite to that of the field due to P, we saw that a current can flow through an isolated conductor in obedience to the force of an outside electric field only until the new field be- tween A and B } created by the charges accumulating at these points, is sufficiently strong to neutralize the initial field, i.e. the fieKTdue to P ; that is, strong enough to reduce to zero the force which was initially causing a current to flow from A to B. This force gone, there can of course be no longer any P.D. between A and B ; for the existence of a P.D. depends upon the existence of an electric force (sects. 9 and 11). It was thus seen that as soon as a current ceases to flow through a conductor in an electric field, all of the points of the conductor must have the same poten- tial. Similarly, in Figure 30, the result of connecting A and B by a conductor is to cause a current to flow between these points until the field strength existing between them is reduced to zero, i.e. until A and B have the same potential. 35. Potential difference between points on the terminals of an electric generator. But if A and B are the terminals of a static machine, or of a galvanic cell, or of a dynamo, or of any generator E (Fig. 32) of electricity which is capable of building up a field of given strength between A and B, i.e. of maintaining, on open circuit, a given P.D. between them, then, when these points are connected by a conductor, the first result must be, as above, that a current flows from A to B, thus tending to discharge these bodies, and to cause the field between them to collapse, i.e. to cause their P.D. to disappear. But the generator E tends instantly to re- charge A and B, and if this recharging takes place at a rate which is very rapid in comparison with the rate at which the 48 ELECTBICITY, SOUND, AND LIGHT connecting conductor discharges them, it is evident that, in spite of the presence of the conductor, a mean strength of field may be kept up between A and B which is almost as great as that kept up when the connecting conductor was absent. On the other hand, if the conductor discharges A and B very rapidly in comparison with the rate at which the generator is able to recharge them, the strength of field maintained between them must be small ; in other words, A and B will have nearly the same potential. Thus, connecting the points A and B by a very good conductor may lower their P.D. almost to zero, never, of course, quite to zero so long as a current is flowing from A to B, while connecting them with a very poor conductor may lower their P.D. an inappreciable amount. Connections of intermediate conductivity must of course produce intermediate lowerings of the P.D. between A and B. It is evident, then, that in all cases a P.D. exists between two points, ivhether on a conductor or off from it, only by virtue of the existence of a static field which makes it necessary to do work in order to carry a charge from one point to the other against the existing electrical force. 36. Work done independent of path. If A and B are the ter- minals of a cell or a dynamo, and if these terminals are connected by a long wire ACDB (see Fig. 33), the assertion that a certain P.D., say 10 absolute units, exists between A and B means simply that it would require 10 ergs of work to carry a unit + charge straight across through the air against the force of the electric field which exists between A and B. It also means that it would re- quire just the same work D to carry a unit charge from 33 B to A by any other path whatever; for the work done is wholly independent of the path (see sect. 12, p. 11). Or, conversely, it means that the field would do 10 ergs of work in driv- ing a pith ball charged. with 1 -+- unit from A over to B; or, since work is independent of path, that it would do 10 ergs of work in carrying one unit of charge from A to B through the wire A CDB. MEASUREMENT OF POTENTIAL DIFFERENCE 49 37. How the work appears. If the pith ball were driven through the air from A to B, the work done by the field would be expended in overcoming the friction of the air and in impart- ing kinetic energy to the ball. This kinetic energy would all be given up, and a corresponding heat energy would appear, when the pith ball struck B. Hence all of the work done in moving the + unit charge from A to B would appear as heat. Similarly if the unit charge goes from A to B through the wire ACDB, the 10 ergs of work expended by the field in carrying it in this way from A to B must in general appear as heat. If any chemical changes took place in the wire because of this passage of current, this conclusion would not hold, nor would it hold if an electric motor were interposed anywhere in the wire ACDB. In these cases the work done by the field would be equal to the heat developed plus the chemical or mechanical work done. But if the heat is the only observable effect produced, then it follows from the principle of the conservation of energy that the work done by the electric field in moving a unit charge from A to B must be equal to the heat developed in the wire ACDB when this heat has been reduced to mechanical units. According to the electron theory the heating is in this case due to the frictional resistance which the wire offers to the pas- sage of the electrons through it. 38. Measurement of P.D. It is evident then from the definition of P.D. that any measure- ment of P.D. must con- sist in measuring the work done in carrying unit charge between the two points whose FlG 34 P.D. is sought. This work , may be obtained by measuring the strength of the field in dynes, and the distance between the points whose P.D. is sought 50 ELECTRICITY, SOUND, AND LIGHT in centimeters. These are the actual measurements which are made in the use of the so-called absolute electrometer. This is an instrument which consists essentially of two plates, A and B (Fig. 34), which are connected by wires to the points a and b between which the P.D. is sought. These plates are so .arranged that the force of attraction between them may be measured. From the force of attraction of the plates the strength of the field between A and B is easily obtained. The product of this field strength by the distance between the plates gives the work necessary to carry unit charge from B across to A. Since all points on a conductor in which no current flows have the same potential (sect. 13, p. 12), it follows that the potential at A is the same as that at a, and that at B the same as that at &, i.e. the P.D. between A and B must also be the P.D. between a and b. Commercial electrostatic voltmeters, which are coming more and more into use, and which are in many respects the most satisfactory of all instruments for measuring P.D., differ but little in principle from the absolute electrometer. They consist of two fixed plates A and B (Fig. 35), which may be connected to the two points whose P.D. is sought, and which carry between them a gold leaf g, or some other movable system, the deflections of which may be taken as a measure of the field strength between the plates. The instrument is empirically calibrated so that given deflections of the movable system correspond to given differences in potential between A and B, and therefore between the points to which A and B are attached. Another method of making an absolute measurement of P.D. con- sists in measuring the number of calories of heat developed by the passage of a known quantity of elec- tricity between the points whose P.D. is sought. Thus suppose that a constant current of strength / is made to flow through a platinum wire AB (Fig. 36) immersed in a calorimeter, and that a number of FIG. 35 FIG. 36 MEASUREMENT OF POTENTIAL DIFFERENCE 51 calories of heat M are thus communicated to the water in t seconds. Then the amount of work done by the current is MJ ergs, where J is the mechanical equivalent of heat, i.e. the number of ergs equivalent to one calorie, namely 4.19 x 10 7 . But since the P.D. between A and B is the number of ergs of work done in carrying unit quantity between these points, the work done by the passage of Q(= It) units of quantity may be written PD x Q = PD x It. Equating these two expressions, we obtain PD It = MJ, or PD = ' (I) It Now the absolute electro-magnetic unit of P.D. is the P.D. which exists between two points when it requires 1 erg of work to carry 1 unit of quantity (measured in the electro-magnetic system) between the two points. The above equation evidently gives the P.D. in such units, provided M is measured in calories, t in seconds, and / in absolute electro-magnetic units of current. This absolute unit of P.D. is so extremely small, however, that it has been decided to use as the unit in practical work a P.D. which is 10 8 absolute units. This unit is called the volt. Hence the P.D. between two points is 1 volt when it requires 10 s ergs of work to carry one electro-magnetic unit of charge between these points. 39. Voltmeters. Neither the calorimetric method nor that which makes use of the absolute electrometer is convenient for the rapid measurement of P.D. ; but either one of them may be used for calibrating a so-called high-resistance galvanometer in such a way that P.D. may be read off upon it directly. The operation' of finding the P.D. between any two points between which a current is flowing will then consist simply in touching the galvanometer terminals to the two points whose P.D. is sought, and observing the deflection produced. FIG. 37 52 ELECTRICITY, SOUND, AND LIGHT Thus suppose a given P.D. is maintained between the terminals C and D of a platinum wire immersed in the calorimeter (Fig. 37) and suppose this P.D. has been found by measuring the rate at which heat is developed in the calorimeter. If now, while the current is still flowing, a galvanometer be connected across C and D, the conductivity of the galvanometer, added to that of the wire CD, will in general discharge the terminals C and D faster than they were being discharged when they were connected by the platinum wire alone. Hence the introduction of the galva- nometer will in general cause a fall in the P.D. between C and D. But if this galvanometer carries a quantity ef electricity per sec- ond which is wholly negligible in comparison with that carried by CD, then the P.D. will remain essentially unchanged by the introduction of the galvanometer. The deflection of the galvanom- eter may then be marked and labeled with the number of units of P.D. which exist between C and D, as measured by the heat developed in the calorimeter. The P.D. between C and D may then be increased by' the use, say, of a different generator, and another deflection labeled with a new P.D., as measured by the calorimeter. In this way the galvanometer may be empirically calibrated throughout its whole range of deflection, so that its readings always indicate the P.D. existing between the points to which its terminals are touched. It is evident that such a voltmeter could not be used for determining the P.D. between two isolated static charges, for the P.D. between these charges would instantly become zero as soon as the galvanometer terminals were touched to them. It could be used for determining the P.D. between the terminals of a dynamo, or of a galvanic cell on open circuit, provided it carried so little current as not to lower appreciably this P.D. by its intro- duction between them. It could be used for determining the P.D. between two points on a conductor which is already carrying a current, provided it did not alter this P.D. by its introduction in shunt with the conductor, i.e. provided it carried a current which is negligible in comparison with that carried by the conductor. Of course after the galvanometer has once been calibrated for P.D. its readings represent under all circumstances the P.D. existing MEASUREMENT OF POTENTIAL DIFFERENCE 53 between its own terminals, whether the current which it carries is negligible or not; but if its current is not negligible, the P.D. between the points to which it is touched is not the same when it is in contact with these points as when it is not in contact with them. Thus a voltmeter is merely a galvanometer which has been calibrated empirically by reference to some absolute measurement of P.D. (Because of the uncertainties of calorimetric measurement, the method given above for calibrating a voltmeter is not the one which is most commonly used (see Chap. V)). EXPERIMENT 4 Object. To test the calibration of a voltmeter by an absolute determina- tion of P.D. ; otherwise stated, to verify the relation PD x Q = MJ. Directions. Connect as in the diagram of Figure 37. B is a generator capable of producing a constant P.D. of at least 20 volts. (Either 55 or 110 volts will serve the purpose admira- bly. ) A is an ammeter with a range of, say, to 10 amperes. al> is a platinum wire (Xo. 30 to No. 34) which is capable of being raised to a red heat by a cur- rent of from 3 to 6 amperes. S is a switch which is to be left open until the instructor has seen that all connections have been properly made, and which is in no case to be closed unless the plati- num wire is immersed in water. It is a variable German silver resistance (Fig. 38) capable of reducing the current from 110 volt mains to from 3 to 6 amperes. N is a calorimeter the inner vessel of which has a capacity of about 300 cc. (Fig. 39 shows cut of the complete calorimetric outfit.) Figure 40 shows the construction of the voltmeter V which is to be calibrated. Fill the calorimeter to within about 1 cm. of the top with water the temperature of which is from 10 C. to 15C. below the temperature of the room, provided this temperature is not below the dew-point. If dew collects on the vessel the initial temperature should be taken a degree or two above that at which this dew is seen to appear. Take a preliminary observation as follows. Stir thoroughly, read the temperature, then throw on the current, and keep it on for just one minute, the ammeter reading ELECTRICITY, SOUND, AND LIGHT being adjusted to 3} or 4 amperes. If the final temperature is not more than two or three degrees above room temperature, the conditions need not be changed. Otherwise alter the current until a run of a minute produces about the final temperature indicated above. Then fill the calorimeter again with cold water, stir it thoroughly, insert the platinum coil, and, ^ while stirring continuously, take four or five successive readings of the temperature of the water, estimating the thermometer to tenths of the smallest division. As soon as possible after the last reading throw on the current, and keep it on for just a minute, stirring vigorously all the time and adjusting R, if necessary, so as to hold the current constant. Stir for at least a minute after the current has been turned off, then take the final tem- perature very carefully. If the experiment is performed in this way, no correction for radiation will in general be necessary. But if the duration of the heating is several minutes, or if the final temperature is 8C. or more above room temperature, the final tem- perature corrected for radiation may be found as follows. Observe the initial temperature precisely as indicated above ; take the time at which the heating begins, the time at which it ends ; then, beginning 1 minute after this ^^^^=^r=^^^^^^ latter time, record the temperature at minute ~~~~ Tfl intervals for at least 5 minutes, the stirring (I being continuous from the instant of throwing on the current. Next plot times, measured from this instant, as abscissas, and tempera- tures as ordinates, in the manner indicated in Figure 41. The full line drawn through these points shows the rate of cooling by radiation, and this line, extended back so as to cut the vertical line drawn through the time at which the current was discontinued, gives the tem- perature of the water at the instant of cessation of the current. This quantity could not have been observed directly because it is impossible, however vigorous the stir- ring, to keep all parts of the water at the same temperature during the heating. Next find from the curve the loss in temperature per minute per degree above room temperature. This quantity, multiplied by the time in minutes during which the current was flowing, and by the number of FIG. 39 MEASUREMENT OF POTENTIAL DIFFERENCE 55 degrees that the average temperature during this time (as found by taking the mean of initial and final temperatures) was above the room tempera- ture, gives the number of degrees to be added to the final temperature in order to obtain the final temperature corrected for radiation. If the aver- age temperature during the heating is below room temperature, obviously heat is gained, arid the correction, found exactly as indicated above, is to be subtracted. As soon as possible after the observations on temperature, weigh on a rough balance the inner vessel of the calorimeter and the contained water, FIG. 40 then the empty calorimeter, then the stirrer. Estimate the volume of the electrodes in cubic centimeters from rough measurement of their dimensions. The water equivalent of the inner vessel of the calorimeter, that is, the number of grams of water which would be raised 1C. by the num- ber of calories which is required to raise the calorimeter 1C. is evi- dently the weight of the calorimeter x the specific heat of brass (.094). The water equivalent of the electrodes is their volume x their density 56 ELECTRICITY, SOUND, AND LIGHT (= 8.4) x .094. Similarly the water equivalent of the stirrer is obtained from its weight and specific heat. You can probably neglect the water equivalent of the thermometer.* The total weight of water plus these water equivalents, multiplied by the observed rise in temperature, evidently gives the number of calories of heat developed during the experiment. One calorie is equivalent to 427 gram meters of work or to 4.19 x 10 7 ergs. If I is the mean current in amperes, the heat developed, w T hen reduced to e"rgs, represents the work in absolute units which has been done in trans- ferring Q(= 7^/10) absolute units of electricity from C to D. Hence the P.D., or the work expended in transferring one unit, is at once obtained. This is reduced to volts by dividing by 10 8 . t EXAMPLE The room temperature was found to be 17.2C. Initial temperature of water was 15.4C. Final temperature (see Fig. 41) was 29.0C. Mean 29 + 15 4 temperature of water during heating was '- = 22.2C. Mean rate of cooling at end of experiment (eleven degrees above room temperature) was .15 per minute. .'. Mean rate of cooling at one degree above room temperature would be .014 per minute. .'. Rate of cooling at 22.2 was .014 x (22.2 17. 2) = . 07 per minute. .'.Final temperature corrected for radiation was 29.07C. .'. Rise in temperature was 13.67C. Weight of water and calorimeter was 282.9 g. Weight of calorimeter was 54.9 g. Weight of stirrer, 7.87 g. .'. Weight of water was 228.0 g. Water equiv- alent of calorimeter was 5.16 g. Water equivalent of stirrer, .74 g. Vol- ume of electrodes, .64 cc. Water equivalent of electrodes, .51 g. Water * To calculate the water equivalent of the thermometer note that the heat capacity of mercury per unit of volume, viz. 13.6 ( = sp. gr.) x .033(=sp.ht.) = .47, is nearly the same as that of glass, viz. 2.5(=sp. gr.) x.!9(=sp. lit.) =.45. Hence to calculate the heat capacity of a thermometer, estimate the volume immersed, and multiply by .46. t The calculation may be freed from the use of powers of 10 as follows. The practical unit of work, the joule, is 10 7 ergs. One calorie is therefore 4.19 joules. If, then, M represents the number of calories developed, 4. 19 M is the heat developed expressed in joules. If PD and I are expressed in volts and amperes respectively, then the work done is (PD) 10 8 I lQ~ l t ergs = (PD) It joules. This gives (PD) It = 4.19 3f, where PD and I are expressed in practical units. A coulomb is defined as the quantity carried by a current of 1 ampere in 1 second. In general, then, the product of volts and coulombs gives joules. Also, since the practical unit of power, the watt, is defined as 1 joule per second, the product of volts and amperes gives the rate of work or power in watts. MEASUREMENT OF POTENTIAL DIFFERENCE 57 equivalent of thermometer was .32 g. . ' . Total water equivalent was 234.7 g. Duration of heating was 60 seconds. Mean current /, 4.08 amperes. There- fore P.D. was 54.90 volts. The reading of the voltmeter was 54.8, which FIG. 41 differs from the calculated P.D. by but .2%. The voltmeter was therefore correct within the limits of observational error. CHAPTEE V THE MEASUREMENT OF RESISTANCE 40. Ohm's law. In 1829 the Berlin physicist G. S. Ohm announced the famous law now called after its discoverer Ohm's law. This law asserts that when a continuous current is flowing through a given conductor, the temperature of which is kept con- stant, the ratio of the P.D. existing between the terminals of the conductor and the current carried by the conductor is a constant, no matter what the value of the current may be. Symbolically, PD PD = constant, or R. (1) It is customary, as above, to denote this constant, the value of which depends only upon the nature of the conductor, by the let- ter R, and to name it the electrical resistance of the conductor. The name is natural, because when the value of R is large, the amount of electricity which flows per second through the conductor as the result of a given P.D. is small, and vice versa. Ohm's law is to be regarded as an empirical law, which may be verified directly by varying the value of the P.D. between the terminals of a given conductor and measuring the corresponding currents. The most convincing evidence for the rigorous correctness of the law is found, however, in the agreement which exists between precise observations, like those made with a Wheatstone's bridge, and cal- culations which depend upon the assumption of this law. So far as the most delicate measurements have been able to show, Ohm's law is not, like Boyle's law, an approximation, but is, rather, a law of rigorous exactness. 41. Units of resistance. The absolute unit of resistance is defined as the resistance of a conductor which conveys one absolute electro-magnetic unit of current when its terminals are maintained at a P.D. of one absolute electro-magnetic unit. The practical unit 58 MEASUREMENT OF RESISTANCE 59 of resistance is the resistance of a conductor which carries one practical unit of current (1 ampere) when its terminals are main- tained at a P.D. of one practical unit (1 volt). This practical unit of resistance is called the ohm. Thus volts = ohms. amperes Since 1 volt = 10 8 absolute units of P.D., and 1 ampere = 10" 1 absolute units of current, it is evident that 1 ohm = 10 9 absolute units of resistance. It appears at once from the definition of R, namely PZ>//, that the resistance of a conductor is determined once for all when an absolute measurement has been made both of the P.D. existing at any time between its terminals and of the corresponding current. Thus the resistance of the conductor ab in the preceding experi- ment is known, at the mean temperature of the calorimeter, from the measurements there made upon PD and /. When such an absolute measurement of the resistance of a wire has been once made, it is very easy to preserve the wire so that no change takes place in its character, and it is also easy to reproduce the temper- ature at which its resistance was determined. Hence with the aid of this wire and an ammeter standardized by comparison with a silver voltameter, it is easy to standardize any voltmeter; for according to Ohm's law the P.D. between the terminals of the wire will always be the product of the resistance of the wire times the current flowing through it, so that, R being known, we have only to vary the current / in a known way and take in every case the product RI in order to obtain PD. This is a method actually used in standardizing laboratories for calibrating voltmeters. When one standard resistance has been found by any absolute method, other standards can easily be obtained by the method of comparison to be described in section 43. Many elaborate investi- gations have been undertaken for determining accurately the length of a mercury column 1 sq. mm. in cross section which will carry exactly 1 ampere of current when its terminals are maintained at a' P.D. of 1 volt, that is, the length of a mercury column 1 sq. mm. in area which has 1 ohm of resistance according to the above 60 ELECTRICITY, SOUND, AND LIGHT definition. The methods employed for measuring the P.D. have not generally been calorimetric methods like the above, but have involved electro-magnetic principles to be considered later. The mean of the results of the most eminent observers gives 1 ohm as the resistance at 0C. of 106.3 cm. of mercury 1 sq. mm. in cross section. Hence, at the Electrical Congress in Chicago in 1893, the resistance of a mercury column 1 sq. mm. in area and 106.3cm. long was adopted as the "international ohm." Stand- ards of resistance are made by comparison with the resistance of such a mercury column. 42. Laws of resistance. Experiment shows that R varies with the length, the area, the temperature, and the material of the con- ductor. It is found to be directly proportional to the length and inversely proportional to the sectional area. For metallic con- d ductors it is found to in- ^\ crease with the tempera- ture, the coefficient of 1 increase per degree being nearly the same for all FIG. 42 J pure metals. If the separate resistances of conductors connected in series are R v R 2 , and R 3 (see Fig. 42), the joint resistance of the three, that is, the total resistance R between the points a and d is easily proved to be Z r (2) For, if PD : represents the potential difference between a and b, PD Z that between b and c, PD S that between c and d, and PD that between a and d, then, since the current which is flowing in all parts of the circuit must be the same, we have ^. (3) But PD = PD l + PD 2 + PD S . (4) ( 5 ) (6) PD PD PD , Hence _i + _* + _. ( 5 ) MEASUREMENT OF RESISTANCE 61 If the conductors are connected in parallel (see Fig. 43), it can easily be shown .that the joint resistance R, i.e. the total resist- ance between the points a and I, is given by the equation R I R 2 R 3 For, if PD represents the potential difference between the points a and b } then, by Ohm's law, PD PD PD "3*3 FIG. 43 R R l R 2 But the total current / between a and b must be the sum of I v J 2 , and I y PD PD PD PD Hence -_ = _. + _. + _.. (8) PD PD PD R* R~ R n R (9) 43. Theory of the Wheatstone bridge. Since at the present time standards of resistance are obtained not from the absolute measurement of P.D. and current, but from comparison with standards already existing, a very accu- rate method of com- paring resistances is of extreme impor- tance. Wheatstone '&. bridge is an instru- ment By means of which such accurate comparisons can be made. It consists of a divided circuit abc and adc (Fig. 44) com- posed of three known resistances P, Q, R, and of the unknown FIG. 44 62 ELECTRICITY, SOUND, AND LIGHT resistance X, the value of which is to be found. The resistances P and Q are in general the parts ab and be respectively of a long German silver wire connecting a and c. As soon as the current has become constant in the circuit, a definite P.D. must exist between the points a and c. Hence the \total fall of potential along the branch abc must be the same as the fall of potential along the branch adc. It is evident, there- fore, that there must exist somewhere on abc a point which has exactly the same potential as the point d on adc. This point can /easily be found by connecting one terminal of a galvanometer at d, /and sliding the other terminal along abc until the point b of no J deflection is reached. This must be the point which has the same ! potential as the point d. When the bridge has been put into this condition of balance, let PD^ be the difference of potential between a and d, PD^ that between d and c, PD S that between a and b, and PD 4 that between b and c. Also let / x and / 2 be the currents flowing in the upper and lower branches respectively. Then, by Ohm's law, PD l PD 2 ~ = and - = ' PI). PD, R PD. Hence ~JT ^' ^ = ^ P Similarly PD PD But since b and d are points of equal potential, TT fl 11 R P V QX Hence, finally, = ' or X = JL (^ * As soon, then, as a condition of balance has been found, X is obtainable at once in terms of the known resistances P, Q, and ^. In the slide-wire form of bridge (Fig. 45) abc is a uniform wire. Since in this case resistance is proportional to length, P/Q is simply the ratio of the lengths P and Q. The heavy connecting strips of brass have negligible resistances. MEASUREMENT OF RESISTANCE 63 44. The mirror D' Arson val galvanometer. The accuracy with which the point b (Fig. 45) can be located, and hence the accuracy with which Xcaii be found, evidently depends upon the sensitiveness FIG. 45 of the galvanometer G. There are two general types of galvanometer. The first, like the tangent galvanometer (Fig. 22, p. 35), consists of a needle suspended in the middle of a coil of wire and hanging parallel to the plane of the coil. When a current flows through the coil the needle tends to set itself at right angles to the plane of the coil. This form of galvanometer becomes very sensitive when the coil is brought very close to the needle and the magnetic field in which the needle hangs is made very weak (see sect. 57). But a weak field means that the galva- nometer is very susceptible to external magnetic changes. For FIG. 46 this reason this so-called Thomson form of galvanometer is much less convenient for general laboratory work than is the D'Arsonval, or moving-coil, form. This form (see Fig. 46, and also Fig. 60, p. 80) 64 ELECTRICITY, SOUND, AND LIGHT differs from the Thomson in that the current is made to pass through a movable coil suspended in the field of a fixed magnet, instead of passing through a fixed coil at the center of which a movable magnet is suspended. The plane of the coil is initially parallel to the hues of magnetic force. As soon as the current traverses the coil it tends to rotate so as to place its plane at right angles to the direction of the magnetic lines. A mirror M is attached to the coil so that, with the aid of a telescope and scale placed at a distance, an extremely minute deflection of the coil can be detected. Since in this galvanometer the magnetic field in which the coil is suspended is very powerful, slight changes in the external magnetic conditions have no observable influence upon the galvanometer readings. EXPERIMENT 5 Object. To test as accurately as possible, by means of Wheatstone's bridge, the laws of series and parallel connections. Directions. I. Setting up the galvanometer. After setting the coil approxi- mately parallel to the direction of the magnetic field, and leveling the gal- vanometer until there is perfect freedom in the swing of the coil, shut out all air currents by covering the galvanometer with a box provided with a glass window in front. Then, setting the scale and telescope at a distance of about a meter from the galvanometer, focus the telescope upon the re- flected image of the scale, as follows. Locate first the image of your eye in the galvanometer mirror and set the telescope pointing approximately along the line of sight. Then, sighting along the telescope, not through it, adjust the scale vertically until its reflected image is seen in the mirror. Next focus the eyepiece of the telescope until the cross hairs appear most distinct. This is accomplished by sliding the eyepiece alone in or out of the telescope tube. Now, leaving the eyepiece as adjusted, focus the telescope until the image of the scale, as seen in the telescope, coincides with the image of the cross hairs. This coincidence is obtained by the method of parallax as follows. The head is moved slightly from side to side while the scale is viewed through the eyepiece. If there is no apparent motion of the cross hairs with respect to the scale, the image of the scale formed by the objective is exactly coincident with the cross hairs, and observations may be begun. If the cross hairs appear to move to the right across the scale when the head moves to the right, the cross hairs are farther from the eye than is the image of the scale, so that the draw tube .of the telescope should be pulled back slightly toward the eye. MEASUKEMENT OF EESISTANCE 65 FIG. 47 If the cross hairs appear to move over the scale in a direction opposite to that of the motion of the head, the draw tube should be pushed in until all apparent motion vanishes. II. Connections. Following Figure 45 connect the unknown resistance x l to the bridge in the place of X. Also connect a known resistance at R. Both of these connections should be made of heavy strips of copper of neg- ligible resistance. The resistance R may be either a single standardized coil, or a resistance box. Figure 47 shows the manner in which the coils in such a box are connected. When a plug is taken out the current is obliged to pass through the resist- ance coil which lies beneath it in the box. When the plug is put in this particular resistance is cut out. K l and K 2 are keys which may be combined, if desired, into a double key of the form shown in Figure 48, but in this case the battery circuit should be arranged to close through the upper contact points, the galvanometer circuit through the lower contact points. This is because the equation for the bridge was deduced from Ohm's law, which holds only for constant currents, not for variable currents such as exist at the instant at which the current of a battery is closed (see Chap. XIV). K z is a damping key which merely short circuits the galvanometer. It is used for bringing the coil quickly to rest (see Chap. XII). Determine first the value of x l as explained in the next para- graph. Then in a similar manner determine x 2 . From these values calculate the com- bined resistance of x 1 and x 2 , first, when connected in series (x s ), and second, when con- nected in parallel (x p ). Then, FIG. 48 j -v using x l and in series as A, determine the value by observation and compare with the calculated value. Make a similar determination and comparison for the parallel connection. III. Readings. If the known resistance R is adjustable, i.e. a resistance box, remove some plug, close the short-circuiting key K s , and, setting the slider about the middle of the bridge, press first the battery key K l and then the galvanometer key K . Notice the direction of the galvanometer deflection. Now try in the same way some widely different resistance for R. If the deflections are in the same direction, both of these values of R lie on the same side of the value necessary for a balance. If the deflections are 66 ELECTRICITY, SOUND, AND LIGHT in opposite directions, the value sought lies between those used. Select some value of R which will bring the position of the slider near the mid- dle of the bridge and complete the adjustment by moving the slider. If the known resistance is not adjustable, try two different positions of the slider to ge't an idea of the necessary position. The short-circuiting key K s was only closed at first in order to prevent violent throws. As soon as a first approximation toward a balance has been made, it should be opened and the accurate setting made with it in this condition. After each deflec- tion the coil may be brought quickly to rest by closing the key 7\ 3 . In order to find the position of the slider for which there is no deflection, find two points which correspond to the smallest observable deflections to the right and to the left, K 3 being open. If these points are more than .2 or .3 mm. apart, the galvanometer is not sufficiently sensitiv3. The point midway between these two is the point of balance. Since imperfect con- tacts in the resistance box may lead to erroneous results, it may be well to try the point of balance when the resistance R is made up of different com- binations of the resistance coils of the box. If a large variation results, clean the plugs by wiping them with a cloth wet with benzine.* Do not push down hard on the plugs ; merely twist them until you are sure that they make good contacts. Be sure that all other contacts are firm. In order to minimize the error due to lack of uniformity in the bridge wire and also to eliminate thermo-electric currents, f take the mean of two determinations of X, making the second determination as follows. Inter- change the positions of R and X and obtain values P' and Q' for the lengths on the bridge wire. EXAMPLE The resistance R was given a value of 20 ohms and remained unchanged throughout the experiment. For x l the point of balance on the bridge wire was found at P = 30.41 cm. and Q = 19.59 cm. When R and x l were interchanged P was found to be 30.49, and Q, 19.51. The average value 19 55 of x l was, therefore, - x 20, or 12.84 ohms. Similarly x 2 was found to 30.45 be 21.07 ohms. Also # was found to be 34.00 ohms, a value which agreed to within .3 per cent with the calculated value of 33.91 ohms, obtained by adding x l and x v The value found for x p was 7.93 ohms, which agreed well with the calculated value of 7.94, found from the law for parallel con- nections by taking the reciprocal of the sum of the reciprocals of x l and x 2 . * The chief source of error in all work with resistance boxes lies in imper- fect contacts at the plugs. Hence it is of great importance to have the plugs scrupulously clean, and to twist each plug in the plug seat at the time of its insertion. t Thermo-electric currents are discussed in Chapters VI and X. This inter- change of R and X eliminates these currents only partially. CHAPTEE VI TEMPERATURE COEFFICIENT OF RESISTANCE; SPECIFIC RESISTANCE 45. Disadvantages in the slide-wire form of Wheatstone's bridge. For accurate work the form of Wheatstone's bridge described in the preceding experiment is open to four serious objections. First, the bridge wire may not be uniform in diam- eter and resistance throughout its length. The exchange in posi- tion of R and X tends to minimize the error thus introduced in assuming P and Q proportional to their lengths; but since, except for a point of balance at the center of the wire, a certain portion of the wire is included in both positions in the larger of the two quantities P or Q, the error mentioned is not entirely eliminated. In general, however, the error due to this cause is small. A second and more serious objection is this. Since the wire is in general not more than a meter in length and must, for mechan- ical reasons, be of considerable diameter, its resistance is small. The fall in potential which takes place along the wire is conse- quently small, so that a small change in the position of the slide causes but a small variation in the P.D. between the terminals of the galvanometer and consequently but a small deflection. A third objection is that thermo-electric differences of poten- tial are introduced which cannot rigorously be eliminated by the interchange of R and X. A thermo-electric difference in potential exists between the two junctions of a metal with a second metal if these junctions are at different temperatures (see Chap. X). The source of this error in the wire bridge is largely the result of the heating by the hand of the junction of the brass slider and the German silver wire. Between this junction and one of the other junctions of the wire with the brass strips of the 67 68 ELECTRICITY, SOUND, AND LIGHT bridge there may exist a P.D., causing a deflection of the galva- nometer even when P y Q, X, and R are in balance. The point of balance actually observed is where this P.D. is counteracted by a small P.D. in the opposite direction, due to having the resistances slightly out of balance. Since the heating due to the presence of the hand is not constant, the assumption of a constant thermal current is not rigorous, and the interchange of R and X does not entirely eliminate the error. Another place where this error may be introduced is at the brass-platinum contact of the key in the galvanometer circuit. Since the platinum contact is thin, and consequently cannot differ appreciably in temperature at its two junctions with the brass of the key, the error here introduced is small. On the assumption of a constant error it would be elimi- nated entirely by reversing the connections of the battery, since this reverses the directions of all the differences of potential in the bridge except those due to the thermal effects. A fourth objection lies in the fact that only resistances of about the same order of magnitude may be compared. For evi- dently the observational error introduced in setting the slider or reading its position causes a minimum error in the determination of the resistance when P and Q are the same, i.e. when R and X are equal. If R and X are widely different, this error may become quite large. 46. The post-office-box form of Wheatstone's bridge. A form of bridge known from its original use by the telegraph department of the British post office as the post-office-box bridge overcomes the first, second, and fourth of these objections, and also reduces largely the errors due to thermo-electromotive forces. It possesses also the advantage of compactness, since the resistances P, Q t and R, as well as the keys for the battery and galvanometer circuits, are contained in the same box. The essential difference between the box and slide-wire forms consists in the substitution for ab and lc (see Figs. 44 and 45) of two series of accurately determined resistance coils. This is shown diagrammatically in Figure 49. The branches ab and lc consist of 1000-, 100-, 10-, and 1- ohm coils. The branch ad is merely the ordinary set of coils to be found in a resistance box. The ratio of MEASUREMENT OF SPECIFIC RESISTANCE 69 P to Q is limited to values of 1000, 100, 10, 1, T ^, T ^, and In the use of this form a fixed ratio is chosen for P/Q, and the resistance 7? is varied until a balance is obtained. For example, 7 w 100 WO woo WOO FIG. 49 in measuring a resistance X, known to lie between 3 and 4 ohms, the ratio P/Q was made 1000, and E for a balance was found to be 3682 ohms, thus giving X= 3.682 ohms. /^~N EH ED d] IZH CDiCZi ; CD EH a cn cula FIG. 50 In practice the boxes are not made in the form shown in Figure 49, which has been introduced as a transition between Figure 44 and Figure 50, in which is shown the ordinary scheme 70 ELECTEICITY, SOUND, AND LIGHT of connections. The dotted lines indicate connections made within the box. Figures 51 a and 51 b show, in diagram and in perspective, a box similar to that represented diagrammatically in Figure 50, OOOOGGGO ,., R TJA. GA. FIG. 51 a the only difference being that in Figure 51 the battery is con- nected between the points b and d of Figure 49, and the gal- vanometer between the points a and c. This has no effect upon FIG. 516 the equation of balance, for in considering the diagram of the Wheatstone bridge it is evident that if the battery and galva- nometer are interchanged, the condition for equilibrium becomes MEASUREMENT OF SPECIFIC RESISTANCE 71 P/B = Q/X. But this is true if P/Q = li/X. That is, the gal- vanometer and battery are interchangeable in position without affecting the relation P/Q = R/X* 47. Temperature coefficient of resistance. As was stated on page 60, the resistance of a conductor depends upon its tem- perature. The temperature coefficient of resistance of a conductor is denned as the ratio between the change in resistance per degree change in temperature and the resistance at 0C. It will be seen that the definition is altogether analogous to that given for the coefficient of expansion of a gas. In symbols, if R t and R Q rep- resent the values of the resistance of any conductor at C C. and 0C. respectively, then the temperature coefficient of resistance a is denned by the equation This coefficient is positive for the metals, and has a value of approximately .0038 for all pure metals. It is negative for car- bon ; that is, the hot resistance of an electric-light carbon is less than its cold resistance. It will be seen for the above equation that the resistance R t of a conductor at any temperature may be written p p n A /9N M t H Q (L+at). (2) 48. Specific resistance. The specific resistance of a substance at any temperature is denned as the resistance between two oppo- site faces of a centimeter cube of the substance. Since the resist- ance of a conductor varies directly as its length, and inversely as * The reason that post-office-box bridges are commonly connected as in Fig- ure 51, rather than as in Figure 50, may be seen from the following rule taken from Maxwell's "Electricity and Magnetism," Art. 349. The deduction of the rule cannot be taken up in a text of this scope. " The rule, therefore, for obtaining the greatest galvanometer deflection in any given system is as follows. Of the two resistances, that of the battery and that of the galvanometer, connect the greater resistance so as to join the two greatest to the two least of the four other resistances." 1 " 1 Since a galvanometer will practically always have a higher resistance than an ordinary battery, and since, in the use of the post-office box (Fig. 49), the resistance E is almost always larger than X, and, therefore, P larger than Q, it will be seen that, in accordance with the rule, the galva- nometer should be connected across ac, and the batteiy across bd, if the highest sensibility is to be obtained. ELECTRICITY, SOUND, AND LIGHT its cross-sectional area, the resistance R of any conductor, at the temperature for which its specific resistance r is known, may be written 7 = , or RA (3) in which / is the length in centimeters, and A the average area of cross section expressed in square centimeters. Knowing, then, the resistance R at a given temperature and the dimensions of a conductor, its specific resistance at that temperature may be readily calculated from (3), and if its temperature coefficient is also known, its specific resistance at 0C. may then be obtained from (2) by throwing it into the form (4) at EXPERIMENT 6 (A) Object. To determine the temperature coefficient of a commercial copper conductor. Directions. I. Connections. The copper wire of which it is desired to find the temperature coefficient of resistance is wound on a wooden frame and supported by a wooden top within a thin, brass tube closed at the lower end. Heavy copper wires pass from the coil through the cover (see Fig. 52). An open- ing in the cover admits a ther- mometer supported so as not to touch the metal. This tube is immersed in a large vessel of water to which the heat is applied. The resistance of the wire is determined by means of a post- office box, first at the temperature of the room, then at a series of higher temperatures. This data is plotted as a curve on coo'rdi- : S=:5B ' nate paper, using temperatures for abscissas and resistances as ordinates (see Fig. 53). The curve, which is a straight line, may be pro- longed backwards, and its intercept on the axis of resistances (i.e. on the MEASUREMENT OF SPECIFIC RESISTANCE 73 ordinate corresponding to the temperature 0C.) taken as R . The slope of this line, namely divided by R Q gives a. II. Observations. Measure the resistance of the coil, including the con- necting wires, at the temperature of the room, as follows. Make P/Q j, Temp. Resist 21.2 0.814 34.0 0.852 47.3 s 0.893 0.935 71.5 3 0.966 .961 -.751 70.0 oc =0.00402 FIG. 53 and vary R until a balance is obtained to within 1 ohm. (For example, 3 ohms too small, 4 ohms too large.) It will be best to provide the gal- vanometer with a damping key as in Experiment 5, and to keep the latter closed during this operation. Now make P/Q iA R will be between 300 74 ELECTRICITY, SOUND, AND LIGHT and 400 ohms. Again find the value to within 1 ohm. Suppose R is found to lie between 368 and 369. Then make P/Q ^?- and find Pi to the nearest ohm. If R lies between 3682 and 3683, take the number which gave the smallest deflection, e.g. 3683. Then X = 3.683. If the galvanometer is sensitive, the accuracy can be pushed at least one place farther by the method given below in (B), but with the accuracy attainable in the temperature readings of this experiment, it is scarcely advisable to attempt greater refinements in the resistance measurements. Keeping P/Q = - 1( \ 00 , raise the temperature of the water about 10 above that of the room. Stir thoroughly, and by regulating the flame of the Bunsen burner keep the temperature of the water as constant as possible. Since the air surrounding the coil is a poor conductor of heat, the coil will be slow in rising to the temperature of the water. Follow its rise by varying the resistance in the post-office box and take a reading of this resistance as soon as it becomes constant. Note the temperature of the thermometer in the inner tube. Next begin to plot your readings as follows. Plot temperatures along the X axis of the coordinate sheet, using, for example, 1 division to repre- sent 2C., or choosing some other scale which will make 70 or 80 degrees occupy nearly the full width of the page. Plot resistances along the Y axis, using any convenient scale. The method of selecting a scale which will make the fullest use of the sheet may be seen from the example presented in Figure 53. The first two observed temperatures were 21.2C. and 34.0C. The increase in resistance corresponding to this increase of 12.8 in tem- perature was .038 ohm. In raising the temperature from 0C. to 70C. it was estimated that there should be an increase of about 5.5 times this amount, or approximately .210 ohm. Now there were on the particular sheet shown in the figure 110 available spaces along the Y axis. Dividing the approximate number of thousandths of an ohm, namely 210, through which the resistance was expected to be raised, by the number of available divisions, gave 1.95 as the number of thousandths of an ohm to be repre- sented by 1 space, if the sheet was to be completely utilized. Since this number was very inconvenient both for plotting the graph and for reading the values from it, the nearest whole number above 1.95, namely 2, was taken as the number of thousandths of an ohm to be represented by 1 division. The general rule for this choice would be as follows. Select the nearest whole number larger than the quotient found as above except where this number would be 3, 6, 7, or 9, in which case it would be more convenient to choose 4, 5, 8, or 10 respectively. Do not call the intersection of the two axes zero resistance, but make it instead some convenient number which is about 10 per cent less than the resistance of the coil at the temperature of the room (see Fig. 53). Having plotted the first two points, raise the temperature by about ten- degree steps, and plot each point as soon as it is obtained. If any point MEASUREMENT OF SPECIFIC RESISTANCE 75 does not lie on, or very near to, the right line joining the first two points, repeat the observation before raising the temperature farther. If results continue to be erratic, wipe all the plugs with a clean cloth moistened with benzine and look to your contacts. For example, turn each plug until it catches. Do not, however, push down hard upon the plugs, as this may spring the lugs and thus injure the box. After carrying the temperature to 70C. or 80C., disconnect the connecting wires from the heater, connect their terminals to each other, and measure the resist- ance of these wires. III. Calculation*. Draw a straight line which will come as near as pos- sible to all of the plotted points. Produce the curve thus found until it cuts the Y axis (the line of zero temperature) , and take this point of inter- section as the resistance of the coil at 0C. Subtract from this value the resistance of the connecting wires to find R Q . Find a by dividing the slope of the curve by R Q (see Fig. 5o) . (B) Object. To find the specific resistance of copper at 0C. Directions. In order to determine a specific resistance the dimensions of the wire must be accurately measured. This condition necessitates a large wire, and hence one of small resistance. But neither the slide-wire nor the post-office form of bridge is suitable, without modification, for the measurement of a resistance which is so small as to be comparable with the resistances of the con- tacts or of the connecting strips. These are often several thousandths of an ohm. A very satisfactory method of eliminating all errors due to contacts, connecting wires, or thermal electromotive forces, and of measuring with considerable accuracy a re- sistance of a few thousandths of an ohm is as follows. Let large connecting wires a and b from the post-office box terminate in two holes 1 and 3 (Fig. 54) bored in a wooden block and filled with mercury. Let an auxiliary wire c connect holes 1 and , and the wire of unknown resistance X complete the circuit by connecting holes 2 and 3. Let the surfaces of contact of all the wires with the mercury be well amalgamated by dipping the ends of the wires into nitric acid, then into mercury, then rubbing dry with filter paper. Let P/Q be made J- ^" and let R be varied until a change of 1 ohm causes a change in the direction of the deflection. Then take the permanent deflection produced by closing K z (Fig. 45) when R has the lower value, say 6 ohms. Then add 1 ohm to R and take the permanent deflection in the other direction. If, for example, the first de- flection were 4.35 mm. and the second 62.70 mm., then the value of R for FIG. 54 o 76 ELECTRICITY, SOUND, AND LIGHT a perfect balance would have been + - = G.065 ohms. Now. 4.35 + 62.70 with as little delay as possible, remove J\T, transfer b from 3 to 2 (see Fig. 55) , and again find precisely as before the value of R which would produce a balance. If this is found to be, for example, 4.107 ohms, then obviously X = .001958 ohm. Repeat the observations and see how nearly the difference between the two values of R can be checked. Changes may occur in the individual values of R because of temperature changes, but the differences should remain very nearly constant. Next take the mean of some ten or more measurements of the diameter of the wire, using the micrometer caliper. Measure with a tape the distance /between the two points on the wire to / " which it was immersed in the mercury. p IG 55 From these data and the temperature coefficient as found in (A) compute the specific resistance of commercial copper at 0C. by the aid of equa- tions (3) and (4). EXAMPLE (A) The curve plotted as above (Fig. 53) was the record of this experiment. Tabulated in one corner are found the various values of the resistance and temperature, the value of 7t , the value of the slope of the line > and the temperature coefficient ex.. (B) Total resistance = .00595 ohm ; temperature = 26 C. ; resistance of connecting wires and contacts = .00403 ohm; length of wire = 137 cm.; mean radius = .205cm.; therefore specific resistance at 0C. =.00000107 ohm = 1.G7 X 10~ 6 ohms = 1670 absolute units. CHAPTER VII GALVANOMETER CONSTANT OF A MOVING-COIL GALVANOMETER 49. Galvanometer constant. In the formula for the current flowing through a tangent galvanometer, namely ffrttmti /= -2^T' it is seen that in general the current is not proportional to the deflection, although it would be so if deflections were always kept so small that tan 6 were approximately equal to 6. In galvanom- eters in which the current is proportional to the deflection, the reduction factor by which the angle of deflection must be multi- plied in order to give the current is known as the galvanometer constant. Thus if /represents the current which produces in such a galvanometer a deflection of radians, then the definition of the galvanometer constant K is given by the equation J=M or A-=|- (1) That is, the galvanometer constant is defined as the constant ratio between the current and the deflection produced "by it. This ratio for small angles is still called in many cases the galvanometer constant, even though the instrument be one with which the ratio may not remain constant for large angles. It is the object of this discussion to find the expression for this constant in the case of a moving-coil (i.e. a D'Arsonval) galvanometer. To do this it will be necessary first to reconsider the definition of unit current given in section 27, page 34. 50. Restatement of definition of unit current. Unit current has been defined as a current which, when flowing in a conductor of unit length bent into the arc of a circle of unit radius, exerts unit force on a unit magnetic pole lying in the plane of the 77 78 ELECTRICITY, SOUND, AND LIGHT 1 c m \ conductor and at the center of the arc. Thus the unit pole at (Fig. 56) is urged from the plane of the paper toward the reader with a force of 1 dyne if the current in the wire has unit strength. Since action and reaction are equal and opposite in direction, the conductor is urged from the reader toward the paper with an equal force. Now the magnetic field due to the isolated unit pole has a strength at all points on the wire of unity, and a direction at right angles to the wire, since its lines of force are straight lines radiating from it in all directions. Instead, then, of stating the definition of unit current in terms of the force which the current exerts on the magnet, we may state it in terms of the force which the magnetic field exerts on the conductor. Thus unit current is that current unit length of which, when flowing in unit field at right angles to the direction of the field, experiences unit force. In symbols, the force F, in dynes, ex- erted on / cm. of length of a conductor carrying / electro-magnetic units of current, by a field of strength &C, at right angles to I is, by o FIG. 56 definition of current, (2) Xieft Hand Magnetic 51. The motor rule. The relation between the directions of the magnetic field, the current in the conductor, and the force exerted on the conductor may be seen from the figure to be stated in the following rule, known, from its especial application to the direct-current motor, as the " motor rule." Extend the thumb, forefinger, and second finger of the left hand in directions at right angles to one another ; let the forefinger point in the direc- tion of the magnetic field, and the second finger in the direction of the current; the thuml) will then point in the direction of the force acting on the conductor (Fig. 57). FIG. 57 MOVING-COIL GALVANOMETER 79 52. Faraday's explanation of electro-magnetic forces. The fact that the force between a magnet and a current does not act along the line connecting the magnet and the conductor, but at right angles to this line, constitutes a striking difference between electro- magnetic forces and the forces met with in the study of mechan- ics ; for these uniformly act along the lines connecting the acting bodies. Further, the forces of mechanics, excepting gravitation, act only as the result of the transmission of stresses through matter. Because of the difficulty of the conception of action at a distance, that is, action assumed to take place without the inter- vention of a medium, and in order to admit of a more perfect visu- alization of the actions of magnets and electrical charges, Faraday conceived of these electrical and magnetic actions as having their seat in the lines of force ; that is, he imagined, merely as an aid to thinking, that these lines of force constituted a sort of system of invisible, stretched, elastic bands endowed w r ith the following prop- erties: (1) a tension in the direction of their length, and (2) a repul- sion at right angles to this direction, so as to cause them to act on one another as if a hydrostatic pressure existed at right angles to their direction. Thus in Figure 58, a, which shows the field existing between two opposite magnetic poles, the lines at the top and bot- tom were thought of as forced out by a repulsion, not balanced, as at the center, by an equal and opposite repulsion due to other lines. Again in Figure 58, b, is shown the field surrounding a wire carrying a current toward the paper, while in Figure 58, c, is the resultant field due to the presence in the uniform field of a of the conductor of b. The resultant field at p is reenforced, while that at q is weakened. The lines are therefore crowded together more closely at p and less so at q. There results from these hypothetical 80 ELECTRICITY, SOUND, AND LIGHT characteristics of the lines an unbalanced repulsion forcing the conductor across the field toward the bottom of the page, in accordance with the demands of the motor rule. This mechanical picture of the action of electric and magnetic forces has so per- meated the literature of electricity and magnetism that a familiarity with it is of importance to the student of the subject. 53. The equation of the moving-coil galvanometer. In Figure 59, a, is shown a diagram representing the magnets and one single loop of the coil of a moving-coil galvanometer (see also Fig. 60). Suppose that a current of /units is flowing through the coil and that the strength of the mag- netic field between N and S is 3{ units. Then the left side of the loop is urged toward the reader with a force of JOT dynes (sects. 50 and 51) and the right side is urged away from the reader with an equal force. Upon the horizontal wires no force acts, for they carry currents parallel to the direction of the field. If the distance between the two sides is d centimeters, the moment of force Fh acting to twist the loop about a ver- tical axis midway between its sides is given by FIG. 59 Fh = II (3) Since Id is simply the area a of the loop, this equation may be written Fh = la&C. (4) If there are n loops of average area a, the total twisting moment FIG. 60 MOVING-COIL GALVANOMETER 81 is of course lancH*', and if WQ call an the total area of the coil and represent it by A, we have ~Fh = IA&C. (5) This is the value of the couple which acts on the coil so long as the plane of its loops is parallel to the direction of the field cK. Under the influence of this moment of force it rotates until it is brought to rest by the restoring moment due to the torsion of the suspending fiber. Suppose that when equilibrium is established the coil has rotated through an angle (see Fig. 59, b). The couple arm will then have changed from d to d cos 0, so that the moment of force producing the deflection is now IA&C cos instead of IA&C, while the restoring moment is TJ9, if we represent by T the mo- ment of torsion of the suspending fiber. Hence the equation of equilibrium of the moving-coil galvanometer is In deducing this expression we have assumed a rectangular coil, but it can very readily be seen that the result is precisely the same whatever its shape. For any irregular coil may be considered to be made up of infini- tesimal rectangular elements (see Fig. 61) and we have just seen above that the moment of force acting on a rectangular coil is IctcK cos0, where a is the area of the rectangular element. The total moment is therefore 2/#<5f cos0 or cos 0, where the total area of the coil, found by summing up the infinitesimal areas, is A. Equation (6) shows that the current / is not proportional to the deflection 6, but rather to - - If, however, is so small that cose' cos may, without appreciable error, be taken as unity, then equation (6) becomes and it is clear that the galvanometer constant K, i.e. 1/6 (see eq. (1)), is given by _ 82 ELECTRICITY, SOUND, AND LIGHT This expression makes it obvious why K is called the galvanometer constant, for it involves only the nature of the suspending fiber ( T Q ), the area of the coil (A), and the strength of the magnetic field (cf(). Keeping clearly in mind the restriction limiting the determina- tion and use of K to small angles, it is possible to find K for any galvanometer by measuring the current which produces a small angular displacement and dividing the current by the displacement. For some purposes it is more convenient to have this reduction factor expressed, not in terms of current per radian, but in terms of the current necessary to produce a deflection of 1 mm. on a scale at a distance of 1 in. The current in amperes necessary to do this is known as the figure of merit of the galvanometer. Since the mirror of a galvanometer actually turns through one half the angle through which the reflected ray is rotated, and since 1 mm. is .001 m., it is evident that the figure of merit k of a gal- vanometer is 7T/2000. Again, instrument makers often express the " sensibility " of a galvanometer in terms of the number of megohms (million ohms) which would have to be placed in series with it in order to reduce the deflection to 1 mm. on a scale 1 m. distant, when a P.D. of 1 volt exists between the ends of this resistance. This is obviously the re- ciprocal of the figure of merit. Thus a galvanometer which has a constant K equal to .000004,7 being measured in amperes, has a figure of merit of .000000002 = 2 x!0~ 9 , and a sensi- bility of 1/2x10 = 500 megohms. 54. Method of determining K. The scheme of connections which is used for determining K is shown in Figure 62. The current from a Daniell cell B is made small by inserting into the circuit a large FIG. 62 ^^ n . .. e resistance R r Only a small fraction of this current is passed through the galvanometer G, which is placed in one arm of a divided circuit made up on the one side MOVING-COIL GALVANOMETER 83 of the small resistance R 2 and on the other side of the large resistance R s plus the galvanometer resistance G. The deflection corresponding to a given arrangement of the resistances is observed and the current passing through the galvanometer is calculated from the reading (P.D.) of the voltmeter and the values of the various resistances. K is then found by dividing the current by the deflection, the latter being expressed in radians. The calculation of the current follows directly from Ohm's law. If, as is usually the case, the resistance of the branched circuit from a to b is negligible in comparison with R v then the current I I in the main circuit is given by Since the potential difference between a and b is the same for both branches, the currents, / through the galvanometer, and / 2 through E 2 , will vary inversely as the resistances of their respective branches. That is (9) Since the sum of these two currents must be the current in the main circuit, we have /+/, = /,. (10) From (9) and (10) we obtain This shows that in general the current which flows through one side of a shunt is obtained by multiplying the total current by the resistance of the other side divided by the resistances of both sides. From (8) and (11) w^e at once obtain I in terms of the various resistances and the voltmeter reading P.D. If this current causes a deflection of d centimeters on a scale D centimeters from the mirror, the angle through which the mirror rotates (one half that through which the reflected beam rotates) is d/1 D. Hence 211) A \ ( G \ \ 84 ELECTEICITY, SOUND, AND LIGHT 55. Resistance of the galvanometer. The resistance of the gal- vanometer can be found most accurately by connecting it as the unknown to a post-office box and measuring it by the Wheatstone- bridge method, using another galvanometer. But if J? 3 is large (compared with G), G may be found very much more conveniently and with sufficient accuracy by connect- ing a Daniell cell B, a voltmeter V, a milliammeter A, and the galva- nometer G as in Figure 63. The read- ing of V in volts, divided by that of A in amperes, gives the resistance in ohms of G and A together, and the resistance of A will, in general, be either negligible or kno\vn. 56. Direct-reading moving-coil galvanometers. If the term A3i cos0 in the expression /= T 0/A&C cos# (see eq. (6), p. 81) can be made constant, the deflection will be proportional to the current for all values of 6. This condition means that the pole pieces must be so shaped that the field strength ctf acting on the vertical wires of the coil in any position shall be inversely pro- portional to the cosine of the angle through which it has been deflected. Instruments in which this condition has been met may be used for the measurement of widely different currents on a scale of uniform divisions. The common, commercial, direct-reading ammeter is an instrument of this sort. The plan of an ammeter is shown in Figure 64. M is a per- manent magnet of which aa f and W are the shaped pole pieces. E is a soft-iron cylindrical core which concentrates the field. The coil c is wound on a rectangular aluminum frame, and turns on pivots in jeweled bearings against the torsion of two flat spiral springs not shown in the figure. The pointer D is attached to the coil and moves over a scale S empirically cali- brated, but having practically constant divisions. The terminals of the coil are attached to heavy wires ww r which connect with the binding posts BB'. Across these wires, in shunt with the coil, are a number of small resistances r which carry the greater MOVING-COIL GALVANOMETER 85 part of the current, allowing only a fraction to pass through the movable coil. The current flows through the coil of the ammeter in a direc- tion such that it produces the magnetic field indicated by the small letters n and s. The coil therefore tends to rotate to the right. The angle 9 is reckoned from a position of the coil parallel to the lines of force NS. Thus 6 has a value of about 45 for the initial position of the coil shown in the figure. Owing to its rotation FIG. 64 o from this position, decreases to zero when the coil is parallel to NS, and then increases to about 45 for a position parallel to a'b. This means that cos increases to unity and then decreases. In order that 3i cos shall be constant 3i must be larger at the tips of the pole pieces a, a', b, and b f than at the center. This is accom- plished by giving the pole pieces the shape shown in the figure. A current-measuring instrument of high resistance may be cali- brated to read units of P.D., e.g. volts, as was explained on page 52. Such an instrument, of the movable-coil type, is the voltmeter. 86 ELECTRICITY, SOUND, AND LIGHT FIG. 65 a It differs essentially from the ammeter of Figure 64 only in the matter of resistance. Instead of the small resistances r in shunt between the binding posts BB' there is a single large resistance in series with the movable coil c. 57. Types of current-measur- ing instruments. Thus far two distinct types of instruments for measuring currents by the mutual actions of two magnetic fields have been described. In the first the current is measured by the deflection of a magnet which as- sumes a position of equilibrium under the action of an external field and the field due to a fixed coil through which the cur- rent to be measured is flowing. The tangent galvanometer is an instrument of this type, the external field being that of the earth. The Thomson galvanometer described in connection with Figure 46 is also an instru- ment of this type, but in it the external field is in general the resultant field due to the earth and a bar magnet attached to the galvanometer in the manner shown in Figure 65 a. This magnet is used to oppose the earth's field at the galvanometer, and, by weakening the resultant field, to increase the sensitiveness of the instrument. In the most sensitive instruments of the Thomson type the effect of the earth's field is reduced practically to zero by the use of a so-called astatic system. This consists of two oppo- sitely directed sets of small magnetic needles ns (Fig. 65 a) mounted in the same plane on mica disks which are attached to a light rigid frame d. The earth's field obviously tends to make one set rotate in one direction and the other in the opposite direction, so that, if the needles were exactly alike, the resultant FIG. 656 MOVING-COIL GALVANOMETER 87. effect would be zero. In order to avoid having the effect of the current 011 one set opposite to its effect on the other set, the two sets are hung in separate coils D and E, through which the current flows in opposite directions as shown in the figure. A commercial form of this instrument is shown in Figure 65 b. The D'Arsonval galvanometer and its practical form, the ammeter, are the chief representatives of the second type of instruments, in which a movable coil rotates in a fixed magnetic field due to permanent magnets. While instruments of this type cannot be made nearly as sensitive as the Thomson galvanometer, they are much more satisfactory for ordinary work because of the fact that they are not influenced by the magnetic disturbances due to the electric currents which flow within or near to most buildings. There is a third form of current-measuring instrument in which the movable coil is placed in the magnetic field due to a fixed coil. The principle underlying many dif- ferent varieties of instruments of this type may be seen from Figure 66 a. A fixed coil AS carrying the current to be measured sets up a magnetic field in which there is suspended by a helical spring T a movable coil CD connected in series to the fixed coil and also carrying the current to be measured. Since the magnetic field due to the fixed coil is proportional to the current, and since the reaction between the current flowing in the mov- able coil and this magnetic field is proportional to this current times the strength of field, it follows that the moment of force tending to rotate the movable coil is strictly proportional to the square of the current, provided that the relative' positions of the coils are kept unaltered. This condition is satisfied by twisting the helical spring until the torque it exerts balances the moment acting on the coil. The angle through which the FIG. 60 a ELECTRICITY, SOUND, AND LIGHT spring is twisted is proportional to this moment and consequently to the square of the current I. That is, IxV0, or I=kV0. (13) The reduction factor k, by which the square root of the angle must be multiplied to give the value of the current, may be found from an observation of the current and the corresponding deflections. Thereafter the current may be found from the relation of equation (13). A commercial form of this "dynamometer" is shown in Figure 6 6 fr. The great majority of alter- nating-current ammeters and voltmeters are instruments of the dynamometer type. For, in view of the fact that any change in the direction of the current takes place simultaneously in. both coils, the direction of the torque remains always the same ; so that, if the alternations are sufficiently rapid, an alternat- ing as well as a direct current will produce a constant deflec- pp , tion which is proportional to the mean square of the current strength. The dynamometer also has the advantage of having no permanent magnets whose magnetism may undergo slow changes with time. This is indeed the chief source of inaccuracy in cali- brated instruments of the D'Arsonval type. EXPERIMENT 7 Object. T find the constant K of a moving-coil galvanometer. Directions. Find the resistance of the galvanometer* by either of the two methods stated above, as directed by the instructor. Connect the *The galvanometer should be the ballistic one which it is intended to use in Experiment 8. MOVING-COIL GALVANOMETER 89 resistances, a voltmeter and a Daniell cell, as in Figure 62. See that the scale is normal to the line drawn from the zero reading to the mirror. Make 7^ = 10,000 ohms and R s = 1000 ohms. Make R 2 successively 1, 2, and 3 ohms, and observe the corresponding deflections. If these values of R v J? 2 , and R 3 do not cause deflections of from 3 to 10 cm. on a scale at 1 meter's distance, choose other values which will keep the deflections within about these limits. Since the chief source of error in this experi- ment lies in imperfect contacts, it is very important to make all connec- tions carefully and to see that the plugs, particularly of the resistance box R 2 , are well cleaned with benzine and carefully inserted. Reverse the direction of the current through the galvanometer and repeat. Read the P.D. on the voltmeter at the time of taking the readings of the deflections.* Calculate the value of K for each value of 7? 2 , using in each case the mean deflection for d. Neglect in the calculation any figures the dropping of which from the result will not introduce into the value of K an error larger than that due to the observational error in reading the voltmeter. Express K in amperes per radian and also in absolute electro-magnetic units per radian. Record also the " figure of merit " and the " sensibility" of the galvanometer. EXAMPLE The galvanometer was found to have a resistance of 536.8 ohms. The resistance R l was made 20,000 ohms, and R s 1000 ohms. When R 2 was 1 ohm the average deflection for direct and reversed currents was 3.10 cm. on a scale distant 143.4 cm. from the galvanometer mirror. As the volt- meter reading across the terminals of the Daniell cell used was 1.05 volts, this deflection corresponded to a current of .0000000342 ampere. Hence K was .00000315 ampere. Similar readings for R 2 = 2 and R 2 = 3 ohms gave 6.3 cm. and 9.48 cm. as values of the deflection, and .00000310 and .00000309 for the corresponding values of K. The average value of K was therefore .00000311 ampere, or .000000311 absolute unit of cur- rent. The figure of merit of the galvanometer was 1.55 x 10~ 9 , and the sensibility 645 megohms. * If the voltmeter is in demand it may be freed from this experiment as follows. Observe carefully one deflection when the voltmeter is connected as in Figure 62, then disconnect the voltmeter. If the deflection changes, the P.D. at the terminals of the cell when the voltmeter is disconnected is the voltmeter reading multiplied by the ratio of the second deflection to the first. Thenceforth use the cell without the voltmeter, but use in the calculations this corrected value of the P.D. CHAPTER VIII THE ABSOLUTE MEASUREMENT OF CAPACITY 58. Definition of capacity. The difference of electrical poten- tial between any conductor and the earth is commonly called simply the potential of the conductor and is designated by the letter V. In other words, the electrical potential of the earth is arbitrarily chosen as the zero from which the potentials of all other conductors are measured. It follows from this convention, and from the definition of P.I), given on page 11, that the electrical potential in absolute units of any conductor A (Fig. 67) on which there is a charge of Q units ^_^ of electricity is equal to the number of (+) ergs of work required to carry unit charge A of positive electricity from the earth E up to A against the force which the charge Q exerts upon this unit charge. 67 Suppose now that the quantity Q were to be doubled. The field strength at all points between the earth and A would obviously be doubled -also, and hence the work required to carry unit charge up to A would be doubled. In other words, under the conditions indicated, the ratio between the charge on A and the potential which this charge imparts to A is a constant. This constant ratio between the charge on a conductor and its potential is called the electrical capacity of the conductor, and is denoted by the letter C. Thus for the defining equation of capacity we may write ph aro . p n Charge = Q = Potential V We may state the definition in words as follows : The capacity of any conductor is the charge which must be placed upon it in order to raise its potential one unit above the potential of the eartli. 90 ABSOLUTE MEASUREMENT OF CAPACITY' 91 59. Factors upon which capacity depends. A charge placed upon an isolated conducting sphere establishes at any point out- side the sphere a held intensity of the same value as would be produced at that point were the whole charge concentrated at the center of the sphere. If, then, the size of the sphere A is increased while the charge upon it is kept constant, the unit charge which we are imagining to be brought up from the earth to A does not need to ap- proach so near to the center of the charge A hi order to be placed upon the sphere, and therefore does not require as great an amount of work. Indeed it may be shown, by a more extended analysis than will be attempted here, that the amount of work required varies inversely as the radius of the sphere. Since the amount of work required represents the potential of the sphere, it follows that the potential of an isolated conducting sphere carrying a given charge varies inversely as its radius, and hence that the capacity of a sphere varies directly as its radius. A second and very important factor upon which the capacity of a conductor may depend is the presence of neighboring con- ductors. Thus suppose that a negatively charged body B is placed near A, as in Figure 68. It is clear that the work required to bring a unit positive charge from the earth to A will now be less than when A was isolated, for the attraction which the negative charge on B exerts upon the unit positive charge which we are imag- ining to be brought from the earth to A will partially neutralize the repulsion due to A. Hence the capacity of A, i.e. the charge required to raise it to a given "*"*" potential, is much greater when B is near .b IG. 69 to it than when B is remote. If the body B is not initially charged, but is simply connected to earth by a conductor as in Figure 69 and then brought near to A, it will acquire a charge by electrostatic induction (see sect. 5, p. 4). That is, A will induce upon B a charge opposite in kind 92 ELECTKICITY, SOUND, AND LIGHT to that upon itself by attracting toward itself an unlike kind of electricity and by repelling a like kind along the wire toward the earth. The conductor B will therefore lower the potential and thus increase the capacity of A, just as well as though the negative charge had been imparted to it before it was brought near to A. In other words, the capacity of a conductor may be very greatly increased ~by simply bringing near it another conductor which is connected to earth. On account of this fact two conductors which are very close together and so arranged that one of them can be connected to earth are said to constitute an electrical condenser. 60. The condenser. It will be clear from the above discussion that if we wish to increase enormously the capacity of a conductor by the presence of an adjacent conductor, we have only to give to the two conductors such shapes that one can be brought almost into coincidence with the other. This condition is realized by making the con- ductors thin metallic sheets, or plates, which are separated from each other by a very thin layer of air, or other insu- lating material. Hence condensers usu- ally take the form of two sheets of tin foil separated by thin sheets of mica, or glass, or paraffined paper. To charge such a condenser we might connect one plate B to earth and join the other plate A to one terminal of any electrical generator 6, for example a static machine or a galvanic cell. But if we wished to raise A to as high a poten- tial as possible above the earth, we should also connect the other terminal of the generator to earth (Fig. 70), for then the full differ- ence of potential which the generator is able to maintain would be built up between the earth and A. Furthermore, since B is also connected to earth, its potential is necessarily zero, i.e. the same as that of the earth, since all points on a conductor in a static condition are always at the same potential (p. 12). Hence the capacity of A in this condition would be simply the charge upon it divided by the P.D. between A and B, i.e. by the amount of work required to carry unit charge from B to A. But it is clear that we could produce the same P.D. between A and B by connecting the ABSOLUTE MEASUREMENT OF CAPACITY 93 terminals of the generator directly to A and B without the inter- vention of the earth, for in either case the P.D. between A and B would be the total number of volts which the generator is able to maintain. Hence it is customary to charge a condenser by con- necting it, as in Figure 71, with a battery b and pressing the key k r It is evident that the two opposite charges upon A and B are neces- sarily of the same size, since A and B are simply the terminals of the generator &, and since positive and negative electricities always appear in exactly like amount whatever be the means by which they are generated. We have, then, the charge Q upon either plate A or B divided by the P.D. between A and B as the capacity of the condenser. We can obtain the P.D. at once with a volt- meter attached to the terminals of the cell b. To obtain the charge Q upon either plate, we discharge the | -L/ the galvanometer G. This is done by open- condenser through ing k l and closing k y We may conceive of the discharge as F IG . 71 being accomplished either by the passage of the total charge on A around to B (this would accord with Franklin's one-fluid theory, which made nega- tive the absence of positive), or by the passage of the total charge on B around to A (this would accord with the electron theory, which makes the negative the mobile kind of electricity), or by the passage of half of the charge on A around to B and half of that on B around to A (this would be in accordance with the old two-fluid hypothesis). In any case, however, the total quantity which would pass through the galvanometer would be the total charge Q on one of the plates. 61. To find Q from the throw 9 of the galvanometer, the gal- vanometer constant K, and the period t of its half vibration. While the quantity Q passes through the coil of the D'Arsonval galvanometer, it constitutes an electrical current which reacts upon the magnetic field of the galvanometer so as to impart 94 ELECTRICITY, SOUND, AND LIGHT an impulse to the coil. If i represent the mean value of this current, then the mean moment of force Fh acting on the coil while this current is flowing is given by ~Fh = i&CA (eq. (5), p. 81). (2) If r represents the time during which this discharge takes place, then the total impulse acting on the coil, i.e. the moment of force times the time, is given by Fhr = i-rWA = QfflA. (3) Now, by Newton's second law, the impulse, or product of the force by the time during which it acts, is equal to the momentum imparted (ft = mat = mv). Similarly, in rotation, the moment of an impulse, or the impulse multiplied by its lever arm, is equal to the moment of momentum, or angular momentum, imparted. If, then, / represents the moment of inertia of the coil and co the angular velocity imparted to it by the impulse, we have Jkr = Ito. (4) Furthermore, we have seen (p. 81) that the constant of the gal- vanometer K is equal to T /A&C. We get, then, by substitution of these new values of Fh r and &CA in (3), /*> = or Q = K^~. (5) A J.Q Since we do not observe directly the initial angular velocity of the coil, but rather the angular distance 6 through which it moves because of this velocity, we must find a way of expressing &) in terms of 6. This is most easily done by equating the initial kinetic energy of the coil, namely |-/ and 6 we tacitly assumed that the vibration of the galvanometer coil takes place altogether without frictional losses of any sort ; that is, we assumed an un- damped swing. The damping which in fact always exists may be taken into consideration as follows : If the line oo' (Fig. 72) represents the posi- tion of 'rest of the galvanometer, and if the amplitudes lt 2 , 8 , etc., of successive swings are represented by the distances of the successive turning points from the line oo' , then it is found by experiment that the law which always holds, FIG. 72 * See " Mechanics. Molecular Physics, and Heat," pp. 87-91. 96 ELECTRICITY, SOUND, AND LIGHT approximately at least, for any damped vibration is that the suc- cessive amplitudes bear to one another a constant ratio. If we call this ratio p, we have That is, 1 = p0 2 , 2 - p0 z , etc. or, in general, m = p n ~ m n . (8) This equation tells us that any amplitude m may be obtained from any later amplitude n by multiplying 6 n by the damping factor p raised to a power corresponding to the number of swings through which the damping acts between 6 m and n . In accordance with this rule the amplitude 6, which would have been attained in the first swing if there had been no damp- ing, may be obtained from the actual amplitude l of the first swing by multiplying 1 by p raised to the power corresponding to the number of swings through which the coil has been damped between the instant at which it starts and the instant at which it reaches r Since this is one half swing, we have 6 = pW r The value of p may be found by observing any two amplitudes, say the first and the twenty-fifth, and then substituting in equation (8), which, for this case, becomes \ a (9) We obtain, then, as the final form of the equation for the determina- tion of the quantity Q in terms of the throw of the galvanometer, (10) 7T If, then, V denotes the potential difference between the plates and the capacity of the condenser, we have ABSOLUTE MEASUREMENT OF CAPACITY 97 63. The ballistic galvanometer. It is evident that we cannot determine the capacity of a condenser by the above method unless we use a galvanometer which oscillates for a long time, that is, one which does not damp down rapidly ; for t cannot be deter- mined accurately unless a considerable number of swings can be observed. Furthermore, the damping law mentioned above obvi- ously cannot hold for a very rapid rate of damping, for if the sys- tem should not make more than one or two swings, we evidently could not say that the ratio of successive swings was constant. Now a galvanometer which is designed to reduce damping to a minimum is called a ballistic galvanometer. The only differences between a ballistic and a nonballistic D' Arson val galvanometer lie in the absence from the former of all mechanical damping devices, and in the fact that the coil is not wound on a conducting frame, for such a frame causes electro-magnetic damping.* 64. Units of capacity. If Q and V are measured in absolute electro-magnetic units, then C will also be obtained in absolute units ; but if Q is measured in coulombs and V in volts, then C will be obtained in practical units. The practical unit of capacity is named the farad in honor of Faraday. It is the capacity of a condenser which acquires a P.D. of 1 volt when it receives a charge of 1 coulomb. Thus . , coulombs 10' 1 farads = = = 10~ 9 absolute units. (12) volts 1 8 The farad is so large a unit that the microfarad (= .000001 farad) is the unit which is now most commonly in use. EXPERIMENT 8 Object. To make an absolute measurement of the quantity of electricity discharged by a condenser, charged to' a known difference of potential, and hence to determine the capacity of the condenser. Directions. I. Set up in the manner indicated in Figure 73 a standard condenser C (between .1 and 1. microfarad), a Daniell cell B, a voltmeter * The conducting frame rotating in the magnetic field of the galvanometer would have induced in it a current in such a direction as to oppose its motion. See Chapter XII. 98 ELECTRICITY, SOUND, AND LIGHT F,* a ballistic D'Arsonval galvanometer G, a commutator c, a damping key A-. 2 , and a discharge key k r This last instrument is merely a special form of two-point key (see Fig. 74). It is best to connect it so that when the tongue is pressed down the battery will charge the con- denser through the lower con- tact points and when the tongue is released the condenser will discharge through the galva- nometer by way of the upper contact points. If a commuta- tor is not available, commu- tate by interchanging the bat- tery terminals at d and e (see Fig. 73). II. See that the scale upon which you observe deflections is set normal to the line of Fi<;. 73 ^_^y sight, then bring the galvanom- eter quite to rest by closing 2 . If a swing of a few millimeters occurs, because of thermal effects, when A* 2 is opened, it will be easy with a little practice to press k 2 tempo- rarily at such instants as to check the slight remain- ing swing and bring the coil to rest even when k z remains open. Record this point as the true zero, then charge the condenser by depressing the tongue. Discharge by releasing it and observe the throw in millimeters.! Reverse the commutator and repeat. The mean of six throws, three to right and three to FIG. 74 * As in Experiment 7, the voltmeter need not be kept attached to the battery throughout the experiment, provided, in calculating capacity (equation 11), the voltmeter reading is multiplied by the ratio between two throws taken, one with the voltmeter disconnected, the other with it connected. t If it is found too tedious to bring the coil exactly to rest when & 2 is open, the throw can be taken quite accurately, even when there are one or two milli- meters of swing, in the following way. Observe the middle point about which ABSOLUTE MEASUREMENT OF CAPACITY 99 left, divided by twice the distance from the scale to the mirror, should be taken as the correct value of in radians. Since the zero may be slightly variable, a new zero reading should be taken for every throw. Further, if the first observations of the deflections do not agree closely, record no readings until successive readings in a given direction do agree.* III. To obtain ?, discharge the condenser again, and with a stop watch take the time of at least 20 half swings, timing your counts at the instants of passage of the coil through its zero position, not through its end positions. IV. To obtain p bring the galvanometer again to rest, charge and discharge the condenser, note carefully the first throw, and, calling this first turning point 1, the next 2, etc., count the turning points (the readings corresponding to them need not be taken) until the amplitude has been reduced to about one third its first value, then note carefully that ampli- tude. Calculate p from equation (9). V. Take K from Experiment 7, or from a value furnished by the instructor. Calculate in microfarads the value of the capacity of the con- denser used, and compare with the value marked upon the condenser. EXAMPLE The condenser used was charged to a potential of V 1.06 volts as given by the voltmeter reading. When discharged through the galvanom- eter it caused a mean deflection of 15.58 cm. on a scale at a distance of 143.4 cm. Reversing the direction of charging, an average throw of 15.56 cm. was observed. Hence = .05430 radians. The time of 20 half vibra- tions was found to average 125.8 seconds, hence t= 6.29 seconds. The damping factor p was found by observing 2 15.7 and 22 = 4.5. Hence p \\-~~ = 1.064. .-. pi =1.031. The galvanometer constant K as \ 4.o found in Experiment 7 was .00000311 ampere. Hence c _ .00000311 x6.29 x .05430 x 1.031 Ti-1.06 = .000000329 farad = .3293 microfarad. The condenser used was marked 1 microfarad by the manufacturer, hence the per cent of difference was 1.2. (This error is not larger than the uncertainty in the calibration of most voltmeters.) the oscillation takes place and call this the zero reading. Then discharge the condenser at the instant at which the coil is passing through one of its posi- tions of rest at the ends of its swing. The deflection will then be the difference between the zero and the extreme reading produced by the discharge. * The reason for this disagreement is that frequently the suspending wire acquires a " set" or tendency to a greater deflection in one direction than in the other. Taking several throws in the same direction will in general, however, result in constancy for that direction of throw. CHAPTEK IX COMPARISON OF CAPACITIES, THE DETERMINATION OF DIELECTRIC CONSTANTS, AND THE RATIO OF THE ELECTROSTATIC AND ELECTRO-MAGNETIC UNITS 65. Comparison of capacities by means of a ballistic galva- nometer. From the definition of capacity (in symbols, C = Q/PD) it follows that the quantities of electricity acquired by two separate condensers when charged to the same P.D. vary directly as their capacities. Since the throws of a ballistic galvanometer are proportional to the quantities of electricity passing through it, the ratio between the capacities of two condensers may be found by charging them to the same P.D. and noting the ratio of the throws they cause whqn discharged through the same ballistic galvanometer. If one of the condensers is a standard of known capacity, the value of the capacity of the second condenser may be found from this ratio. 66. The bridge method of comparing capacities. A comparison of capacities may also be made by a method which is analogous to the Wheatstone-bridge method of comparing resistances, i.e. a method in which a condition of balance is indicated by no deflec- tion of the galvanometer (a so-called " zero method "). The two condensers to be compared are connected as in Figure 75. C l and C z represent their capacities. R and S are two resistance boxes. The key K permits of the charging or discharging of the condensers. A galvanometer is connected between a and c. When the galva- nometer shows no deflection on charging or discharging, there exists a condition of balance for which, as is shown in the next paragraph, the following relation holds : /*T Of 1 = .^. C z R 100 COMPARISON OF CAPACITIES; Since by the assumption of no current through the galvanom- eter the points a and c are always at the same potential, it follows that during the whole time of discharge, for example, the P.D. between b and a is always equal to that between b and c. Now it follows from Ohm's law that if the P.D. between b and a is the same as that between b and c, the currents flowing hi the branches R and S are inversely proportional to the resistances of R and S. Since, however, the time during which the condensers are discharg- ing is the same for both, for otherwise a current would have to FIG. 75 flow through the galvanometer, it follows that the quantities dis- charged through R and S by the condensers 1 and 2 are propor- tional to these currents, and hence inversely proportional to the resistances R and S. That is, =! (1) Q 2 R But since a and c are always at the same potential, the condensers must always be charged to the same P.D., and hence the quantities which they hold must be proportional to their respective capacities. Substituting for QjQ^ the equal ratio CJC Z we obtain, as the con- dition which must exist for no galvanometer deflection, = 4- (2) 102 , SOUND, AND LIGHT 67. The bridge method, using an induction coil. No essential difference is introduced if the charging and discharging of the condensers in the method just described is accomplished, not by the key, but by the use of an alternating current. To this end the points b and d in Figure 75 are connected to the secondary of an induction coil, to the primary of which is connected the battery. The effect of the automatic make and break of the current in the primary circuit is to set up in the secondary a current increasing and decreasing in alternate directions. This charges the condensers, permits their discharge as it decreases, and, recharging in an oppo- site direction, again permits their discharge. If the balance is not perfect, there will be a rapidly alternating P.D. between a and c. In the place of the galvanometer a telephone receiver is therefore used, and the test for a balance becomes the absence of a buzz- ing sound in the telephone.* 68. The dielectric constant. When the capacities of con- densers which differ from one another only in the nature of the medium between the plates are compared by either of the methods given above, it is found that they differ widely. Hence Faraday introduced the term " specific inductive capacity " to denote the constant of the medium which is measured by the ratio between the capacity of a condenser which has the given medium between its plates and the capacity of the same condenser when air (or more strictly a vacuum} is betiveen the plates. Thus the specific in- ductive capacity of ordinary glass is from 2.3 to 2.8, that of hard rubber is from 2 to 3, that of paraffin 1.8 to 2.3, that of mica 2 to 4. This constant relates, of course, only to insulators or dielectrics. Hence it is now commonly called the dielectric constant. * Precisely the same equations as were developed in section 66 hold, not only for the scheme of connections mentioned in this section, but also for that shown in Figure 76, where the telephone Tand the induction coil have been iiitercharged. FIG. 76 DIELECTRIC CONSTANT 103 69. Dielectric absorption. The results of a comparison of two condensers by the bridge method, described in section 67, will often not agree closely with the results of a comparison by the method of ballistic throws. Particularly is this true if the condensers com- pared are made of quite different insulating materials, e.g. mica and paraffin. The reason for such a disagreement is as follows. The capacity of a condenser has been denned as the constant ratio between the charge given it and the P.D. to which it is charged. But when we measure a capacity by finding the ratio of the quantity discharged to the charging P.D., we tacitly assume the equality of the quantity of charge and that of discharge. As the result of a phenomenon known as dielectric absorption, such an equality does not always exist. The insulating medium between the condenser plates, called the dielectric, appears to absorb a portion of the charge. Thus if a discharged condenser be allowed to stand for a short time on open circuit, a second or even a third charge may be taken from it. This residual charge is practically negligible, however, in the case of the ordinary mica condenser. The amount of the absorption for any given condenser is some com- plicated function of the time consumed in charging and discharging. Methods of comparison, therefore, which employ markedly different times will not in general agree closely. 70. Laws for the combination of condensers. Using either of the methods of comparison described above, it is possible to test the laws for the combination of condensers. These laws _j_ Cy Q -^2. ^ ^[_ 3 follow at once from the de- nning equation of capacity. Thus in Figure 77 are shown three condensers of capacities C lt C 2 , and C 3 respectively, connected in parallel. Obviously they are all charged to the same potential. By definition, the quantities they receive, Q v Q 2 , and Q s , are then expressed in terms of their common P.D. as follows : Q, = C,(PD), Q. 2 = C 2 (PD), Q S = C,(PD). (3) The total charge Q which they receive is the sum of these separate charges. That is, Q = Q,+ Q z + Q y (4) 104 ELECTRICITY, SOUND, AND LIGHT Now if C represents their joint capacity, Q= C(PD), and there- fore, by substitution in (4) of the values of Q, Q v Q 2 , and Q s , we obtain That is, for condensers in parallel the joint capacity is the sum of the several capacities. In Figure 78 are shown these same condensers connected in series. PD V PD 2 , and PD 3 represent their respective differences of potential. In this case the total potential difference PD between the ends of the series is necessarily equal to the sum of the several P.D.'s. . That is, PD = PD^ + PD 2 + PD 3 . (6) The quantities of electricity held by all these condensers are the same. For obviously the + charge 011 condenser 3 equals the charge on condenser 1, since the generator B must develop equal amounts of positive and negative electricity. But since the + charge and the charge on the plates of a condenser PDo PO,. Ill 7 V 3 TB r are equal, the charge on con- denser 1 equals the + charge on the opposite plate. And this + charge is the same as the charge on the plate of condenser j, for the 4- charge 011 condenser 1 and the charge on condenser 2 are simultaneously produced by electrostatic induction, and, accord- ing to the laws of induction as discovered by Faraday, + and charges always appear in equal amount. From these considera- tions it is clear that But, by definition, Q = C(PD), Q^C^PDJ, Q Z = C Z (PD 2 ), and Q S = C S (PD) S . (8) By substitution in (6) of the values of PD, PD V P> 2 , and PD S , given in (8), it follows, in consideration of (7), that +- RATIO OF THE UNITS 105 That is, for condensers in series the reciprocal of the joint capacity is the sum of the reciprocals of the several capacities. Hence the law for capacities in series is similar to the law for resistances in parallel, and vice versa. 71. Standard condensers. Condensers carefully constructed of tin-foil plates and sheets of mica as the dielectric are arranged in boxes similar in form to resistance boxes. In Figure 79 is shown a standard form of subdivided condenser. The binding posts b and b' are connected to the brass strips M and N respectively. Between M and N, and capable of connection by plugs to either M or N, are other strips A, B, C, D, E, and F. The condensers are con- nected between these as indicated by the dotted connections. To connect between b and V any single capacity, for example that of the condenser between E and F, we connect M to E by a plug M -~ ! b 9 c A i i H } i : v^- B * f C - - r^^- D .^-^ L ^r E j~^ i -v_A F ./-^ i 6' i N ' FIG. 79 and also N to F. If it is desired to have the sum of the capacities of the three condensers CD, DE, and EF between the binding posts, since capacities in parallel are added, we connect the con- densers in parallel by plugging D and F to M, and also C and E to N. Since the sum of the reciprocals of several capacities in series is the reciprocal of their joint capacity, to obtain a small fraction, we connect, for example, A to M and C to N, thus putting con- densers AB and BC in series between b and b'. 72. The ratio of the electro-magnetic and the electrostatic units. As stated in section 28 (p. 35), the electro-magnetic unit of quantity is found to be equal to 3 X 10 10 electrostatic units, and the fact that this number is precisely the same as the velocity of light in centimeters has been one of the chief factors in the estab- lishment of the electro-magnetic theory of light. 106 ELECTRICITY, SOUND, AND LIGHT This relation between the units is made the object of an experi- mental verification in the experiment following this chapter. It is not, however, a quantity of electricity which will be directly measured in the two systems, but rather the capacity of a con- denser. From this latter measurement the former can be easily calculated, as will appear from the following considerations. If the electro-magnetic unit is in fact 3 x 10 10 electrostatic units, it follows that when a given quantity of electricity is measured in electrostatic units we should get a number 3 x 10 10 larger than when we measure the same quantity in electro-magnetic units. Furthermore, since the unit of P.D. in either system is the P.D. between two points when it requires 1 erg of work to carry unit quantity, measured in that system, between the points, it will be seen that a P.D. measured in electro-magnetic units should be 3 X 10 10 times as large a quantity as the same P.D. measured in electrostatic units. Now since C = Q/PD, it follows that C electrostatic Q electrostatic PD electro-magnetic C electro-magnetic Q electro-magnetic PD electrostatic = 9 x 10 20 . That is, if the ratio of the electro-magnetic unit of quantity to the electrostatic unit of quantity is 3 x 10 10 , the capacity of a condenser measured in electrostatic units should be 9 X 10 20 greater than its capacity when measured in electro-magnetic units. In order, then, to determine the ratio of the units of quan- tity, we have only to measure in electro-magnetic units, by the method of Chapter VIII, the capacity of a condenser of simple geometric form for which the capacity in electrostatic units may be calculated according to relations which will be established in the next section. Since, however, the capacity of a condenser of form suitable to this calculation is very small, we shall deter- mine its capacity in electro-magnetic units by comparison with a larger condenser, the capacity of which has already been measured in absolute electro-magnetic units by the method discussed in the preceding chapter. RATIO OF THE UNITS 107 73. The calculation of the capacity of a plate condenser in elec- trostatic units. We can compute C in electrostatic units at once, if we know the area of the condenser plates, the distance between them, and the dielectric constant of the medium. To see how this is done, let us first find the field strength in dynes which exists between two condenser plates in air when each of these plates is charged with Q electrostatic units. Let a represent the amount of electricity upon each square centimeter of each plate. c Th^s quantity cr is known as the density of charge. Consider the force / which the charge upon any little element of surface ds (Fig. 80) exerts upon a unit charge at any point a be- tween the plates. The number of units of charge upon ds is ads. If we represent the distance of this charge from a by r, then by the law of electrostatic force given in section 2, page 2, we have ds FIG. 80 ads (10) Now the component of this force which is normal to AB is the only one with which we are concerned, since it is evident that, except at the very edges of the plates,* the resultant force which the whole charge on AB exerts upon a unit charge at a is directed along the normal from AB toward CD. If f represents the value of the normal component of /, we have ,, o-ds f = y- cos 6. (11) * The error introduced by assuming that the lines of electrostatic force between the plates are everywhere normal to the plates, even at the very edges, is of the same order of magnitude as the ratio of the distance between the plates to their diameter. In the experiment which follows this chapter the first distance will be, perhaps, . 1 mm. and the last 20 cm. or 30 cm. Hence no appreciable error will be here introduced by assuming the lines to be everywhere normal to the plates. 108 ELECTRICITY, SOUND, AND LIGHT But if ds r represents the projection of ds upon a plane normal to r and passing through ds (which it will be remembered is an element of infinitesimal area), then we have (12) Hence f = T' ( 13 ) Now the solid angle subtended by ds at the point a is by defini- tion ds'/r 2 . If then we call this solid angle u, we have f'=(TU. (14) That is, the force, normal to the plate, which each element of charge upon the plate exerts upon a unit charge at any point between the plates, is equal to the density of charge upon the element, multiplied by the solid angle subtended by the element at the point consid- ered. Now if the density of charge is the same at all points upon the plate, then the total force F' which all the little elements of charge on AB exert on the unit charge at a is a times the sum of all the solid angles subtended by all the elements of surface on AB] i.e. it is a times the total solid angle subtended at a by the whole plate AB. But since the distance between AB and CD is very small as compared with the dimensions of these plates, the total solid angle subtended at a by AB is practically the solid angle subtended by a hemisphere at its center. The solid angle about a point is by definition the surface of a sphere of radius r about that point as a center divided by r 2 ; that is, it is 4 ?rr 2 = 4 TT. The solid angle subtended by a hemisphere at the center is therefore 2 IT. Hence we have f"=2iro: (15) Since the charge upon the plate CD exerts a like force upon the unit charge at a, the total field strength F between the condenser plates is given by Hence, if d is the distance from CD to AB, the P.D. between the condenser plates, i.e. the work required to carry unit positive DIELECTRIC CONSTANT 109 charge from one to the other, is 4 iro-cL But if A is the total area of one plate, the charge Q upon it is ACT. Hence the capacity of the condenser (7, which is by definition Q/PD, is given hy A(T A If we replace the air by a medium of dielectric constant K, we have, by definition of K, EXPERIMENT 9 (A) Object. To test the laws for the combination of condensers in series and in parallel by the comparison of several such combinations with a standard condenser. Directions. Using for C\ a subdivided condenser and for C 2 a single standard condenser of say ^-microfarad capacity, connect as indicated in Figure 75, except that a telephone receiver replaces the galvanometer G and a small induction coil replaces the discharge key K. R and S should be resistance boxes capable of a range of from 1 to 10,000 ohms. Give to R a fixed value and vary S until a balance is obtained. Approach the value of S which corresponds to a balance from the side of too large a resistance and also from the side of too small a resistance. Record both observations and use the average in the calculation. Arrange the plugs in the sub- divided condenser box so as to make the following combinations of condensers : (1) series connections, .2 and .05 microfarad ; (2) parallel connections .05, .05 and .2 microfarad. Compare these combinations with the J-microfarad condenser. Calcu- late the joint capacity from the laws of combination of condensers and compare with the observed value. (B) Object. To determine the dielectric constant of mica. Directions. Place a thin smooth sheet of mica between the metal plates of the parallel-plate condenser shown in Figure 81. Determine the capacity of the condenser so formed by the method used in (A), being careful to select a value for C 2 which is as -p IG near as possible to that of the unknown (say, .05 or .1 microfarad). Cut off three small pieces (3 to 5 mm. square) from the corners of the sheet of mica and, separating the plates by these, 110 ELECTRICITY, SOUND, AND LIGHT form an air condenser of the same thickness as that of the mica.* Measure its capacity in the same manner and determine the dielectric constant from the ratio of the capacity with mica as the dielectric to the capacity with air as the dielectric. (C) Object. To determine the ratio of the units of capacity, and hence of the units of quantity, in the electrostatic and the electro-magnetic systems. Directions. The capacity in electro-magnetic unit?- of a parallel-plate air condenser has already been determined (Exp. 9, (B)). It only remains to calculate its capacity in electrostatic units from the relation of equation (17). Measure carefully the thickness of the small mica strips used in Experi- ment 9, (B) to keep the plates separated. Take the average of at least ten observations for the value of d. Measure the radius of the condenser plates and from this data calculate the capacity. Determine the ratio of the E.S. to the E.M. value of this capacity, remembering that one micro- farad is 10~ 15 absolute electro-magnetic units (see p. 97). Extract the square root of this ratio to find the ratio of the electro-magnetic unit of quantity to the electrostatic unit of quantity. Compare this value with the value of v (the velocity of light), namely, 3 x 10 10 . EXAMPLE (A) The resistance S was kept constant at 1000 ohms. The value of the standard condenser was microfarad. The .05 and .05 microfarad in the multiple condenser box were connected in series. For a balance R was 74 ohms. Hence C = = .0247. The calculated value found by the relation = - -\ -, namely .0250, differed from the observed value by 1.2 per cent. Similar observations with .05 and .05 in parallel gave for a balance R = 1000, ,9=301; that is, C= .1003 observed value and .1000 calculated value, a difference of .3 per cent. (B) The average of thirty readings of the thickness of the mica was .0112 cm. The average of ten readings on the small strips used in sepa- rating the plates for the air condenser was .0119 cm. Using a .1 micro- farad and making It = 2000 ohms, S was found to be 130 ohms when the condenser contained mica. Therefore C mica = .00650 microfarad. Keeping 72 = 2000 ohms but using .05 microfarad, S was found to be 113 ohms when the condenser contained air. Therefore C air = .002825 microfarad. The balance in both cases was so exact that a change of 1 ohm in S disturbed it. Since the mica and air condensers differed in thickness, it * If the mica plate is not of uniform thickness, a correction must be made as is shown in the example following this experiment. RATIO OF THE UNITS 111 was necessary to increase the capacity of the air condenser to the value which it would have had if it had been of the same thickness as the mica. This value is evidently ^i | x .002825, or .00300. Hence the dielectric .00650 constant of mica was found to be ; = 2.17. .00300 (C) In Experiment 9, (B), the capacity of the air condenser had already been found to be .002825 microfarad for a thickness of .0119 cm. The radius of the plates was 10.9 cm. Hence the area yl=7r!0.9 2 , and S Ti-10.9 2 ., the capacity E.S. C- = - = 2496 electrostatic units. Hence - r^ : 4 ird 4 TT .01 1 9 the capacity E.M. -=8.84xl0 20 , and therefore the ratio of the electro- .002825 x 10- 15 magnetic unit of quantity to the electrostatic unit was found to be 2.973 x 10 10 . This determination differs from 3 x 10 10 by .9 per cent. CHAPTER X ELECTROMOTIVE FORCE AND INTERNAL RESISTANCE 74. Definition of electromotive force. Whenever two dissimilar substances are brought into contact one of them is found to take up a potential higher than the other. Thus, if copper and zinc are joined as in Figure 82, the zinc is found to acquire a slight positive charge, and the copper a corresponding negative charge. In order to assign a cause to this phenomenon, it is customary to assume that there exists at the surface of contact of copper and zinc an agent which we call an electromotive force (E.M.F.), which lias the ability to drive positive electricity from the copper to the zinc ; that is, in terms of the electron theory, the ability to drive negative electrons from zinc to copper. In obedience to this agent a current flows, according to our conventions (see p. 31), from Cu Zn the copper to the zinc until the action ^ ,.[4. ~ ^? of the agent is brought to rest by the F 82 back pressure, that is, the P.D. cre- ated between the zinc and the copper because of the charges which they acquire. We then define an electromotive force as any agent which is able to cause the ap- pearance of positive and negative electrical charges ; that is, any agent which" is able to set up a P.D. And we measure the strength of an E.M.F. by the P.D. which it is able to maintain. The relations of E.M.F. and P.I), are then simply the relations of cause and effect.* * It is to be noted that in the above the phenomenon of contact E.M.F. has merely been described, and the term E.M.F. defined. None of the present theories as to the nature and cause of this phenomenon have as yet been gen- erally accepted. Helmholtz imagined the phenomenon to be due to differences in the specific attraction of different kinds of matter for the two kinds of elec- tricity. In terms of the electron theory this view reduces to a difference in the specific attraction of different substances for the free electrons. If, however, there is a larger number of free electrons in one substance than in another, 112 ELECTEOMOTIYE FOKCE 113 75. Ther mo-electromotive forces. Perhaps the easiest way of showing the existence of electromotive forces of contact between metals is to place a sensitive galvanometer in a circuit consisting of two different metals, for example copper and zinc (Fig. 83), and then to heat one of the junctions. So long as the temperature of the two junctions a and b is the same, the E.M.F. from copper to zinc at a is necessarily exactly equal and opposite to the E.M.F. from copper to zinc at b, and hence no current can possibly flow through the circuit. But as soon as one junction is heated even a fraction of a degree above the other, a current is at once found to appear in the galvanometer. This is because the value of the Zn E.M.F. of contact between any two substances changes with the temperature. Hence if one junc- tion undergoes a change in tem- perature, while the other does not, the balance between the E.M.F.'s at the two junctions is destroyed and a current flows through the circuit which is proportional to the resultant E.M.F. of the circuit. If this current flows from a over to b through the zinc, and from b back to a through the copper, then there will of course be a continuous fall of potential from a to b in the zinc and from b to a in the copper. Furthermore, the sum of these falls in potential through the zinc and copper must of course be exactly equal, numerically, to the resultant E.M.F. in the circuit, for otherwise there would be either a cause (an E.M.F.) without.an effect (a P.D.), or else an effect (a P.D.) without a cause (an E.M.F.). 76. The E.M.F. of a galvanic cell. In a galvanic cell (Fig. 84) there is an E.M.F. at every surface of contact of two dissimilar substances, and the total E.M.F. of the cell is simply the algebraic sum of all these E.M.F.'s of contact. It is measured by the total P.D. which the cell is able to maintain between its terminals A there would, of course, be a tendency to diffuse more rapidly in one direction than in the other, and this fact would have to be taken into account, in connection with the specific attractions, in order to determine which of two substances would have an excess of negative electrons. 114 ELECTRICITY, SOUND, AND LIGHT a FIG. 84 and B on open circuit. That is, the resultant E.M.F. of contact causes a positive charge to accumulate on A and a negative charge on B until the back pressure through the cell, set up by the accumulation of these charges, just balances the total E.M.F. which is producing it, and thus brings to rest the action of this E.M.F. The water analogy of the galvanic cell is presented in Figure 85. The weight W corresponds to the E.M.F. of the cell. It is the cause of the movement of the water from B to A. Furthermore, just as the weight W is brought to rest as soon as the back force against the paddle wheels, due to the difference in levels in A and B, becomes equal to the forward force which W is able to exert, so the actions at the sur- faces of contact of the dissimilar substances in a galvanic cell are brought to rest by the P.D. which the E.M.F. of the cell sets up be-, tween its terminals. 77. Internal resistance of a galvanic cell. Connecting the ter- minals A and B of the galvanic cell by a conductor (Fig. 86) causes the P.D. between them at once to fall, just as connecting the tanks A and B by a pipe mn (Fig. 87) causes the difference in level be- tween them to diminish. Further, just as the weight W at once be- gins to cause the wheel to rotate and thus to restore the lost differ- ence in level, so the E.M.F. of the cell begins at once to act to restore the lost P.D. between the ter- minals of the cell as soon as they are connected by a conductor. FIG. 85 ELECTROMOTIVE FORCE 115 A ff! D e FlG. The fact that a current flows through the circuit composed of the conductor AB, the dissimilar metals Cu and Zn, and the liquid in the cell, means that there is a fall in potential from A to B in the external, and from B to A in the internal, portion of the circuit. The total fall in potential through- out all of the conductors of the circuit (Fig. 86) is equal to the resultant E.M.F. of contact of the dissimilar substances in the circuit, just as in the case of Fig- ure 83 total P.P. equals resultant E.M.F. Similarly, in the case of the water analogy of Figure 87, the water flows through the external circuit mn, and through the in- ternal circuit po ; and the sum of the falls in pressure in the external and the internal portions of the circuit is equal to the differ- ence in pressure produced by the wheel. In the case of a galvanic cell this E.M.F. is the result of a difference in the energy of combi- nation of the zinc with the liquid (for example, sulphuric acid) and the energy of combination of the copper with the liquid. In the case of the thermal E.M.F.'s the result- ant E.M.F. was the direct result of energy supplied to one junction in the form of heat. How near to the original P.D. the E.M.F. of the cell is able to maintain the terminals A and B (Fig. 86) when they are connected by the conductor depends wholly upon the relative amounts of diffi- culty which the conductor, on the one hand, has in discharging A and B, and which the E.M.F., on the other hand, has in forcing FIG. 87 116 ELECTRICITY, SOUND, AND LIGHT new charges through the cell up to these terminals ; that is, it depends upon the relative resistances of the internal and the external portions of the circuit (see p. 47). Similarly, the differ- ence in water level maintained by W between A and B (Fig. 87) will depend wholly upon the relative capacities of the pipes mn and op for carrying water. If the resistance of the conductor connecting A and B (Fig. 86) is very large in comparison with the resistance of the liquid be- tween Z and C, then the P.D. maintained between A and B will be practically the same as upon open circuit. But in any case the total E.M.F. of the cell must be equal to the sums of the falls in potential in the external and internal portions. Thus if I denote the current furnished by the cell, R its external and r its internal resistance, and if PD^ and PD 2 denote external and internal falls in potential respectively, then, by Ohm's law, PD 1 = RI, and PD 2 = rl\ or, since PD 1 + PD 2 = EHF, we have EMF = I(R + r). (1) Otherwise stated, while Ohm's law applied to any part of a circuit is PD/I = R, as applied to a complete circuit it is 78. Polarizing and nonpolarizing cells. A cell which consists simply of two dissimilar metals, for example zinc and copper, immersed in a conducting liquid like sulphuric acid, will, in gen- eral, have neither a constant E.M.F. nor a constant internal resist- ance ; for, since the positive ions which are already in the liquid are of hydrogen,* as soon as the zinc begins to go into solution in the form of positively charged zinc ions which repel the positive hydrogen ions away from the zinc plate and toward the copper plate, these repelled hydrogen ions begin to accumulate about the copper plate and there to change into neutral hydrogen mole- cules, and thus to alter completely the character of the surface of the plate. In effect the plate of copper is replaced by a plate of * H 2 S0 4 + H 2 -* H 2 + 211 + S0 4 (see Chap. XVI). ELECTEOMOTIYE FOECE 117 hydrogen. This alters completely the E.M.F. of contact at the surface of this plate, and it also alters the internal resistance of the cell. A cell of this sort is said to be a polarizing cell, because the current which it furnishes diminishes rapidly as the hydrogen accumulates. A nonpolarizing cell is one in which the plate toward which the ions are urged is immersed in a solution of a salt of the same metal as the plate ; for example, a copper plate in a copper sulphate solution. The character of the surface of this plate is then quite unchanged by the passage of the current, for the sub- stance deposited is the same as the substance of which the plate is composed. The most familiar of the nonpo- larizing cells is the Daniell cell (Fig. 88). It consists of a jar partly filled with a copper sulphate (CuS0 4 ) solution in which is placed a porous earthenware cup containing a solution of zinc sulphate (ZnS0 4 ). In the zinc sulphate is a zinc plate and in the copper sulphate a copper plate. The porous cup is simply to keep the liquids from mixing and yet to permit the passage of the ions, which travel with relative ease through its pores. Such a cell, if freshly set up, will have both an E.M.F. and an internal resistance which are very nearly constant. Since the copper is continually passing out of solution, the compartment S is usually kept filled with copper sulphate crystals so as to keep the solution saturated. 79. Methods of measuring the E.M.F. and internal resistances of Daniell cells. One of the simplest and most satisfactory ways of measuring the E.M.F. and internal resistance of a Daniell cell is as follows. First connect with, say, No. 18 copper wire, the terminals of the cell directly to the terminals of a voltmeter and read the P.D. indicated. Then replace the voltmeter by an am- meter and read the current furnished. If R v and It represent FIG. 118 ELECTRICITY, SOUND, AND LIGHT the resistances of the voltmeter and ammeter respectively (quan- tities which may be assumed to be known), PD the voltmeter reading, and / the ammeter reading, then, by Ohm's law, the cur- rent sent through the voltmeter in the first of the above obser- vations is PD/R V . But this same current is also equal to Hence our first equation connecting EMF and r is F EMF_(R v +r) PD = R v The equation which corresponds to the second observation is <^) = (4) We have only to solve (3) and (4) in order to obtain both EMF and r. When r is small as compared to R v , equation (3) gives EMF=PD (the voltmeter reading) as a first approximation. Now for the determination of a quantity so little con- stant as r it will, in general, be suffi- ciently accurate to make the assumption that the E.M.F. of the cell is the volt- meter reading, and then to solve equa- tion (4) for r. The value of the E.M.F. correct to a second approximation, which will generally be as accurate as the volt- meter can be read, may then be obtained by substituting the value of r as thus found "Fir 89 in equation (3), and solving for EMF. A second method of measuring internal resistance, and one which has the advantage of making it possible to determine whether or not, with the cell employed, r varies largely with the current used, is the following. The cell B, a voltmeter V, a resistance box R, and a key K are connected as in Figure 89, the resistances of contacts and connections between K and R being made as small as possible. The voltmeter reading when the key is up is approximately the E.M.F. of the cell. The voltmeter reading when the key is down is the P.D. across the INTERNAL RESISTANCE 119 resistance R. If R is small in comparison with the resistance of the voltmeter, then the current furnished by the cell when the key is down is the reading of the voltmeter (which we shall call simply PD) divided by R. But this same current is also given by dividing the E.M.F. of the cell by R + r. Hence we have EMF PD (R + r) R R(EHF-PD) In the use of this method, if results correct to even so much as 5 per cent, for example, are to be obtained, R should never be made more than a twentieth of the resistance of the voltmeter. Furthermore, instead of using the voltmeter reading when the key is open as the E.M.F. of the cell, it is, of course, more accurate to raise this reading to the true E.M.F. by substituting in equation (3) the approximate value of r, known by a rough preliminary observation either by this method or its predecessor. EXPERIMENT 10 Object. To determine the E.M.F. and the internal resistance of a Daniell cell. Directions. I. Clean the zinc of the Daniell cell carefully so as to remove any deposit of copper or copper oxide which may have accumulated because of the diffusion of the copper sulphate through the porous cup, and which ivS a fruitful source of polarization in the cell. Clean also the porous cup and put in a fresh solution of zinc sulphate. II. Determine the resistance of the coarser register of the milliammeter either by the Wheatstone-bridge method or else by sending a current from the cell through it and the voltmeter arranged in parallel. In the latter case the reading of the voltmeter divided by the reading of the ammeter will be the resistance of the latter. If the current is too large for the ammeter, reduce it by inserting a few feet of German silver wire. III. Determine the resistance of the voltmeter either by the bridge method, or else by putting the milliammeter (low register) and the volt- meter in series in the circuit of the cell. The reading of the voltmeter divided by the reading of the milliammeter will be the resistance of the voltmeter. 120 ELECTRICITY, SOUND, AND LIGHT IV. Connect the high register of the milliammeter directly to the ter- minals of the cell, or insert, if necessary to keep the deflection on the scale, a small length of German silver wire of known resistance per meter and read the current ; then replace the ammeter by the voltmeter and read the P.D. Compute r and EMF as indicated in section 79, equations (3) and (4), subtracting, of course, the resistance of the German silver wire if it is used. V. Connect the cell, a resistance box, a key, and a voltmeter, as indicated in the second method of section 79, using a short, thick wire (e.g. No. 16) to connect the key to the resistance box, and seeing to it that the contacts at all binding posts and plugs are thoroughly good. Compute r from equation (6) for at least three different values of R, e.g. 1, 2, and 5 ohms, and decide whether or not the internal resistance of a Daniell cell varies appreciably with the current taken from it. EXAMPLE I. The ammeter reading was .382 ampere when the reading of the volt- meter shunted across it was .04 volt, hence RA = .10 ohm. II. A current of .0027 ampere, as measured by the low register of the milliammeter, was passed through the voltmeter, which read 1.04 volts. Hence R v = 385 ohms. III. The voltmeter connected directly to the Daniell cell used read 1.05 volts. The ammeter when connected directly read .427 ampere. Hence r = 2.35 ohms, and EMF= 1.065 volts. IV. A P.D. of .33 volt was observed across the terminals of a resist- ance of 1 ohm. Substitution of these values and the value of the E.M.F. found above gave 1.065 - .33 00 , r = = 2.23 ohms. Similarly f or R = 2 ohms, PD = .85 volt and r = 2.39 ohms. Also for R = 3 ohms, PD = .60 volt and r = 2.32 ohms. CHAPTEE XI THE COMPARISON OF ELECTROMOTIVE FORCES 80. E.M.F.'s of polarizing cells. The methods described in the preceding chapter can obviously be employed only with cells, like the Daniell. which do not polarize. But having once found the E.M.F. of any nonpolarizing cell, the E.M.F.'s of any other cells, polarizing or nonpolarizing, can be very easily and very accurately determined by the use of a comparison method in which the first cell is taken as a standard.* We shall consider two such methods. 81. The ballistic galvanometer method. Since the E.M.F. of a cell is the P.D. which it maintains between its terminals on open circuit, it is only necessary to connect the terminals of any cell to the plates of a condenser in order to charge these plates to a P.D. *The cell most commonly used as a standard for the com- parison of E.M.F.'s is some modification of the Clark cell shown in section in Figure 90. When made under certain standard conditions it has an E.M.F. expressed by the following relation where t is the tem- perature. E = 1.434 x [1-. 00077(^-15)]. Platinum Wire Zn. Amalgam Zn S0,-m S0 4 Solution ZnSO, = Paste ofHg 2 SO, --g -Platinum Wire Pure Hg. FIG. 90 The cell, however, is incapable of sup- plying an appreciable current without considerable polarization. It is there- fore serviceable for accurate work only if kept on open circuit. It' will be seen that it may be used according to either of the two methods of this chapter. 121 122 ELECTRICITY, SOUND, AND LIGHT equal to the E.M.F. of the cell. Since, further, the quantity of charge on the plates of a given condenser is proportional to the P.D. between them (Q = Cx PD), we have only to charge the same condenser successively by means of different cells in order to obtain charges which are strictly proportional to the E.M.F.'s of the cells. Since, finally, the throws of a given ballistic galvanom- eter are proportional to the charges sent through it (Q = > see p. 96), we have, if O l and 2 are the respective throws pro- duced by discharging a given condenser through a ballistic gal- vanometer when the condenser has been charged, first by a cell of E.M.F. E 19 and second by a cell of E.M.F. E v (1) 82. The potentiometer method. The only disadvantage of the preceding method is that it is a deflection rather than a zero method. The fol- i i lowing method is just as faultless theoretically, and it can be made ac- curate to the fifth or sixth place of decimals if tem- perature and other conditions can be kept constant enough to make such a degree of accuracy desirable. A storage battery B, or any combination of batteries having an E.M.F. which is con- stant and somewhat higher than that of any of the cells to be compared, is connected as in Figure 91 to the ends of a wire ab of sufficiently high resistance to prevent a current from flowing which is large enough to heat the wire appreciably. The resist- ance of ab must also be so high in comparison with the internal resistance of B that the P.D. maintained by B between a and FIG. 91 COMPAKISON OF ELECTEOMOTIVE FORCES 123 I is greater than the E.M.F. of either E^ or E v the two cells to be compared. These cells are connected as in the figure, so that their negative terminals are joined to the same point a to which the negative terminal of B is connected, their positive terminals being connected to the contact points m and n of a double switch S, through which either cell can be put into connection with a resistance box R, a galvanometer G y and a wire w which can be touched at any point along the wire ab. The comparison of the E.M.F.'s is made as follows. Suppose that the switch S is turned so as to touch the contact m and thus put the cell E l into the galvanometer circuit. If now the free terminal of the wire w were to be touched to any point on ab, for example c, a current would always flow through G and R from right to left provided there were no cell in the circuit cGEma. The cell E 1 which is in this circuit, however, tends to force current in the opposite direction, namely from left to right through R and G. If, then, the P.D. which already exists between c and a, when the wire is touched at c, is greater than the E.M.F. E v a current will actually flow through G and R from right to left; but if the E.M.F. of the cell E l is greater than the P.D. which is maintained between the points c and a by the battery B, then a current will flow through the galvanometer from left to right. If the P.D. between c and a is exactly equal to the E.M.F. E v then no current what- ever will flow through the circuit of the cell, that is, through G. We have then only to find the point on ab which can be touched by the free end of the wire without producing any galvanometer deflection whatever, in order to obtain the point such that the P.D. between it and a is exactly equal to the E.M.F. of the cell. Suppose now that the switch S is turned so as to make contact with the terminal n of the other cell. If it is now found that some other point d is the point for which the galvanometer shows no deflection, then, if the wire ab is uniform, we have E l _ PD from c to a _ length ac E^ PD from d to a length ad While the point of no deflection is being found, the resist- ance R should be made very large (e.g. 20,000 ohms), for then no 124 ELECTRICITY, SOUND, AND LIGHT appreciable current will flow through the cell circuit, and hence this cell will not polarize, even if it be one of the polarizing kind. After the point of zero deflection is found, the resistance R may be varied, or in fact entirely removed, without altering the point of balance, for obviously at this point the cell is in exact equilib- rium with the P.D. between c and a; that is, it is virtually on open circuit. The only reason for introducing R at all was to prevent the cell from polarizing while the point of balance was being found, and to protect the galvanometer from too violent deflec- tions. Varying the value of R will then alter nothing save the sharpness with which the point of zero deflection can be located. EXPERIMENT 11 (A) Object. To find the E.M.F. of a Leclanch6* cell by comparing it with a Daniell cell by the ballistic-galvanometer method. Directions. First determine the E.M.F. of the Daniell cell E l by the method of Experiment 10. Then set up E v the Leclanche" cell E 2 , the double switch 5, the condenser C, the discharge key K, the galvanom- eter G, and the damping key, in the manner shown in the diagram (Fig. 93). Be very careful in so doing not at any time to short circuit the Leclanche" cell, for if it is once allowed to become polarized, it may take it hours to recover entirely its original E.M.F. Take in succession the throws of each cell. From the known E.M.F. of the Daniell cell and the mean of at least six throws, compute the E.M.F. of the Leclanch6 cell. * The Leclanch4 cell, shown in Figure 92, is of wide commercial use for cir- cuits closed only intermittently. It consists of zinc and carbon electrodes in a solution of ammonium chloride (NH 4 C1). The carbon electrode is commonly in a porous cup filled with manganese dioxide (Mn() 2 ). The action of the cell is as follows. The zinc goes into solution in the ammonium chloride in the form of posi- tively charged zinc ions. This gives a positive charge to the solution about the zinc plate, and hence the posi- tive ions already in solution (NH 4 ) are driven toward the carbon plate. Instead of going out of solution, how- ever, they decompose the water (H 2 O) about this plate and form ammonium hydroxide (NII 4 OH) and free II ions, which are driven out of solution at the plate in the form of hydrogen gas. The function of the MnO 2 is to dispose of this hydrogen by uniting with it to form Mn 2 O3 (manganic oxide) and water. This depolarizing action of the MnO 2 is quite slow, however, so that the cell polarizes rapidly on short circuit, but recovers completely when left to itself for a few hours. FIG. 92 COMPARISON OF ELECTROMOTIVE FORCES 125 (B) Object. To compare the E.M.F. 's of the Daniell and Leclanche' cells by the potentiometer method. Directions. Connect as in Figure 91, being very careful not at any time to short circuit the Leclanche cell. Let II be a resistance of from 10,000 FIG. 93 to 100,000 ohms while the point of balance is being found. If the point of no deflection is not sharply marked, reduce R after the point of balance has been approximately located. Compare results with those obtained in (A). EXAMPLE (A) The voltmeter reading for a Daniell cell was 1.080 volts. The volt- meter resistance was 197.5 ohms and the internal resistance of the cell as found by the first method of section 79 was 2.48 ohms ; hence the E.M.F. of the Daniell cell was 1.094 volts. A condenser of .1 -microfarad capacity charged by this E.M.F. gave a throw of 3.00 cm. on the galvanometer used. The same condenser when charged by the Leclanche cell caused a throw 4 os of 4.28 cm. Hence the E.M.F. of the Leclanch cell = x 1.094 = 1.563 volts. :W)0 (B) For the Daniell cell a balance was obtained when ac was 50.06 cm., for the Leclanche" cell when ad was 71.10 cm. Hence the E.M.F. of the Leclanche' cell = * ' x 1.094 = 1.556 volts. The values found by the 50.06 two methods differed by .45 of one per cent. CHAPTER XII ELECTRO-MAGNETIC INDUCTION 83. The principle of electro-magnetic induction. Thus far our discussion of electrical phenomena has been centered about Oersted's discovery that an electric current is surrounded by a magnetic held. We now come to the discussion of the applications of an exceed- ingly important discovery, published for the first time by Faraday in 1831, although the discovery itself was doubtless made a year earlier by Joseph Henry, at that time a high-school teacher in Albany. It is to the discovery of the principle of electro-magnetic induction that we owe the modern dynamo, the induction coil, the transformer, the telephone, and most of the other practical appli- cations of electricity to-day. This principle may be stated as fol- lows : Whenever a conductor moves in a magnetic field in such a way as to cut lines of magnetic force, an E.M.F. is induced in it. Thus when the vertical wire ac, shown in Figure 94, is moved across the field, an E.M.F. is induced in it, and since, in this case, the circuit is completed by the wire abc, this E.M.F. causes a cur- rent to flow about the circuit. No current whatever is found to be induced when the conductor moves parallel to the lines of force. These statements can be easily verified by moving a single wire, to the ends of which a sensitive galvanometer is attached, first in a direction perpendicular to, then in a direction parallel with, the field of a strong magnet. 84. Direction of the induced E.M.F. Since energy is expended when an electrical current flows in a wire, it follows from the principle of the conservation of energy that work must be done in inducing this current, Now we have already seen in sections 50 and 51 (p. 78) that any current in ac (Fig. 94) interacts with the magnetic field in such a way as to tend to push the wire across the field. If, then, work must be done in inducing the current in ac, 126 ELECTRO-MAGNETIC INDUCTION 127 it follows that the direction of the induced current must be such that the mechanical force with which it is urged across the field constitutes the reaction against which the work is done ; i.e. this mechanical force must be opposite to the direction of motion. For if the induced current could be in the opposite direction, the wire would be urged across the field in the direction in which it is going and electrical energy would be created indefinitely with- out any expenditure of mechanical energy whatever. The induced Field FIG. 94 current must then be in a direction such as to call into play a force which opposes the motion inducing it. This statement, which is seen to follow from the principle of the conservation of energy, is known as Lenz's law. A current, of course, cannot flow unless the circuit is complete, but an E.M.F. which would cause a current to flow in a direction such as to oppose the motion inducing it, is nevertheless always induced when a conductor cuts lines of force, and this E.M.F. causes positive and negative charges to appear upon the ends of 128 ELECTKICITY, SOUND, AND LIGHT the open circuit just as the E.M.F. of a cell causes such charges to appear upon the terminals of the cell. It is possible to formulate for induced currents a rule connect- ing the directions of the motion, of the magnetic field, and of the current, similar in form to the motor rule, but making use of the right hand instead of the left. Thus if the thumb and the first two fingers of the right hand are extended in directions at right angles to one another, and if the forefinger is made to point in the direction of the magnetic lines and the thumb in the direction of the component of the motion of the conductor which is at right angles to these lines, then the second finger will point in the direc- tion of the induced current. It will be obvious from a consideration of Figure 94 that this rule is only another form of statement of Lenz's law. From its especial application to the dynamo it is known as the dynamo rule. 85. The value of an induced E.M.F. The quantitative expres- sion for an induced E.M.F. may be found by a consideration of Figure 95. A and B represent heavy conducting bars of negli- gible resistance, between f the terminals of which . L . . . is connected the coil D. ! The line ab represents a sliding bar, also of negli- gible resistance. The dots represent the cross i section of a uniform rnag- i , * netic field of intensity d F 95 ofC perpendicular to the plane of the paper, in which this metal framework is supposed to lie. A force F acts on the slider ab, moving it with uniform speed through a small distance dx to a new position a'b f . This motion induces an E.M.F. which causes to flow around the circuit a'Db' a current of / absolute units. The reaction of the field &C on the / cm. of length of the slider is a force equal' to F and of value II 3i, dynes (sect. 50, p. 78). The mechanical work done in moving the coil dx cm. against this reaction is Ildidx ergs. If E ELECTKO-MAGNETIC INDUCTION 129 represents in absolute units the E.M.F. causing this current to flow, and if dt is the time during which the motion is taking place (i.e. the time during which we are considering the flow of current), then the electrical energy expended by the current in this time is Eldt ergs (see Chap. IV). Now since no part of the original mechanical energy is transformed into forms other than electrical (e.g. chemical), these two expressions, the one for the work done and the other for the electrical energy developed, must be equal. That is, II &djs = Eldt, (1) IcKdx E=. (2) Now since Idx represents the area across which the slider has moved, and since cK represents the number of lines per square centimeter, it is clear that Icftdx is the total number of magnetic lines cut in the time dt, and hence that an induced E.M.F. is numerically equal to the rate at which magnetic lines of force are cut. That is, one absolute electro-magnetic unit of E.M.F. is induced in a conductor which cuts one magnetic line of force per second. Since a volt is 10 8 . absolute units, a volt is induced in a conductor when it is cutting lines at the rate of 100,000,000 per second. The total number of magnetic lines which pass through any area is known as the magnetic flux through that area, or simply as the flux. Algebraically stated, if represents flux, eft field strength, and A area, then, by definition, in case the direction of the field is normal to the area, $> = A$C. (3) In case the field makes an angle 6 with the normal to the area, then < = AW cos 0. (4) That is, the flux across any area is the product of the field strength by the projection of this area upon a plane normal to 130 ELECTRICITY, SOUND, AKD LIGHT the direction of the field. In the expression for E just found, loKdx evidently represents the small increment of flux added (algebraically) during the element of time dt to the flux already inclosed by the circuit. Denoting the increment of flux by d<&, we have That is, the E.M.F. in absolute units induced in any closed circuit is the rate of change of the included flux. To be adaptable to this expression, Lenz's law may be restated as follows : The E.M.F. induced in any closed circuit will cause a current to flow in a direction such as to oppose the change in flux inducing it. 86. General statement of Lenz's law. As was seen above, the one and sufficient condition necessary for the electro-magnetic induction of an E.M.F. in a conductor is that the conductor shall cut magnetic lines of force. This cutting may take place, how- ever, in a variety of ways, and it is of some importance to deduce FIG. 96 a statement for Lenz's law which will be applicable to all cases. Thus if instead of moving a wire through a magnetic field, as in the above illustration, we push a magnet N/S from left to right up to a coil B, as in Figure 96, some of the lines of force which exist in the space surrounding the magnet, and move with it, are cut by the coil, and consequently a current flows in it in such a direc- tion that the field it sets up about the coil opposes the approach of the magnet ; for this case is obviously merely the converse of that considered in the preceding paragraph. Furthermore, pulling the coil away from the magnet must cause a cutting of lines in ELECTRO-MAGNETIC INDUCTION 131 the opposite direction, and hence a current in such a direction as to attract the receding magnet ; that is, to oppose the separation. Again, if instead of pushing up the magnet NS we set near B a second coil A (Fig. 97), which is connected to the circuit of a battery D through a key K, and then close K, we obviously cause B to be cut by the lines of force which spring into existence about A, precisely as though we had thrust NS up to B as above. The induced current in B must then be in such a direction as to cause repulsion between A and B\ that is, it must thrust lines of force through B in a direction opposite to that in which the rising current in A is thrusting lines through B. Conversely, when K is opened, the current induced in B must be in such a direction as to cause attraction between A and B] that is, in such a direction \ I \ ( ^ ' / itJ-^L. ' * / /XZIZIZ.Z , FIG. 9.7 as to thrust lines through B in the same direction as that of the lines which are disappearing from B because of the decay of the current in A. In its most general form Lenz's law may now be stated thus : Whenever an E.M.F. is induced by the relative motion of a conductor and magnetic lines, the induced current will always floiv in such a direction as to oppose the change which is induc- ing it. If the change is a mechanical one, the induced current is in such a direction as to oppose the motion ; if the change is a mag- netic one, the induced current is in such a direction as to oppose the magnetic change which is taking place. 87. The ideal dynamo. The most practical application of this principle of electro-magnetic induction is to the dynamo, a machine for converting mechanical energy into electrical energy. In its simplest form it consists of a rectangular coil of wire, rotating 132 ELECTRICITY, SOUND, AND LIGHT about the axis of its length in a uniform magnetic field, the direc- tion of which *is perpendicular to its axis. Suppose the coil to consist of one turn of wire and to be represented in cross section by the points A and B of Figure 98. The field of strength o) sin d. (10) If the coil consists of N loops in series instead of a single loop, the same E.M.F. is induced in each, and these separate E.M.F.'s must be added to get the total value of E Q . That is, E e = N&a>Bm0. (11) This is the E.M.F. in absolute units. Expressed in volts it is E e = N<&cosm0W-*. (12) Since, for any given case, N and are constants, if o>, the angular velocity, is kept constant, the induced E.M.F. varies directly as the sine of the angular displacement of the coil from a position at right angles to the field. A method for showing this relation will now be described. 134 ELECTRICITY, SOUND, AND LIGHT 88. Experimental analysis of dynamo induction. The ideal dynamo shown in Figure 99 is arranged so that at every revolu- tion of the crank the ratchet is lifted, allowing a spring to rotate the coil through 10. The terminals of the coil are connected to two separately insulated copper rings attached to the shaft. Copper strips or "brushes" bearing upon this ring are connected to a moving-coil galvanometer. As the coil moves through any ten-degree interval, an E.M.F. is induced which corresponds to the mean angular displacement of the coil from its original position. Since the total resistance through which this E.M.F. causes a cur- rent to flow remains the same throughout the experiment, the quantity of electricity passing through the galvanometer, and there- fore the galvanometer deflection, is proportional to this E.M.F. If FIG. 99 the observed deflections are plotted as ordinates and the corre- sponding values of the angular displacements as abscissas, the smooth curve drawn through these points will in general have practically a sine form. 89. The molecular theory of magnetism. The phenomenon of an induced E.M.F. as a result of the cutting of magnetic lines may be made use of to determine the magnetic condition of iron, for example the distribution of magnetism in a bar magnet. Before considering the method of doing this, it is desirable to consider the general nature of magnetism. It is a matter of experimental knowledge that into however small parts a magnet is divided each part still retains magnetic ELECTRO-MAGNETIC INDUCTION 135 properties. The assumption which naturally follows is that the mol- ecules of magnetic metals are themselves magnets. This hypothesis offers an easy explanation of the phenomenon of induced mag- netism, that is, the phenomenon of the acquisition of mag- netic properties hy a bar of iron, or of steel, as a result of its presence in a magnetic field. For we have only to assume that in an unmagnetized bar the molecular magnets have various orien- tations, so that their magnetic effects neutralize one another, and consequently the bar as a whole shows no magnetic properties. It is not improbable that the molecular magnets are arranged in small groups, as shown in Figure 100, the magnets of each group being in equilibrium. When the bar is under the influence of a magnetic field this condition of equilibrium is disturbed, and FIG. 100 FIG. 101 equilibrium does not again exist until some, at least, of the mole- cules have formed new groupings, oriented in the general direction of the impressed magnetic field. If the field is very intense, the alignment might perhaps be well-nigh perfect, as in the ideal diagram of Figure 101, which corresponds to a condition of entire saturation; but in an ordinary magnet the alignment would be very imperfect, and doubtless, also, some of the closed molecular groups 136 ELECTKICITY, SOUND, AND LIGHT would still exist. Figure 102 is an attempt to picture something more or less similar to this condition. When the directive force of the field is withdrawn, some of the molecules probably form new closed groups, but a large number retain the direction into which they have been rotated by the field. The fact that causes which facilitate a rearrangement of the molecules, such as heating or jarring by blows, facilitate also either the induction of magnet- ism in a bar placed in a magnetic field, or its demagnetization when withdrawn from the field, may be taken as evidence in favor of this hypothesis as to the molecular nature of magnetism. 90. The distribution of magnetism. Since through the space surrounding a magnet the magnetic lines of force are known to pass from its N end to its S end, it is in accordance with the I ^N^5/Tj)) v ^^^^^ FIG. 102 molecular hypothesis to assume that these lines form closed curves passing through the iron from the N end of each molecule to the S end of the next molecule (see Fig. 101). Wherever, then, lines of force leave the large magnet there are free molecular N poles, i.e. N poles of molecules not in immediate conjunction with neighbor- ing S poles (see Fig. 102). And similarly, wherever magnetic lines enter the large magnet there are free S poles. In the case of a bar magnet these lines of force leave the magnet along the entire N half and enter it along the entire S half after the manner shown in Figure 102. The distribution of these lines may then be taken ELECTRO-MAGNETIC INDUCTION 137 as a measure of the distribution of magnetism along the bar. Within the bar the actual number of lines is of course greatest at the center where the largest number of molecules are aligned in the direction of the axis of the magnet. But the magnitude of any action at a point in the immediate neighborhood of the bar depends upon the number of magnetic lines entering or leaving the bar at that point, i.e. upon the distribution of magnetism. The distribution is not uniform along either half of the bar, and except for a bar which is carefully magnetized it is also asymmetrical with respect to the center of the bar. 91. Experimental determination of the distribution of mag- netism. The distribution of magnetism in a bar magnet may be found by the use of a " test coil." This is a small coil c (Fig. 103) FIG. 103 which fits closely over the bar and which may be slipped along it, thus cutting the lines which enter or leave the magnet. The ter- minals of the coil are connected to a moving-coil galvanometer, the deflections of which are proportional to the quantity of induced electricity which flows through the galvanometer, and hence to the number of lines entering or leaving the magnet in the space over which the coil has moved. These deflections are plotted as ordinates with a direction up or down depending on the direction of the galvanometer throw (see Fig. 104), the abscissas corre- sponding to these ordinates being the distances from one end of the magnet to the middle of the space over which the test coil is 138 ELECTRICITY, SOUND, AND LIGHT moved. A smooth curve is then drawn through these points. The intersection of this curve with the axis of distances gives the dis- tance from one end of the bar to the point of no free magnetism, that is, to the magnetic center of the bar. 92. Location of the poles in a bar magnet. Now the moment of force which would act upon the magnet if placed at right angles to a uniform magnetic field is by definition proportional to the magnetic moment of the magnet (p. 18). It is the sum of the N.Pole: 3= abs.x ord. = 2120. , ord. = 270 .-. 2=7.9 S.Pole:* abs.x ord. = 2310. . * ord. = 283 length of magnet = 23 cm. distance between poles = 1C.2 cm. " " " = .71ensrth Test coil Deflection Coil OX + 46. mm. 12.5 0.5-1.5cm.+ 37. " 13.5 1.5-2.5 " 34. " 14.5 15.5 27. 16.5 24. 22. 18.3 15.5 10. 5.6 1. FIG. 104 moments due to all the separate quantities of magnetism in the bar ; that is, it is proportional to the sum of the products of all these quantities of magnetism by their lever arms taken with respect to an axis at the magnetic center of the bar. Now the ordinates of the curve are proportional to the quantities of magnetism. There- fore a number proportional to the magnetic moment may be found by adding all the products of these ordinates by their correspond- ing lever arms as found from the curve. The distance from the ELECTRO-MAGNETIC INDUCTION 139 point of intersection of the curve with the axis of abscissas to the foot of any orclinate is the lever arm corresponding to the quantity represented by that ordinate. But by definition the mag- netic moment is the product of the pole strength of the magnet and the distance between the poles. Now the sum of all the ordinates on one side of the magnetic center is proportional to the pole strength. Therefore the distance between the poles may be found by dividing the number found as above to be proportional to the magnetic moment by the number proportional to the pole strength. This distance between the poles of a magnet may be any fraction whatever of the length of the bar. In a permanent magnet it never exceeds five sixths of the length and is usually much less than this. EXPERIMENT 12 (A) Object. To plot the curve showing the variation of the E.M.F. induced in the coil of an ideal dynamo with the angular displacement of the coil. Directions. By connecting to the galvanometer terminals a battery which is short circuited through a piece of copper wire, find the direction of the current which corresponds to a given direction of deflection. Connect the galvanometer to the dynamo and allow the coil of the latter to turn through a ten-degree interval. Applying to this motion the dynamo rule, note the verification of the rule. Then following the method outlined in section 88, note the deflections of the galvanometer for ten-degree intervals during one complete revolution of the dynamo coil. Choose as the axis of abscissas a line about the middle of the sheet of coordinate paper and in the direc- tion of its length. Choose the scale for plotting the curve as large as the size of the sheet will allow. Tabulate the observed throws and the corre- sponding angular displacements in one corner of the sheet and let this sheet be the record of the experiment (see Fig. 105). (B) Object. To plot the curve representing the distribution of the mag- netic lines leaving a bar magnet and to find the actual distance between the poles of the magnet. Directions. The magnet mm' to be used is contained in a slotted brass tube on which a scale is ruled or pasted (see Fig. 103). The test coil c may be moved suddenly between the stops e and f of the frame. These should be so set as to admit of a motion of the coil of about 1 cm. For convenience of manipulation this distance may be made an even sub- t multiple of the length of the magnet (e.g. T ^). For the first deflection set the frame so that e and f are equidistant from the right end of the magnet. Now remove the stop / and the test 140 ELECTBICITY, SOUND, AND LIGHT coil from the tube. Then note the deflection caused by suddenly slipping the test coil back over the end of the magnet to a position against the stop e. This motion cuts all the lines that leave the end of the magnet, and the corresponding deflection is therefore a measure of the magnetism at the end of the bar. Plot it from the point on the X axis of your diagram which corresponds to the end of the bar. Replace the stop f and shift the frame until when the coil is against / it is in exactly the same position as when it was against e before. Place e so that the coil moves, say, 1 cm. between stops, then move it quickly from /to e and plot the throw at the point on the X axis which corresponds to the mid-point of its motion. . 3 J :g it -.-.i| : armature : ::::::::::::: o*-10' 10mm. 140' 7mm. 280' 18.5 ::: " 10'-20 14 150' 3.5 290" 16. :: ::Jj:::::::::::: :5::::::::::::::::|::^i 41|lilii| ;!Si!llffi!ij!!il!;i 60' 22. 19010.5 330' 4. - 90' 21. 220' 19. 360* 7.0 -- IN 120" 14. 250' 22. ::: 130' 10. 260 '21.5 ^ : 2f : ~- < : : - ^ : : H 2 Of ;!!!$ '2- 0! : |j j& : : iffi : : ) ! : : ii. ^|::::::^:X^|:::|::::::||::|^:: FIG. 105 Following the method outlined in section 91, continue until the opposite end of the magnet is reached. Then remove the stop e and take the deflec- tion caused by slipping the test coil entirely off the tube. Plot the curve showing the distribution of magnetism, utilizing in so doing the full size of the sheet of coordinate paper. Tabulate upon the same sheet the values used in plotting the curve. Also indicate on this sheet the sum of the products of the ordinates by their corresponding abscissas, the sum of the ordinates, and the value determined for the distance be- tween the poles. Let the curve and these tabulated values be the record for the experiment (see Fig. 104). CHAPTEE XIII THE CONSTANTS OF THE EARTH'S MAGNETIC FIELD 93. Direction of the earth's magnetic field. A magnetic needle placed in the earth's magnetic field, and free to rotate only in a horizontal plane, i.e. about a vertical axis, is not found, in general, to set itself exactly in coincidence with the geographical meridian. The angle by which the direction of the needle differs from the geographical north-and-south line is known as the variation at the point at which the needle is placed. Again, if the needle is free to rotate about a horizontal axis, it is not found to assume a horizontal position, but in the northern latitudes the N end dips downward. The angle which H the needle makes with the horizontal at any point is known as the magnetic dip at that point. For most localities within the United States the angle of dip is between 60 and 75. Let the intensity of the earth's field be denoted by /. Since, in most of the instruments in which a magnetic needle is employed, such, for example, as the tangent galvanometer, the needle is suspended so as to be free to turn only in a horizontal plane, it is usually only the component of / parallel to the horizon which it is necessary to determine. In the experiment which follows, however, we shall measure both the horizontal component H and the vertical component V of the earth's field, and from these measurements we shall deduce the absolute intensity I of this field in the direction in which it acts, and the angle which the direction of this field makes with the horizontal. The relations which exist between these four quan- tities, and by means of which these deductions can be made, are obviously the following (see Fig. 106)- 141 FIG. 106 142 ELECTRICITY, SOUND, AND LIGHT V I sin 8. (1) (2) (3) The quantities IT, V, and B, with the variation, are known as the magnetic constants of any given locality. 94. Absolute measurement of H and V by means of an earth inductor. The most rapid and convenient and, on the whole, one of the most satisfactory methods of measuring H or V is based upon the principle of electro-magnetic in- duction. It requires a ballistic galvanom- eter, a resistance box, and a "test coil" known as an "earth inductor" (see Fig. 107). Such an instrument consists simply of a circular coil of large area and large number of turns, so mounted that it may be rotated very suddenly through 180. The principle involved is that stated in section 85 (p. 130), namely, that the E.M.F. in- duced in a con- ductor is equal to the rate at which the conductor cuts lines of magnetic force. Thus consider the coil of the earth inductor to have an average area of a square centimeters and to be composed of n loops of wire. Let it be placed so that at" the beginning of the motion its plane is at right angles to the horizontal component of the earth's field. In Figure 108 the heavy line represents the coil in this position. As then it rotates through one fourth revo- lution about a vertical axis lying in the plane of the coil, each loop cuts Ha lines. Its position is then represented by the dotted lines of the figure. In the next one fourth revolution each loop FIG. 107 CONSTANTS OF EARTH'S MAGNETIC FIELD 143 also cuts Ha lines, but in such a manner that the E.M.F. induced is in the same direction as that induced by the first one fourth revolution. In one half revolution, then, each loop cuts 2 Ha lines. If T represents the time of this half revolution of the coil, the average E.M.F. induced in each loop is - - Hence the average E.M.F. for the coil of n loops is - The product an will be denoted by A and known as the total area of the coil. If the terminals of the coil are connected to a ballistic gal- vanometer circuit such that the total re- sistance of the circuit is R C.G.S. units, this E.M.F. causes to flow a current of 2 HA average value - The quantity Q of electricity caused to pass RT through the galvanometer is T times this current. Hence i i FIG. 108 1HA R C.G.S. units. (4) If the time r is so small, as compared with the period of the gal- vanometer, that this quantity Q may be assumed to have passed before the coil turns appreciably, then Q may be obtained from the throw B H and the galvanometer constants, as explained in sec- tion 61 (p. 93). Thus Q = Lz From this and equation (4) 7T the value of If is obtained in the form RQ 27TA (5) Both p, the damping factor, and t, the half period of the gal- vanometer, must be determined under the conditions prevailing in the circuit at the time of the observation of Q H \ for the ratio p [= OJO^\ will not have at all the value which it had in Experi- ment 8, where the galvanometer was swinging on open circuit. In the present case the circuit of the galvanometer coil is at all times closed through the resistance R. It is desirable to make R 144 ELECTEICITY, SOUND, AND LIGHT so large, through the insertion of resistance boxes into the circuit, that the coil will make at least ten or fifteen swings before com- ing to rest. Otherwise neither p nor t can be determined with sufficient accuracy. If the coil is turned over so that in its original position its plane is perpendicular to the vertical component of the earth's field, a similar set of observations will of course give the value of V. Thus If we do not care to determine either H or V absolutely, but wish to know simply the value of the angle of dip S, we have only to take the throws 6 H and Q v corresponding to the motions of the coil which cause it to cut the horizontal and vertical components respectively. We obtain, then, from equations (5) and (6), V _Q V r\ and from equation (3), S = tsm~ 1 ~- (8) "H EXPERIMENT 13 (A) Object. To find the angle of dip. Directions. Connect in series the ballistic galvanometer, a resistance box, and the earth inductor, the last instrument being placed in the position occupied by the tangent galvanometer in Experiment 3. Adjust the resist- ances of the box until the throw produced by cutting the horizontal com- ponent of the earth's lines is from three to six centimeters. Take the mean of six throws, three to the right and three to the left, as the correct value of H . Take similarly six observations upon O v . Then compute 8 from equation (8) above. Change the value of the resistance in the box and see how well you can duplicate the ratio 6y/0 n . (B) Object. To determine the absolute values of V and //. Directions. Insert as much as 10,000 ohms into the circuit. Set the coil so as to cut the vertical component of the earth's field and take observa- tions upon Oy, t, and p. Measure the resistance of the earth inductor and the galvanometer, either by the bridge method or by determining the E.M.F. of a Daniell or a storage cell and observing with a milliammeter the current which this cell sends through the inductor and the galvanometer in series (see Fig. 63, CONSTANTS OF EABTH'S MAGNETIC FIELD 145 p. 84). Either determine K or take its value from the results of Experi- ment 7. Measure the mean area a of the inductor and multiply by the number of turns to find A. Compute V from equation (6), remembering that R in C.G.S. units is R in ohms x 10 9 , and that K in C.G.S. units is K in amperes x 10" 1 . Having found V, compute H by multiplying V by the value of 6 H /0 V found in (A). From V and 8 compute 7. EXAMPLE (A) The earth inductor was placed in the position occupied by the tangent galvanometer in Experiment 3. When there was no additional resistance in the galvanometer and earth-inductor circuit, the throws cor- responding to V and H were 20.51 cm. and 6.53 cm. Hence tan 8 3.139. When there was 4000 ohms additional resistance, making the total 4919 ohms, the throws were 6.35 and 2.00 cm. Hence tan 8 = 3.175. The mean value of 3.157 corresponds to an angle of 72 28' 30", the angle of dip. (B) Making R now equal to 10,919 ohms, and placing the scale 146.7 cm. from the galvanometer mirror, an average throw of 3.30 cm. was observed when the vertical component of the earth's field was being cut. Hence Q v .01125 radians. The period and damping factor of the galvanometer found under these conditions were t 3.20 seconds and pi = 1.103. The constant K was found to be, in amperes per radian, .00001055. The average area of the earth inductor was 208.7 sq. cm. and there were 600 turns. Hence = 10919 x 10 9 x .00001055 x 1Q- 1 x 3.20 x .01125 x 1.103 _ 5g23 27rx600 x 208.7 Also H = -^- = '-^ = .1845 and 7 = -^=.6108. tan 8 3.157 sin 8 The value of H as found by the earth inductor agreed, therefore, to within .6 per cent with the value .1856 determined in Experiment 2, for the same locality, by using a magnetometer. CHAPTEE XIV SELF-INDUCTION 95. Nature of self-induction. In section 83 (p. 126) the neces- sary and sufficient condition for the electro-magnetic induction of an E.M.F. in a conductor was found to be the cutting of magnetic lines of force by the conductor. It follows, then, that in general any con- ductor which lies in a magnetic field will have an E.M.F. induced in it by any change in the intensity of this field, for such a change must cause lines of force either to appear or to disappear, and in so doing they will, in general, sweep across the conductor. Now since a straight conductor carrying a current lies in its own magnetic field (see Fig. 109, w T hich represents a section of the field about a conductor carrying a current into the plane of the paper), and since the strength of this field varies as the current varies, it is to be expected that an induced E.M.F. will be set up in every element of this conductor by any change in the current which it carries. This inference is completely confirmed by experiment. We may picture this so-called E.M.F of self-induction to arise as follows : when the field shown in the figure collapses, i.e. when the current in the wire dies out, the circular Lines of force may be thought of as shrinking to points at the center of the wire, and in so doing sweeping across the conductor from outside to inside. Similarly, when the current rises we may imagine these lines as spring- ing from the center outward, and in so doing sweeping through the conductor from inside to outside. In each case, in accordance with Lenz's law, the induced E.M.F. must be in such a direction as to oppose the change which is inducing it, i.e. in such a direction as 146 SELF-INDUCTION 147 to oppose the growth of the current when the latter is rising and to retard its decay when it is falling. We may reach this conclusion also by the direct application of the dynamo rule, remembering that the direction in which the conductor is cutting the lines is opposite to the direction in which the lines are moving across the conductor. It will be seen that the self-induction of an electrical current is exactly analogous to the inertia of a mechanical system, for, just as inertia is the property of such a system by virtue of which it resists any attempt to change its condition of rest or of motion, so self-induction is the property of an electrical system by virtue of which it resists any attempt to change its existing state. 96. Self-induction in various forms of circuit.- Since self- induction is due to the cutting of the circuit by its own mag- netic lines, it will be seen that we have it in our power to increase or decrease it almost B \ at will, for we can easily give Sw the circuit such a form that ^-^ its own magnetic lines will sweep across it many or few times. Thus if we wish to remove self-induction entirely from a circuit, we have only to arrange it, as in Figure 110, so that the outgoing and incoming currents from the generator B travel over practically the same path. In this case there will be no magnetic field what- ever about the conductor, since the fields due to the outgoing and 1 1 incoming currents are everywhere equal and opposite. |LT Otherwise stated: if the current changes, the E.M.F. in- duced in each element of the outgoing conductor by the cutting of its own lines is exactly neutralized by the oppo- site E.M.F. due to the cutting of the oppositely directed lines of the adjacent element in the other conductor. The coils used in resistance boxes are always made nonin- ductive in this way. The wire is first doubled on itself and then wound on the spool as a double strand, in the manner shown in Figure 111. On the other hand, if we wish to make a circuit of very large self-induction we have only to arrange it in the form of a closely packed coil of many turns, through all of which the current passes in the same direction (see Fig. 112), for in this case the lines of force which the rise of the current in each loop would thrust through that 148 ELECTRICITY, SOUND, AND LIGHT loop are also thrust through all the other loops, so that the E.M.F. of self-induction which opposes the rising current is enormously increased. If, for example, the coil has 100 turns, the lines of force due to any one turn sweep across the en- tire 100 turns. Across any turn, then, there sweeps 100 times the number of lines which it itself produces, and therefore, other tilings being equal, there is induced in each turn 100 times the E.M.F. which would be in- FIG. 112 , , . . , duced in a single turn. But since there are 100 turns in series, the E.M.F. of self-induction in the whole coil is 100 2 times as large as it would be for a coil of a single turn. 97. Units of self-induction. It is evident from the last section that the total number of cuttings by lines of force which a given conductor undergoes because of the appearance or disappearance of a given current within it depends upon the form of the con- ductor. For a given conductor it is also directly proportional to the current, since the total number of lines, i.e. the magnetic field strength, about the conductor is proportional to the current within it. The factor by which the current must be multiplied to give the number of cuttings is then a constant of the conductor. It is called the coefficient of self-induction of the conductor. If L repre- sents this coefficient for a given conductor, / the current flowing through it, and N the total number of cuttings by lines of force which the conductor experiences when this current is stopped, then the definition of the coefficient of self-induction is given by the equation N=LI or = (1) In words, the coefficient of self-induction of a conductor may be defined as the total number of times that this conductor is cut l>y lines of force when the current which it carries undergoes an increase or a decrease of one unit. From equation (1) it is obvious that L is 1 when I is 1 and N is 1 ; i.e. the absolute unit of self-induction is the self-induction of SELF-INDUCTION 149 coefficient a conductor which is cut once by a line of force when the current within it undergoes a change of one absolute electro-magnetic unit of current. The practical unit of self-induction, the henry, named in honor of the American physicist Joseph Henry (1799-1850), is taken as 10 9 times this unit. It is the self-induction of a conductor which is cut by lines of force 10 9 times when the current within it undergoes a change of one absolute unit, or by 10 8 lines of force when the current undergoes a change of one ampere (see also sect. 98). If this cutting takes place in just one second, the mean E.M.F. of self-induction developed is evidently one volt (see sect. 8 5, p. 129). 98. The measurement of self-induction. One of the simplest ways of determining the coefficient of self-induction, of a coil of wire for which this coefficient is not too small is the following. The coil 8 (Fig. 113), the coefficient of which is sought, is joined in series with a variable noninductive re- sistance r x , and S and r l are then made one arm of a Wheatstone's bridge, the other arms of which, M, 0, and T, are also noninductive resistances. After suitable values have been given to M, 0, and T, r 1 is adjusted until, when K^ is first closed and then K 2 , the galva- nometer shows no deflection. This is of course merely the balanc- ing of resistances for steady currents, such as has already been done in connection with the Wheatstone's bridge. If, however, while 7T 2 is closed, K^ is suddenly opened, the condition of steady currents no longer holds. Since r l , T, 0, and M are noninductive resistances, the decaying current can cause no cutting of lines by these con- ductors. In the case of S, however, the interruption of the current causes the field about S to collapse, and there is thus induced in S FIG. 113 150 ELECTRICITY, SOUND, AND LIGHT an E.M.F. which causes a momentary current to flow through the circuit. This circuit is composed of the series resistances S, r v M, and the divided circuit, one branch of which is G and the other T + 0. If we let R represent the total resistance of this divided circuit, then the total resistance of the circuit through which the self-induced current flows is S + i\ + R, + M. If r represents the time required for the disappearance of these magnetic lines in S, then, since the induced E.M.F. is numerically equal to the rate of cutting of these lines of force, we have the mean E.M.F. of self-induction = = -^-- the mean current of self-induction = LI (2) If then Q l represents the total quantity of electricity which flows through the circuit when these L I lines of force cut it because of the stopping of the current /, we have, since quantity is the product of current by time, T r /) _ L /q\ ^1 Tf /> i ~T\T \ / The fraction of this quantity which passes through the galva- nometer G is of course determined solely by the character of the conductors T, 0, and G. The coil S is now replaced by a con- denser C (Fig. 1 1 4) , across which is shunted a noniii- ductive resistance U. U is then given such a value that a balance is again obtained when K v is first closed, then /C. This means, of FIG. 114 course, that U_ = S+r r The quantity Q of electricity which is now upon the condenser plates is the capacity C of the condenser times the SELF-INDUCTION 151 RD. to which it is charged. But this P.D. is by Ohm's law U X /. When now K l is opened, A" 2 being kept closed, this quantity UC1 discharges partly through U and partly through R and M. By the law of parallel connections (see eq. 11., p. 83), the fraction Q 2 of Q which follows the latter path is given by R+M U + lt (4) Since the fraction of Q 2 which passes through the galvanometer is the same as the fraction of Q l which passed through it in the dis- charge of the coil, and since these fractions are proportional to the two throws l and 2 of the galvanometer, we have e, = Q l = LI U+X + M. 2 Q, S+r^ + X + M U*C1 or, since S + ?\ has been made equal to U, we have z = r'c|- (6) We have then but to know the shunt resistance U, the capacity C, and the two throws 6 l and 2 , in order to determine the coefficient of self-induction L. If U and C are expressed in absolute electro-magnetic units, then L will also be expressed in absolute units of self-induction. If U and C are expressed in practical units, i.e. U in ohms and C in farads, then L will be expressed in practical units of self-induc- tion, i.e. in henrys. Since an ohm is 10 9 and a farad 10~ 9 abso- lute units, it is clear from equation (6) that a henry is 10 9 absolute units ; i.e. that a henry is the self-induction of a conductor which is cut by 10 8 lines of force when there is a change of 1 ampere in the current carried by it (see also sect. 97). 99. Practical illustrations of self-induction. One of the most striking experimental proofs of the existence of an E.M.F. of self- induction is the following. If a 100-volt lamp L is shunted as in Figure 115 across an electro-magnet U, which is energized by a 25-volt or even a 50-volt battery, the lamp will not glow appre- ciably so long as the key K is closed, but when K is opened the 152 ELECTRICITY, SOUND, AND LIGHT E.M.F. of self-induction will force a large enough momentary cur- rent through the lamp to make it glow brightly. Furthermore, if K is opened very suddenly, as by a blow struck with a mallet, the brilliancy of the momentary glow will be very greatly increased. The experiment furnishes very satisfac- tory demonstration of the fact that an induced E.M.F. is proportional to the rate of cutting of lines of force. For when K is opened slowly the E.M.F. of self-induction causes a spark to jump between the opening contact points, and as these points draw farther and farther apart, this spark is drawn out into an arc of gradually increasing resistance. Since, on account of the formation of this arc, the current dies out slowly rather than suddenly as K is opened, the induced E.M.F. is small. Increasing the suddenness of the break decreases the time required for the magnetic field to collapse, and hence increases the rate of cutting of the circuit by lines of force ; i.e. it increases the E.M.F. of self- induction. 100. The induction coil. One of the finest illustrations of self-induction is found in the operation of an induc- tipn coil. A primary coil P (Fig. 116) is wound upon a soft iron core and connected into the circuit of a battery B through the contact point o. A secondary coil S is wound about the same core as the primary. A condenser C is connected across the spark gap o. When the cur- rent starts in the primary it magnetizes the core and draws the iron FIG. 115 ,( FIG. 116 SELF-INDUCTION 153 hammer Ji away from the contact point o. This breaks the circuit and the spring sp then restores contact at o. The operation then begins over again. At every break of the primary at o a spark passes between the terminals of the secondary, but no spark passes at make. The cause of this unidirectional character of the dis- charge between n and n is found in the fact that at make the effect of the self-induction of the primary is simply to retard the growth of the primary current, and hence to render the growth of the field within the secondary slow. This means, of course, a small induced E.M.F. in the secondary. At break, however, the presence of the condenser causes the current to fall to zero in an exceedingly short time. For as soon as the contact points begin to separate, the self-induction of the primary, which would normally create an arc between the points, now drives its induced current into the condenser for an instant, and thus gives the contact points time to get so far apart that, by the time the condenser is charged, the spark can no longer leap across the gap and set up an arc. The function of the condenser is then to prevent sparking at o and thus to make a sudden, rather than a slow, collapse of the field. Thus, while the same number of lines of force cut the secondary at make as at break, they take, at make t perhaps a thou- sand times as long to do it, and hence the E.M.F. at make is much too weak to force a spark across the terminals of the secondary, unless these terminals are extremely close together, for example .01 of an inch. 101. The transformer. The transformer is a modified form of the induction coil. In it the core to be magnetized by the primary current is given some shape such that the magnetic lines of force have a continuous iron path instead of being obliged to push out into the air, as in the induction coil. Further, it is an alternating rather than an intermittent current which is sent through the primary of a transformer. The effect of such a current is first to magnetize the core in one direction, then to demagnetize it, then to magnetize it again in the opposite direction, and so on. These changes, of course, induce in the secondary an alternating current similar to that which is being sent through the primary. The E.M.F. induced in the secondary is obviously proportional to the 154 ELECTRICITY, SOUND, AND LIGHT number of turns of wire upon it, for each turn is threaded by all of the lines which pass through the core. In practice the transformer is largely used as a " step-down transformer " ; that is, the mean potential at which the primary is supplied is many times higher than the mean potential which is induced in the secondary. In this case the primary coil has a large number of turns and the secondary coil has a smaller num- ber. For example, if the main conductors are kept at a mean P.D. of 1100 volts, while the lamps of the secondary require 110 volts, there must be 10 times as many turns upon the primary as upon the secondary. Transformers are usually kept connected in parallel to the high potential mains which connect with the alternating-current dynamo, n / i \ ( Dynamo a feP ^ \ Transformers *gii -o ft Low Potential Mains with Lamps FlG. 117 -o -o -o and lamps in the secondary circuit are turned on as needed (see Fig. 117). It may seem at first sight as if, with this arrangement, large currents would always be flowing through the primaries, whether any were required in the secondaries or not. This is, however, not the case, for when all the lamps on a given secondary are turned out, so that no current is delivered by it, the self-induc- tion of the primary " chokes off " practically all current from this primary itself. For, since the effect of self-induction in any circuit is simply to retard the growth of a rising current, or the decay of a dying current, when an alternating E.M.F. is applied to a circuit of sufficiently large self-induction, the current due to the impressed E.M.F. in one direction scarcely gets started before the reversal conies ; and this reversed E.M.F. in its turn only begins to change SELF-INDUCTION 155 this current when another change in direction occurs. So that, in general, a circuit of sufficiently large self-induction behaves toward an alternating E.M.F. just as a body of sufficiently large mass behaves toward a rapidly alternating mechanical force; i.e. in neither case is any appreciable effect produced upon the system. But suppose that lamps are turned on in the secondary of such a transformer circuit. The currents which are induced in the secondary are, by Lenz's law, in such direction as at all times to thrust lines of force through the core in a direction opposite to the direction in which the primary E.M.F. is attempting to thrust lines through the core. The actual number of lines sent through the core by the rise of a given current in the primary will then be less if the secondary is closed than it will be if the secondary is open. Closing the secondary circuit is therefore equivalent to withdrawing a certain amount of self-induction from the primary. Hence a larger current flows through the primary when a lamp is turned on in the secondary than when the latter is open. Turning on two lamps in the secondary withdraws twice as much self- induction from the primary and thus increases in like amount the current which the impressed E.M.F. sends through it, and so on. Thus the current taken from the mains by the primary of a trans- former automatically adjusts itself to the demands of the second- ary. If no energy is taken from the secondary, the primary takes practically no current from the dynamo ; but if a large amount of energy is demanded by the secondary, a large current is taken from the dynamo by the primary. The transformer is therefore merely a device for transferring energy from one circuit to another without the intervention of any mechanical motions of any sort. Its efficiency is commonly as high as 97 per cent, the remaining 3 per cent being transformed into heat in the core and coils of the transformer. EXPERIMENT 14 Object. To find the coefficient of self-induction of a coil. Directions. In order to avoid the necessity of obtaining a perfect balance for steady currents, a tedious operation at best, modify as follows the ideal arrangement described above. Insert a double key K i as in the 156 ELECTRICITY, SOUND, AND LIGHT diagram (Fig. 118),* so that when K 1 is depressed and the switch K 2 closed, the battery circuit is closed through the bridge, and at the same time the galvanometer is short-circuited ; and also so that when K 2 is closed, lifting the finger from 7i x opens the short circuit on the galvanometer an instant before it opens the battery circuit. Give the resistances AT, 0, and T val- ues which are of about the same order of magnitude as the galvanometer resistance. Observe the zero of the galvanometer, then close in succession K v the lower contact of K v and then the upper contact of K r Adjust r 1? all these contacts being kept closed, until the galvanometer stands approxi- mately at its zero. If now opening K z produces a large deflection, adjust FIG. 118 r l still further until this deflection is reduced to not more than a centi- meter or two. In order to effect this adjustment it is usually necessary to make the wire which connects r l to T a few feet of bare German silver wire (say No. 22), and to slip this along through the binding post of r l in such a way as to include a greater or smaller length of it in the branch which contains r r When the balance has been made correct to within a centimeter close K 2 permanently, take the reading, then open K l and read the throw. Repeat several times ; then, in order to eliminate thermal effects, reverse the battery terminals with a commutator or otherwise, and read the throw in the opposite direction. Take the means of several * This scheme of connections is due to Fleming. SELF-INDUCTION 157 throws in opposite directions as 6 r This throw will be practically all due to the self -inductive current from S, since the interval between the open- ing of the galvanometer short circuit in K l and the battery circuit is so brief that the galvanometer coil has no appreciable impulse given to it because of the want of exact balance in the bridge arms. This statement may be tested by seeing whether the same throw is not obtained when the balance is exact as when it is inexact. If 1 is not a throw of several centimeters at least, the number of dry cells in B should be increased. Now replace S and r l by C and U. Make U the same as M, 0, and T, and adjust still further with the German silver wire, if necessary, until as good a balance is obtained as before. Take the throw 2 precisely as l was taken, and substitute in equa- tion (6) to obtain the coefficient of self-induction of S. This will be expressed in henrys if C is in farads and U in ohms. It is, of course, desirable to make O l and 2 of the same order of magnitude. If 2 is too small, use a condenser of larger capacity. If it is too large, place C in shunt with a fraction only of U and use the number of ohms across which C is shunted as the resistance to substitute in equation (6). EXAMPLE The coil, the self-induction of which was determined, was 3.7cm. high, 9.9 cm. in external diameter, 4.1 cm. in internal diameter, and was wound with about 4000 turns of No. 25 copper wire. The resistance of the coil was about 29 ohms, that of the galvanometer about 600 ohms. The resistance in each of the four branches of the bridge was made 1000 ohms. The mean value of 1 was 27.9 mm., the current being furnished by three dry cells. The condenser used was of -'-microfarad capacity and produced a deflec- tion, when U was 1000 ohms, of 45.3 mm. Therefore L = .205 henrys. This differed by less than half of one per cent from the value marked upon the coil, which had been previously.obtained by a careful comparison with a standard of self-induction by a method similar to that described in section 67 for comparing condensers. CHAPTER XV MAGNETIC INDUCTION IN IRON 102. The test coil. In section 94 a method was described for measuring the intensity of a magnetic field by the use of a ballis- tic galvanometer and a test coil, known in that experiment, from its special application to the measurement of the earth's magnetic field, as the " earth inductor." The principles involved may be applied to a general study of magnetic induction and are here re- stated with particular reference to the experiment which follows this section. Consider a magnetic field which may be es- tablished or destroyed at will by making or break- ing the circuit of a con- ductor C. In the neigh- borhood of this conductor (see Fig. 119) there is placed a small coil of wire c, the test coil. Let the average area of this coil be such that when multiplied by the number of turns of the coil the total area is A square centimeters. The coil is connected to a ballistic galvanometer, the constants of which are known. The combined series resistance of the galvanometer and coil, in C.G.S. units, will be represented by R. If in the time r the number of magnetic lines passing through the test coil is changed from zero to &C lines per square centimeter by the establishment of the magnetic field, then the mean E.M.F. induced in this coil during the time r is cKA/T C.G.S units. The quantity of elec- tricity Q which this E.M.F. causes to flow through the " galva- nometer is given by the relation Q = &CA/R. Q can be found 158 FIG. 119 MAGNETIC INDUCTION IN IKON 159 as shown in section 94 from the throw of the galvanometer. Transforming the expression gives that is, QR/A is a measure of the number of magnetic lines per square centimeter passing through the test coiL 103. Magnetic field-intensity. Now the number of lines passing through the test coil as determined from the galvanometer throw is found to depend upon the medium within the coil. If the medium is air, the number of magnetic lines per square centimeter is known as the magnetic field-intensity. And even if the medium is not air, the number which would exist were the medium replaced by air is still known as the magnetic field-intensity and is com- monly denoted by the symbol di . 104. Diamagnetic and paramagnetic substances. The number of magnetic lines per square centimeter determined as above is for most substances very slightly less than it is for air. Substances for which this is true are called diamagnetic. That the difference is very slight may be seen from the fact that for bismuth, the most highly diamagnetic substance known, the ratio of the num- ber of lines established to the number which would be established in air by the same cause is approximately .9998. Those substances for which the number of lines, other things being equal, is greater than it is in air, are called paramagnetic. Most of these substances also differ but very slightly from air. The exceptions, iron, nickel, and cobalt, with certain of their com- pounds or alloys, admit of such exceptionally large values for the number of magnetic lines that they really constitute a separate group. In comparison with members of this group, which will be known as magnetic, all other substances may be considered as magnetically indifferent, and for practical purposes identical in their behavior with air. 105. Magnetic induction. The name magnetic induction is given to this phenomenon of the establishment in a magnetic substance of a number of magnetic lines different from the number which would exist in air from the same cause. To study this phenomenon 160 ELECTKICITY, SOUND, AND LIGHT consider two identical coils of wire, the first wound about an annu- lar ring composed of some magnetic material (e.g. soft iron), and the second about an identical ring of some magnetically indifferent material (e.g. wood) (see Fig. 120). These two coils are connected in series so that identical conditions of current, and consequently of magnetic field, exist for both coils. The magnetic conditions within these rings are observed by similar test coils S and S r . Upon completing the circuit through the primaries the number of magnetic lines within the wood core, as measured by the expres- sion QR/A, gives the magnetic field-intensity &C in lines per square centimeter. The test coil wound over the iron core has induced in FIG. 120 it a quantity of electricity Q', which is larger than Q, and which, when substituted in the expression Q'R/A, gives a measure of the number of lines per square centimeter through the iron. This number will be represented by the symbol cB and will be known as the induction in the iron due to the magnetic field of intensity c?f. That is, induction is the total number of lines per square centimeter passing through a substance as a result of its presence in a mag- netic field. Induction is also frequently known as flux density, since it follows from its definition that it is the flux (or total num- ber of lines through the iron) divided by the area of cross section normal to their direction. MAGNETIC INDUCTION IN IRON 161 106. Magnetization. If a larger current is used, a larger value of cK is obtained, and a larger value of cS. In this way it is possi- ble to study the variation of the induction in iron with the mag- netic intensity of field which causes it. The curve representing the general nature of the variation is shown in Figure 121. Now, lines per square centimeter in the air core, since there are <. there must be 3V of the lines in the iron core which are due to the current. The dif- ference, cB cTf , represents, then, the number of lines per square centimeter which are due solely to the presence of the iron in the magnetic field in- side the circular coil. It is taken as a measure of the magnetization of the iron. The technical definition of magnetization, commonly denoted by o7, is this number of lines divided by 4 TT. Thus, by definition, / fT> 7/*\ (2) = Field-Intensity FIG. 121 4-7T The reason for the introduction of the factor 4 TT will be discussed in a later paragraph. 107. Three stages of magnetization. When successive values of c7 are computed from the last equation and successive values of cB and cfC as given in Figure 121, and when then the successive values of 3 are plotted as ordinates with the corresponding values of c?f as abscissas, a curve of the type shown in Figure 122 is 162 ELECTRICITY, SOUND, AND LIGHT obtained. This latter curve shows that the process of magnetiz- ing iron may be divided -in to three stages which are represented respectively by the portions of the curve from & to c, from c to d, and from d to e. During the first stage, which corresponds to very small values of the magnetizing force e?f, the magnetization increases in nearly direct proportion to c?f ; then during the second stage, namely from c to d, the magnetization of the iron increases exceed- ingly rapidly, and during the final stage, from d to e y it becomes practically constant, showing that the magnetization of iron can- not be pushed higher than a certain limit, however strong fields may be used. e The explanation of this behavior in terms of the molecular theory (see sect. 89,p. 134) is that while the field is still too weak to break up any of the existing molecu- lar groups, the molecular mag- nets turn slightly against the mu- tual magnetic actions of the surrounding molecules, much as a solid body which is not dis- torted beyond its elastic limits yields under the influence of an external force. But as the magnetizing force c/if increases, a point is reached at which the molecular groups begin to break up, and the molecules to set themselves parallel to the direction of the field &C. The part of the curve between c and d represents the cor- responding values of d and cK when this breaking down of the mo- lecular groups is taking place most rapidly. It will be seen that within this region a very small increase in di produces a very large .increase in the magnetization. As cK is still further increased the = Field-Intensity FIG. 122 MAGNETIC INDUCTION IN IKON 163 magnetization enters upon its third stage represented by the por- tion de of the curve. In this stage the molecular groups are all broken up, but the molecules have not yet all assumed exact par- allelism with the field. As c7f still further increases, the molecules gradually assume the exact direction of the field, and the iron is then said to be magnetically saturated, since any further increase in cTf produces no increase in &. 108. Origin of the factor 4?r. The reason that the magnetiza- tion 3 is not taken as numerically equal to the number of lines due to the iron, namely cB cK , but rather to (oB e/f )/4 TT, is as follows : Since, by definition, unit magnetic pole, when placed at the center of a sphere of unit radius, produces a magnetic field of unit strength at every point on the surface of the sphere, and since a field of unit strength is represented by one line of force per square centimeter, it is clear that one line of force from the pole must be thought of as piercing each of the 4 TT sq. cm. on this unit sphere. Hence we must imagine 4 TT lines of force as emanat- ing from every unit N pole, and 4 Trm lines of force as emanating from any N pole which contains m units of magnetism. Now we imagine that in an ideal magnet which is fully saturated there are no free poles except at the very ends of the magnet, so that all of the lines which are associated with the north pole emerge from the face itself, pass around in closed curves to the S face, and then return through the magnet to the j'Vface (see Fig. 101, p. 135). If the strength of the magnet's poles is m, then, as has just been shown, the number of these lines is 4 Trm. The number of unit poles in one square centimeter of the face is then not equal to the number of lines which pass out of this face, but is rather equal to this number divided by 4 TT. Now it has been decided to regard magnetization as the number of unit poles per square centimeter, rather than as the number of lines per square centimeter. Hence it was that we divided the number of lines per square centimeter due to the iron, namely cB cK , by 4 TT in order to obtain the numerical value of 3 inside the ring. Even in an ordinary unsatu- rated magnet in which lines emerge all along the sides, the inten- sity of magnetization at any point within the iron is defined as the number of lines per square centimeter there present divided 164 ELECTRICITY, SOUND, AND LIGHT by 4 TT. That is, it is the number of units of molecular magnets which we must imagine to be present in this particular square centimeter in order to account for the number of lines found in this square centimeter because of the magnetization of the iron. 109. Hysteresis. The curve Icde (Fig. 122), representing the values of the magnetization corresponding to a magnetic field of constantly increasing magnetization, is shown again in Figure 123. As now the magnetic field is allowed to decrease to its original value of zero, the magnetization assumes successive values repre- sented by the curved portion ef. The explanation, in terms of the FIG. 123 molecular theory, of the fact shown by the curve, that the succes- sive values of the decreasing magnetization do not lie on the origi- nal curve of increasing values, is as follows : When the magnetizing field &C is withdrawn or diminished, some of the molecules return under their mutual actions to their original groupings, but many of them retain their alignment until this is broken up by jars or other outside forces. In hard steel the difficulty which the mole- cules experience in moving out of any positions which they have MAGNETIC INDUCTION IN IKON 165 once assumed is very large, while in soft iron it is small. In general, then, the magnetization is said to " lag behind " the field-intensity which induces it. This phenomenon, known as hysteresis, is an important characteristic of all magnetic substances. When the field has been decreased to zero the value of the mag- netization retained by the iron is known as the residual magneti- zation. In practice, however, the knowledge of this quantity is of little value. It . is in general more desirable to know the ratio which the residual magnetization, represented by the ordinate 5/, bears to the maximum value of the induced magnetization, repre- sented by the ordinate en. This ratio is defined as the retentivity and is usually stated in per cent. In order that the magnetization shall be again zero, the direc- tion of the field must be reversed, and the value of its intensity caused to increase in this opposite direction. Under the action of this reversed field, the values of which will for convenience be called negative and plotted to the left of the vertical axis, the value of the magnetization rapidly decreases in a manner repre- sented by the curve fg. The value of the negative intensity which must be applied to reduce the residual magnetization to zero is known as the coercive force. Numerically, it is represented by the abscissa bg. It is evident that if this oppositely directed field is still further increased in intensity, there must result a magnetization opposite in direction to the original magnetization. Now let the maximum value of the original intensity, shown in the figure by the abscissa bn, be represented by the symbol +<$f m . It is of interest to consider the effect of hysteresis upon the successive values of the magneti- zation corresponding to a cyclic change of c?f, in which cK passes continuously through all the values from +<2f m to 3i m and back again to +&C m . After a sufficient number of repetitions of this cycle, the values of 3 are found to lie on a closed curve of the gen- eral form represented by efghije and known as the hysteresis loop.* * In the commercial transformers described in section 101, page 153, the iron core undergoes such cyclic changes in its magnetic state. Work is done in producing these changes, and this loss of energy due to hysteresis is of practical importance in the design of transformers. 166 ELECTRICITY, SOUND, AND LIGHT 110. Formula for field strength within a long solenoid. It is generally desirable in experiments on cB and 3K to determine cB by a test-coil method, applying the principles of section 105 in a manner which will be explained in the next section ; but instead of using a wooden ring for the determination of ef{ it is customary to calculate this quantity from the number of turns z per centi- meter of length of the solenoid and the current / flowing about it. The relation between these quantities is as follows : v _ 4 TTZl ~~' The origin of this formula may be seen from the follow- ing considerations. ' The force which a unit mag- FIG. 124 net pole placed at a point p (Fig. 124) on the axis of a long solenoid experiences because of a current / flowing around it is, of course, simply the resultant of all the magnetic forces exerted upon it because of the magnetic fields surrounding all elements of the wire constituting the solenoid. Consider first the value of the force exerted upon the pole at p by the elements of current which lie on a small element of area A/As of the surface of the solenoid. If there are z turns per centi- meter, there will be zbl elements of current in the distance A/. Each of these has a length As. The force which each element exerts on a unit magnet pole at the point p a distance r from the element is, by definition of unit current (see sect. 30, p. 36), ^- > and the total force / due to all the elements z&l is given by Since, from considerations of symmetry, it is obvious that the resultant force exerted on the pole at p by the whole solenoid must be parallel to the axis of the solenoid, we are here concerned only with the component of / in the direction of the axis. Calling this /', we have i^lAscosff _ J '2 \ ' MAGNETIC INDUCTION IN IRON 167 But precisely as in section 73, page 108, - is the solid angle subtended at p by the elementary area A/As. Hence, representing this solid angle by u, we have f = Izu. (5) That is, the magnetic force exerted parallel to the axis by any ele- ment of the solenoid is the solid angle subtended at the point by the element, multiplied by the current and by the number of turns per centimeter of length of the solenoid.* The total force 3i acting upon the unit pole at p is the sum of the forces due to all the elements of the surface, i.e. di = 2lzu. Since / and z are the same for all elements, we may write this in the form cK = Iz2u, and if the solenoid is so long that we may neglect the solid angle subtended by its open ends in comparison with the solid angle subtended by its surface, we have, since the solid angle about a point is 4 TT, W = 4 -irzl. (6) If / is expressed in absolute units of current, the field strength Si will be expressed in gausses. If / is in amperes, in order to obtain di in gausses we must write -*? Since this analysis holds for any point within the solenoid which is -not too close to its ends, it is obvious that the field strength within a long solenoid is uniform. If the solenoid is a ring, the deduction is rigorously correct for all points within the ring. If it is a cylinder, the general rule is that the length must be as much as twelve times the diameter in order that equation (7) may be applied without appreciable error to find n> ' ' On the other hand, if the dissociation were uninfluenced by the dilution, but if the resistance which the ions meet in moving through the solution were proportional to the viscosity of the solu- tion, then obviously the ionic speeds would be inversely propor- tional to the viscosities at the dilutions considered. That is, if 77 and 77^ represent the viscosity coefficients at the given dilution and at infinite dilution respectively, then we should have moo 77 If both the dissociation and the viscosity change with dilution, we have, by combining the two variations expressed in equations (1) and (2), m L = ^ L ^_ moo n 77 ' n m TJ * or - = --- - (4) Woo moo 7700 If at infinite dilution the dissociation is complete, then obviously n/n< represents the fraction of all the molecules introduced which are split into ions at the dilution corresponding to m. This is usually represented by the letter a. We have, then, = -5L-5-.t (5) m< 7700 As a matter of fact the dissociation of electrolytes as computed from equation (4) does not agree perfectly with the dissociation * The molecular conductivities m and m^, as well as the viscosity coefficients 17 and 77^, all refer to the same temperature. t Arrhenius omitted the term V^, and wrote merely a = m/m^ for any given temperature. Indeed, since for dilutions greater than 8 the ratio y/ri^ is found to differ but little from unity, it is not improper to write the dissoci- ation equation in this form, provided its restriction to dilutions larger than about 8 is understood. ELECTKOLYTIC CONDUCTION 183 estimated from freezing-point determinations, although in general there is a fair agreement, and in some cases an exceedingly close one. In view of these uncertainties it must be said that the dis- sociation theory as here developed is not yet fully established. 120. Methods of measuring molecular conductivity. In prac- tice molecular conductivities cannot be measured in the simple manner described in section 119, in which the immersed area of the plates increased proportionally to the dilution, and hence the conductivity changed only because of a change in the molecular conditions due to the dilution. The same result is obtained by measuring the resistance of the solution between plates 1000 sq. cm. in area and 1 cm. apart, and multiplying the reciprocal of this resistance by the dilution. If the molecular conductivity is to be obtained in absolute units, plates of 1 sq. cm. area are taken 1 cm. apart ; that is, the absolute values of molecular conductivity are -joVo of those obtained as described above. Since the resistance of a centimeter cube is called specific resistance (see p. 71) and since specific conduc- tivity is the reciprocal of specific resistance, it will be seen that molecular conductivity in abso- lute units is the dilution divided ly the specific resistance of the solution under consideration. The vessel most commonly used for molecular- conductivity experiments is shown in Figure 132. It consists of two platinum electrodes supported, as shown, in a vessel made of a kind of glass which is especially insoluble in water. If only comparisons of molecular conduc- tivities are desired, they are made by introducing the different solutions to be compared succe'ssively into the vessel and divid- ing the given dilutions by the observed resistances. On account, however, of the fact that in this vessel the current is conducted to some extent by the portion of the solution which is outside the edges of the plates, an absolute determination is usually made on a standard solution in a vessel of another type, and then com- parisons are made in the vessel showTi above. However, absolute FIG. 132 184 ELECTRICITY, SOUND, AND LIGHT specific resistance determinations of approximate correctness may be made with the cell of Figure 132 by multiplying the observed resistance by the area of the plates and dividing by the distance between them. The measurement of the resistance is made by making the vessel one branch of a Wheatstone's bridge, as in Figure 133. Although the electrodes are of a different substance from that of the metallic ion of the electrolyte, difficulties due to the back E.M.F. of polari- zation are eliminated by using a source of alternating current in place of the battery of the ordinary bridge scheme. This obviously FIG. 133 allows no permanent separation of the elements of the electrolyte at the electrodes. The source of alternating current is the second- ary of a small induction coil C. A telephone receiver replaces the galvanometer of the ordinary bridge arrangement. The connec- tions of Figure 133 differ from those of Figure 45, page 63, in the fact that the position of the generator and of the detector are inter- changed. But, as we have seen, this does not alter the equation of balance.* It is sometimes difficult in an arrangement of this sort to get a sharp minimum. This is probably because a thin film of some insulating material, presumably gas, collects on the platinum * That this is so is evident from the discussion in section 46. The reason for this interchange is given in the footnote on page 71. In this case the gener- ator resistance is that of the secondary of the small induction coil and is higher than the resistance of an ordinary telephone receiver. ELECTROLYTIC CONDUCTION 185 electrodes, and makes each of them act as a condenser, having the gas film as a dielectric, the platinum and the solution as the two opposite condenser plates. Such condensers in series with the electrolytic resistance would have no effect upon the current, pro- vided their capacities were so large that the opposite sides of the condensers did not charge to any appreciable P.D. before the applied P.D. of the generator reversed its direction. If this were not the case, the introduction of such condensers would cut down the alternating current flowing through the branch containing them and hence destroy the balance. Hence these difficulties may be eliminated by using platinum plates of very great surface, for then the capacities are very large. The method usually employed for increasing the size of the plates is to cover the surfaces of the electrodes with spongy deposits of platinum called "platinum black." * EXPERIMENT 16 Object. To compare the molecular conductivities of some salt, for example sodium chloride (XaCl), at different dilutions. Directions. Set up a cell, similar to that of Figure 132, in a large bath of water at the room temperature. Connect as in the diagram of Figure 133, using a meter bridge of which the wire is Xo. 36 German silver. Prepare a normal solution of XaCl either by weight or by hydrometric measurements with a Mohr's balance, f using the values of the density given in table 5 of the Appendix. Introduce two pipettes full of this solution into the cell. Measure the resistance and take its reciprocal. Carefully remove one pipette full of solution and replace by a pipette of distilled water at room temperature. The dilution is then 2. Stir by moving the electrodes up and down, but be very careful not to alter in any way their distance apart. Then measure the resistance again. Similarly introduce another pipette full of water, and measure the resistance at the dilution 4. Continue in this way until a dilution of 2048 is reached. * To platinize the electrodes place them in a 3 per cent solution, by weight, of platinum chloride to which is added a drop or two of lead acetate solution. Pass a current through this cell of such a value that the bubbles rise freely from the solution, then reverse its direction occasionally. Wash in distilled water and allow the electrodes to soak in the water for some time before using them. t See "Mechanics, Molecular Physics, and Heat," p. 175. 186 ELECTRICITY, SOUND, AND LIGHT Empty and rinse the cell and determine the resistance of the distilled water used. Take its reciprocal and subtract from the reciprocals of the resistances obtained above, then multiply by the corresponding dilutions to obtain quantities proportional to the molecular conductivities. Assume the number thus corrected which corresponds to a dilution of 2048 as j , and find a for the other dilutions used from equation (5) on page 182. The values of y and TJ X may be taken from table 6 of the Appendix, and the ratio assumed correct for the temperature at which the experiment was performed. EXAMPLE The distance between the plates of the cell used was about 1 cm., and the diameter of the plates about 2.5 cm. The resistance of the distilled water used was found to be 11,143 ohms. Hence its conductivity was pro- portional to .0000897. This correction was neglected for dilutions less than 64. The salt used was ZnSO 4 . The values observed and calculated follow. Dilution Resistance Corrected Conductivity 1? 'VoO or the % Dissociation 1 2.27 ohms .480 1.362 .844 2 3.88 .515 1.166 .848 4 7.16 .559 1.080 .852 8 13.64 .586 1.040 .859 16 26.17 .611 1.020 .879 32 50.41 .635 1.010 .904 64 97.57 .655 1.005 .928 128 189.6 .671 1. .946 256 368.1 .673 1. .949 512 706.7 .679 1. .958 1024 1293. .701 1. .989 2048 2294. .709 1. 4096 3747. .709 1. oc .709 temperature 22 C CHAPTEE XVII VELOCITY OF SOUND IN AIR 121. Sound. Sound is the name applied to the sensations of the auditory nerves. It is the result of the propagation through the air to the ear of a disturbance produced by the sounding body. Thus when a pistol is fired the layer of air surrounding it is sud- denly pushed outward and compressed by the expanding gases which emerge from the pistol. This compression is transmitted from layer to layer and constitutes a compressional pulse traveling outward from the pistol. It is this pulse which, giving up a por- tion of its energy to the eardrum, results in a stimulation of the nerves of the ear and the sensation of sound. For the purposes of physics the subject of Sound is commonly limited to the study of the causes and nature of these pulses and the mechanism of their transmission. 122. Velocity of a compressional pulse in any elastic medium. Consider, then, the velocity of propagation of a compressional pulse in an elastic medium. For convenience the medium will be assumed to be contained in a tube of one square centimeter cross section and of infinite length. Let the density of the medium be \\\\\\\\\ O ) 1 '\Ll 2 3 4 5 6 7 FIG. 134 represented by p and the pressure under which it stands before any compression is produced by p. For convenience of analysis let the medium be conceived to be divided into centimeter cubes, 1, 2, 3, 4, 5, etc. (Fig. 134). Let one end of the tube be closed by a frictionless, weightless piston. Now let the piston be suddenly 187 188 ELECTRICITY, SOUND, AND LIGHT started forward by the application to it of a constant pressure slightly greater than p, viz. p + dp. There will thus be started down the tube a compressional pulse which will travel through the medium with a velocity which will be represented by S. It is this velocity which it is desired to determine. As soon as the piston begins to move forward, cube 1 begins to be compressed. The pressure inside of this cube rises. When this pressure has reached the value p + dp, the cube will cease to be compressed and will thenceforth merely transmit pressure to cube 2. Call dv the amount of compression, measured in fractions of a cen- timeter, which cube 1 thus experiences. Under the action of the pressure p + dp, which is thus transmitted by cube 1 to cube 2, the latter will also be compressed an amount dv and will thereafter merely transmit the pressure p + dp to cube 3. Similar reasoning may be applied to the remaining cubes in their numerical order. Since the cross section of the tube is 1 sq. cm., while any particular cube, such as 6, is being compressed the amount dv cc., the piston will move forward dv cm., and each of the cubes previously com- pressed, namely 1, 2, 3, 4, and 5, will move forward dv cm. . Thus, as the pulse moves down the tube, the piston, and with it all the compressed cubes, will move uniformly forward. Since S repre- sents the velocity of the pulse, and since the cubes were chosen one centimeter long, it follows that in one second each of S cubes w T ill experience the compression dv. During that second the piston will therefore have moved forward Sdv cm. That is, the velocity of the piston is Sdv cm. per second. Evidently this is also the expression for the velocity with which all of the cubes which have been compressed are moving forward at the end of a second. Now the velocity with which the pulse moves forward may be found by an application of Newton's principle of work * as follows. In the above operation the acting force is the pressure applied to the piston, viz. p + dp. The work done by this force in one second is (p + dp) Sdv. The effects of the action of the force are of two kinds : (1) the compression of all the gas contained in S cubes * This method of analysis is found in Edser's " Heat for Advanced Students." VELOCITY OF SOUND IN AIR 189 from a condition in which it exerts a pressure p to one in which it exerts a pressure p + dp ; and (2) the communication of a veloc- ity Sdv to all the mass contained in S cubes. The mean force overcome in the first operation is half the sum of the initial and final pressure, namely p + dp/2, and the work done against this force is (p + dp/ 2) Sdv. In the second operation, since the mass of each cube is p, and since in one second S cubes are set in motion with a velocity Sdv, the kinetic energy imparted per second is \- pS (Sdv) 2 . Now by the scholium to Newton's third law of motion, the work done by the acting force is equal to the work done against the resisting forces ; i.e. it is equal to the sum of the potential and kinetic energies imparted to the gas. Hence we have (p + dp) Sdv = p + Sdv + pS(Sdv) 2 . (1) dp _ pS 2 dv .'. S' 2 = -t-, or S = (2) pdv \J p Now since we are dealing with unit cubes the quantity dp/dv is the force applied per unit area divided by the change in volume per unit volume, and this is by definition the volume modulus of elasticity E of the medium. Hence, finally, we have for the velocity of a compressional pulse in any medium, Since this expression involves only the constants E and p of the medium, it is evident that a pulse once started will travel on and on down the tube at a rate which has nothing whatever to do either with the size or shape of the tube, or with whether the pis- ton continues to move forward or not. In other words, we have deduced a general expression for the velocity of a compressional pulse in any medium. The expression must hold for the velocity of propagation of a sound pulse which originates at a point within any medium and spreads radially from the center of disturbance ; 190 ELECTRICITY, SOUND, AND LIGHT for, in this case, as in the case just discussed, the pulse is one of pure compression, since the particles are free to move only in one direction, namely along radii emanating from the point of disturbance. It will be interesting to see how well the results obtained by the use of this formula, which has been deduced from purely theo- retical considerations, agree with the results of direct experiment. In 1893 Aiuagat in Paris made some experiments on the compres- sibility of water, and found that at 10C. a change in pressure from 1 atmosphere to 50 atmospheres produced a change in volume from 1 cc. to .99757 cc.* From this data we get as the theoretical value of the velocity of sound in water, all quantities being expressed in absolute units, S = 149x76x13.6x981 =1 N lx.00243 The result of direct measurement made at 8C. in Lake Geneva in 1827 by Colladon and Sturm gave 1435 meters per second. J The difference between the two values is well within the limits of observational error. 123. A train of waves. If in the case of the tube and piston of Figure 134 the applied pressure had been p dp instead of p + dp, the piston would have started back instead of forward, and cube 1 would then have expanded until its pressure reached the value p dp, which expansion would have been followed by a similar expansion of cube 2, etc. Thus a pulse of rarefaction in- stead of one of condensation would have traveled down the tube, and reasoning in every respect identical with the above shows that the velocity of this pulse also would have been ^/E/p. It is important to observe that in a pulse of condensation tlie par- ticles are always moving in the same direction as the pulse ; whereas in a pulse of rarefaction the direction of motion of the particles is always opposite to the direction of propagation of the pulse. * Wiillner, "Experimental Physik," Vol. II, ed. 5, p. 100. t 981 is the value r df the, acceleration of gravity in Paris. \ Wiillner, " Experimental Physik," Vol. I, ed. 5, p. 955. VELOCITY OF SOUND IN AIR 191 If the piston is moved alternately forward and back at regular intervals, a succession of compressions and rarefactions will follow one another down the tube. Such a succession of compressions and rarefactions is called a train of waves. In this case it is evi- dent that the motions of all of the particles of the medium follow, in succession, exactly the motions of the piston ; that is, each par- ticle moves forward during just the interval of time during which the piston is moving forward, and back during just the interval during which the piston is moving back. If then the piston is replaced by the vibrating prong of a tuning fork or by any body which vibrates under the influences of its own elasticity, the back- ward motion will begin at exactly the instant at which the forward motion ends, and hence at the end of one complete vibration of ABC c a FIG. 135 the prong, i.e. at the end of the time required for the prong to go from A to C and back again to A (Fig. 135), the whole of the medium between B and some point a to which a sound pulse travels during the period of one vibration, may be divided into two equal parts, ac and cB, such that all the layers between c and a are moving forward and are in a state of compression, while all the layers between c and B are moving back and are in a state of rarefaction. The relative velocities with which the layers are moving forward or back at the various points between B and a are represented by the lengths of the arrows in the figure. As the fork continues to vibrate the whole region about it becomes filled with a series of such waves, each consisting, as above, of a conden- sation and a rarefaction (see Fig. 135). The distance between the beginnings of two successive condensations or two successive rare- factions, or in general the distance bet\^Jen any two particles which are in the same condition or phase of vibration, is called a wave 192 ELECTRICITY, SOUND, AND LIGHT length. It is obvious that if S represent the velocity of sound, n the number of vibrations of the fork per second, and X the wave length, then the following relation holds : S = n\. (3) The only distinction between a musical note and a mere noise is that the former consists of a train of waves, while the latter con- sists either of single pulses, or of irregularly timed pulses. 124. Newton's deduction of the velocity of sound in air. If dp represent the change in pressure which is applied to any body of volume F, and if dV be the change in I p > volume produced by dp, then, by defini- tion, the volume modulus of elasticity E is given by E=L TT/ ( 4 ) 1 T 1 FIG. 136 Let the body be a gas of volume V and pressure p, and let the applied force change the volume of the gas to V' and its pressure to p' (see Fig. 136). Then , dp _p f -p dV V-V But if the heat which is developed by the compression is allowed to pass off, so that the final temperature is the same as the initial (i.e. if the change is isothermal), then Boyle's law may be applied. This gives p/p' = V / V, or V = p V/p'. Substitution of V in (5) gives r ._ ^ (6 ) Now if, as is the case with sound waves, p f differs from p by an immeasurably small quantity,* then it is possible to replace p' byj?. * Lord Rayleigh estimates that for certain feeble, though distinctly audible, sounds dp is not more than 6 x 10~ 9 atmospheres. VELOCITY OF SOUND IN AIE 193 In other words, the volume coefficient of elasticity of a gas for isothermal changes is the pressure under which the gas stands. Hence if a sound wave produces no temperature changes in a gas we have velocity = . (7) N density If the gas is air under ordinary conditions, then p is simply the barometric pressure, which must, of course, be reduced to dynes if S is to be determined in centimeters. This theoretical value for the velocity of sound in air, first deduced by Newton (1687), does not agree with experimental determinations, for the simple reason that the compression produced by a sound wave does not take place isothermally, as Newton assumed it to do. 125. Correct formula for velocity of sound in air. It was a hundred years after Newton's time when La Place first pointed out that a gas is heated by a compressional wave, and on account of the great velocity of sound the heat thus developed has no time to diffuse before the wave is past. When all the heat developed by compression is retained in the compressed body the change in pressure is said to take place adiabatically. Since bodies expand with heat, it is evident that the change in volume produced in a gas by a given change in pressure will be greater if the heat of compression is allowed to pass off than if it is retained. That is, d V/ V is less if the change is adiabatic than if it is isothermal. Hence the quantity E = (dp/d V] V is greater for an adiabatic change. Analysis which is beyond the scope of this text shows that the bulk modulus of a gas for an adiabatic change is equal to its bulk modulus for an isothermal change multiplied by a factor 7. This factor 7 is the ratio of the specific heat of the gas at con- stant pressure and its specific heat at constant volume. It is always greater than unity ; for air it has a value 1.403 ; for carbon dioxide (CO 2 ), a value 1.30. Hence the theoretical value for the velocity of sound in a gas is 194 ELECTRICITY, SOUND, AND LIGHT 126. Effect of temperature upon the velocity of sound. If the pressure is changed without a change in temperature, then by Boyle's law the density change is directly proportional to the pressure change ; that is, p/p = constant. Hence the velocity of sound in gases is wholly independent of pressure. The dependence of the velocity upon the temperature of the gas may be found by the following reasoning. Suppose the pressure of the gas to remain constant. Let S t represent the velocity at a temperature of t C., and p t the corresponding density. Similarly, let S and p represent the velocity and density at 0. The veloc- ity at the temperature t is given by the equation S t = ^yp/p t , the velocity at by the equation S =^/yp/p . Therefore (9) Now by Gay-Lussac's law the density of a given gas is inversely proportional to its absolute temperature. Hence S t in which T is 273 and T is the temperature of the gas on the Centigrade scale plus 273. By substituting in this formula 274 for T and for S the best determination of the velocity of sound at 0, namely 33,127 cm., it will be seen that the velocity of sound increases about 60 cm. for each degree Centigrade of rise in temperature. 127. Reflection of wave trains produced by a change in the density of the medium. Eeturning to the consideration of the tube and piston, suppose the latter to move forward suddenly a short distance and then stop. A compressional pulse will start down the tube. The progression of this pulse is due to the com- munication from layer to layer of the motion which is imparted by the piston to the layer next it. The case is precisely analogous to that of the collision of perfectly elastic equal balls. So long as VELOCITY OF SOUND IN AIR 195 FIG. 137 the pulse travels in a medium of uniform density each layer of particles gives up all its motion to the next layer, which is pre- cisely like it, just as a moving elastic ball striking a stationary elastic ball of the same size gives up all of its motion to the stationary ball and itself comes to rest. But if the pulse at some point A (Fig. 137) strikes a denser medium D, the case becomes analogous to the impact of p A a lighter ball upon a heav- ier one. The lighter layer CZZ adjoining A on the left, in- stead of coming to rest in the impact, reverses its mo- tion and starts back. This backward motion is communicated from layer to layer so that a reflected compressional pulse travels from A back toward P. Let the long arrow represent the direction of the pulse and the short one the direction of motion of the par- ticles in the pulse. It is evident that the direction of motion of the particles with respect to the direction of motion of the pulse is the same in the returning as in the advancing pulse. In other words, a pulse of condensation is reflected from a denser medium as a pulse of condensation. But suppose the medium to the right of A is less dense than that in the tube (see R, Fig. 138), then the case is analogous to the impact of a heavy ball upon a light one. Until the pulse reached A each layer in the tube gave up its motion to the next P A and itself came to rest ; but the layer at A, instead of coming to rest after the im- pact, continues to move for- ward and thus produces a rarefaction, i.e. a diminution of pressure at A. The excess of pressure in the layer to the left of A then drives particles toward the right. Thus a pulse of rare- faction moves back from A toward P. In other words, a pulse of condensation is reflected from a rarer medium as a pulse of rarefaction. In the pulse approaching A the particles move in the direction of propagation of the wave. In the pulse receding FIG. 138 196 ELECTEICITY, SOUND, AND LIGHT from A the particles move in a direction opposite to that of propagation of the wave (see arrows, Fig. 138). Since the pulse changes instantly in the reflection from a condensation to a rare- faction, and since in a train of waves a rarefaction always follows just one half wave length behind a condensation, a wave is said to experience a loss of one half wave length in reflection from a rarer medium. The same process of reasoning shows that a wave of rarefaction is reflected from a denser medium as a wave of rarefaction, but from a rarer medium as a wave of condensation. 128. Resonance of vibrating air columns. When an air wave traveling along a pipe reaches the open end it experiences the same sort of reflection as though it passed from a denser to a rarer medium. This statement can be easily proved experimentally (see sect. 130). It 2 2 1 i is also evident from the theoretical con- sideration' that as soon as the wave > < > < 1> B reaches a point at FIG. 139 which lateral ex- pansion is possible, the forward movement of the particles is greater than inside the tube, where lateral expansion is not possible. This increased for- ward movement at the end of the tube means a wave of rarefac- tion starting back in the tube. Consider now a train of waves approaching the open end of a pipe the other end of which is closed (Fig. 139). In the conden- sations of the advancing train of waves the motions of the parti- cles are all in the direction of motion of the wave, i.e. from left to right. In the rarefactions they are in the opposite direction. Suppose the pipe to have a length of exactly one fourth wave length. Then the condensation which is marked will move down the pipe and be reflected at the closed end as a condensation, i.e. as a motion of the particles now from right to left. It will obvi- ously return to the open end at the exact instant at which the rarefaction which is marked J, and which also consists of a motion VELOCITY OF SOUND IN AIR 197 of the particles from right to left, reaches the open end of the pipe. Since the reflected condensation which is returning from the closed end of the pipe now undergoes reflection at the open end as a rare- faction, i.e. as a motion of the particles from right to left, the newly reflected wave of rarefaction which starts back down the pipe unites with the rarefaction marked J- which is just entering the pipe, and a wave of rarefaction of increased amplitude is the result. This wave is reflected at the closed end as a rarefaction (motion from left to right), and again at the open end as a con- densation (motion from left to right), exactly in time to unite with the condensation marked 1 as it enters the pipe. Thus by this process of continuous union of direct and reflected waves the motion in the pipe becomes larger and larger until it may be hun- dreds of times as large as the motion of the particles in the original wave. In fact, there would be no limit to the amplitude of the waves traveling up and down the pipe if at each end the motion were not partially given up to the outside air. The pipe thus becomes in a way the source of sound. The phenomenon is called resonance. If the pipe had been only a trifle longer or shorter than one fourth wave length, the error in the coincidence of the first reflected wave with J- would have been but slight. Since, however, the reflected waves are now obliged to travel, each time they go up and down the pipe, a distance a little too great or too small, it takes them but a short time to get completely out of step with the advancing waves. In this condition the direct and reflected waves interfere with rather than assist each other, and no reso- nance is possible. This explains why, when the length is but a trifle more or less than the right amount, very little resonance is obtained. If, however, the pipe is continually lengthened, other resonant lengths will be reached. The length above chosen was one which permitted to return to the mouth of the pipe exactly in time to unite with \. This length is manifestly the shortest possible resonant length. It is clear that the next possible resonant length is one which permits, to return exactly in time to unite with f. Since | is one wave length behind \, the second resonant pipe 198 ELECTRICITY, SOUND, AND LIGHT length must be one half wave length greater than the first; i.e it must be three fourths wave length. Similarly, it is possible to obtain resonance when the pipe length is five fourths wave length, seven fourths, nine fourths, and so on. Experiment shows that the theoretical values of the pipe length required for the fundamental resonance, namely one fourth wave length, is slightly too large. The discrep- ancy is explained as follows. If the reflec- \ tion of a wave traveling down the open / pipe ac (Fig. 140) took place exactly in the plane cd, the resonant length would FIG. 140 be exactly one fourth wave length ; but in reality the point of free lateral expan- sion is not reached at the instant at which the wave reaches cd ; hence the reflection does not occur until the wave has pushed out a short distance beyond cd. Thus the true pipe length is slightly greater than the apparent length. The amount of this correction which must be applied at the open end is estimated by Eayleigh as the radius of the pipe, but in glass pipes it is somewhat less than this. EXPERIMENT 17 Object. To find the velocity of sound in air from resonance experiments upon a closed pipe, and to find the correction which must be applied to the open end of a pipe. Directions. The pipe to be used consists of a long glass tube A (Fig. 141), one end of which is closed by water admitted by a tube from the vessel B. The length of the column of air contained in this pipe may be varied by changing the height of B. A rubber band around A may be slipped into any position along its length. A train of waves from a tuning fork of known rate is caused to enter the pipe by holding the fork over the mouth of the tube. Set the fork T in vibration by striking it with a rubber mallet, and hold it over the mouth of the tube. It is not important in this part of the experiment that the fork be held farther than three or four milli- meters from the end of the tube, provided it is always held in precisely the same position. Taking especial care about this point, raise or lower VELOCITY OF SOUND IX AIR 199 the vessel B and thus change the level of the water in A until the note of the fork is strongly reenforced. By causing the water to rise and fall rapidly several times in the vicinity of the position of reenforcement, the length of maximum resonance can be fairly accurately obtained. Set the rubber band at the level of the water in the tube and measure to it from the top of the tube. In this way determine two or three successive reenforcement points. If then l v 1 2 , and l s represent the first, second, and third measured resonant lengths, x the unknown cor- rection which must be applied because the reflection at the open end does not take place exactly in the plane no, the preceding theory gives (1) 1 1 + x = i wave length, (2) 1 2 + x = | wave length, 1 (3) / 3 + x - f wave length. Eliminating x both from (1) and (2), and from (1) and (3), gives n o - A S - ^=T=J 1 f 3 anc 0) and =~_ 1=7 Frc the ma] tioi this bar FIG. 141 From the mean of these two values of A. and from e of the fork n, which should be found marked upon it, determine with the aid of the equa- tion S = n\ the velocity of sound in air. Compare this with the theoretical value, deduced from the barometer reading and the density of air for the existing temperature and pressure, by substitution in the equation c_ yp & \ i \ p Xext repeat very carefully the determination of the first resonant length, the fork being now held at least as far away from the end as the radius of the tube. By subtracting this length from a true fourth wave length, as determined above, find the correction which must be applied at the open end of a pipe to make the first resonant length ^X. This might have been found from the preceding measurements had not the fork been held closer to the end than the radius of the tube. This was done for the sake 200 ELECTRICITY, SOUND, AND LIGHT of making it easier to locate accurately the second and third resonance points which otherwise would have shown but feeble reinforcement. Express the correction x as a fractional portion of the radius of the pipe. EXAMPLE The experiment was performed at a temperature of 24.0 C. and a cor- rected barometric pressure of 74.63 cm. The density of air at this tem- perature and pressure was found from Table 2 in the Appendix to be .001167. The pressure p, expressed in dynes per square centimeter, was found by multiplying 74.63 by 13.596, the density of mercury at 0, and I~~J by g. Hence S, computed from the theoretical relation S = -i/y ,was 34,570 cm. per second. The resonant lengths observed were 15.9, 49.8, and 84.1 cm. Hence the average | wave length was 34.0 cm. The fork was marked n = 500 ; but when it was sounded simultaneously with a standard 512 fork, and the two held close to the ear, it was found that there were four beats a second. Furthermore, loading the standard fork slightly with wax was found to decrease the number of beats. Hence the correct rate of the fork was 508. Therefore, since S = nX, the observed value of 5 was 1016 x 34.0 = 34,544 cm., a value which differs from the theoretical result by but .07 per cent. When the fork was held 3 cm. above the end of the pipe the first reso- nant length was found to be 15.7 cm. Hence x = 17 15.7 = 1.3 cm. The radius of the pipe was 1.8 cm. Hence this correction was .71 of the radius. CHAPTEE XVIII THE MUSICAL PROPERTIES OF AIR CHAMBERS 129. Notes to which a closed pipe will respond. It has been shown in section 128 that resonance is possible in a pipe closed at one end when the corrected length of the pipe (which will be denoted by P) is an odd multiple of one fourth of the wave length X of the train of waves entering the pipe ; i.e. when P = \ X, or when P = I X, or when P = | X, and so on. If the pipe length is kept constant while the wave length is changed, it follows at once that a succession of wave lengths, \, X 2 , X 3 , X 4 , and so on, will be found to which the pipe will respond, and that Xj = 4 P, X 2 = ^ P, X 3 = | P, and X 4 = ^ P, etc. These wave lengths bear the ratios 1, 1, l, l, etc. Hence, since vibration numbers, n, vary inversely as wave lengths (for S = n\), the vibration numbers of the notes which are able to produce resonance in a closed pipe must bear the ratios 1, 3, 5, 7, 9, and so on. The note of longest wave length to which a given pipe can respond is called the fundamental of the pipe ; the notes of higher frequency which give resonance are called its overtones. 130. Notes to which an open pipe will respond. If it is true, as stated in section 128, that a wave which is traveling down a pipe is reflected, upon reaching an open end, just as it would be if it had come to a new medium of smaller density than that within the pipe, then it ought to be possible to obtain the phenomenon of resonance with open as well as with closed pipes ; and, furthermore, the shortest length of an open pipe which should produce the reenforcement of a given train of waves should be twice as great as the shortest resonant length of a closed pipe. For, if a pulse of condensation (Fig. 142), in which the particles are moving from left to right, is reflected upon reaching B as a pulse of rare- faction, i.e. as a motion of the particles of the returning wave 201 202 ELECTRICITY, SOUND, AND LIGHT from left to right, it is at once evident that the shortest length of AB for which there can be a union of the direct and reflected waves in the same phase is that which permits to return to A just in time to unite with 1. This means obviously that the shortest resonant length must ~be one half the- wave length of the train, instead of one fourth, as in the case of a closed pipe. The fact that a length of open pipe can indeed always be found which will respond just as' loudly to a given note as any closed pipe, and that this length is twice as great as that of the shortest resonant closed pipe, may be taken as complete experimental demonstra- tion of the statement made in section 128 as to the nature of the reflection occurring when a wave reaches the open end of a pipe. If the open pipe is gradually lengthened, there should obviously again be resonance when returns to A just in time to unite with FIG. 142 2 ; i.e. when the pipe length has been increased by one half wave length, and again when returns in time to unite with 3, etc. In a word, an open pipe should produce resonance when its length is a,ny multiple whatever of J-X; i.e. when the pipe length bears to the wave length any of the ratios \, f , |, |, |, f, etc. This is equiva- lent to saying that the notes which will produce resonance in a given open pipe of fixed length must bear the frequency ratios 1, 2, 3, 4, 5, etc. In other words, while the fundamental and over- tones of a pipe closed at one end must bear the frequency ratios represented by the odd numbers only, the fundamental and over- tones of an open pipe should bear the frequency ratios represented by all the numbers, even and odd. This is often stated thus : In closed pipes only the odd overtones are possible ; in open pipes all the overtones, even and odd, are possible. MUSICAL PROPERTIES OF AIR CHAMBERS 203 131. Natural periods of pipes. Not only will a given pipe, open or closed, intensify, as explained above, trains of waves of certain definite wave length which present themselves at its mouth, but a single pulse entering such a pipe must be returned, by virtue of successive reflections at the ends, as a succession of pulses follow- ing one another at equal intervals. In other words, a single pulse must be given back by the pipe as a musical note, of very rapidly diminishing intensity, it is true, but of perfectly definite wave length. Furthermore, this wave length must be the wave length of the train which is capable of producing the fundamental reso- nance of the pipe. For, if the pipe is closed, for example, at the lower end, then the first time the pulse returns to the mouth after reflection at the closed end it will produce an outward motion of the particles near the mouth, the next time an inward motion, the next time an outward motion, and so on ; i.e. the pulse must travel four times the length of the pipe in the interval between the appearance of two successive condensations at the mouth. The length of the pipe is thus one fourth of the wave length of the note given off FlG 143 by it, and this is the relation which ex- ists in the case of a train of waves producing the fundamental resonance. The pipe is therefore said to have a natural period, or to be capable of producing a note of wave length four times as great as its own length. If the pipe is open instead of closed at the farther end, a single pulse (a condensation) entering at a (Fig. 143) will emerge at b first as a motion of the particles from left to right. The reflected portion will then travel back through the tube as a motion of the particles from left to right (a rarefaction), which will in turn be reflected at a, still as a motion from left to right ; and thus, after traveling the length of the tube twice, the pulse will again emerge at 5 in its original direction (a condensation). Thus, in this case, the wave length of the train of waves into which the pipe has transformed the single pulse is twice the length of the pipe ; i.e. the note given off by the pipe has, as before, the same wave length as that which will produce the fundamental resonance in the pipe. This note is, 204 ELECTRICITY, SOUND, AND LIGHT of course, an octave higher than the note given off by a closed pipe of the same length. It is this ability of a pipe, open or closed, to pick up irregular pulses and transmute them by successive reflections into notes of definite pitch which explains the contin- uous humming in definite pitch which is heard when a tube, a sea- shell, or any sort of cavity of sufficient size is held close to the ear. 132. Production of the fundamentals of pipes by air jets. In order, however, that a pipe may be made to give forth its funda- mental note distinctly, it is necessary to do more than to start a single pulse in at one end ; for the energy of this pulse is dissipated so rapidly in the successive reflections and transmissions that only when the pipe is placed very close to the ear can anything which resembles a musical note be recognized at all. If, however, a gen- * tie current of air is directed continuously against one edge of the pipe, as in Figure 144, the fundamental note can be made, with suitable blowing, to come out very strongly. In order to understand this action, consider first a pipe closed at the lower end, and sup- pose that the original current of air is so directed as to strike at a just inside the edge (see Fig. 144). A condensation starts down the pipe and is reflected, when it reaches the bottom, as a condensation, i.e. as an upward motion of the particles. When this con- densation reaches the mouth it pushes the current of air outside of the edge. This starts a rarefaction down the pipe which, upon its return to the mouth as a rarefaction, draws the current of air inside the edge again. Thus the current is made to vibrate back and forth over the edge, the period of its vibration being controlled entirely by the natural period of the pipe; for between two instants of emergence of the jet from the pipe a rarefaction must travel twice the length of the pipe and then a condensation must do the same ; i.e. a sound pulse must travel four times the length of the pipe. Hence the wave length of the emitted note is the same as that which corresponds to the natural period; i.e. it is four times the length of the pipe. The FIG. 144 MUSICAL PROPERTIES OF AIK CHAMBERS 205 source of the musical note is to be found, then, in the vibration of the air jet into and out of the end of the pipe. The pipe itself may be looked upon merely as a device for enabling the jet to send pulses to the ear with perfect regularity. The theory of the open pipe differs only slightly from that of the closed. If the jet is directed just inside the edge, a condensa- tion starts down the pipe, and at the same time, as is indeed also the case with the closed pipe, the pressure within the upper end of the pipe begins to rise because of the influx of air. If the blow- ing is of just the right intensity, this pressure may force the jet outside the edge at just the instant at which the original conden- sation reaches the lower end and starts back, in this case as a rarefaction. When this returning rarefaction reaches the mouth it draws the jet inside again. At this instant the rarefaction which started down the pipe when the jet first swung outside has just reached the lower end of the tube. Upon its return to the mouth as a condensation it drives the jet again outside, and thus the jet is alternately forced back and forth over the edge, its period being controlled entirely by the natural period of the pipe, for it will be seen that between two successive emergences of the jet from the mouth of the tube a sound pulse travels down the tube and back. If the blowing is not of just the right intensity so that the pres- sure reaction near the mouth throws the jet out for the first time at just the instant at which the first condensation reaches the lower end, then the pulses reflected from the lower end do not reach the mouth at the right instants to set up regular vibration of the jet over the edge, and consequently no note is produced. 133. Production of the overtones of pipes by air jets. If, in the case of the open pipe, the violence of the blowing is increased to just the right amount, the pressure within the top of the pipe may be increased so rapidly that the jet is thrown out in just one half its former period. In this case the reflected pulses will get back to the mouth in just the right time to keep the vibration going, but the note given forth will be the first overtone of the open pipe, namely the octave of the fundamental. Similarly, still harder blowing of just the right intensity will cause the jet to swing out in just one third its former period, and the returning 206 ELECTRICITY, SOUND, AND LIGHT pulses will then get back to the mouth in just the time to keep the jet vibrating in the period of the second overtone, the fre- quency of which is three times that of the fundamental, etc. Blowing of intermediate intensities will produce no notes at all, since the times of return of the reflected pulses are then such as to interfere with the period of vibration which is starting, instead of to keep it going. The production of overtones in closed pipes is precisely similar, save that in order to produce the first overtone the blowing must be so hard as to cause the jet to swing out of the pipe in one third of the time required for the first con- densation to travel to the bottom and back, for the first overtone of a closed pipe has a fre- quency three times that of the fundamental, the second five times, etc. (see sect. 129). By blowing with varying degrees of violence across either open or closed tubes, it is generally easy to produce three or four notes of different pitch which are found to have precisely the frequen- cies demanded by the above theory. If the pipe is long and narrow, it may be quite impossible to produce the fundamental for the reason that the jet is forced out by the increased pressure long before the first pulse returns from the remote end. 134. Types of wind instruments. The above theory explains the action of nearly all wind in- struments. In organ pipes (Fig. 145) the current of air is forced through the tube ab into the air* chest C, thence through the narrow slit de, into the embrocliure E or mouth of the pipe, where it passes as a narrow jet toward the thin edge or lip fg. As a result of small differences in pressure inside and outside of the embrochure, the jet is caused to deviate to one side or the other of the lip. When the pipe is sounding, it vibrates back and forth across the lip precisely as the air jet vibrated FIG. 145 MUSICAL PROPERTIES OF AIR CHAMBERS 207 FIG. 146 back and forth across the edge of the pipe in the discussion of section 133. Flutes and whistles of all sorts are precisely similar in their action to organ pipes. In any of them the air chamber may be either open or closed. In flutes it is open ; in whistles it is usually closed ; in organ pipes it is sometimes open and sometimes closed. In pipe organs there is a different pipe for every note, but in flutes, fifes, etc., a single tube is made to produce a whole series of notes either by blowing over- tones or by opening holes in the side, an opera- tion which is equivalent to cutting off the tube at the hole, since a reflected wave starts back as soon as a point is reached at which there is greater free- dom of expansion than has been met with before. In the case of some instruments, like the clari- net (Fig. 146), the end of the pipe against which the performer blows is almost closed by a reed I which is loosely pivoted at the base and free to swing, under the influence of an outside pressure, so as to close the opening entirely. When the performer blows upon this mouth- piece, a pulse of condensation enters the tube and at the same time the reed closes the open- ing. This pulse after reflection from the open end of the clarinet as a rarefaction, and a sub- sequent reflection at the mouthpiece (closed by the reed), also as a rarefaction, is again reflected at the open end, but now as a condensation ; and therefore, after traveling the tube four times, the original condensation returns and forces the reed open, admitting a new pulse. The overtones which may be produced in such an instrument are evidently those of a closed pipe. It is evi- dent that the vibration frequency is independent of the reed and depends only upon the effective length of the clarinet. In the case of instruments like the trumpet and other brass wind instruments the current of air enters a mouthpiece similar FIG. 147 208 ELECTEICITY, SOUND, AND LIGHT to that shown in Figure 147. The lips of the performer act as a double reed. A pulse of condensation enters, the lips closing when the reaction of its pressure equals that of the air in the rnouth of the performer. This pulse, reflected as a rarefaction from the open end of the trumpet to the lips, reduces the pressure at that point and a new pulse enters. The fundamental depends then only upon the length of the instrument. The overtones are produced exactly as in an organ pipe, by blowing more suddenly and to some extent by increasing the tension of the lips. The possible overtones are those of an open pipe. 135. The musical scale. The physical basis of harmony in music lies in the simplicity of the ratios of the vibration frequen- cies of the notes which are sounded together. When a note and its octave are sounded together the result is recognized by the ear as agreeable, and the two notes are said to be consonant. The ratio of the frequencies of the two notes, known as the interval between them, is in this case ^. The next most consonant interval is that between do and sol (C and G). It is found to be the next most simple physical interval, having a value |. It is known as a fifth because G is the fifth note above C in the sequence of eight notes which constitutes the octave of the ordinary musical scale. If the note G and the octave of C are sounded together, the inter- val is |, as is evident from the fact that | of 1 is | of 2. These three notes do, sol, and the octave of do thus utilize the three simplest frequency ratios, namely -|, |, and -|. The next most conso- nant interval is that between do and mi (C and E). It is known as the third and represents the vibration ratio |. The notes do, mi, sol (C, E, G) sounded together are known as the major chord. It will be seen from the above that their relative vibration fre- quencies are as 4 : 5 : 6. The so-called major diatonic scale is made up of three major chords. The absolute vibration number taken as the starting point is wholly immaterial, but the explanation of the origin of the eight notes of the octave, commonly designated by the letters C, D, E, F, G, A, B, C/ may be made more simple if we begin with a note of vibration number 24. The first major chord do, mi, sol, or C, E, G would then correspond to the vibration numbers MUSICAL PROPERTIES OF AIR CHAMBERS 209 24, 30, 36, or, as explained above, the vibration ratios 4, 5, 6. The second major chord starts with C', the octave of C, and comes down in the ratios 6, 5, 4. The corresponding vibration numbers are 48, 40, 32, the corresponding notes do', la, fa, and the correspond- ing letters C', A, F. The third chord starts with G as the first note and runs up in the ratios 4, 5, 6. This gives the vibration numbers 36, 45, 54, the syllables sol, si, re, and the letters G, B, D'. Since the note D' does not fall within the octave, its vibration number being above 48, the note D, an octave lower and having a vibration number 27, is taken to complete the eight notes of the major diatonic scale. The chord do-mi-sol is called the tonic, sol-si-re the dominant, and fa-la-do the subdominant. The relations between the notes of the octave are given below in tabular form. Notes C D E F G A B C' Frequencies 24 27 30 32 36 40 45 48 Intervals with C . . . 1 f J f -^ 2 major major fourth fifth major seventh octave Name of interval . . . , second third sixth Intervals with next note f * if f -V f Jf major minor half Name of interval. . . Any scale the notes of which are separated by these intervals is known as a scale of just temperament. The scale adopted by physicists starts with middle C = 256. That adopted internation- ally for musical purposes has A = 435. This gives a series of values slightly higher than that of the physical scale. The scale formed on C = 256 follows. Absolute names . . . C D E F G A B C' D' E' F' G' A' B' C" Syllables do re mi fa sol la si do re mi fa sol la si do Relative frequencies .Iff -^ 2 f \ f 3 Af- - 1 , 5 - 4 Absolute frequencies 256 288 320 341.3 384 426.6 480 512 576 640 683 768 853 960 1024 It is often desirable in music to adopt some other note than C as the fundamental note in the scale, i.e. as the keynote. Thus the scale formed on G as a fundamental and having intervals of just temperament would be do re mi fa sol la si do 384 432 480 512 576 640 720 768 210 ELECTEICITY, SOUND, AND LIGHT If, then, a piece of music is to be played in just temperament in the scale of G, the instrument used must be capable of producing two notes in addition to those found in the major scale of C, namely the notes corresponding to the vibration numbers 432 and 720. Similar investigation of the other possible scales of A, B, D, E, and F show occasion for many other notes, so that a piano which could be played in just temperament in all the keys demanded by modern music would require about 50 notes in each octave. Since, how- ever, the introduction of all these notes would make the manipu- lation of the instrument a physical impossibility, there has been devised, for keyed instruments of the piano type, another scale which is known as the scale of even temperament. The origin of this scale may be seen as follows. A careful comparison of all the notes necessary for the various scales (found as above in the case of G) shows that many of these notes differ so slightly that a single note may do satisfactory duty for several of almost the same frequency, provided we are will- ing to content ourselves with slightly imperfect intervals. Twelve notes are accordingly chosen to replace the fifty, and these twelve are, as a matter of fact, made to divide the octave into twelve exactly equal parts. It is for this reason that the scale is called the scale of even temperament. Since there are twelve equal intervals between a note and its octave, each interval is of value v^, or 1.059. This scale is written below, starting with C = 256 for the fun- damental note. For purposes of comparison the frequencies of the notes in the scale of just temperament are also given. Notation C CtforDb D DJorEb E F FflorGb G GflorAb A AJorBb B C Even 256 271.3 287.4 304.8 322.7 341.7 3G2.2 383.8 40G.G 430.7 456.5 483.5 512 Just 256 288 320 341 384 - 427 480 512 It is evident from the table that although only C and its octave retain their old values in the scale of just temperament, the dif- ference between the frequencies of any note in the two scales is so small as not to be in general noticeable.* * The difference would, of course, be at once noticeable because of the phe- nomenon of beats (see sect. 149, p. 234), if the same note were sounded simultane- ously in the two scales. In general this would not occur, for all instruments with MUSICAL PROPERTIES OF AIR CHAMBERS 211 EXPERIMENT 18 Object. To find the overtones possible in open and closed pipes. Directions. Figure 148 shows an arrangement consisting of a rotating table T, a siren W, and a five-foot open pipe P of diameter 5 or 6 cm. A current of air is forced through the glass tube g by a large bellows or other arrangement. When W is rotated the current of air from the bellows sends a pulse of condensation into the pipe every time an opening in W comes over the tube g. When W is rotated with a frequency such that the number of these pulses entering the pipe is equal to the natural vibration frequency of the pipe, resonance takes place and the pipe gives forth loudly its fundamental note. In the same way, if W is rotated with the proper uni- form speed, P may be made to give forth loudly any desired overtone. Determine exactly the number of holes which pass the ori- fice of the pipe for one revolution of the wheel T. Cause the pipe to give forth its fundamental* and hold the speed con- stant while a second observer takes, with a stop watch, the time of 15 revolutions of T. Measure the pipe length. From the data thus obtained and the correction which it was found necessary in Experiment 17 to apply to the open end of a pipe, deduce the velocity of sound in air. Repeat the determination, using successively the first, second, third, and fourth overtones instead of the fundamental. Transform P into a closed pipe by inserting a cork in the upper end, and determine as above the vibration numbers corresponding to the fun- damental, the first, and the second overtones. fixed keyboards or frets are tuned to the scale of even temperament. It is, how- ever, worthy of note that many prominent violinists, when playing without accompaniment by an instrument tuned in even temperament, instinctively play in just temperament and so satisfy the demands of the human ear for intervals represented by the ratio of simple whole numbers. * Difficulty may be experienced in detecting the frequency corresponding to the fundamental because of the fact that the pipe may give off its fundamental note weakly for a number of pulses smaller than that corresponding to its natural frequency (see sect. 131). This response will scarcely be noticeable except when the number of pulses is one half that corresponding to the natural period, when it may be quite pronounced. In general, however, the funda- mental resonance will be so much louder that it can scarcely be mistaken. If there is doubt, multiply at once twice the pipe length by the number of pulses and see whether the product is about 34,000 cm., as of course it should be. FIG. 148 212 ELECTKICITY, SOUND, AND LIGHT EXAMPLE The plate W had 40 holes and made 9}-|- revolutions for each revolution of T"; hence 374 pulses entered the tube P for each revolution of T. The pipe had a length of 113 cm. and a radius of 3.8 cm. ; hence the corrected length was 115.6. The average of three determinations of the number of seconds necessary for 15 revolutions of T when the open pipe was emitting its fundamental note was 37.4 seconds. The average found for 30 revo- lutions when the first overtone was sounding was 37.2 seconds ; for 45 revolutions with the second overtone it was 37.6 ; and for 60 revolutions with the third overtone, 37.4. The corresponding frequencies were 150, 300, 450, and 600 ; that is, they were in the ratio of 1, 2, 3, 4. The aver- age speed of sound from these values of the frequency and pipe length was 347.6 m. per second. The correct value as found from the temperature 21. 5 C., and the relation of section 126 was 344.1 m. per second. The difference was about 1 per cent. A cork placed in the upper end of the pipe shortened its corrected length to 113.7 cm. The average time for 15 revolutions when the funda- mental of this closed pipe was sounding was 74.3 seconds. The time of 15 revolutions with the first overtone was 24.7 ; with the second overtone it was 14.8 seconds. The corresponding frequencies were 75.5, 227, and 380; that is, the frequencies were in the ratio of 1, 3, 5. The average value of the speed of sound from these observations was 344 m. This determina- tion with a closed pipe differed therefore from that made with an open pipe by 1.2 per cent. CHAPTEK XIX LONGITUDINAL VIBRATIONS OF RODS 136. Velocity of waves in thin rods of elastic material. The analysis of section 122 shows that if there is no possibility of lateral expansion of the medium, the velocity of a compressional wave depends only upon the bulk modulus of elasticity and the density of the medium. This condition is realized when a disturb- ance originates in the midst of an elastic medium of great extent in all directions. But when the wave travels along a thin rod there is a slight lateral expansion of that portion of the rod which is undergoing the compression. Hence if we imagine a rod of 1 sq. cm. cross section divided into centimeter cubes after the fashion of section 122, and if we imagine a small pressure dp to be applied by means of a piston p at one end (Fig. 134), then while each cube is undergoing the voluminal compression dv, the piston will move forward, not now dv, but some distance ds numerically a trifle larger than dv. From reasoning identical with that given on page 188, an -equation results which differs from equation (2), page 189, namely $ 2 = dp/pdv, in no respect save that ds replaces dv. We obtain, then, dp But ds is the change in the length of the rod per unit length. Hence dp/ds is Young's Modulus (Y).* Thus in thin rods com- pressional wavks move with a velocity which is given by the equation *See "Mechanics, Molecular Physics, and Heat, p. 67. 213 214 ELECTRICITY, SOUND, AND LIGHT 137. Natural periods of free rods. A rod surrounded by air is in every respect analogous to an open pipe, for the reflections at the ends are such as occur when a wave passes from a denser to a rarer medium. Thus such a rod will respond to a train of waves if it is of such length that a pulse (Fig. 149) just entering the rod at A as a condensation will, after reflection at B, return to A and be again reflected as a condensation at the precise instant at which pulse 1 reaches A. The length %AB is then the distance which a pulse travels in the rod before the succeeding pulse 1 enters the rod. This is, by definition, one wave length of the note in the rod. If one single pulse strikes the rod, the successive reflections of this pulse at A and B will cause a train of waves to be given off at each end. Thus the rod will emit a musical note the wave length of which in the rod is twice the length of the rod. The wave length in air of this note obviously bears the same relation to its wave length in the rod as the velocity of the wave in air bears to its velocity in the rod. If the rod be clamped in the middle, it will respond to and give off precisely the same note as though it were free, for the com- pression produced by the clamp at the middle produces at that point the same sort of a reflection as occurs at the boundary of a denser medium ; hence the rod is equivalent to two closed pipes, each of which gives off the same note as would an open pipe (i.e. a free rod) of double the length. In order to set a rod into longi- tudinal vibrations of this sort, it is customary, instead of striking one end, to clamp it in the middle and stroke it with a rosined cloth if it is of metal, or with a wet cloth if it is of glass. The mechanism of the tone production in this case will be more fully discussed in section 140. 138. Comparison of the velocities of sound in two solids. The above theory suggests an extremely simple and satisfactory means of comparing the velocities of sound in two solids. Thus we have FIG. 149 LONGITUDINAL VIBRATIONS OF RODS 215 only to find the vibration frequencies (the pitches) of two notes produced by stroking steel and brass rods of the same length, in order to find the relative velocities of sound in steel and brass. For with rods of equal length the number of pulses communicated to the air per second by the traveling of pulses up and down the rods is obviously proportional to the velocities of sound in the two rods. Thus if S 8 and S b represent these velocities in steel and brass respectively, and n s and n b the corresponding frequencies produced by the rods of equal length, we have (3) The frequencies n s and n b can be determined in a variety of ways; for example, by changing the length of a given sonometer wire until it is in tune, first, with the note from the steel, and then with that from the brass. Since the frequencies of the notes produced under these circumstances are inversely proportional to the lengths (see Chap. XX), we have, if 1 8 and l b are the lengths of the same wire which are in tune with the steel and brass respectively, s. i,, S, L (4) 139. Nodes and loops in pipes and rods. A careful considera- tion of the resonance of pipes which are giving off the first or higher overtones reveals effects which have thus far been over- looked. For example, it was shown that when a pipe has its FIG. 150 second resonant length, a condensation (Fig. 150) must return to the mouth of the pipe at the instant at which the rarefaction | reaches the mouth; but, in the return after reflection, must somewhere in the pipe collider with the advancing condensation 1. Since at the instant of the reflection of 0, 1 is one wave length 216 ELECTRICITY, SOUND, AND LIGHT behind 0, it is evident that this collision must take place just | wave length from the end of the pipe, namely at n. Such a col- lision of two oppositely moving condensations is entirely analogous to the collision of two oppositely moving perfectly elastic balls. These are shown simply to exchange motions,* the effect being the same as though each ball passed through the other without experiencing any effect whatever from it. Thus the waves may be thought of as passing through one another, and their mutual effects may be ignored. As a matter of fact, it is of course 1 which returns to the left after the collision and unites with f at the mouth, while is forced back again toward the closed end of the pipe. One half period after the collision at n (Fig. 150) of the con- densations 1 and (-^- -*-) there will occur at n a collision of the rarefactions |- and J- (-*- --). Thus the particles near n are first pushed together by opposing forces, then pulled apart by opposing forces. The result is that they do not move at all. The matter FIG. 151 about n suffers alternate compression and expansion, but the par- ticles at n can never move either to left or to right, because they are always being urged in opposite directions by the oppositely moving waves. The .point n is called a node. The points between the nodes where the disturbance is greatest are called loops. If the length of the pipe is -|, J,|, etc., wave length, it is evident from considerations precisely like the above that there will be nodes at n 1 , n", n'", etc. (Fig. 151). In other words, in any reso- nant closed pipe (and it is to be remembered that such a pipe is * See "Mechanics, Molecular Physics, and Heat," Chapter VII. LONGITUDINAL VIBRATIONS OF KODS 217 resonant when and only when its length is an odd number of fourth wave lengths) the first node is one fourth wave length from the open end, and other nodes follow at intervals of one half wave length. The conventional method of representing nodes and loops in pipes is that used in Figure 151. Since the first resonant length of an open pipe is one half wave length, and since a condensation is reflected as a rarefaction, it is evident that will collide in the middle of the pipe with J-. Hence an open pipe responding to its fundamental has a node in n' FIG. 152 the middle. Similarly an open pipe responding to its first over- tone has nodes at n 1 and n" , each one fourth wave length from an end and one half wave length apart. Similarly for the higher overtones (see Fig. 152). Although the preceding discussion has been limited to pipes, yet, since by section 137 a rod surrounded by air acts in every respect like an open pipe, the above conclusions hold also for rods. 140. Kundt's tube experiment. A rod mn (Fig. 153), to one end of which is attached a light cork piston B, is supported by a clamp C at its middle point. The piston B fits very loosely in a long tube AB, one end of which is closed by a tightly fitting piston A. The rod mn is set into longitudinal vibration by drawing along it a cloth covered with rosin. It would seem at first thought as though the slipping of the cloth along the rod were so irregular that no musical note could be produced. As a matter of fact, however, the slipping is controlled by the natural period of the rod in much the same way as the vibrations of the air jet at the 218 ELECTRICITY, SOUND, AND LIGHT mouth of an organ pipe* are controlled by the natural period of the pipe. Thus the first slip starts a pulse down the rod which, because of the reflections at the ends, returns to the starting point at stated intervals. Of course the tendency to slip is greatest at the instant of the return of the first pulse, so that succeeding slips take place at the instants of return of succeeding pulses. Thus the rod gives off loudly the note corresponding to its natu- ral period. The rod mn is clamped in the middle, but it was shown above f that the natural period in this case is precisely the same as when the rod is free. The wave length of the note produced in the material of which the rod is composed is there- fore twice the length of the rod. The piston A is so adjusted in position that the air column AB is of such a length as to be resonant to the note of the rod. Nodes are then formed which may be brought into evidence by placing FIG. 153 along the bottom of the tube a layer of light cork filings. The cork dust will be collected into ridges at the points of maximum disturbance, i.e. at the loops. The explanation of the fact that each loop is marked not by a single ridge, but by a series of ridges, demands an analysis which is beyond the scope of this text. The wave length of the note given off by the rod may be obtained at once by measuring the distance between two successive loops. If, then, S r represents the velocity of the wave in the rod, \ r its wave length in the rod, and n the frequency, we have S r = n\ r . Also, if the tube contains air, and S represents the velocity of sound waves in air, and X the wave length of a note of frequency * See section 132, page 204. t See section 137, page 214. LONGITUDINAL VIBRATIONS OF RODS 219 7i, then *S' = n\. Combining these two equations we obtain as the expression for the velocity of compression al waves in the material of the rod, ~ Sr = ^S, (5) in which X r is merely 2 mn and X is twice the distance between nodes in air. By similar reasoning, if some other gas is substituted for the air in the tube and S g and X^, represent the velocity and wave length of the same note in that gas, then S g = n\ g> or c< \<7 o /f\ ,S, = -A. (6) Since S has already been determined, and X and X^ may be ob- served, the velocity of sound in any gas may be found. EXPERIMENT 19 (A) Object. To find the velocity of compressional waves in steel. Directions. Following the method of the Kundt's tube experiment de- scribed in section 140, adjust carefully the sliding piston A (Fig. 153) until a maximum of agitation of the cork dust at the loops is produced when a steel rod mn is stroked with a rosined cloth. Measure the distance between A, which is a node, and the node most remote from A, say the nth, then that between A and the (n l) st node, then between A and the (n 2) d , etc. Make two vertical columns, one of measured distances, the other of the corresponding numbers of half wave lengths. The sum of the first column divided by the sum of the second gives the most accurate value of -| wave length which is obtainable from this sort of an observation. Shake up the cork dust and obtain a second set of readings, measuring this time first between the first and last loops, then -between the second and next to the last, etc., and averaging as described above. If S r represents the velocity of sound in steel, X r the wave length in steel of the note produced, and if S and X represent the corresponding quantities when the waves have passed over into air, then S r can be found at once from the relation S r = (\ r /\)S. Then from S r and X r the frequency n may be obtained if desired. Compare the observed value of the velocity in steel with the theoret- ical value deduced from Young's Modulus, and the density, by use of the relation S = ^Y. 220 ELECTRICITY, SOUND, AND LIGHT (B) Object. To find the velocity of sound in CO 2 . Directions. Replace the air in the Kuudt tube by CO 2 by permitting a gentle current from a charged cylinder to pass in at o (Fig. 153) and out at o' for two or three minutes. Then determine, precisely as above, the wave length in CO 2 of the note given forth by the steel rod. Thence deduce the velocity of sound in CO 2 and compare the result with the theo- retical value obtained from the barometer height, the density of CO 2 (viz. 1.53 x the density of air), and the value of y (= 1.30). (C) Object. To compare the velocities of sound in brass and steel. Directions. Take two equal rods three or four meters long, one of steel and one of brass. Set them successively into longitudinal vibrations by clamping them in the middle and stroking with a rosined cloth. By means of a sliding bridge vary the length of a small wire of a sonometer (or violin) until, when picked transversely, it produces first a note in tune with that of the steel rod, then with that of the brass rod. Determine the rela- tive velocities by the method of section 138 and compare with the relative values obtained by taking the values of Young's Modulus and the appro- priate densities from a table and substituting in equation (2), page 213. EXAMPLE (A) The observations of the distance between nodes made according to the directions given above were as follows. Number of i Number of A Distance in cm. Distance in cm. wave lengths wave lengths' 7 65.3 6 56.1 6 55.7 4 36.5 5 47.5 _2 17.7 4 38.0 12 110.3 3 28.5 .-.IX =9.2 cm. 2 19.0 J. 9.5 28 263.5 .-. IX = 9.4 cm. The average value of A in air was thus found to be 18.6 cm. The tem- perature of the room was 26 C. The velocity of sound in air at this temperature is 331.27 + 26 x .6 = 346.9m. per second. The length mn was 136.5cm., thus making \ r = 273cm. 73 Hence S r for steel was ^ - 346.9 = 5073 m. per second. 18. o Using the value of Young's Modulus previously found from experiments on a steel wire, namely 19.7 x 10 11 , and taking 7.8 as the density of steel, LONGITUDINAL VIBRATIONS OF RODS 221 the theoretical value of the velocity of sound in steel, as calculated from the formula S A/ was found to be 5026 m. per second. The difference between the two values is .9 per cent. (B) When the tube was filled with CO 2 at atmospheric temperature the average wave length produced in the CO 2 by the vibrations of the steel rod was 14.5. Hence the velocity in CO 2 was 346.9 x ii^ = 270.4 m. The 18.6 corrected barometric pressure was 74.22 cm. The value of the velocity calculated from this pressure, together with the density of carbon dioxide, namely .001152 x 1.5:3, and the factor y = 1.30, was found to be 269.4. The difference was thus .4 per cent. (C) Rods of steel and brass 290 cm. long were tuned to lengths of 21.2 cm. and 30.5 cm. respectively on a given sonometer wire. Hence | ' The theoretical value obtained by using 7.8 and 8.4 respectively for the densities of steel and brass, and 19.7 x 10 11 and 10.2 x 10 11 for the corre- sponding values of Young's Modulus, was 1.442, a difference of 0.3 per cent. CHAPTER XX WAVES IN STRINGS 141. General characteristics of wave motion. In the preceding sections we have discussed only compressional, or longitudinal, wave motion, and have found this to be characterized by the fact that the particles which transmit the wave move in the line of propagation of the wave itself. This is the only sort of an elastic wave which is possible in substances which do not possess rigidity.* But in substances which possess rigidity another type of elastic wave motion is possible, namely transverse wave motion. This is characterized by the fact that the particles of the medium move in paths which are perpendicular to the direction of propagation of the wave. The waves which travel along a rope when one end is caused to vibrate by the hand are of this sort. Before this second type is discussed it is of importance to have clearly in mind the general characteristics of wave motion. These may be seen from a consideration of Figure 154, which represents the effect on the particles of a medium of an oscillatory motion of the piston P. The unit cubes of Figure 134, section 122, page 187, are here replaced by vertical lines. Thus Figure 154, , rep- resents the state of the medium before the piston has begun to move. Figure 154, 5, represents its state when the piston has un- dergone its greatest displacement to the right and is ready to return, a condition which is represented by the double arrow (< ). In the upper line of arrows each arrow represents the direction of displacement of the layer toward which it points. The small arrows below the vertical lines show the direction of the motion of the layers. The zero below any line indicates either that the medium is there at its mean (original) position, or is just changing the direction of its motion at the end of its path. The succeeding * The large waves on the surface of a body of water are gravity waves and have nothing to do with the elasticity of matter. 222 WAVES IN STRINGS 223 figures show the progression of the initial condensation and the subsequent rarefaction for the indicated positions and directions of motion of the piston. Obviously a condensation exists, for ex- ample, in Figure 154, 6, at 4, since the layers are there crowded together, and conversely a rarefaction exists in Figure 154, d, at 4, since the layers are there separated. From the figures it is evident that the layers, or particles, of the medium are in vibration, and that at any instant these particles possess a definite configuration, for example that of particles 4 to Z J 4 S 6 78 9 10 II 12 '3 14- IS 16 I 2 3 4 S 6 7 / 2 3 4 S 6789/0 / 2 J 4- S 6 7 8 9 10 // /2/J J 4 S (> 7 8 9 10 II / 2 13 14- /S /6 FIG. 154 16 in Figure 154, f. This configuration is known as the wave form, and it travels from left to right through the medium, although the particles themselves are merely vibrating back and forth across their original positions. If the motion of the piston is simple har- monic, then the motion of the particles must be simple harmonic also. As a matter of fact practically all vibrations which arise from the elasticity of matter are of this type. The amplitude, or maximum displacement, of the particles depends upon the ampli- tude of the motion of the piston, that is, upon the intensity of the 224 ELECTRICITY, SOUND, AND LIGHT disturbance which the particles are propagating. The difference between the times at which any two particles of the medium pass through the middle points of their paths, divided l>y the period of the vibration, is called the phase difference between the particles. As has already been indicated in Chapter XVII, the distance between two successive particles which are in similar states of motion at the same time is called a wave length. Thus a wave length is the dis- tance between particles 4 and 16, or 1 and 13, in Figure 154,/. 142. Transverse waves. The conception of wave motion just given will now be considered more analytically in connection with the transmission of transverse waves. Thus if all the particles in the line XX' (Fig. 155, a) are in some sort, of rigid connection, and ft*. f -I >> . FIG. 155 if particle 1 is given a displacement in a direction perpendicular to XX', then this displacement will be successively communicated to particles 2, 3, 4, 5, etc. Further, if 1 is made to vibrate with simple harmonic motion across XX 1 , then all the particles 2, 3, 4, 5, etc., will in succession take up this simple harmonic motion across XX'', i.e. a series of transverse waves will travel along XX'. Let the amplitude of the motion of each particle be represented by A and the period by T. Then it may be shown that the ver- tical displacement y z of any particle, such as 2, expressed in terms of A, T, and the time t since that particle left its original position, is as follows : y a =^sm27r-~ (1) WAVES IN STRINGS 225 This will be evident from a consideration of Figure 156; for a particle P moving with simple harmonic motion along the ver- tical path of length 2 A has at any instant a displacement from the center which is represented by the distance from to the projection P upon that path of a point P' which moves uniformly in a time T about the circumference of a circle which has 2 A as its diameter.* The angular speed of the point P' about is 2 TT/T radians per second. After t seconds the particle P r has moved through an angle of 2 jrt/T radians. The distance OP is then A times the sine of this angle, i.e. y 2 =^sin27r - The displacement at this instant of another particle such as 3 (Fig. 155,/) is, of course, different. Thus, suppose that this par- ticle leaves its mean position t' seconds after particle 2 has left its mean position, i.e. let t'/T be the phase difference between the two particles. The displacement of 3 at the instant con- sidered is obviously = A sin 2 TT t-t r T r And similarly the displacement of any particle is represented by an expression of the form y = A sin 2 TT t-t' T 2A - FIG. 156 where t'/T is the phase difference between the particle under con- sideration and the reference particle. The particle for which t' = T will be in a state of motion similar to that of the reference particle 2, and therefore distant from it by a wave length X. It is also evident that while the first disturb- ance which is imparted to particle 2 is traveling forward this dis- tance X (for example, in Fig. loo,/, from 2 to 14), the particle 2 *See "Mechanics, Molecular Physics, and Heat," p. 88. 226 ELECTRICITY, SOUND, AND LIGHT makes one complete vibration ; that is, the wave travels X centi- meters in T seconds, or has a velocity S which is given by S = ~ (4) Returning now to a consideration of the motion of a particle distant x from particle 2 (for example, 3, Fig. 155), it is evident x t r Tx that - = Substitution of the value t 1 = in the expression A< JL A, for the vertical displacement of any particle gives (5) The several particles along the line XX' are therefore displaced from their original positions according to a sine function. The angle represented by 2 TTX/\ is called the angle of phase differ- ence between the particle under consideration and the reference particle. Figure 155, fr, c,f, represents transverse displacements equal hi amount to the longitudinal displacements of Figure 154, b, c, f. It is also evident that the confusion resulting in the figure for com- pressional waves from the fact that the displacements are parallel to the direction of the wave motion may be obviated by plotting those displacements vertically. With that convention in mind Figure 155, b, c, f, may be taken to represent the progression either of a compressional or of a transverse wave. In the discus- sion of nodes and loops in pipes (sect. 139, pp. 215-217) this con- vention has already been used. 143. Stationary waves. The phenomenon known as stationary waves is the result of the action upon a series of particles of two equal trains of waves traveling in opposite directions. Figure 157, a, shows two such trains of waves, namely A traveling toward the left, and B traveling toward the right. The waves are represented at an instant at which the crests of A are opposite to the troughs of B, and vice versa. The heavy line shows the resultant displace- ment of the series of particles. Obviously in this case each one of the particles transmitting the motion is under the action of two WAVES IN STRINGS 227 disturbances which tend to produce equal and opposite displace- ments, and as a result the particles suffer no displacement at all. In Figure 157, b, is shown the case in which each of the wave trains has progressed an eighth of a wave length ; that is, the waves have become displaced a quarter of a wave length with respect to one another. The heavy line represents the form assumed by the row of particles at this instant as a result of the superposition of the two disturbances. Similarly, when one wave train has moved a half wave length past the other the resultant is again zero at every point. From the figures it is evident that for any particle the resultant displacement is the algebraic sum of the displacements produced FIG. 157 at that point by the two wave motions. For certain points, e.g. N, N' y N", distant from one another by a half wave length, the resultant displacement is always zero. These points are the nodes, and correspond to the nodes for compression al waves which have been previously discussed. Between the nodes the particles are in constant vibration, but all pass through their mean positions at the same time. There is therefore no phase difference between successive particles. It is for this reason that the phenomenon of the combination of two oppositely directed trains of waves on the same particles is known as the phenomenon of stationary ivaves. The amplitudes of the vibration of successive particles vary from a maximum at the loops to zero at the nodes. On opposite sides 228 ELECTKICITY, SOUND, AND LIGHT of a Dode the displacements at any instant are in opposite direc- tions. In Figure 158 are shown forms assumed by the particles between the two nodes drawn for successive instants of time. The successive positions assumed by the particles are numbered in their order. 144. Equation for a stationary wave. The ideas developed above from a study of the diagrams may also be obtained from a consideration of the equation of a wave motion. For let / 1 x\ y = A sin 2 TT / - --- I represent the equation for the displacement \T X/ given to the successive particles by the direct wave. The reversed wave must be one for which at some given instant of time (e.g. t = 0) the displacements given to the same series of particles will be equal / 1 x\ and opposite. The equation y 1 = A sin 2 TT ( -- 1 I satisfies this condition and represents the equation of the reverse wave. The resultant displacement Y is the sum of y and y l . That is, = . sn ^ - sn + By expansion * and addition we have TT- I C\ A C\ ^ \ * C\ V Y = ( 2 A cos 2 TT - I sm 2 TT A,/ JL (6) *For and sin (0 -f 0) sin cos + cos 6 sin 0, sin (0 0) = sin cos cos 6 sin 0. WAVES IN STBINGS 229 Now consider the fundamental equation for a wave motion, (t x\ - I > and notice that it consists of two 1 A./ parts. The first part is the amplitude A and the second a sine func- tion of the time t plus a phase constant x/\. Now in the expression for Y just obtained sin27r is the sine function of the time. And evidently, since there is no phase constant in this expression, the particles must all be in the same phase of vibration. Simi- larly 2 A cos 2 TT - represents the amplitude. But \ since x represents the distance of the particle under consideration from the reference particle, it is evident that the amplitude varies for successive /v particles. Also since cos 2 TT is zero when -x is an \ odd multiple of X/4, it follows that there are nodes, or points of zero amplitude, at points differing succes- sively by half wave lengths. Further, the algebraic rvi sign of cos 2 TT changes at these same points. A, 145. Melde's experiment. A capital illustration of stationary waves in strings is furnished by what is commonly known as Meldes experiment. One end of a light cord is attached to one of the prongs B (Fig. 159) of a tuning fork, while the other end carries a weight W. The waves which start down the cord from the vibrating fork are reflected at W, so that two trains F IG . 159 of waves moving in opposite directions become super- posed upon the cord. In accordance with the principles of the last section this condition tends to give rise to stationary waves, the positions of the nodes being at distances from W correspond- ing to exact multiples of a half wave length of the train sent down the cord from the fork. Since, however, the upward-moving train is again reflected at B, the condition for stationary waves in which the nodes are at distances from B corresponding to exact 230 ELECTRICITY, SOUND, AND LIGHT multiples of a half wave length is also established. It is obvious that both of these conditions can be met, and permanent stationary waves set up in the string, only if the length L of the string is an exact multiple of a half wave length.* Instead of varying the length of the string so as to fulfill this condition*, it is customary to vary the wave length by varying the load W. For the wave length X is connected with the vibration rate n of the fork and the speed of propagation S of the train of waves along the string by means of the relation S = n\ and the speed S is connected with the tension T in the string and its mass p per centimeter of .length by means of a formula which is very similar to that given in the last equation on page 189. It is, * This statement is only approximately correct, since the end of the fork is not exactly at a node, but rather just as near to a node as a point near some other node which has the same amplitude of vibration as the fork. t This formula is most satisfactorily deduced with the aid of the calculus, but it may also be obtained as follows. Let the curve mno (Fig. 160) represent a portion of the cord over which the deformation is being propagated. Let ee' be an element of the cord so small that it may be considered as the arc of a circle of radius R. If the string is wholly devoid of rigidity, then the only force which is urging the element toward the center c arises from the tension T in the string, and this may be regarded as a pull acting upon each end of the arc ee'. FIG. 160 The directions of these two pulls are the directions of the tangents to the curve at the points e and e' respectively (see Fig. 160). The component of each of i gg/ these pulls which is urging ee' toward c is T sin 6 = T 2 The total force / c C6* CO* which is urging ee' toward c is therefore 2 T = T But since the defor- 2 .R R mation mno is propagating itself unchanged in character along the string, each element of the string must assume in succession the positions occupied at any instant by all the other elements of the curve mno. In other words, at the instant which we have been considering the elem'/nt ee' is not moving at all in the WAVES IK STRINGS 231 so that by varying the tension T it should be possible to find a whole series of values of X which will give rise to permanent stationary waves in the cord. Thus when L = X/2 the string should vibrate in 1 segment, when L = 2 X/2 it should vibrate in 2 seg- ments, when L = 3 X/2, in 3 segments, etc. Or, in general, since by combining equation (7) with the equation we obtain (10) it is evident that the equations of condition for 1, 2, 3, 4, etc., segments may be written 1 (4T P These equations are obtained by substituting in (10) the above relations between L and X/2. Equations (11) show, since n, L, and p do not change, that the product of the stretching force by the square of the number of segments should be a constant, and that this constant should represent the tension when the entire string is vibrating in one segment. By substituting this constant in the first of equations (11), T having been expressed, of course, in dynes, it should be possible to find the vibration rate n of the fork, pro- vided p and L are known. The fact that all of these relations are found by experiment to hold, constitutes complete experimental \T proof of the correctness of the formula S= \l > for the case of strings which have no rigidity. direction of c, but is instead moving into the position of the adjacent element on mno; that is, it is moving with a velocity S along the circumference of the circle which has R for its radius. Hence we may apply to its motion the law of centripetal force deduced on page 102 of " Mechanics, Molecular Physics, and Heat," namely, ,02 fc=~ (8) But if p is the mass per unit length of the string, m = ee'p. Hence (8) becomes f r = R But since f c is also equal to Tee'/R, we have - = , or S = \ \ (9) R R \P 232 ELECTRICITY, SOUND, AND LIGHT 146. Fundamentals and overtones in strings. If a stretched string is plucked in the middle, the deformation travels in opposite directions to the two ends, is there reflected, and, since the two reflected portions returning to the middle unite in like phases at this point, the net result of the propagation of the disturbance back and forth over the string is a vibration of the string as a whole in the manner indicated in Figure 158, page 228, in which the various lines represent some of the successive positions of the string. A string vibrating in this way imparts successive con- densations and rarefactions to the air in which it moves, and these, being transmitted to the ear, give rise to a note of a definite pitch which is called the fundamental note of the string.* Since the time elapsing between the instant at which the string is in the position AcB (Fig. 158) and the instant at which it assumes the position AclB is the time required for the deformation to travel over the paths cBd and cAd, it will be seen that during the time of one half vibration of the string the disturbance travels on the string a distance exactly equal to the length of the string. Hence during the period of one complete vibration of the string the dis- turbance travels twice the length of the string. Thus we arrive, from a wholly different point of view, at the conclusion of the pre- ceding section, namely, that when a string is vibrating as a whole, Le. in one segment, its length is one half the wave length of the waves which are traveling back and forth over it. If the string is clamped in the middle as w T ell as at the ends and plucked one fourth of its length from one end, each half vibrates precisely as the whole string vibrated in the preceding case ; but since the speed of propagation is the same as before, while the dis- tance between reflections is one half as great, the period of vibration of each half of the string must be one half as great as the preced- ing period. Hence the note communicated to the air is the octave of the original note, and the wave length of the note is the length of the string. The note thus produced by the string is called its * Practically, of course, the sound thus derived is of small intensity, and in most musical instruments the greater magnitude of sound is due to synchronous vibrations which the string impresses upon its supports and through them upon sounding-boards and resonant volumes of air. WAVES IN STRINGS 233 first overtone. If the string is not clamped in the middle, but is plucked one fourth of its length from one end, it still tends to vibrate as above in two segments, but this vibration is super- posed upon the vibration of the string as a whole, so that the fundamental and the first overtone can be heard simul- taneously. Figure 161 is an FlG 161 endeavor to show the appear- ance of a string which is vibrating so as to produce its funda- mental and first overtone. Similarly, if the string is plucked one sixth of its length from one end, it tends to vibrate in three segments and the second over- tone will be heard with the fundamental. Thus the string is capable, under suitable conditions, of vibrat- ing in any number of segments and of giving out a series of notes whose frequencies bear to the fundamental frequency the ratios 2, 3, 4, 5, 6, 7, etc. In general, in the case of the strings of musical instruments several of these overtones are produced simultaneously with the fundamental, which ones are present depending chiefly upon where the string is struck "--"----'.'JJ.."', or bowed. It is to differences in the number or relative promi- nence of the overtones that all ---"" """"'^ >''' differences in the qualities of ~"~ - -'" % ^-. . . different notes of the same pitch are assigned. ^ .^ 147. Transverse waves in c ^,-- : l e: -.._^ ,_-"'' : * : ^ rods. In the case of rods the wave travels as the result of the rigidity of the substance of a "":y --y- : - '.'-"y^"" which the rod is composed. 'A FIG 1G2 consideration of the velocity of propagation of the wave is, how- ever, beyond the scope of this text. The form assumed by vibrat- ing bars may be seen from Figure 162. A bar clamped at one end gives off its fundamental note when vibrating in the form 234 ELECTRICITY, SOUND, AND LIGHT shown in Figure 162, a. If struck more sharply and nearer the free end, it may be made to give off its first overtone ; in this case it vibrates in the form shown in Figure 162,&. The relation between the frequencies of the fundamental and its various over- tones are not, however, simple numbers, as is the case with pipes or strings. If the rod is supported at two points, as in Figure 162, c, it will vibrate in the form shown in that figure when yielding its funda- mental note. The form assumed by the rod when yielding its first overtone is shown in Figure 162,c?. In this case also the relation of the frequencies is not a simple number. 148. The tuning fork. If the rod shown in Figure 162, c, is bent, it is found that the nodes are brought closer together. If it has the form of Figure 163, the nodes will occur at the points marked NN. The higher overtones are then very difficult of production and are very much less in intensity than the fundamental. A bar bent into this form and supported at P is known as a tun- ing fork. Because of the purity of its note that is, the absence of overtones it has been adopted for use as a convenient standard of frequency. Any given fork must, of course, be rated first by some absolute method and then it may be used for comparison with other sources of sound. 149. Beats. The case of stationary waves (see sects. 143-144) is a special case of interference. Another case of especial interest in sound is the interference of the wave trains from two musical sources of sound of almost the same pitch (Le. frequency). Obviously if a crest due to one source reaches the ear at the same time as the trough from the other source, there will be destructive interference. But since one source is vibrating slightly more rapidly than the other, an instant later two crests (or two troughs) will be in coin- cidence at the ear. There results then a reenforcement of the vibration. These alternations in the intensity of the sound at any point will obviously occur as many times per second as the fre- quency of one source exceeds that of the other. That is, if m and n represent the frequencies of the two sources, there will be m-n WAVES IN STRINGS 235 such coincidences per second. These alternations in sound intensity are known as beats. The phenomenon of beats is useful in explaining the physical basis of discords. So long as the number of beats produced by sounding two notes together is not more than five or six per second the effect is not particularly unpleasant. From this point on, however, the beats begin to become indistinguishable as sepa- rate beats and pass over into a discord. The unpleasantness be- comes worst at a difference of about thirty vibrations per second. Thus the notes B and C 1 , which differ by thirty-two vibrations per second, produce about the worst possible discord. When the difference reaches as much as seventy, the difference between C and E, the effect is again pleasing or harmonious. But in order that two notes may harmonize it is necessary not only that they themselves should not produce an unpleasant num- ber of beats, but that their overtones also should not do so. Thus C and B are very discordant although they differ by a large num- ber of vibrations per second. The discord arises in this case between B (vibration number 480) and C', the first overtone of C (vibration number 512). Thus if two notes are to be consonant, neither they nor any of their overtones can fall close enough together to produce an unpleasant number of beats. EXPERIMENT 20 V^T , and to determine the rate of a given tuning fork. Directions. Set up an electrically driven fork having from about 100 to 300 vibrations per second and connect as shown in Figure 159. Connect to the binding posts of the fork a single storage cell.* To one prong of the fork attach a light string, for example a piece of oiled fish line or linen thread about four feet in length. To the other end of this string hang a light pan for holding weights. Xow vary the tension by adding weights until the string breaks up into some number of vibrating segments. The adjustment may be made exact by varying the tension until the nodes are * If it is necessary to supply more energy to the fork in order that it may not be damped down by the tension on the string, a larger number of cells may be used. To prevent excessive sparking at the break a condenser may be shunted across it, as is clone in the induction coil. 236 ELECTRICITY, SOUND, AND LIGHT most sharply defined. It is desirable to use values of the tension for which there will result not more than seven half waves. If the tension necessary to produce this result is sufficient to stop the vibrations of the fork, follow the instructions of the footnote on the preceding page, or shorten the length of string used. Vary the tension and note the weights corresponding to at least three different wave lengths such that the numbers of half wave lengths in the length of the string are successive numbers ; for example, 3, 4, 5. See how nearly a constant number you obtain by multiplying the tension by the square of the number of segments (see sect. 145). Weigh the string and measure its length. Calculate its linear density p. Express in dynes the tension necessary to cause the string to vibrate in one segment. Calculate from these data the number of vibrations per second made by the fork (see eq. (11)). (B) Object. To find the vibration frequency of the fork used in (A) by the method of beats. Directions. It was the conclusion of the discussion of section 149 that if two sources of sound differ slightly in their vibration frequencies, there results in a second of time a number of alternations in the intensity of the resultant sound that is equal to the difference between the frequencies of the two sources. If one of the sources is. of known frequency, for exam- ple, a standard fork, the frequency of the second source may be deter- mined by observing the number of these " beats." The unknown fork to be used is that of the preceding part of this experiment.* Select by ear a standard fork of about the same note as the unknown. Always set the fork in vibration by striking it with a felt-covered hammer like a piano hammer, or with a rubber mallet. The latter is conveniently made by placing a rubber stopper on the end of a rod. Now, using a stop watch, count the number of beats for several seconds. If the number is large, it will be found easier to count the beats in groups of three or four, e.g. one, two, three ; one, two, three, etc. Now make the frequency of the unknown less by attaching to one of its prongs a small piece of soft wax. Count again the number of beats. If this number is less than before, obviously the unknown has been brought nearer the standard by weighting ; that is, its vibration frequency is larger than the standard by the number of beats first observed. If the act of weighting the unknown fork increases the number of beats, then the number of vibrations per second of the unknown i^f smaller than that of the known. Hence the number of beats per second must be added to the vibration number of the standard to obtain that of the unknown fork. * Or it may be the fork of the falling-body apparatus, or of the inertia disk of Experiments 1 or 10 of "Mechanics, Molecular Physics, and Heat." Either of these should be rated in the supporting frame and not removed. WAVES IX STRINGS 237 EXAMPLE (A) The tension for 3 segments was 162 g. ; for 4 segments, 94 g. ; for 5 segments, 60 g. ; for 6 segments, 42 g. ; for 7 segments, 30 g. The products of the tension in grams and the square of the number of segments were 1458, 1504, 1500, 1512, and 1470 respectively. The average product was 1488. The length of the string was 95.4 cm.; its mass, .390 g.; its linear density, .00409. Multiplying 1488 by 980 to reduce the tension to dynes and then substituting the above values of T, p, and L in the first of equations (11), gave n = 99. The fork used was marked by the maker 100 vibrations per second. (B) Using the fork and a standard of frequency 100, one beat in two seconds was observed. Weighting the unknown increased the number of beats; hence the rate of the unknown was 99.5 per second and the per cent of error in (A) was .5. CHAPTEE XXI DIFFRACTION OF SOUND AND LIGHT WAVES 150. Two theories of light. In Sir Isaac Newton's day (1642-1727) two rival theories of light were struggling for recog- nition. The one, the wave theory, fathered and championed by the Dutch physicist, Christian Huygeiis (1629-1695), regarded light, like sound, as some sort of a wave motion, the chief differ- ence between the two being, according to this theory, that, while sound is propagated through the agency of ordinary matter, light is a wave motion in some all-pervading medium to which the name of " the ether " was given. The rival theory, called the corpuscular theory, regarded light as due to the emission from all luminous bodies of minute cor- puscles which travel in straight lines and with enormous velocities through space and produce the sensation of light when they impinge upon the retina of the eye. This theory had its most famous and most brilliant advocate in Sir Isaac Newton himself. Newton's chief reason for rejecting the wave theory lay in the fact that he was unable to understand why, if light is a wave motion, it is always propagated in straight lines past the edges of opaque objects, instead of undergoing diffraction, that is, being bent around such objects, as are sound waves, water waves, and all the other types of waves with which Newton was familiar. What is com- monly regarded as the decisive test between the two theories was made in the year 1800 by Thomas Young, and consisted in show- ing that it is possible to produce with light waves the diffraction phenomenal which are to be discussed in the later sections of this chapter, and which it does not seem possible to account for from the standpoint of the corpuscular theory. It is the object of the present discussion and of the succeeding experiments to show both theoretically and experimentally that, under suitable conditions, sound does not bend around corners, as 238 DIFFKACTION OF SOUND AND LIGHT WAVES 239 it is commonly supposed invariably to do, and that light, on the other hand, does under suitable conditions bend around corners, as it is commonly supposed not to do. More explicitly stated, our aim will be to show that the phenomenon of straight-line propa- gation is characteristic of any and all types of wave motion, pro- vided only the aperture through which the waves pass is large in comparison with the wave length of the waves. If this proposition can be proved, it will be evident that the fact of the straight-line propagation of light does not furnish any argument against the wave theory, provided the wave length of ordinary light waves is very minute in comparison with the dimensions of ordinary aper- tures. Before proceeding to this proposition it is necessary to con- sider further the nature of a wave motion in a medium of indefinite extent, and the conditions for interference in such a medium. 151. Definition of wave front. Consider S in Figure 164 to be the point source of a \vave motion in an isotropic medium ; that is, a medium in which the disturbance is propagated with equal speed in all directions. When the disturbance which originates at S has just reached a, it has also then just reached all other points, such as b, c, and d, which are at the same distance from S. The spher- d \ ical surface passing through these points is known as the wave front of the disturbance. In general, the wave front may be defined as the surface passing through all the particles which are in the same phase of vibration. The form of the wave front under the conditions just mentioned is spherical, but it will be shown later that conditions may arise in which it has not this form. Further, it will also be shown that under proper conditions a spherical wave may be converging, i.e. concave toward the direction in which it is traveling, instead of diverging, as in the case just considered.* If the source is far enough away (rigorously, at an infinite distance), the wave front will obviously be plane. * See section 155, page 245. 240 ELECTRICITY, SOUND, AND LIGHT 152. Construction of a wave front. Any particle in the wave front of a disturbance may be considered as a point source from which is spreading out a spherical wave. Thus consider the par- ticles , b, c, and d in the plane wave represented in Figure 165. A short time after the disturbance has reached these particles let the spherical wave surfaces due to them have the forms shown in the figure. If the number of these new centers is very large, it is evident from the figure, where for clearness only four have been represented, that the disturbance along the surface AD is very much greater than at any other points. In fact it may be shown by a mathematical analysis that these small spherical waves destroy each other by interference except at the surface AD. From its geometrical nature as a surface tangent to all the smaller surfaces, AD is known as their envelope. This envelope is then the new wave front of j, , 1C5 the disturbance. In general it may be said that at any instant the wave front of a disturbance is the envelope of all the secondary- wave surfaces which are due to tlie action as separate sources of all the various particles that at some previous instant constituted the wave front. 153. Conditions for interference of two wave trains in a medium of indefinite extent. Let A and B (Fig. 166) be two particles vibrat- ing in the same phase from each of which is propagated a disturb- ance having a spherical wave front. Let similar- wave fronts be constructed for each particle. Thus the circular arcs a and a' represent the position of particles at the same distance from their respective sources and therefore in the same phase. The arcs I and y represent the wave fronts when the disturbances have traveled one half a wave length farther; that is, each of them represents the locus of a series of particles which are exactly opposite in the phase of their vibration to the particles of a and a f . The arcs c and c f represent the wave fronts when they have traveled a whole wave length beyond a and a 1 . Their particles are in similar phase to those of a and a' and opposite to those of b and I'. DIFFBACTION OF SOUND AND LIGHT WAVES 241 The particles in the line determined by the points marked x, x z , x z have superimposed upon them vibratory motions of the same phase from both sources. Along this line there is therefore a reenforcement, or a maximum disturbance. Along the line deter- mined by the points marked o v 2 , 3 , on the other hand, the vibra- tions superimposed are opposite in phase, and there is interference, or a minimum disturbance. Further, along the line determined by the points x v there is again reenforcement. From the construction of the figure it is evident that the con- dition for a maximum at any point is the existence of a difference in length of path between the point and the sources A and B respectively of some in- tegral multiple of a whole wave length. Thus at x the difference in path is zero wave lengths, at x l it is one wave length, etc. Similarly, for a mini- mum the difference in distance must be an odd multiple of a half wave length. At o v o 2 , o g , etc., it is ^ wave length. Ad- ditional maxima and minima may be found by extending the lines a, It is important to notice that the lines of minimum disturbance o v 2 , 3 , etc., move farther and farther away from the central line of maximum disturbance x, x z , x z , the smaller the distance AS becomes in comparison with a wave length. Thus if AB is very large in comparison with a wave length, the line o v o 2 , o s is very close to the line x, x z , x s , and similarly the line x v # 4 , x 5 is close to the line o v o fl , o s . But as AB becomes smaller and smaller these lines diverge more and more. When AB is just FIG. 166 6, V, etc. 242 ELECTRICITY, SOUND, AND LIGHT equal to a wave length the line x v # 4 , x & is in the prolongation of AB, since it is only points in this line which can then differ by one wave length in their distances from A and B respectively. When AB is equal to a half wave length the line o v o 2 , o 3 is in the prolongation of AB, and there are then no points of quiescence at all to the right of AB. When AB is less than a half wave length there are no points of quiescence anywhere. These con- siderations will now be applied to the discussion of the rectilinear propagation of wave disturbances through openings in screens. 154. The propagation of wave motions through apertures. Consider the case of a train of short waves, which, proceeding from a distant source, pass through an opening ac (Fig. 167) and fall upon a screen mn. Assume that the length ac is large as compared with the wave length of the train of waves. A distant source is chosen so that the wave front of the disturbance which reaches the FIG. 167 aperture ac may be practically a plane, and thus admit of the consideration of all the particles lying in the plane of the aperture as being in the same phase of vibration. The lines ao and cr are drawn from the source, assumed to be a point, past the edges of the opening ac to the screen ; i.e. they are the lines which mark the limits of the geometrical learn. Suppose that the wave length and the opening ac are so related that the point p 2 on the screen, for which the distance cp 2 is exactly one wave length greater than the distance ap z , falls outside the limits of the geometrical beam, i.e. above the point o. Then the particles a and b will differ in distance to p 2 by a half wave length. Hence the vibrations produced at^> 2 by these two particles mutually neutralize DIFFBACTION OF SOUND AND LIGHT WAVES 243 each other. Similarly the disturbance originating in the first par- ticle below a will at p 2 be just one half wave length ahead of the disturbance coming from the first particle below b. Thus every particle between a and b may be paired off with a corresponding particle between b and c such that the effects of the two particles neutralize each other at p 2 . Hence the total effect at p 2 of the disturbances coming from the portion ab of the opening is com- pletely neutralized by the effect of the disturbances coming from the portion be of the opening. Consider next a point p 4 which is so situated that the distance cpt is two wave lengths more than the distance ap 4 . The opening ac may now be divided into four parts, ae, eb, bf,fc, such that eb neutralizes at p the effect of -ae, since ep is one half wave length more than ap, and fe neutralizes the effect of bf, since fp is one half wave length more than bp. There is therefore no disturbance at all at p. At some point p 3 , between p 2 and p 4 , the distance cp s will be one and a half wave lengths more than ap s . If we now divide ac into three equal parts, the effect of the upper third will be completely neutralized at p s by that of the next lower third, but the effect of the lowest third has nothing to neutralize it at p 3 ; hence there is a disturbance at p z which is due simply to one third of the particles between a and c, and even the effects of the particles in this third partially neutralize one another at p s , since they differ somewhat in phase. It is obvious that between p% and p^ the disturbance increases from zero at p 2 to a maximum at p 3 , and then falls grad- ually to zero at p ; that, further, there are other points of zero disturbance, p Q , etc., so situated that the distance from c to the point in question is any even number of half wave lengths more than the distance from a to this point; and that between these points of zero disturbance are points of maximum disturbance, p 5 , etc., so situated that the distance from c to the point in question is any odd number of half wave lengths more than the distance from a to this point. But it will also be noticed that the successive maxima, p%, p 5 , etc., diminish rapidly in intensity, since, while but two thirds of the particles between a and c completely neutralize one another's effects at p%, four fifths of these particles neutralize 244 ELECTRICITY, SOUND, AND LIGHT one another's effects at p 5 , six sevenths at p 7 , etc. Hence it is not necessary to go a great distance above o in order to reach a region in which there are no points at which there is any appreciable disturbance. Further, if we consider wave lengths which are shorter and shorter in comparison with 'ac, the points of maximum disturbance p%, p 5 , etc., draw closer and closer together, and soon some of them begin to fall inside the limits of the geometrical beam, i.e. below the point o. Hence those that are left above o are weaker and weaker members of the series. It follows, there- fore, that when the wave length becomes very short in comparison with ac, the disturbance will have become practically zero at a very short distance above the point o. In other words, a wave motion should be propagated in straight lines through an opening, or past an obstacle, and should not bend around appreciably into the region of the geometrical shadow, when and only when the wave length is very minute in comparison with the size of the opening ; for in this case the disturbances from the various elements of the opening must interfere in such a way as completely to destroy one another at practically all points outside the limits of the geometrical beam. The analysis of the conditions which, exist inside the limits of the geometrical beam when p^, p 4 , etc., fall below o will not here be taken up, since we are not concerned at this point with showing what happens inside of or so much as with proving that practically nothing happens outside of or. Suffice it to say that experiment and theory both show that under the conditions assumed there is practically uniform disturbance within the region or. Now since ordinary sound waves have a wave length of from 1 to 8 feet, it will be seen from the above analysis that in passing through a window or any ordinary opening they may be expected to spread out in all directions beyond the opening, as in fact we know that they do. Indeed, if the aperture is less than one wave length in width, it should be impossible to find any point of quiescence whatever on the side of the screen which is away from the source. It is clear, then, that we must produce extremely short sound waves, if we are to hope to observe with any ordinary openings the diffraction phenomena presented in the above theory. DIFFKACTION OF SOUND AND LIGHT WAVES 245 In light, however, since, as we shall presently see, the average wave length according to the wave theory is only about .00005 cm., we should expect that with ordinary openings the maxima p 8 , p 5) etc., would lie so near to the edges o and r of the geometrical beam as not to be easily discernible, so that, in order to bring them into evidence at all, we should expect to be obliged to work with exceedingly small openings. There is a still further condition, how- ever, which must be met in order to bring out with maximum clearness, in the case of either light or sound, the diffraction bands Pi> Ps> etc - of the preceding theory. It is that the wave experi- mented upon be converging instead of plane or diverging, as assumed in the preceding discussion. Before presenting, therefore, experiments on diffraction it is desirable to consider the methods by which the form of a wave may be altered. 155. Formation of images by changes in wave form. For the sake of clearness let us take a concrete case and imagine, with Huygens, that light is a wave motion, and that a light wave originating in a point S E (Fig. 168) beneath a sur- v / face of water spreads from that point as a spherical wave. Let us imagine further that light travels faster in air than it does in water. Then it can easily be seen that the wave will undergo a change in curvature in passing ,, , ,, . , , FIG. 168 through the interlace mn into air. For if there were no change in the velocity of prop- agation in going from water into air, then the wave front which at one instant had reached the position cod would an instant later have reached the position mo'n, so drawn that cm = oo' = dn. But if oo" represents the distance which light travels in air while it is traveling the distance oo' in water, then the wave, upon emergence into air, should occupy some position mo"n instead of mo'n. In 246 ELECTKICITY, SOUND, AND LIGHT other words, the waves should bulge upward upon passing from the water to the air, so that an eye placed at E should receive a portion of a wave which actually has its center at some point 8' instead of at S. This means, of course, that the point S should ap- pear to be, not at its real position, but at some higher position S'. This is precisely what everyday experience teaches is the case ; for objects under water always appear closer to the surface than they actually are. The point S f is called the image of the point S. It is further called a virtual rather than a real image, because, although the center of the wave which reaches the eye is at S f , the actual center of disturbance is at some other point, namely at S. Again, when a spherical wave from a point S strikes the bound- ary mn (Fig. 169) of a new medium, it is to be expected that a portion of it will undergo reflection, and that, at the instant at which the ad- vancing wave front would have been in the position mo'n, had there been no re- flection, the reflected wave will actually be in the posi- tion of the circular arc mo"n, so drawn that oo" =00'. This means, of course, that an eye at E, or at any point above mn, would receive a wave which has its center at S r instead of at S. As above, 1<' IG . 109 S f is called the image of S. In general, when for any reason a spherical wave originating at a point in space is so modified that the wave appears to have originated at a different point, this second point is called the image of the first point. It will be obvious that in this case, as in that preceding, S' is a virtual image of S. Since in Figure 169 the chord mn is common to the two arcs mo'n and mo"n, it is evident that the image of a point in a mirror DIFFRACTION OF SOUND AND LIGHT WAVES 247 FIG. 170 is on the perpendicular drawn from the point to the mirror, and as far behind the mirror as the point is in front of it. The figure also shows that the light which comes to the eye by reflection from the mirror must follow the law, angle of incidence Srp = angle of re- flection prE; for since oS = oS r , Z rSo = Z rS'o /_prS = /.prE. These theoretical deductions D from the wave theory are pre- cisely the laws which experi- ment shows to be those which govern the reflection of light from plane mirrors. A third method of modify- ing the shape of a wave, and one which finds large application in optical instruments, consists in causing the wave to pass through a lens in the manner shown in Figure 170. If the speed of propagation of the wave is less in the material of the lens, for example glass, than it is in air, then, since the portion of the wave which passes through the middle of the lens is retarded more than the portions which pass through the edges, it is obvious that the lens will tend to reverse the direction of curvature of the wave. If the source S is close to the lens, the curvature of the wave front which has passed through the lens may not be reversed, but it will be diminished ; that is, the wave will actually be flattened so that its center will lie to the left of S. In this case an eye placed to the right of the lens and looking toward S will see a virtual image of S at the point p at which this center is located (see Fig. 170). If the distance of S from the lens is that of the so-called principal focal plane F, the emerging wave will be plane (Fig. 171); but if S is to the left of the principal focal plane F, the emerging wave will be reversed ; that is, it will be concave toward the direction in which it is traveling (Fig. 172). It cannot, in general, be assumed that the emerging wave front will be strictly spherical, but if a diaphragm ac is introduced, as FIG. 171 248 ELECTRICITY, SOUND, AND LIGHT in Figure 172, so as to cut out those portions of the wave which have passed through the edges of the lens, the remaining wave front may, with a suitable lens, be made practically spherical. In this case the disturbances starting from every point on the spher- ical surface reach the center p of the sphere at precisely the same instant. Since these disturbances will all be in the same phase, s- I) ( i > FIG. 172 they will unite to produce a disturbance of very great intensity at p. In ordinary language the light from S will be focused at p. In technical terms a real image of S will be formed at p. Such an image is distinguished from the virtual images which have thus far been discussed in that the point p now becomes an actual P FIG. 173 center of disturbance from which waves spread out to the right of p as though the source S were itself at this point (see Fig. 172). A second method of producing a wave which is concave toward the direction in which it is traveling is to allow a diverging wave of suitable curvature to be reflected from a concave mirror. If the source S (Fig. 173) is closer to the mirror than the principal focal DIFFRACTION OF SOUND AND LIGHT WAVES 249 plane F, the reflected wave will still be convex toward the direc- tion in which it is traveling, as is shown in the figure which is constructed precisely as was Figure 169, oo" being made equal to oo'. But if S is farther from the mirror than the principal focal plane, then the reflected wave will be concave, and a real image will be formed at p (see Fig. 174). In the cases illustrated in Figures 170, 172, 173, and 174, the points S and p are ' ' p - i i j | called conjugate foci. In Figure 171 the con- jugate focus is obvi- ously at infinity. 156. The nature of a real image of a point source. An appli- cation of the reasoning of section 154 to the case of a converging wave shows that the image of a point source of waves should not be a single point of disturbance, but, instead, a series of max- ima and minima of disturbance. For consider, as in section 154, a point p 2 (Fig. 175) far enough to one side of p so that ap z is one wave length more than cp z . The disturbance from the portion */> FIG. 174 FIG. 175 ab of the wave front dbc will completely neutralize at p z the dis- turbance from the portion be. Furthermore, the contrast between the disturbance at p and the absence of disturbance at p 2 will be much more pronounced in this case than it was in the case illustrated in Figure 167, since in the case of Figure 175 all of the disturbances from the wave front ac are concentrated at the center p, thus making it a point of extreme brightness, while in the case illustrated in Figure 167 no such concentration occurs. 250 ELECTRICITY, SOUND, AND 'LIGHT Precisely as in Figure 167, at p 3 , p & , and other points for which the distance from a to the point in question is any odd multiple of a half wave length greater than the distance from c to this point, there will be maxima of disturbance ; but these maxima will decrease rapidly in intensity, p s being about a sixtieth as . intense as p, p 5 one two hundredth as intense as p, etc. Moreover, p 3 , p 5 , etc., will draw very close together and very close to p as the wave length \(=ad) becomes small in comparison with the aperture ac. In other words, the distance apart of the maxima p, p s) etc., will depend simply upon the ratio between the wave length X and the aperture ac. If, then, as the wave theory demands, the wave length of light is very minute as compared with the diameters of ordinary lenses or mirrors, the maxima p a , p 5 , etc., will be so close to p as to be indistinguishable from it; so that for practical purposes common lenses or mirrors should form, in the case of light, essentially point images of point sources, and at all points outside the limits of the geometrical beam, that is at all points not included within the region inclosed by the lines ap and cp, the light waves should mutually destroy one another. Nevertheless, that the image of a point is not in reality a point even in the case of light waves, but consists of a series of maxima and minima, as the preceding theory demands, may be easily demonstrated by reducing the opening ac until it becomes more nearly comparable with a light wave. Thus a pin hole in a piece of cardboard held immediately in front of a bright flame and viewed at a distance of two or three feet through a small pin hole in another card held very close to the eye will appear, not as a point, but as a central bright disk surrounded by one, two, or even more black rings which correspond to the points p z , p, etc., of Figure 175. Again, if a slit, say a half millimeter wide, be made to replace the first pin hole, and if it be viewed at a dis- tance of a few feet through another slit, say one tenth millimeter wide, which is held very close to the eye, the first slit will appear as a central bright band flanked by a series of dark bands which correspond to the points p 9 , p 4 , etc., of Figure 175. In these experi- ments the remote pin hole or slit corresponds to the point S, the DIFFRACTION OF SOUND AND LIGHT WAVES 251 lens of the eye to L, the retina of the eye to the screen upon which the points p, p v p. z , p s , etc., are observed, and the pin hole or slit held very close to the eye to the aperture ac, which is indeed in this case on the other side of the lens, but this fact does not alter in any way the theory of the phenomenon. The experiment may be made more striking by throwing a beam of direct sunlight through a half-millimeter slit covered with red glass, and then observing the slit through a telescope placed in the path of the beam. So long as the lens has its normal aperture the image of the slit as seen in the telescope will be sharply denned, if the tele- scope is properly focused.* But when the aperture is made small by slipping a second slit over the objective of the telescope parallel to the first, the series of light and dark bands p, p<,, p 3 , etc., will at once appear. Furthermore, as the width of the slit over the objective is made larger and larger these bands will draw closer * The way in which a lens or mirror forms an image of an extended object, such as a slit, may be readily seen by regarding the object as an assemblage of points. If the diameter of the lens or mirror is very large in comparison with a wave length, each point A or B (Fig. 176) of the object will have what is practically a point image of itself formed at a or 6. Further, the points a and b will lie, approximately at least, in the prolongations of the lines drawn from A and B respectively to the center C of the lens, as is evident from the approximate symmetry of the lens, and hence of the retardation produced in the incident wave by it, about the lines AC /] and BC. Similarly, the image of any point on the object will be in the prolongation of the line connecting this point with the center of the lens or mirror (see also Figs. 170 and 173). When, however, we consider the clearness with which detail in the object is brought out in the image, we must even here remember that the images of points are not points but, instead, bright centers, each of which is surrounded by its series of bright and dark rings. If two of these centers are closer together than the radius of the first dark ring about each, they will evidently not be sep- arated in the image by a dark region, and will therefore not be distinguishable as separate points. That is, a lens cannot produce an image which will make two close bright points appear as separate points if the angle subtended at the lens by the two points is less than X/ac. This expression X/ac, or the wave length divided by the aperture, is therefore known as the limit of resolution of the lens. FIG. 170 252 ELECTRICITY, SOUND, AND LIGHT and closer together and finally be lost in the central image p of the first slit, precisely as the above theory demands that they should. In order to demonstrate the existence of the same phenomena with sound waves it is obvious that, if the mirrors or lenses are to be of ordinary size, we must find a way of producing sound waves of unusual shortness. Such waves may be produced by the whistle described in the next section. They must be detected and measured, however, by a special appliance called a sensitive flame. 157. The sensitive-flame apparatus. The sensitive flame is a very high but very narrow flame produced by igniting a jet of gas as it issues from a tank under high pressure. The burner has a pin-hole opening and may be conveniently constructed by draw- ing out a piece of glass tubing with a long taper until the orifice has a mean diameter of about half a millimeter. The opening is in general made slightly elliptical and the flame is most sensitive when the shorter axis of the ellipse lies in the direction of the approaching disturbance. The entering gas is regulated until there results a long flame usually twenty or thirty centimeters high, which is just on the point of flaring. Small disturbances of high frequency produced in the air sur- rounding the mouth of the burner cause the flame to flare and shorten very perceptibly, the reason being that very sudden vari- ations in the pressure at the orifice of Compressed ^ ne burner, such as are produced by Air very rapid vibrations, cause correspond- ing variations in the rate of emission of gas, and hence corresponding variations in the height of the flame. The flame is very sensitive to exceed- ingly short waves such as are produced, for example, by jingling a bunch of keys. To the ordinary notes of audible sound it is wholly insensitive, because such notes do not produce sufficiently sudden changes in pressure at the orifice of the burner. There is used with the flame, therefore, as a source of waves, a small whistle shaped as shown in cross section in Figure 177, the dis- tance between the two small openings being from one to three millimeters. The whistle is blown by compressed air and acts as DIFFRACTION OF SOUND AND LIGHT WAVES 253 a very short organ pipe,* the emission of pulses from each of the two openings being controlled by the period of the reflected pulses which bound back and forth between the two ends of the minute air chamber. Ordinarily no recognizable note is heard, for the reason that the note produced is above the limits of audition. EXPERIMENT 21 (A) Object. To show experimentally that with sound waves of very short wave length sharp shadows are cast by ordinary objects. Directions. Set up the sensitive flame described in section 157. Turn on the gas until the flame is from ten to fourteen inches high and just on the point of flaring, and also until a slight rattling of a bunch of keys will cause it to flare strongly. Locate the side of the flame which is most sen- sitive to this noise, and in all future use of the flame turn that side toward the direction from which the disturbance comes. , FIG. 178 Set up the whistle W in the manner shown in Figure ITS, so that it is in, or just a trifle beyond, the principal focal plane F of a concave mirror of about twenty centimeters focal length f and twenty centimeters aperture * A constant air pressure is desirable, since otherwise overtones varying with the changes of pressure will be produced. A pressure of about 15 cm. of water is usually sufficient. t If the focal length .F of the mirror is not known, and if the latter is made of plaster of Paris, or of some other material which renders a determination by optical means impossible, then determine F by measuring d and ac (Fig. 178), and substituting in the formula F ac 2 /l6 d. This formula is obtained from a consideration first of the fact that ac/2 is a mean proportional between d and the other segment of the diameter of the circle of which the mirror is an arc ; second, of the fact that, since d is very small, this other segment may be taken as the diameter itself ; and third, of the fact that the focal length of the mirror is one half its radius of curvature (see Chap. XXIII). 254 ELECTRICITY, SOUND, AND LIGHT (ac, Fig. 178). Place the sensitive flame G in the principal axis* of the mirror and at a distance from it of about eight feet. Slowly increase the air pressure with which the whistle is to be operated until the flame begins to flare strongly. Turn the tripod rod R which supports the mirror and whistle about a vertical axis and observe that under the conditions pre- scribed a sharply outlined beam of sound is produced which causes the flame to flare when it strikes it, but leaves it burning undisturbed when turned in another direction. Turn the beam again upon the flame ; then interpose between it and the whistle a strip of cardboard, or some other object a few centimeters wide, and observe that when in the right position it causes the flaring to cease, thus showing that with these short sound waves even small objects cast sharp shadows. Next let a beam of sound SC (Fig. 179) be sent from the mirror through an opening about a foot in diameter be- tween two screens db and ce?, disposed as in the figure, and let this beam be reflected by a flat smooth board C to the sensitive flame placed at F. It will be found that the flame will flare strongly only when the board is so turned that the FIG. 179 angle of incidence SCNis equal to the angle of reflection NCF. (B) Object. To find the wave length of the note emitted by the whistle by locating nodes and loops in front of a reflecting wall. Directions. Set up a smooth planed board behind the flame G, arranged as in Figure 178, and let the plane of the board be at right angles to the line connecting the flame and whistle. Attach a base to the board so that, as the latter is moved back, its plane remains parallel to itself. Starting with the board but a centimeter or so from the flame, move it a few milli- meters forward or back until a position is found in which the flame ceases to flare. Mark the position of the board with a piece of chalk ; then slide it slowly back from the flame and count the nodes, i.e. the positions of no flaring, as each is passed. Mark the position of, say, the twenty-fifth node * The principal axis is a line drawn through the center of the mirror and its center of curvature C. DIFFRACTION OF SOUND AND LIGHT WAVES 255 and find the wave length of the note given off by the whistle from a consid- eration of the fact that the distance between successive nodes is one half wave length. (C) Object. To find, by means of diffraction experiments, the wave length of the note used in (B). Directions. Place the whistle W (Fig. 178) in a position on the princi- pal axis of the mirror 2 or 3 cm. from its principal focus. Then from the distance /(= MS, Fig. 180) from this point to the mirror, and the focal length F, compute the distance /' ( = pM, Fig. 180) from the mirror to the conjugate focus p by means of the formula 1+1=1* / /- f Place the mirror and whistle so that the orifice of the flame G (Fig. 178) is accurately at this conjugate focus p] then slowly rotate, about its own axis, the vertical rod R which carries the mirror and whistle, and thus locate, on either side of the central region p (Fig. 180) of intense disturb- ance, at least one point p. 2 of no disturbance, and one more point p 8 of FIG. 180 maximum disturbance. Attach a horizontal index of length I (for exam- ple, 50 cm.) to the vertical rod R and measure the length of arc a traced out by the end of this index as the mirror is rotated from the position in which the flame is burning quietly at /> (Fig. 180) to the position in which it is burning quietly at the point p<, which is on the other side of p. It will be evident at once from Figure 180 that the angle through which the mirror must be rotated to cause the point p to move over to the posi- tion of p 2 is pp 2 /pM. But this angle is also \/ac (see Fig. 180). Hence the equation pM But obviously, Hence 2 X - ac PiPi _ ^ or pM la \=--ac. 11 * See Chapter XXIII for the derivation of this formula. 256 ELECTRICITY, SOUND, AND LIGHT Compare the value of X thus found with the result obtained in (B).* (D) Object. To determine, by diffraction experiments, the average wave length of red light. Directions. At a distance of about 3 in. from a vertical slit s (Fig. 181) about .5 mm. wide, set up an ordinary reading telescope T. Throw a beam of sunlight through s in the direction sT, cover s with a piece of red glass so that only red light reaches the telescope T from s ; then focus T carefully upon the slit s, and thus obtain a sharply defined image of s in the eyepiece of T. Now place over the objective of T a cap con- taining a slit not more than .5 mm. wide, and rotate this slit until it is parallel to the slit s. We have then precisely the conditions given in Figure 175 for the formation of diffraction bands. In a manner identical with that used in (C) we may now turn the telescope T on its vertical sup- port and observe the angle p 2 p$/pL (Fig. 175) through which it must pass in order to cause the cross hairs to move from one of the two dark bands p 2 , adjacent to the central bright one p, to the other. In order to measure this angle accurately, attach to the telescope support a mirror m and observe its rotation with a second telescope and scale t (see Fig. 181). If d is the deflection observed on the scale of t, and if D is the distance from the mirror to the scale, then d/2 D is obviously the angle sought. To obtain as accu- rate a determination as possible of this angle, take the mean of the distances P^Pzi PPi/~ anc ^ PsPQ/S- Measure with a micrometer microscope the width ac of the slit over the objective, then compute the wave length A, of red light from the relation A d and compare the result thus found with the value .000067 cm., which is about the mean value of the wave lengths of the light transmitted by ordinary red glass. * Rigorously, the analysis given in section 150 applies only to a single plane, that is, to a rectangular aperture or slit, the dimension of which at right angles to the plane of the paper is very great in comparison with the dimension ac. The complete analysis for a circular aperture such as that here used is too complicated to be within the scope of this book. Suffice it to say. that such analysis leads to the result that pp 2 for circular apertures is 1.22 times its DIFFRACTION OF SOUND AND LIGHT WAVES 257 EXAMPLE - (A) The predictions as to sharp shadows formulated in the directions were verified by experiment. (B) By the method of locating nodes in front of a reflecting wall twenty half wave lengths were found to be equal to 17 cm. Hence A. = .85 cm. (C) The wave length was determined by the method of diffraction as follows. The diameter ac of the mirror was 20.4 cm., the sagitta d was 1.30 cm. Hence R = To72 2 / 2 (1.3) = 40 cm. and F = 20 cm. The whistle was fastened to a rod attached to the vertical axis with which the mirror rotated. It was placed a distance / of 22.1 cm. from the center of the mirror along the principal axis. The sensitive flame was placed at a distance/' of 210 cm. measured along this axis from the mirror, the value of f having been found by the relation 1/22.1 + I/ 'f = 1/20. An index 47 cm. long was fastened to the vertical rod and turned with the mirror. As the mirror turned through the angle between p 2 and p 2 the end of the index moved over 4.65 cm. Hence (see note, page 256) X = - = .83 cm. This value agreed with that of (B) to within 2.4 per cent. (D) Four dark bands on either side of the central bright image were easily observed through an aperture of width ac = .0404 cm. A series of readings was taken on all these bands with a telescope and scale placed 162 cm. from the mirror ra attached to the first telescope T. The differ- ences in readings on this scale for settings on the bands were as follows. From p 2 to p 2 , 1.09cm.; from jo 4 to p 4 , 2.16 cm.; from p e to j , 3.30cm.; and from p 8 to /? 8 , 4.40 cm. Hence the average distance on the scale cor- responding to the angle between p 2 and p 2 was 1.095 cm. Hence, since X = , we have X = - = .0000685. This is in agreement with the mean value of .000067 to within 1.7 per cent. value for a rectangular opening. Hence the value of X found as above must be divided by 1.22 in order to obtain an accurate comparison with the value found in (B). Rigorously, then, we have ~2 pM CHAPTER XXII THE DIFFRACTION GRATING 158. The principle of the grating. The phenomena of diffrac- tion are most strikingly exhibited with the aid of an instrument devised in "1821 by the celebrated German optician Fraunhofer, and known as the diffraction grating. Such a grating consists essentially of an opaque screen in which are placed at regular in- tervals small parallel slits for the transmission or reflection of light. Thus in Figure 182 qq' represents a cross section of the grating, and the openings m, p, s, etc., represent the slits, which are thought of as extending at right angles to the plane of the page. For the sake of con- venience in analy- sis these slits will at first be regarded as exceedingly nar- row in comparison with their distance apart, that is, each opening will be thought of as a mere line. Let the source of light be so far distant that the wave surface ww 1 which falls upon the grating is practically plane. It has been shown in the preceding chapter that if no grat- ing were present, a lens L interposed in the path of the wave ww' would form an image of the distant source S at some point a, while at all other points on the screen uv there would be destructive interference and therefore total darkness. Let us see how this con- clusion would be modified if we take out certain portions of the wave 258 FIG. 182 THE DIFFRACTION GRATING 259 front ww' by means of the grating. When the wave ww' reaches the grating qq' the points m, p, s become new sources of spherical waves, and if we draw the envelope to all these waves after the disturbance has traveled a small distance forward, we shall obtain precisely the same surface AB which we should have had if the grating had not been present, the only difference being that the in- tensity of disturbance in the plane AB is much less than before, since now but a few points, namely m, p, s, etc., are sending out spherical waves to AB, while before all the points in qq' were so doing. As was shown in section 155, the lens will take this plane wave AB, consisting of vibrations all of which are in the same phase, and bring it to a focus at a, so that an enfeebled image of the distant source S will be formed at this point. Thus far, then, the only effect of the grating has been to diminish the intensity of the image at a. But AB is not now the only surface wiiich can be drawn to the right of the grating so as to touch points all of which are in the same phase of vibration, for a surface mD, so taken that the dis- tance from p to it is one wave length, that from s two wave lengths, and so on, satisfies this condition quite as well as does the surface AB. Hence mD may be regarded as another plane wave, which, after passage through the lens, will be brought to a focus at some point c in the line drawn perpendicular to mD through the center C of the lens,* in precisely the same way in which the plane wave AB was brought to a focus at a. It follows, then, that an image of the source should be formed at c as well as at a. Precisely the same line of reasoning will show that another image of the distant source should be formed at c' as far below a as c is above it. But a, c, and c f are not the only points at which images of the source will be formed, for it is possible to pass a plane through m such that the distance from p to this plane is 2 X instead of X, that from s, 3 X, etc. It is obvious that all points in this plane will be in the same phase of vibration, and hence that the resulting plane wave will be brought to a focus at some point d on the perpendicular drawn from the plane through C, and at the same distance from C as are a and c. * See footnote on page 251. 260 ELECTKICITY, SOUND, AND LIGHT Similarly, there will be other images whose direction from C is determined by the simple condition that the successive distances from the slits to the wave front differ by a whole multiple of a wave length ; thus po = \, 2 X, 3 X, 4 X, 5 X, etc. The first image, namely that at c, where this difference is one wave length, is called the image of the first order. Similarly, that at d is the image of the second order, and so on. In a word, then, a lens and grating disposed as in Figure 182 should produce a whole series of equidis- tant images of any distant source of light. This means, of course, that under these conditions light waves will bend far around into the region of the geometrical shadow and be discernible at a large number of different points instead of simply at a. These theoretical deductions from the wave theory of light are completely confirmed by experiment. Furthermore, the experi- ments illustrating them are so simple and so much a part of everyday experience that the wonder is that they escaped detec- tion and explanation for so long a time. Thus if one looks through a handkerchief held close to the eye at a distant arc light, gas flame, or bright star, one can always see nine and some- times as many as eighteen or more images of the light. These are due to the two sets of gratings formed by the two sets of threads which run at right angles to each other. It is usually possible to see as many as three distinct images by simply squinting at a dis- tant light through the eyelashes, which act in this case as a very imperfect grating. In these experiments the retina of the eye takes the place of the screen uv, and the lens of the eye the place of L. In its simplest practical form the grating consists of a plane piece of glass upon which are ruled with a diamond point, say, a thousand lines to the centimeter. The grooves cut by the diamond point constitute the opaque spaces in the grating, for the light which falls upon these grooves is scattered in all directions, so that a negligible part of it passes through in the direction in which the light is traveling. The clear glass between the rulings corre- sponds to the openings m, p, s in the screen of Figure 182. If such a grating is held immediately before the eye and a source of mono- chromatic light viewed through it, the series of images formed at a, c, d, etc., on the retina are apprehended by the observer as a THE DIFFKACTION GEATIXG 261 series of images of the source lying in the prolongations of the lines aC, cC,dC, etc. The images may be thrown upon a screen, if screen, lens, grating, and source are given the relative positions shown in Figure 182. 159. The determination of wave length by means of the grating. If a grating is held immediately in front of the eye and a source of monochromatic light viewed through it, the wave length of the light vibration may be determined as follows. In Figure 183 let the angle between the grating and the FIG. 183 direction of the wave front mD, which forms the image of the first order, be denoted by 6. Let the wave length be denoted by X, and let d represent the distance between successive openings, the grating space, as it is called. Now in the triangle pmh the obvious relation exists, a) 4 sin d Similarly, for the image of the second order, 2JV d' sin 0'= (2) And, in general, if n represents the order of any image and 6 the angle between the grating and the direction of the wave forming that image, then -. o 71 A, , n \ sin = (3) 262 ELECTRICITY, SOUND, AND LIGHT The determination of the angle 6 may be easily made as follows. An illuminated slit is placed at s (see Fig. 183) in front of a scale fg two or three meters away from the grating and the eye. The positions s\ s", s^, s" l} etc., at which the successive images formed by the grating appear, are then marked in some way upon the scale fg. If the grating has been placed so that the line of sight sE is perpendicular to it, then, since s'E is perpendicular to the wave front mD, we have the angle sEs 1 = 6, and sin = ss'/s'E. Therefore ~ , , X ss' , ss' - = _, or X = rf (4) The distance s'E can easily be measured with a tape. To obtain ss f it is customary to measure s f s f and divide it by 2. If it is the images of the second or third order which are located on the scale instead of those of the first order, then the relation evidently becomes , 2X = rf . or 3X = rf (5) The distance d is always obtainable from the maker of the grating. 160. The grating spectrum. If the source sends out, not mono- chromatic light, but, instead, white light, the series of sharply defined images of the source is found to be replaced by a single central image of the source in white light at s, bordered on either side by broad bands of colored light. In the first band the end farther from the source is red. From the red the color grades into orange, yellow, green, blue-green, blue, and finally into violet at the end nearer to the source. The band of light thus produced is called a spectrum, and the phenomenon of its production is known as dispersion. The explanation of the dispersion produced by a grating is as follows. Just as in sound the peculiar tone quality of any note depends upon the combinations of wave lengths (i.e. the various overtones) which enter into its composition, so in light the tint or color quality of a light depends upon the wave lengths of the light vibrations which compose it. Thus pure white light contains vibra- tions of all the wave lengths which are capable of exciting the nerves of the eye. Color is then entirely dependent upon wave THE DIFFRACTION GRATING 263 length and corresponds to pitch in sound. There are in fact as many pure colors as there are wave lengths for visible vibrations, and, in addition, there are an infinite number of combinations of color, just as there are of combinations of tones. The action of the grating in producing dispersion is then easily seen ; for since the position of every image except the central one is determined by the condition sin 6 = n\/d y it is evident that there is a different value of 6 corresponding to each value of X. Now the wave lengths which compose white light vary from about .000076 in the red to about .000039 in the violet. Hence when the source is white light the image of each order as it appears with monochromatic light is replaced by a series of adjacent images in different colors, each image corresponding to a particular wave length or color. This series of adjacent images constitutes the colored band or spectrum of each particular order. The central image is white and sharply defined because the wave front AB (Fig. 182) which gives rise to this image is at the same distance from each of the openings, and in consequence this wave front is the same for all wave lengths. The spectrum of the first order is the only pure spectrum which a grating can produce, for it can be shown that the spectra of higher orders overlap. Thus, since for the red of the second order sin 6 = 2 x .00007/6?, approximately, and since for the violet of the third order sin 0'= 3 x .00004/d, it will be seen that sin 9 is greater than sin 6', and hence that a part of the third violet over- laps a part of the second red. It is on account of this overlapping that one never sees more than two or three spectra on a side, for in the higher orders the overlapping is so complete as to reproduce white light. A grating spectrum is called a normal spectrum, because the distance of each color from the central image is directly propor- tional to its wave length, so long as 6 is small. 161. The dispersive power of a grating. It will be evident at once from Figure 182 that the smaller the distance between openings, the farther apart will be the successive images a, c, d ; in other words, that the angular separation of different orders produced by a grating, and hence, also, the angular separation 264 ELECTRICITY, SOUND, AND LIGHT of different colors in the same order, increases as the distance between the lines of the grating decreases. This may be seen even more clearly by considering the equation sin 9 = n\/d ; for if we subtract from this equation, as applied to one color of wave length \, namely sin 6 l = n\/d, the same equation as it appears when it is applied to another color of wave length X 2 , namely sin0 2 = n\/d, we obtain If we consider that the angles 6 l and 2 , being in general small, are approximately proportional to their sines, we see that the angular separation 6^ # 2 of any two colors X x and X 2 is inversely proportional to the grating space d and directly proportional to the order n of the spectrum. This last result means, of course, that the spreading apart of the colors is twice as great in the second spectrum as in the first, three times as great in the third as in the first, etc. 162. The resolving power of a grating. When, however, we consider the sharpness with which the outline of any particular image in a particular color is formed, we find that this depends, for a given order of spectrum, not at all upon the closeness of the lines, but solely upon the number of lines constituting the grating. Thus, if there are but few openings, the individual images, a, c, d (Fig. 182), are found to be very indistinct in out- line, but if there are a large number of openings these images become very sharply defined. In order to appreciate the reason for this let us consider a plane drawn through m(Fig. 182), so as to make a very slight angle with mD, for example so that po, instead of being exactly equal to X, is just a trifle less than X, say -j 9 Qp^ X. The disturbances which at any given instant have reached this plane will an instant later have passed through the lens and been brought together to the point c" on the perpen- dicular to this plane through C. Now, although the disturbance which comes from p differs but very slightly in phase from that which comes from m, the disturbance from s differs twice as much from that which comes from m, etc., so that the disturbance from the fiftieth opening is just one half wave length behind that from THE DIFFRACTION GRATING 265 m, and therefore completely neutralizes it at c". If, then, there are 100 openings, there will be complete interference at c", although c" is a point whose distance from c is but yl^ of ac. Just below c" there will be a succession of maxima and minima in precisely the same way in which, there are maxima and minima about a f as explained on page 249 ; but, as has been already shown, these maxima will decrease rapidly in intensity, so that we may say that practically there is complete interference between c" and a point as far above a as c" is below c. If there are 1000 open- ings, then the first point below c at which complete interference occurs, will be but y-oVo ^ ^e distance ac, instead of yl-^ of this distance. Thus the larger the number of openings the closer does the first point of complete interference approach to c, and there- fore the more nearly does the grating become able to produce at c a point image of a point source ; i.e. in technical terms, the higher becomes the resolving power of the grating. If to the successive orders c, d, etc., of a given grating (Fig. 182) we apply precisely the same reasoning as that given above, we see that, since the difference in path po corresponding to the image d is 2 X instead of X, the first point of complete interference near to d is but half as far from d as is the case with the correspond- ing point in the neighborhood of c. Hence we see that with a given grating the resolving power is proportional to the order of the spectrum observed. All of the efforts, therefore, which have been made toward increasing the resolving power of gratings have been directed either toward increasing the number of lines, or else toward making it possible to work in spectra of a very high order. The object of producing gratings of a very high resolving power is, in general, to obtain spectra of very great purity, so that colors of but the slightest difference in wave length may yet stand out as distinct colors, that is, as separate spectral lines. The largest and most perfect gratings thus far made, namely those ruled by Professor Michelson in 1908 at the University of Chicago, contain nine inches of ruled surface, the length of the lines being four and a half inches, and the number of lines to the inch 12,700. The total number of lines on these gratings is therefore 114,300. 266 ELECTRICITY, SOUND, AND LIGHT 163. Effects of finite width of slit on series of images produced in monochromatic light. In the use of a grating one often observes that the image of the third order, for example, will be brighter than that of the second, that of the fifth brighter than that of the fourth, etc. The cause of this lies in the finite width of the open spaces which have heretofore been considered to be mere lines. In order to understand the effect of a finite width in the openings upon the relative bright- ness of the successive images, consider Figure 184, in which is shown on a large scale a section of a portion of a practical grating, mo, pr, and su rep- resenting the finite openings, and op, rs, etc., the opaque spaces. The points m, p, and s correspond to the line openings of the preceding discussion (Fig. 182). The line mD represents the wave front which gives the image of the first order. It is drawn as the envelope to the spherical waves due to the particles m, p, and s. The disturbances produced by these particles at the points m, li, and k are all in the same phase of vibration, for each point differs from the next in its distance from its respective slit by a whole wave length. The disturbances due to these points will then reenforce each other at the image c formed by the lens (Fig. 182). Similarly, particles such as n, q, and t, which bear the same relation in position to m, p, and s respectively, will also produce disturbances on the line mD at points which differ successively in their distances from these particles by a whole wave length. The disturbances due to these points will therefore reenforce each other at c. And similarly, for all the other points in the opening mo there will be corre- sponding points in the other openings which will reenforce these vibrations. But it is now to be noticed that the disturbances winch start from the different parts m, n, and o of the same opening are not in quite the same phase of vibration when they reach the plane mD, and further that the wider the openings the greater becomes this phase difference. Suppose, then, that the open spaces mo, IMG. 184 THE DIFFRACTION GRATING 267 etc., are just equal to the opaque spaces op, etc., and that we are considering the image of the second order. We have seen that the condition which must hold for this image is that ph = 2\, sk ph = 2 X, etc. Since, then, mo = op, we have of = X. But when this condition exists the disturbance from n is one half wave length behind that from m, and therefore completely destroys it at c. Similarly, the disturbances from all the points between n and o destroy at c the disturbances from the points between m and n.* Similarly for all the other openings. Thus the image of the second order will be entirely missing, and also, for exactly the same reason, the images of the fourth, sixth, etc., orders. If the opening is one third of the grating space instead of one half, the missing images will be those of the third, sixth, ninth, etc., orders. If conditions of this sort are only approximately fulfilled, as is usually the case, the images considered will be simply weakened but not entirely cut out. 164. Reflection gratings. By far the greater part of spectro- scopic work is now done with the aid of gratings, but in actual work reflection gratings are much more common than transmission gratings like that which has been studied. These reflection grat- ings are made by ruling very fine lines on a reflecting metal sur- face, rather than on a transmitting glass surface. The grooves destroy the light, while the spaces between them reflect it regu- larly. The light from any white source which is reflected from such a grating and then brought to a focus by means of a lens shows a central white image at a position such that the angle of incidence equals the angle of reflection (see sect. 155). On either side of this central image are found spectra of the first, second, and third orders, precisely as in the transmission grating. The theory of the two gratings is in all respects identical, for it obvi- ously makes no difference how the lines m, p, s, etc., of Figure 182 become sources of disturbance, whether by reflecting or by trans- mitting a disturbance from some other source. A form of grating which has rendered possible some of the most important of recent advances in spectroscopy is the concave grat- ing invented by the late Professor Henry A. Rowland of Johns * See section 154, page 242. 268 ELECTRICITY, SOUND, AND LIGHT Hopkins University. The essential difference between this and other gratings is that the lines are ruled upon the surface of a concave spherical mirror of large radius of curvature, for example 20 feet. Under such conditions the mirror itself forms the series of images corresponding to a, c, d, etc. (see Fig. 182, and also Fig. 180), so that it is not necessary to interpose a lens. This eliminates all difficulties arising from the absorption of the waves by the lens, difficulties which are especially pronounced in the ultra-violet and infra-red regions of the spectrum. EXPERIMENT 22 Object. To determine the wave length of sodium light by means of a diffraction grating. Directions. Over a Bunsen burner place a sheet-iron chimney C (Fig. 185) in which is cut a narrow vertical slit .)/2. To obtain r for this case, consider Figure 193, from which it is evident that + A = lSQ and + 2 r = 180 ; hence r =A/2. Substituting these values in the expression for the index of refraction, we obtain sin J (A + D) sin J- A (9) 173. The spectrometer. The spectrometer is an instrument of the form shown in Figure 194. Its essential features are repre- sented diagrammatically in Figure 195. A circular table K, the FIG. 194 edge of which is graduated in degrees, is supported upon a mount- ing which carries also a telescope T and a so-called collimator C. The latter consists merely of a tube carrying a slit s so mounted that it may be placed in the principal focal plane of a lens L'. The object of this arrangement is to make it possible to regard s as an infinitely distant source of light, for waves which originate at s become plane waves after passing through the lens L'. The telescope T is mounted so as to rotate about the axis of the table K. The angular position of the telescope with reference to the 278 ELECTRICITY, SOUND, AND LIGHT graduations on the table may be read with the aid of a vernier V attached to the telescope. Attached to the circular table is a second smaller circular plate called the prism table, which may be leveled by means of the level- ing screws E (see also Fig. 198). This table carries the prism P. The telescope may be clamped to the mounting, and the cir- cular table, with the attached prism table, rotated. The rotation with ref- erence to the fixed telescope may then be read with the aid of the vernier V. A small piece of plane transparent glass m is inserted in the eyepiece e so as to make an angle of 45 with the axis of the telescope. The purpose of this ar- rangement is to make it possible to illuminate the cross hairs at x by throwing a beam of light from a flame or other source / into the eyepiece through the circular open- ing 0, and thence, after reflection from the surface of m, down the axis of the telescope tube. An eyepiece provided with the opening and the glass plate m is called a Gauss eyepiece. 174. Measurement of the angle of a prism. The prism whose angle A is to be measured is placed upon the prism table as shown in Figure 195. Now if the illuminated cross hairs x are in the focal plane of the lens L, then the light which passes from them down the tube may be reflected from the prism face P so as to return into the telescope, in which case it will be brought again to a focus in the focal plane of the lens L. Hence an eye looking into the eyepiece should see side by side two images of the cross hairs, one due to light which comes directly from the cross hairs at x to the eye, the other due to light which has passed down the tube and been reflected back again into the tube from the face P. If the FIG. 195 THE REFRACTION OF LIGHT 279 prism table is turned until these two images of the cross hairs are brought into exact coincidence, we may know that the axis of the telescope is strictly perpendicular to the face P. If now the prism table is rotated until a reflected image of the cross hairs is obtained from the face P', and if this image is brought into coincidence with the direct image, the axis of the telescope will be perpen- dicular to P'. The angle through which the prism has been turned, as read upon the vernier and scale, is the supplement of the angle of the prism, that is, it is 180 A. The second method of measuring the angle of a prism is one which makes use of the collimator. Suppose that the slit s has been placed in the focal plane of the lens L 1 ', and that the cross hairs x have been placed in the focal plane of the lens L. Then, when the light from the slit s is reflected from a prism face into the telescope as in Figure 196, an image of s will be formed in the telescope in the plane of the cross hairs. Between L' and L the light waves which form this image are plane, or, to express the same idea in different terms, the beam of light between L' and L consists of a bundle of parallel rays. If now the prism is set so that a part of this beam is reflected from one face of the angle A into the telescope when the latter is in the position M, and another part is reflected from the other face of the angle A into the telescope when the latter is turned into the dotted position M', then the image of the slit s can be seen in the telescope when the latter is in either of these positions. If the cross hairs of the telescope are set upon the middle of this image at M, and the reading of the vernier and scale taken, and if then the telescope is rotated into position M ' and a similar setting and FIG. 196 280 ELECTRICITY, SOUND, AND LIGHT reading taken, a little consideration will show that the angle A is one half of the angle through which the telescope has been rotated. This follows at once from the fact that all of the lines drawn from L' to the prism are parallel, and that " angle of incidence equals angle of reflection." 175. Measurement of the angle of minimum deviation! Since the passage of white light through a prism breaks it up into a band of colored light, the angle of deviation must be different for different colors. To obtain the angle of minimum deviation for a particular color, we first set the cross hairs in the focal plane of L, and the slit in the focal plane of L', and then set up at o a source of monochromatic light, for example, a sodium flame. We next dispose the prism, telescope, and collimator in the manner shown in Figure 197, and observe in the telescope the im- age of the slit formed by light which has undergone refraction in passing through the prism. To make sure that the image seen is formed by refraction rather than by reflection from the face BC, for example, we have only to place a source of white light at o and see whether the ob- served image of the slit is replaced by a broad band of color. .Having found the refracted image, we rotate the prism table slightly, observing meantime the image of the slit in the telescope; and if this image moves in the direction of decreasing deviation, i.e. toward R, we follow it with the telescope until, without changing the direction of rotation of the prism, the image begins to return toward L. If the image first begins to move toward L, WQ at once reverse the direction of rotation and follow with the telescope until the position of minimum deviation has been reached, i.e. until the beam emerging from the face A C of the prism makes the smallest angle which it is found possible for it to make with FIG. 197 THE REFRACTION OF LIGHT 281 the direction of the beam incident upon the face AB. We then set the cross hairs upon the middle of the slit, the latter being now in this position of minimum deviation, and read the vernier and scale. We next remove the prism, rotate the telescope into the position N y and set the cross hairs upon the image of the slit formed by this undeviated beam.* The difference between the two settings gives the angle of minimum deviation D. EXPERIMENT 23 Object. To find the index of refraction of glass for sodium light by a measurement (I) of the angle of the prism, and (II) of the angle of mini- mum deviation. Directions. I. Set the prism P (Fig. 198) upon the prism table in such a position that each of the edges is immediately above one of the leveling FIG. 108 screws E, E' ', E." Loosen the set screw *S\, so that the prism table can rotate by itself. Set up an electric light behind a ground-glass screen about a foot away from the opening in the Gauss eyepiece, and on the *It is frequently undesirable to disturb the adjustments of the spectrometer by removing the prism. In that case enough light will generally pass the edges of the prism to make possible the setting in the position N without removing the prism. If this is not the case the telescope may be turned into a position corresponding with that already shown in the figure, but on the other side of N, the prism reversed, and the position of minimum deviation determined. The position N then lies halfway between these two points, or, in other words, the angle through which the telescope has turned is 2 J). 282 ELECTRICITY, SOUND, AND LIGHT perpendicular drawn through to the axis of the telescope (see also Fig. 195). Clamp the telescope to the frame by tightening the set screw S 2 . If the cross hairs are not perfectly distinct, slip the eyepiece alone forward or back until they are in sharp focus. Then rotate the prism table until the light which enters O from the source /(Fig. 195), and is then reflected down the telescope tube by the plane glass m, is reflected back into the tube from the face P of the prism. If the prism is being rotated somewhat rapidly, this reflection will appear simply as a flash of light across the field of view when the prism face P passes through the position in which it is at right angles to the axis of the telescope. If this flash cannot be found at first, change the inclination of the telescope by means of the screw S 3 until it appears. Keeping this reflected light in the field of view, move forward or back the draw tube which carries the eyepiece and cross hairs until a second image of the cross hairs is seen in this field of reflected light.* The cross-hairs must then be in the focal plane of the objective lens L (Fig. 195), since light waves originating in the plane of the cross hairs return to a focus in this same plane after passing through L and being- reflected by a plane surface back through L. To make the adjustment perfect, turn the prism so as to bring the two images of the vertical cross hair very nearly into coincidence, and move the head from side to side to test for parallax. If this motion causes the two images to appear to move with reference to each other, focus again until such apparent movement is eliminated. The telescope is now said to be focused for parallel rays. This adjustment might also have been made by merely turning the tele- scope upon some very distant object and focusing until there was no parallax between the cross hairs and points on the object. To make the second adjustment, which consists in setting the instru- ment so that the axis of the telescope is strictly perpendicular to the axis of rotation of the prism, turn the thumbscrew S s until the two images of the horizontal cross hair, found as above, coincide; then loosen the set screw S 4 , clamp 5 1? and rotate the graduated circle, with the prism, until one of the other faces, say P' (Fig. 198) instead of P, reflects the light back into the telescope. Unless the axis of rotation is already perpendicular to the axis of the telescope the horizontal cross hairs will now 110 longer coincide. Restore coincidence by taking up one half of the distance between the two images of the cross hairs with S 3 , and the other half with the prism table screw E which is opposite to the face P'. Then rotate the graduated circle until P is again perpendicular to the telescope, and again take up half the * If the prism has three polished faces, there will, in general, be three reflected images of the cross hairs instead of one, two of them being due to double reflec- tions from the rear faces of the prism. To avoid the confusion arising from these double reflections, it is recommended that one face be temporarily painted with a mixture of, say, whiting and alcohol, or lampblack and alcohol. The three polished faces are needed for Experiment 24. THE REFRACTION OF LIGHT 283 difference in coincidence by means of S 3 and half by means of E" , the prism table screw which is opposite to P. After two or three repetitions of this process the -two images of the horizontal cross hairs should coincide, no matter which face of the prism is turned so as to reflect light back into the telescope. If exact coincidence cannot be obtained, it is because the edges of the prism itself are not quite parallel. In this case it is customary to distribute the error in coincidence between the faces. Xext mark carefully the particular angle of the prism which you wish to measure ; then obtain the reflected image of the cross hairs from one of the faces of this angle. See that the clamp S 2 is tight, so that the telescope will henceforth remain quite stationary, rotate the graduated circle until the two sets of vertical cross hairs are in approximate coincidence, and, to make this adjustment perfect, clamp S 4 and complete the coincidence by means of the fine-adjustment, slow-motion screw, S & . Take the exact reading, in degrees, minutes, and seconds, of the vernier and scale; then loosen S 4 , rotate the graduated circle with the prism until the other face of the angle to. be measured is perpendicular to the telescope, clamp $ 4 , make a final setting, as before, with 6 , and then take a second reading of the vernier and scale. The difference between the two readings should be the supplement of the angle of the prism. As a check upon this determination, measure the same angle by the second method of section 174. To do this, first see that the telescope is still focused as above for parallel rays; then remove the prism, loosen , and rotate the telescope so as to see through it into the collimator. Level the collimator approximately by means of S 7 until the cross hairs are in the middle of the slit ; then loosen the tube which carries the slit of the collimator, and slip it in or out until the image of the slit observed through the telescope is in sharp focus. To make the adjustment perfect, move the slit until there is no parallax between the cross hairs and the image of the slit. Xow complete the leveling of the collimator by adjusting S 7 again. Then set the prism as shown in Figure 196, clamp $ 4 , loosen S 2 , and place the telescope first in position AT, then in position A/', and see if a sharp reflected image of the slit can be seen in both positions. If not, adjust the position of the prism until both images appear ; then make the slit very narrow by means of S 8 , turn the telescope into position A/, clamp 5 2 , and by means of the fine-adjustment screw S 6 set the vertical cross hair upon the middle of the slit, and then take the reading of the vernier and scale. Loosen ,, turn the telescope to position A/"', and repeat. The difference between the two readings should be twice the angle A of the prism. II. To measure the angle of minimum deviation, first clamp S 4 and loosen both A^ and S 2 . Then set up a sodium flame at o (Fig. 197), and, by rotating both the prism and the telescope, find the refracted image of the slit in the telescope when the known angle A is placed as in the dia- gram. If the image cannot be found at first, it is probable that the prism 284 ELECTRICITY, SOUND, AND LIGHT needs to be rotated somewhat farther in the direction of the arrow. Having found the refracted image, proceed as indicated in section 175, setting first approximately upon the position of minimum deviation, then clamping both S l and S 2 , and setting the cross hairs exactly on the middle of a very narrow slit by means of S Q . This done, rotate the prism by turning 5, and see if the slit can be moved by it at all in the direction of decreasing deviation. If so, move the cross hair up to the new position by means of S 6 . Finally, compute the index of refraction from equation 9. EXAMPLE I. The scale of the graduated circle of the spectrometer used was divided into degrees and I degrees ; that is, the smallest scale division was 20 minutes. The vernier was divided by long lines into 20 divisions, each of which was divided into two parts. These 40 vernier divisions corresponded to 13 degrees, that is, to 39 scale divisions; hence one vernier division was equal to f of a scale division, that is, each vernier division lacked ^-Q of being a whole scale division. Hence the vernier read to ^ of 20 minutes, or to 30 seconds, the long lines on the vernier indicating minutes. The readings for the setting on the face P were 315 46' 30" on vernier , and 135 46' 30" on vernier b. The readings for the setting on face P / were 75 44' on a, and 255 44' on vernier b. The prism table was "therefore rotated through 119 57' 30". Hence A = 60 2' 30". The reading on vernier a at M (Fig. 196) was 330 36' 30", and that at M' 210 32' 30". The difference is 120 4'. Hence A = 60 2'. The mean yalue of A was therefore 60 2' 15". II. The readings for the angle of minimum deviation gave D= 53 6'. A + D 60 2' 15" + 53 6' Hence i = ^ = - ^- - = 56 34' 8", A 60 2' 15" and r = = = 30 1' 8". Therefore n = = 1.6679. sin r CHAPTEE XXIV TOTAL REFLECTION 176. Form of the wave front in the second medium. In con- nection with the discussion of the formation of an image by a lens (see sect. 155), the statement was made that, in general, a wave which is spherical in one medium is no longer spherical after it has passed into a second medium. The correctness of this assertion FIG. 199 can be convincingly shown by the following experiment. If a pencil is thrust normally into a body of water and viewed by an eye which looks nearly along the surface (Fig. 199), the portion of the pencil which is beneath the surface will appear extremely short, while, if the eye is slowly raised from position 1 to position 5, the length of the submerged portion will appear to become greater and greater. This means, of course, that the center of 285 286 ELECTRICITY, SOUND, AND LIGHT curvature of the wave front ABC is different for different points on ABC\ that is, the curvature is not constant. In other words, the wave front ABC is not spherical. The same conclusion is reached by a consideration of the law of refraction deduced in the last chapter, namely, velocity in air sin i n = - V- velocity in water sm r This equation shows that whenever light travels from a medium of lesser speed to one of greater, sin i is greater than sin r, that is, the ray emerging into the medium of greater speed bends away from the perpendicular. Hence, as r is increased, i must reach 90, and the emerging ray lie flat along the surface before r reaches 90. If, then, this last ray enters an eye in the surface, the point beneath the surface from which the ray comes must, of course, appear to lie in the surface itself. That is, as the eye is raised from a position in the surface through positions 1, 2, 3, and 4 (Fig. 199), the point from which the light comes must appear to sink from the surface to a greater and greater depth beneath it, precisely as it was observed to do in the preceding experiment. 177. Caustics. This fact of the modification of a wave front by reflection or refraction so that it is no longer spherical, and therefore has not a single definite focus, or center, after such modification, is the cause of a group of phenomena known as caustics. These can be observed most easily with -p IG 200 lenses or mirrors of large curvature. One of the most familiar of such caustics is that formed by a cylindrical lens such as a tumbler of water. When the sun's rays fall upon such a vessel, the rays which pass through the outer portion are brought together at points much nearer the tumbler than are those which pass through the central portion. In other words, the changes in TOTAL REFLECTION 287 the form of the wave front which take place, both when the wave enters the lens and when it leaves it, are such that the outer edges of the wave which has passed through the tumbler have their centers of curvature relatively close to the tumbler, while the center of curvature of the central portion of the wave is more remote (see Fig. 200). The caustics are the curved lines ab, cb, which represent the envelopes of the radii of curvature of the different portions of the wave (see also ab, Fig. 199). Fig- ure 201 shows a caustic FlG 2 oi formed by reflection from a spherical mirror of very large aperture. It will be seen from the figure that a spherical mirror cannot be said to have a definite focus unless its aperture is small, that is, unless the angle sub- tended at F by the mirror does not exceed fifteen or twenty degrees. The only general case of the modification of a spherical wave by reflection or refraction, in which the modified wave has a strictly spherical form, is the case of reflection from a plane surface (Fig. 169, p. 246). Nevertheless, for a given Ly form of incident wave, for example a plane wave, .^ it is always possible to FIG 202 give such a shape to the reflecting or refracting surface as to make the resulting wave front spherical. Thus when a plane wave falls upon a parabolic mirror the reflected wave front is strictly spherical, and hence no caustic is formed (see Fig. 202). Conversely, when a wave originates at the focus of a parabolic mirror it is reflected as a rigorously plane wave, that is, as a parallel beam. Hence the use of parabolic mirrors in search lights. 288 ELECTRICITY, SOUND, AKD LIGHT The lack of sphericity in the waves which have passed through ordinary lenses bounded by spherical surfaces is seen in the fact that it is impossible with such lenses to obtain images in which there is sharp definition ; for the rays which pass through the edges of the lens are brought to a focus nearer to the lens than are the rays which pass through the middle portion. This phenomenon of caustics, as shown by ordinary lenses, is usually known as spherical aberration. It may be reduced by decreasing the aperture of the lens, but this also decreases its resolving power (see note *, p. 251). It is elimi- nated entirely by special combinations of convex and concave sur- faces, or by grinding the surfaces of a single lens into special form. 178. Total reflection. From the law of refraction stated in sec- tion 176, it will be seen that when a ray of light travels from a me- dium of lesser speed to one of greater, if we continue to increase r after i has become 90, there can be no refracted ray at 'all. Experiment completely confirms this conclusion and shows that light which is incident upon the upper medium (Fig. 203) at an angle greater than that for which t = 90is totally reflected in accordance with the usual law of reflection, namely, the angle of incidence r equals the angle of reflection r'. Figure 203 shows the course of various rays starting at different angles from the point / beneath a surface of water. The value of r for which i = 90, and hence the value of r beyond which total reflec- tion takes place,is called the critical angle (see c,Fig. 203). It depends upon the relative index of refraction of the two media. Thus since sin i FIG. 203 n = smr it is evident that, since sin 90 = 1, for c, the critical angle, there is the following relation : j sine TOTAL REFLECTION 289 Thus for water, for which n = 1.33, sin c = 1/1.33, or c = 48.5. This shows that no ray of light which comes from water to a surface separating water and air so as to make with the normal an angle greater than 48.5 can pass out into the air. 179. Total-reflection prism. An application of the principle of total reflection is made in the construction and use of the so-called total-reflection prism, a device for changing the direction of a beam of light by 90 without sensibly diminishing its intensity or pro- ducing in it dispersion, if it is a beam of white light. The index of refraction of practically all forms of glass is more than 1.5. Now for a substance for which n is 1.5, c is 42. Hence if a beam of light op (Fig. 204) enters a right-angled prism at normal incidence, it will strike the face AB at 45, that is, at an angle greater than the critical angle. It will therefore be totally reflected and pass out normally through the face AC without refraction or dispersion. 180. Determination of the index of refraction from measure- ments upon the critical angle. One of the most satisfactory methods of demonstrating the facts of total reflection, and at the same time of determining the index of refraction of a substance which can be put into the form of a prism with three polished faces, is as follows. Let MON (Fig. 205) be any broad source of monochromatic light, CDE a prism so set that light from MOX is refracted and reflected to the telescope T in the manner shown in the figure. The different rays which come to T from different points on MON will obviously strike the surface CD at different angles. Those rays which strike CD at an angle greater than the critical angle will be totally reflected, while those which strike at an angle less than the critical angle will be partially reflected and partially transmitted. If OpqS is the ray which strikes exactly at the critical angle, then all of the light which comes to T from the portion Cq of the face CD will have struck CD at angles greater than the critical angle, and hence it will have undergone total reflection. But all of the light which comes to T after reflection 290 ELECTRICITY, SOUND, AND LIGHT at qD will have struck CD at angles less than the critical angle, and hence it will have undergone only partial reflection, the other part having been transmitted. Hence the surface CD should appear to consist of two parts of unequal illumination, the portion Cgr, which corresponds to total reflection, appearing brighter than the portion qD, which corresponds to partial reflection. The line of junction of these two portions is quite sharply marked, so that the precise point q at which total re- flection begins can be accurately determined. Suppose that the cross hairs of the telescope T are first set upon. this point and the telescope then rotated until it is at right angles to the face ED, that is, sup- pose that the angle i is accurately measured. The index of refraction may then be obtained from i and the angle A of the prism as follows. We have FIG. 205 sin i sinr I sin c A = r + c. (2) (3) From these equations we have only to eliminate c and r in order to obtain n in terms of i and A. From (1) and (3) we obtain sin i sin sin (A c) sin A cos c cos A sin c (4) TOTAL REFLECTION 291 From equation (2) sin c = 1/n, and also cosc=vl sin 2 c = Substitution of these values in equation (4) gives sin i (5) sin A - V?t' 2 1 cos A n n 2 /sin i + cos A\* or 7i 2 -l = ( -). (6) \ sin A / ^ I /sin i + cos A\ 2 Therefore n = A ( - -1+1. (7) N \ sin A I EXPERIMENT 24 Object. To find by a total-reflection method the index of refraction of the prism used in Experiment 23. Directions. Focus the telescope for parallel rays and set its axis at right angles to the axis of rotation by the method of Experiment 23. Then set up a broad sodium flame AION (Fig. 205) on the level of the telescope of the spectrometer and at a distance from it of two or three feet. Turn the telescope and prism into about the relative positions shown in the figure, or until the image of the flame reflected from the face CD can be seen in the telescope. Then rotate prism and telescope together until the yellow line which divides the field into two parts of unequal intensity is seen. Set the cross hairs upon this line, and read the vernier and scale. Without moving the prism rotate the telescope into a position at right angles to the face DE, as determined by the coincidence of the two images of the cross hairs as seen in the Gauss eyepiece ; read again the vernier and thus deter- mine the angle i. From this value of i and from the value of A as obtained in Experiment 23, find n by equation (7). Compare the value of n thus found with that obtained in Experiment 23. EXAMPLE The reading of the vernier for the setting on the line of total reflection was 303 54'. The reading for normal incidence was 262 46'. Hence i = 41 8'. The angle A as found in Experiment 23 was 60 2' 15". Hence + 1 = 1.6683. sin 60 2' 15" The value found in Experiment 23 was 1.6679. The difference is .02 per cent. CHAPTEE XXV PHOTOMETRY 181. Laws of illumination. When a surface is illuminated by light from a luminous point we may define the intensity of illu- mination as the quantity of luminous energy which falls upon the surface per second divided by the area of the surface, that is, as the quantity of light per unit area. It will be obvious at once from this definition that if we consider two surfaces at a given distance from the point, the one normal to the direction of propa- gation of the light and the other so inclined that its normal EIG. 206 makes an angle 6 with this direction, then the illumination 7 2 of the inclined surface is related to the illumination / x of the normal surface by the equation (1) For, since the same energy E falls upon the inclined surface s 2 (Fig. 206) as upon the normal surface s lt we have I l = E/s l and I = Es hence Again, the illumination upon two surfaces, s 1 and s 2 , at dis- tances P! and r 2 respectively from a given point source, and making the same angle with the direction of propagation of the disturbance, are related by the equation i __ r 2 . /o\ / ~v ; ^2 ' \ 292 PHOTOMETEY 293 for, since the same energy E falls upon the surface s 2 (Fig. 207) as upon the surface s lt we have 1^ = E/s v and / 2 = E/s 2 . Hence The two results thus obtained may be stated as follows : Intensity of illumination is inversely proportional to the square of the dis- tance from the point source, and directly proportional to the cosine of the angle which the normal to the illuminated surface makes with the direction of the incident light. This conclusion has followed simply from the definition which we have given to the word illumination. If this definition is to have any practical value in photometry, it is necessary that two surfaces appear to the eye equally bright whenever they are illu- minated with equal intensities, as here defined. Experiment shows FIG. 207 that this is the case when, and only when, the two surfaces are illuminated by lights of the same, or nearly the same, color. For example, a screen placed at the distance of one meter from a single candle appears exactly as bright as a similar screen placed at a distance of two meters from four precisely similar candles placed very close together. 182. Intensities of different sources of light. The example just given suggests at once a method of comparing the quantities of light emitted by different sources of the same color. For we have only to arrange the sources so that each illuminates at the same angle one of two adjacent surfaces, and then to vary the distances of the two lights from their respective screens until these two screens appear to have equal illumination. It will then be seen at once from the example of the preceding section that the inten- sities of the two sources, that is, the quantities of light emitted by them, are directly proportional to the squares of their respective distances from the equally illuminated surfaces. Algebraically 294 ELECTRICITY, SOUND, AND LIGHT stated, if L l and L 2 represent the intensities of the sources, and d l and d 2 their respective distances from equally illuminated screens, then we have , 183. Photometric standards. The unit of light-emitting power first used for a comparison of the intensities of different sources of light was the candle. Thus, a light of sixteen candle power is one which produces the same intensity of illumination at a dis- tance of four meters as does a single candle at a distance of one meter. Since candles of different composition and size, burning under different conditions, differ widely in the amounts of light which they emit, it is obviously necessary to specify the type of candle to be used as a standard. The so-called normal candle is a candle of paraffin, 2 cm. in diameter and burning with a flame 50 mm. high. In Great Britain the legal standard of light is a sperm candle which burns 7.776 g. of spermaceti per hour. In Germany the standard candle has been replaced by a special form of lamp invented by Hefner- Alteneck. It burns amyl acetate, and, when regulated so as to have a flame 40 mm. high, emits one so-called Hefner unit of light. This is equivalent to .81 of a normal candle. Another special form of oil lamp, called the Carcel standard, is in use in France. It is equivalent to about 9.4 normal candles. For many purposes it is very convenient to use a carbon-filament glow lamp as a standard, since with suitable precautions it emits a very constant light. 184. The Lummer-Brodhuii photometer. The most approved modern instrument for comparing the intensities of different sources of light is the Lummer-Brodhun photometer. The sur- faces the illumination of which by the two sources S l and S z (Fig. 208) is made the same, are the opposite sides of a white opaque screen AB. These surfaces are viewed by an eye at E with the aid of two plain mirrors M^ and M z . In order to bring the two sides of AB into immediate juxtaposition, as seen by the eye at E, the principle of total reflection is made use of in the construction of the prism CD. This consists of two right-angled prisms, CGH and DGH, pressed very firmly together along the PHOTOMETRY 295 faces GH, which are made so as to come into perfect contact in certain places, "but not to come into contact in other places. Now the light which conies to E through the portion of the interface GH in which the surfaces are in perfect contact is light which comes from the left side of AB, undergoes reflection at M v and then passes without change of medium through the prism from the face CG to the face HD. On the other hand, the light which comes to E from the portions of the face GH which are not in perfect contact, that is, from places at which an air film exists between the two surfaces, is composed entirely of rays which have come from the right side of AB by way of the mirror J/ 2 , and have then undergone total reflection at the surface of the air film. Hence, if the two sides of AB are exactly similar sur- faces, and if M^ and M are exactly simi- lar mirrors, it is only necessary to set AB at such a point be- tween the sources S l and $, that the whole surface GH, as seen from E, is of uniform illumination, and then to apply equation (3). In order to eliminate any possible inequalities in the two sides of AB, or in the mirrors M^ and J/ 2 , the whole instrument is usually rotated through 180 about an axis passing through AB. This interchanges the two sides of AB and also the mirrors J/ x and J/ 2 . The mean of the settings before and after reversal is then taken as the correct setting. Figure 209 shows a horizontal section of the instrument. 185. Photometric values of lights of different colors. As stated in section 181, such an instrument as that described in the last section is capable of yielding concordant results only when the 296 ELECTBICITY, SOUND, AND LIGHT two sources are of the same color. If the colors of the sources differ, the impression of color contrast is so strong that the setting for equality of illumination becomes an estimate which even the same person cannot duplicate closely, and upon which different persons will have widely different judgments. If we are to con- tinue to define illumination, as in section 181, as the quantity of luminous energy per unit area, it would, of course, be possible to compare the intensities of ejnission of lights of different color, such as red and blue, by allowing the radiations from the different sources to fall at a given distance upon equal surfaces which com- pletely absorb them both, and then measuring the relative amounts of heat developed in these sur- faces per second. This would presuppose, of course, the elimi- nation of the heating effects due to the nonluminous radi- ations, which, in general, ac- company luminous radiations. This might be accomplished by passing the lights from both sources through a prism, and allowing only the visible por- tions of the spectra to fall upon the comparison screens whose change in temperature was to be observed. Since, however, even within the limits of the visible spectrum, surfaces illuminated with equal energy do not, in general, appear equally bright, it is obvious that such a comparison would give us no information as to the relative values for the purposes of vision of different lights. Photometry, then, as an accurate science, is limited by the very nature of the eye to the comparison of lights of approximately the same color. In the case of complex lights, such as most commer- cial lamps produce, we must, in general, be content with rough approximations in our estimations of relative candle power, for such lamps generally differ considerably in color. Nevertheless, as suggested above, it is always possible to pass both of the lights to be compared through a prism and thus to separate each into its 209 PHOTOMETRY 297 constituent colors, after which the relative intensities of any par- ticular color in the two sources may be accurately compared. This process is called spectro-photometry. 186. The laws of radiation. The fact that even when a body is placed in the best obtainable vacuum its temperature continually falls when it is surrounded by a colder body, such, for example, as liquid air, shows that all bodies are at all temperatures continually radiating energy in the form of ether waves. It follows that when a body is at constant temperature it must be absorbing energy at precisely the same rate at which it is radiating energy. This prin- ciple is known as Prevost's law of exchanges. How the total amount of radiated energy varies with the tem- perature, and how it is distributed among the waves of different wave length are questions which have been made the subjects of many important investigations, both experimental and theoretical. In 1879 Stefan, of Vienna, discovered experimentally that when a black body, such as a carbon filament, is heated to different tem- peratures the total intensity of emission is directly proportioned to the fourth power of the absolute temperature. This is known as Stefan's law of radiation. It has since been deduced from purely theoretical considerations. It holds strictly only for black bodies, that is, bodies which absorb all radiations which fall upon them, but is approximately correct for most solids. "With reference to the distribution of the emitted energy among the different wave lengths some information may be obtained from very familiar experiments. It is a matter of common observation that as the temperature of any solid is continuously raised it at first emits only heat waves, that then visible waves of a very dull red color make their appearance, and that the color then changes first to orange, then to yellow, and finally to a brilliant white. This behavior shows that as the temperature is raised shorter and shorter wave lengths are added to the emitted light. It must not be supposed, however, that a white-hot body emits less red light than does a red-hot one. In general the intensity of emission of all wave lengths increases rapidly with temperature, but the rate of increase is more rapid in the case of the shorter waves. If the light is passed through a prism and the energy of radiation 298 ELECTRICITY, SOUND, AND LIGHT measured in the different portions of the spectrum hy means of its heating effect upon a suitable thermometer, the wave length at which the heating is a maximum continually shifts toward shorter and shorter wave lengths as the temperature rises. A law known as the, displacement law has been brought to light by the most careful experimental and theoretical investigation. It asserts that the wave length \ m , at which the heating is a maximum, so shifts with rising temperature that the product of \ m by the absolute tem- perature T is a constant. Thus at 820 absolute the wave length at which the radiated energy is a maximum is .00356 mm., while at 1640 absolute \ m is .00178. In symbols the law is \ m T = constant. The enormous increase, however, in the intensity of radiation of any particular wave length with temperature is shown by the fol- lowing example. If the intensity of the red light ( A, = . 000656 mm.) emitted by a body at 1000 C. is called 1, at 1500 C. the intensity of the same wave length is over 130, and that at 2000 C. over 2100. All bodies in which the radiation of heat and light is unaccom- panied by permanent chemical or molecular changes of any sort are found to become visible at the same temperature, namely at about 525 C., and to become white hot at about 1200 C. Never- theless the total intensity of emission depends not only upon the temperature, but also upon the nature of the radiating body. At a given temperature, however, no body emits any wave length in greater intensity than does a black body. These statements apply only to radiations produced by temperature alone. When the radi- ation is produced by molecular or chemical changes it is of a different type, called luminescence. It is illustrated by the electri- cal discharge in vacuum tubes, by the glowworm light, and other similar phenomena. In cases of luminescence there is often a strong emission of light with very little heat. 187. Optical efficiency. It will be seen from the preceding paragraph that if we are to use temperature radiation for the purposes of commercial lighting, then the chief requisite of the incandescent body. is that it be capable of withstanding a very PHOTOMETRY 299 high temperature. Thus the chief advantage of the arc light over the incandescent electric light is that in the former the carbon is in such form that it can be raised to the extreme temperature of 3800 C., while the slender filament of the latter permits a tempera- ture of only about 1900C. Similarly, the high efficiencies of the "VVelsbach and the Nernst lights are due to the fact that in both of them the incandescent body is raised to a temperature of about 2300C.* The optical efficiency of a source of light is defined as the ratio of the luminous energy radiated per second to the energy required to maintain the light for this time. In the ordinary oil or gas light not more than 1 per cent of the total heat energy produced by the combustion is represented in the luminous radiations. The electric light is much more efficient. An ordinary incandescent lamp con- sumes about 3.5 watts per candle power. The luminous energy radiated per second by a light of one candle power is about 1.3 X 10 6 ergs. Since a watt is 10 7 ergs per second, the efficiency e of the incandescent lamp is given by 1.3xl0 6 In the best arc lamps the efficiency is as high as one- third watt per candle power, or approximately ten times that of the incandescent light. This means that in such an arc light, if we neglect the con- sumption of the carbons,f as much as 37 per cent of the total energy expended is utilized in the production of light. These figures relate to the new flaming arc produced between carbons which are im- pregnated with some salt like calcium fluoride, and in which the light comes largely from the incandescent vapors in the arc itself. The efficiency of the ordinary arc, in which the light comes chiefly from the luminous center of the positive carbon, is in general not greater than 1.2 watts per mean spherical candle power. The efficiency of the Nernst lamp is about twice that of the incandescent ; that of the Cooper-Hewitt light about six times that * It should be said, however, that the arc light is probably not an example of pure temperature radiation. The light is due in small part to luminescence. t The neglecting of this factor obviously renders the result quite uncertain. 300 ELECTRICITY, SOUND, AND LIGHT of the incandescent. The new tungsten and tantalum lamps have about twice the efficiency of the carbon filament, the gain being wholly due to the higher temperatures employed. In rating the efficiency of an electric light it is not customary to reduce to absolute units, as was done above, but merely to give the ratio of the candle power produced to the watts consumed. EXPERIMENT 25 Object. To plot the efficiency curve of an incandescent lamp. Directions. Arrange a Lummer-Brodhun photometer P, a 16-candle- power, 50-volt lamp /, an ammeter A, reading to 5 amperes, a voltmeter F, a variable resistance R, and a battery or dynamo B, as in Figure 210. Then connect a 16-candle-power, 110-volt lamp L to the lamp socket, and, using L as a standard, find the candle power of I for a series of values of the P.I), across its terminals. Beginning with a P.D. of 40 volts, decrease R and thus increase P.D. by about 3 volt steps until 65 is reached, taking- readings on the ammeter, voltmeter, and photometer at every change. / L 50 If f> no v. <$> V n 1 IS5T> 770 V. Compute in every case the number of watts (volts x amperes) required to produce one candle power of illumination, and then plot a curve in which abscissas represent volts and ordinates represent optical efficiencies (candle power divided by watts). Tabulate the data in one corner of the sheet and let the graph constitute the record of the experiment. EXAMPLE The record of this experiment is shown in Figure 211. It is seen that the efficiency continuously increases with the P.D. and would doubtless continue to do so until the lamp burned out. The reason that the lamp is not run at a higher P.D. than that marked upon it is that the increased optical efficiency is more than offset by the decreased life of the lamp. PHOTOMETRY 301 Ds. C.P. Eff. 122. 6.54 .145 115.5 8.60 .180 106.5 12.33 19.54 .309 91.0 23.42 .345 82.0 33.12 .443 73.5 47.39 .579 58.5 93.61 54.6 93.5 63.3 105.0 67.8 109.0 1.25 74.7 118.0 81.9 126.5 1.45 101 .5 141.5 FIG. 211 CHAPTEE XXVI DISPERSION AND SPECTRA 188. Newton's experiments on dispersion. It was in the year 1669 that Newton, at the age of twenty-five, published his justly celebrated experiments on the analysis and synthesis of white light, experiments which during more than two centuries formed the basis of all explanations of the phenomena of color. These experiments consisted in admitting light through a small aperture A (Fig. 212) into a darkened room and observing that the round image BC of the sun which was produced on the wall, before the prism P was placed in the path of the beam, became replaced upon the interposition of the prism by the band of colors RV. This band was red at | the end which corresponded to the smallest amount 7 A FIG. 212 of refraction, and changed through yellow, green, and blue, into violet at the other end. Newton further placed a second prism in the path of the colored band in the position indicated by P' of the figure, and observed that the colors w r ere perfectly recornbined on the wall into white light. He also showed that it was impos- sible by means of a second prism to further decompose any one of the spectral colors into more elementary parts. 302 DISPERSION AND SPECTRA 303 In view of these experiments it has been customary, since the time of Newton, to regard white light as composed of a mixture of elementary colored lights of every conceivable wave length between that of the longest red and that of the shortest violet. As a matter of fact, Newton's experiments show, not that white light actually consists of all of these colored lights, but merely that white light is decomposed by a prism into these colored lights, and that by recombining these colors we do actu- ally reproduce upon the retina the effect of white light. How- ever, we are led into no conclusions which are at variance with experiment if we adopt Newton's view point as to the nature of white light, and we shall therefore make this viewpoint the basis of much of our reasoning. We shall, however, return to a more critical analysis of this subject in a later section (see sect. 195). Since violet light is refracted more than red light, and since the amount of refraction is a measure of the change of velocity in going into a new medium, it is clear that the shorter visible waves undergo a greater change of velocity in going into glass than do the longer waves. In other words, the velocity of propagation of violet light through glass is less than that of red light. The exact ratio of these velocities is the ratio of the indices of refraction of glass for red and violet. This for flint glass is about 1.62/1.67. 189. Pure spectra. Since Newton's spectrum consisted merely of a row of circular images of the sun in different colors, and since these images overlapped, as is seen from a consideration of Figure 212, it is evident that the color at any given point of this spectrum w T as a combination of two or more colors ; that is, this spectrum did not consist of colors each of which corre- sponded to one particular wave length. It was on account of this fact that Newton failed to notice some of the most interesting characteristics of the solar spectrum ; such, for example, as the Fraunhofer lines (see sect. 192). In order to obtain a pure spectrum it is necessary to avoid this overlapping of the images of the aper- ture in different colors. But since a small aperture of any shape whatever will always produce a round image of the sun at BC, in order to obtain a pure spectrum two alterations must be made in Newton's arrangement. First, the aperture must be a very 304 ELECTEICITY, SOUND, AND LIGHT narrow slit ; and second, a lens must be placed at such a point between the aperture and the screen as to form upon the screen, not an image of the sun, but an image of this narrow slit. In no other way can the image in any particular color be made a mere line. When, however, the slit and the screen are at conjugate foci of a lens, as in Figure 213, the spectrum becomes simply a row of adjacent line images in different colors of the line source. A spectrum formed in this way is called a pure spectrum. When a spectroscope is in such adjustment that an image of the slit is formed in the focal plane of the eyepiece of the tele- scope, it is evident that a pure spectrum may be obtained. The best way of ascertaining whether or not this >^ condition exists is to illuminate the slit with monochromatic light, such as that produced by incandescent sodium vapor, and then to see whether or not a sharply denned image of the slit is formed in the eyepiece of the telescope. 190. Normal and prismatic spectra. In Chapter XXII it was shown that the spectrum produced by a grating is one in which the angular separation of any two colors is directly proportional to the difference in wave length between these colors, provided 6 is small (see p. 2 64) ; for then This means that two photographs made with two different grat- ings, but reduced to the same size, are identical in the arrangement and proportion of their colors. Because of the ease of comparison resulting from this fact the grating spectrum has been adopted as the standard, or normal, spectrum. DISPEKSION AND SPECTKA 305 FIG. 214 Iii general, the spreading of the different colors produced by the passage of light through a prism is not at all proportional to the difference in wave lengths, nor indeed does a substance which produces a large mean refraction always produce a correspondingly large spreading or dispersion of any two colors. In other words, dispersion is not proportional to refraction, although Newton sup- posed, from his early investigation of the subject, that such propor- tionality existed. In general, pris- matic spectra differ from normal spectra in that the reds and yellows are relatively little separated, while the blues and violets are abnormally spread out. If, then, a photograph of a spectrum, produced by a prism is made to the same scale as one produced by a grating, the different colors will not occupy at all the same positions or the same relative spaces. Nor, indeed, are the spectra of prisms made of different materials found to agree with one another, the red and yellow, for example, suffering a larger relative separation in one case than in another. It is on account of this so-called irrationality of prismatic dispersion that it is possible to construct direct-vision spectroscopes. These instruments consist of prisms so combined as to produce dispersion without producing any mean deviation of the beam (see Fig. 214). It is also on account of the irrationality of dispersion that it is possible to produce so-called achromatic lenses. These will be considered in the following section. 191. Chromatic aberration and the achromatic lens. The fact that a glass lens produces dispersion is responsible for a phenome- non observable with all simple lenses and known as chromatic aberration. When white light falls upon such a lens, since FlG - 215 the violet waves are refracted more than the red ones, the focus for the violet waves must obviously be closer to the lens than is that for the red. If v and r (Fig. 215) represent these two foci respectively, then the foci of the colors v r 306 ELECTRICITY, SOUND, AND LIGHT intermediate between violet and red will obviously occupy positions intermediate between v and r. It is because of this phenomenon that the image formed by a simple lens is in general indistinct and fringed with color. If a card is held nearer to the lens than the mean focal plane, the outer edge of the image is fringed with red, since the red rays, being least refracted, are here on the outside, as shown in the figure. If the card is moved to a position just beyond the mean focal plane, the image is fringed with violet, since the violet rays, after crossing in the focal plane, are here on the outside. The problem of eliminating chromatic aberration is obviously the inverse of the problem of constructing a direct-vision spectro- scope, for in the latter it is necessary to produce dispersion with- out producing mean deviation, while in the former it is necessary to produce refraction without producing dispersion. The problem is solved by combining into one lens a convex lens of crown glass and a concave lens of flint glass in the manner shown in Figure 216. FIG 216 The nm t-glass lens then overcomes practically all the dispersion produced by the crown-glass lens, without, however, overcoming the refraction. Such lenses are called achromatic lenses. They are used in the construction of all high-grade optical instruments. 192. Continuous and discontinuous spectra. In general when a pure spectrum is formed of the light emitted by an incandescent solid or liquid, it is found that there are no breaks whatever in the band of color which constitutes the spectrum, no matter how narrow the slit may be. Looked at from Newton's standpoint this means that the light emitted by a white-hot solid or liquid con- tains every conceivable wave length between the longest red and the shortest violet. All that we are able to assert with certainty, however, is that the light which has passed through the prism contains all conceivable wave lengths within these limits. When, however, a gas or vapor is brought to incandescence, its pure spectrum is found to consist of a number of separate images of the slit, each, of course, in its own color. This means that an incandescent gas or vapor emits only radiations of certain definite DISPERSION AND SPECTRA 307 wave lengths. Spectra of this type are called bright-line spectra. They are produced by vaporizing metallic salts in a hot flame, like that of the Bunsen burner, or by sending electrical discharges through tubes containing, in rarefied form, the gases to be ex- amined. The characteristic spectrum of sodium, for example, is usually formed by placing common salt or some other compound containing sodium, for example an ordinary glass rod, in a Bunsen flame. It consists of two bright yellow lines very close together. In ordinary spectroscopic work it is seen simply as a single line, but if the slit of the spectrometer is made exceedingly narrow, the original broad image of a broad slit is found to separate into two narrow images of the very narrow slit, thus showing that the images were not originally distinguished as two merely because they overlapped. The spectra of other gases and vapors are not so simple as that of sodium, and the lines are in general scattered through the whole range of visible wave lengths. The fact that the spectrum of an ordinary gas flame is of the continuous rather than of the bright-line type is due to the fact that the incan- descent body in a gas flame is not a gas at all, but is rather solid carbon particles suspended in a nearly colorless flame like that of the Bunsen burner. 193. Absorption spectra. In addition to the bright-line spectra discussed above there is another type of discontinuous spectrum, namely the so-called dark-line or absorption spectrum. When a pure spectrum is formed by the light from the sun after it has passed through a sufficiently narrow slit, it is found, upon exami- nation, to consist of a continuous spectrum crossed by a large num- ber of fine dark lines. These lines were first noticed by Wollaston in 1802, but were afterwards rediscovered in 1814 and investigated by Fraunhofer, who located about 700 of them, and after whom they were named. The existence of these lines in the solar spec- trum shows clearly that certain wave lengths are either absent from the sunlight which has passed through the prism, or, if not entirely absent, are at least much weaker than are their neighbors. When the solar spectrum is compared with a sodium spectrum formed by the same spectrometer, it is found that two dark lines in the former are exactly identical in position with the two sodium 308 ELECTRICITY, SOUND, AND LIGHT lines. A similar comparison of the spectra of other elements with the solar spectrum has resulted in the identification in position of many of the dark lines of the latter with the bright lines of the former. An explanation of this phenomenon is suggested by the following experiment. When a pure solar spectrum is formed by means of any instru- ment, it is found that if the intensely yellow flame which is pro- duced by burning metallic sodium is placed anywhere in the path of the beam of sunlight which falls upon the slit of the spectrom- eter, the dark lines of the solar spectrum which correspond in position with the bright lines of the sodium spectrum are intensified, that is, they are very much darker than before. This seems to show clearly that the two prominent lines in the yellow part of the solar spectrum are due in some way to sodium vapor through which the sunlight has somewhere passed on its way to the spec- trometer, since making it pass through more sodium vapor increases the prominence of these lines. Now we know that whenever the waves from a sounding tuning fork fall upon another fork of exactly the same pitch, the latter is set into sympathetic vibrations ; in other words, the second fork absorbs the vibrations emitted by the first. We know further that this phenomenon of sympathetic vibrations cannot be produced unless the two forks have precisely the same natural periods. It is customary to assume, therefore, in explanation of the two dark lines in the yellow portion of the solar spectrum, that the extremely intense radiation of all wave lengths, due to the extremely hot solid nucleus of the sun, has had some of its wave lengths weakened by absorption as it has passed through the cooler sodium vapor in the gaseous envelope (the chromosphere) of the sun, but that the only wave lengths so weakened are those which correspond to the exact periods of vibration which the absorbing vapor itself is capable of emitting. A similar explanation holds for the other Fraunhofer lines. These lines are not then, in general, devoid of light, but merely appear dark in the solar spectrum because of the very much greater intensity of the light of adjacent wave lengths which have not been weakened by absorption. Thus the same sodium vapor which, when viewed by itself in the spectroscope, appears as two bright DISPEESION AND SPECTKA 309 lines, appears as two dark lines when viewed against the brilliant background of the solar spectrum. In other words, the darkness is merely a matter of contrast. Figure 217 shows the location of the main lines of the normal (see p. 263) solar spectrum. The corresponding wave lengths are given in. so-called Angstrom units or 10~ 10 meters. Some of these lines, such as A and B, are known to be due to absorption which takes place in the earth's atmosphere. A and B are in fact due to the oxygen of our air. The group b is due to magnesium vapor in the sun. C, F, and h are due to hydro- gen in the sun. Those of the lines which are due to absorption in the sun are distinguished from those due to absorption in the earth's atmosphere by the fact that when the light from that edge Red Orange Yellow Green Blue Green Violet FIG. 217 of the sun which, in view of the sun's rotation, is moving toward the earth, is thrown into the slit of a spectroscope, the lines which have their origin in the sun are slightly displaced toward the violet end of the spectrum in accordance with the principle, known as Doppler's principle, that when a vibrating body moves toward an observer the wave length is shortened. Lines due to our atmosphere obviously could not show such displacement. 194. The reversal of the sodium lines. The phenomenon of the absorption of the light emitted by sodium vapor when this light is passed through cooler layers of sodium vapor may be shown as follows. If the light produced by burning metallic sodium in the Bunseii flame is observed through the spectrometer, the spectrum will at first consist simply of the two brilliant sodium lines, but as the burning continues a point is reached at which the hot sodium vapor in the interior of the flame has become surrounded 310 ELECTRICITY, SOUND, AND LIGHT by dense masses of its own vapor at a lower temperature. At this moment the absorption occurs and the centers of the two lines suddenly turn black. This phenomenon can be observed in any dense incandescent vapor. It is called the phenomenon of reversal. 195. Theory of bright-line and continuous spectra. The fact that gases and vapors give forth only certain definite wave lengths is easily explained from the standpoint of the kinetic theory of matter. For since, according to this theory, the molecules of ordi- nary gases are for the most part outside the range of one another's influence, it is to be expected that when their constituent parts, for example, their electrons, are once set. into vibration by any cause, they will continue to vibrate in their natural periods quite undisturbed during the whole interval between two impacts. Hence these vibrating electrons will send out relatively long trains of perfectly definite wave lengths. Furthermore, if the atom is a very complex structure, it is entirely possible that a given atom might send forth a large number of different wave lengths. This picture of the mechanism of light emission by incandescent gases requires, indeed, a very great complexity in some atoms, such, for example, as that of iron, the spectrum of which contains sev- eral thousand different lines. The explanation of the continuous spectrum of incandescent solids offers greater difficulty. A theory which was in vogue up to about 1895 was developed in view of the three following facts. (1) If the density of a gas is increased, its spectral lines grow broader. (2) When liquid solutions show absorption spectra the absorption bands are never fine dark lines, as in the case of gases, but are in general broad bands. (3) Theory shows that although a body which vibrates without damping must have a perfectly definite and unchanging period, and must therefore emit a homogeneous train of waves, a vibration which occurs with considerable damping is one of constantly changing period, the limits of which are larger and larger the greater the damping. If, then, we assume that the closely packed molecules of liquids and gases are capable, on account of their mutual influences, of producing only strongly damped vibrations, DISPERSION AND SPECTRA 311 we can account for the fact that matter in the rarefied condition shows narrow absorption bands and emits bright-line spectra, while in the dense condition it has broad absorption bands and emits, with increasing density, broader and broader lines which finally run together into the continuous spectrum. It has been pointed out, however, especially by Gouy in France and Rayleigh in England, that it is not necessary to assume any periodicity at all in the source which emits white light. For it can be shown by mathematical analysis that it is possible to resolve an irregular jumble of pulses, such as might be communicated to the ether by atomic shocks, into such a number of homogeneous vibrations, having periods which lie very close together, as is found in the continuous spectrum. Indeed, if we assume that the particles of an incandescent body do vibrate in an infinite num- ber of different periods, it is clear that, since these vibrations must all be transmitted simultaneously to the eye by the same ether, the resultant disturbance of any particular point or particle of the ether would be very irregular, and even according to Newton's view point this irregular disturbance must be resolved by the prism or grating into the extremely close series of regular wave lengths which the continuous spectrum shows. It is therefore not neces- sary to assume that white light ever consisted of anything but the jumble of irregular pulses which, in any case, the prism or grating is obliged to resolve. It is then very easy to see how a body like a solid or liquid, in which the molecules make extremely short excursions between impacts, might emit such a jumble of ether pulses as this theory requires for the constitution of white light. 196. Spectroscopic analysis. Since a given substance in a gaseous condition always has a characteristic spectrum, and since, furthermore, the spectrum of an elementary substance like hydro- gen is in general found to appear in the spectrum of any compound containing hydrogen, it will be seen that the observation of the character of the spectrum of a substance of unknown composition furnishes a very satisfactory method of testing for the presence of certain substances in the compound. Since gases alone have characteristic spectra, the method of spectroscopic analysis is obviously limited to the observation of the spectra of vaporized 312 ELECTRICITY, SOUND, AND LIGHT substances. There are, in general, three ways in which gaseous spectra are compared. (1) The first is represented in Figure 218. The image of an illuminated scale L is formed in the focal plane of the eyepiece of the telescope T by means of a reflection from one face of the prism P. At the same time some substance, for example, FIG. 218 lithium chloride, is vaporized in the flame $, and the character- istic spectrum of lithium is therefore also formed in the focal plane of the eyepiece 0. The exact positions of the lithium lines on the scale are recorded. Some unknown compound which is to be tested for the presence of lithium is then vaporized in the flame $, or in a spark tube which replaces $, and the presence of lines on the illuminated scale in exactly _ the position just occupied by the lithium lines is looked for in the spectrum of this compound. (2) In the second method the slit of the spectroscope is covered through half its length by means of a small total-reflecting prism P (Fig. 219). Light from the '/I t FIG. 219 DISPERSION AND SPECTRA 313 compound is then caused to enter the upper half of the slit from the flame a, while light from a lithium flame b, for example, is caused to enter the spectroscope through the lower half of the slit by means of total reflection within the prism P. If, then, the characteristic lines of lithium which appear in the upper half of the field of view of the telescope T are found to continue clear across the field of view, then lithium must be in the compound which is being vaporized at a, since the spectrum of this compound occupies only the lower half of the field of view. This method evidently renders the scale L unnecessary. (3) The third, and perhaps the most satisfactory method, consists in replacing the telescope T by a photographic camera and taking a photograph of the unknown substance which is being vaporized at a with the photographic plate partially covered, and then replacing the unknown substance at a by the substance for which the test is being made, and, without altering at all the position of the plate, taking another photograph when only the .previously covered portion of the plate is exposed. It is then only necessary to see whether the char- acteristic lines obtained by the last exposure coin- cide with those obtained in the first exposure. Figure 220 is a copy of a photograph so taken, the upper and lower portions representing the bright-line spectrum of iron, and the middle por- tion the dark-line spectrum of the sun. It is obvious that iron is present in the sun. Some of the other substances which have been identified in this way in the solar spectrum are calcium, oxygen, hydrogen, aluminum, nickel, magnesium, cobalt, sili- con, carbon, copper, zinc, cadmium, silver, tin, and lead. The lines characteristic of the element now known as helium were observed in the sun before the element was known to exist on the earth. FIG. 220 314 ELECTRICITY, SOUND, AND LIGHT EXPERIMENT 26 (A) Object. To become familiar with the spectra of different substances. Directions. Using a spectroscope either of the form shown in Figure 194 or Figure 218, see that the prism is in approximately the position of minimum deviation for sodium light (see sect. 175, p. 280). Using a slit about half a millimeter wide, focus the telescope until the image of the slit is in sharp focus. Then replace the sodium light by a white light and make a rough chart, similar in form to that shown in Figure 222, of the distribution of the light in the spectrum, indicating on the chart by brackets or otherwise what portions of the total length of the spectrum are occupied by the red, the yellow, the green, the blue, and the violet respectively. Replace the white light by sodium light, and reduce the width of the slit until the yellow sodium line is seen to be in reality two distinct lines very close together. If it does not appear as such at first, make the slit still narrower and focus the eyepiece more sharply. If it is not even then double, it is probable that the spectroscope has not a sufficiently high resolving power. Now make a chart of the sodium spectrum; that is, draw two fine lines, or one, if but one is seen, beneath the central portion of the region marked " yellow " in the chart above and label it " sodium." If you are using a spectroscope of the kind shown in Figure 218, indicate the exact numerical position of the sodium lines upon the scale. Using a relatively wide slit (.5 mm.), introduce successively with different platinum wires into the flame of a Bunsen burner S (Fig. '218) the chlorides of lithium, strontium, calcium, barium, and potassium, and make a chart of the spectrum of each. In the case of strontium notice particularly the isolated line in the blue; in the case of potassium, the line in the extreme red. Some of these lines are persistent, w r hile others appear only for an instant after the salt is introduced into the flame. In a similar way make charts of the spectra of nitrogen, mer- cury, and hydrogen, obtaining these spectra by means of an induc- FIG 221 tion-coil discharge through a vacuum tube (see Fig. 221) placed in front of the slit. (B) Object. To analyze a mixture for the presence of various elements. Directions. By either of methods (1) or (2) in section 196 analyze an unknown mixture furnished by the instructor. (C) Object. To observe the phenomenon of the reversal of the spectral lines of sodium. Directions. Use again, if possible, a sufficiently narrow slit to bring out the sodium spectrum as a close double line. Burn a piece of metallic sodium about twice as large as a pea in a Bunsen flame placed a foot or so in front of the slit, and observe the reversal of the sodium lines. DISPERSION AND SPECTRA 315 (D) Object. To show the existence of sodium and hydrogen Jn the sun and to compare a prismatic with a normal solar spectrum. Directions. With a lens of about the same focal length as that of the collimator throw an image of the sun upon the slit of the spectroscope. Make this slit very narrow, focus the eyepiece until the Fraunhofer lines are very distinct, then identify as many as possible of the lines A, a, B, C, D, E, b, F, and G, of Figure 222, which shows a pris- matic spectrum. Xote that the distances apart of A and D, D and F, and F and G are approximately the same, and also that the distance from B to D is about the same as that from D to b. Place a sodium flame before the slit and note that when the sun- light is cut off the yellow sodium lines ap- pear in the exact position of the D lines. In the same way show that C and F are hydro- gen lines. Replace the prism by a reflection grating, or take another instrument in which a grat- ing is in place, and set this grating in such a position that light coming through the slit falls upon the grating at an angle of inci- dence not exceeding 45 degrees. Turn the telescope so as to take in the light reflected from the grating face in accordance with the law, " angle of incidence equals angle of re- flection." This light should produce in the field of the telescope an uncolored image of the slit. This image corresponds to the cen- tral uncolored image produced by the trans- mission grating of Experiment 22. Focusing, as above, the sun's rays upon a very narrow slit, adjust the leveling screws of the grating, or grating table, until the uncolored image of this slit is in the middle of the field of view. Then turn the telescope and observe the spectra of the first and second orders on either side of the central image. Observe and record the distances apart in this normal spectrum of A, D, F, and G, and of B, Z>, and b. Com- pare these distances with those shown in Figure 217. See if the distance apart of the D lines of the sun's spectrum, or the bright lines of the sodium spectrum, is not much greater in the spectrum of the second order than it is in that of the first (see sect. 162, p. 264). CHAPTEE XXYII POLARIZED LIGHT 197. Polarization by reflection. All of the phenomena of light which have been thus far studied have been found to be expli- cable upon the basis of the same wave theory whicli applies to the phenomena of sound. In other words, so far as the fundamental facts of reflection, refraction, diffraction, emission, and absorption are concerned, sound and light are identical in all respects except in the lengths of their waves and in the nature of the media which act as their carriers. There is, however, a class of phenomena, known as the phenom- ena of. polarization,. which differentiate light completely from sound, and show that light waves are not compressional waves at all as are sound waves, but are instead transverse waves similar to those which elastic solids are able to propagate by virtue of their rigidity. These phenomena are so far removed from ordinary observation that they will be here presented in connection with a series of qualitative experi- ments. The facts presented in the first experiment were discovered in 1810 by the French physicist Malus (1775-1812). Experiment 1. Set the plane glass re- flector m of the so-called Norrenberg po- lariscope of Figure 223 so that its plane makes an angle of about 33 with the vertical. Adjust the position of a hori- zontal slit s (about 5 mm. wide) and a sodium flame / so that when you remove the black glass mirror m' and look vertically down upon the middle of m you see a portion of the flame. 316 FIG. 223 POLARIZED LIGHT 31' In this experiment the mirror n may be covered with a piece of black paper. Place m' in position and turn it so that it is exactly parallel to m, that is, so that its plane also makes an angle of 33 with the vertical. Place the eye at E in such a position that when you look at the middle of in you see the twice-reflected image of the sodium flame. Then rotate in' in its frame about a vertical axis and observe the image of the flame as you do so. When you have turned m' through 90, that is, into the position shown in Figure 223, 2, the image of the flame will have com- pletely disappeared. The experiment shows that light waves cannot be longitudinal, for if the particles of the medium which transmits the light from m to m 1 vibrated in the direction of propagation of the light, then the conditions of symmetry would demand that the wave be reflected in precisely the same way after m' has been rotated through 90 as before. But if light consists of waves in which the direction of vibra- tion of the particles of the medium is always transverse to the direction of propaga- tion of the waves, and if, in a very short interval of time, the vibrating particles which give rise to light waves change their direction of vibration many times, then the above phenomena can be very easily understood. For suppose that in Figure 224 db t ccl, ef, gli, etc., represent successive directions of vibration of the particles of the medium across the path of the ray sm (Figs. 223 and 224). All of these vibrations can be resolved into two component vibrations, the one perpendicular to the plane of inci- dence smm'j that is, at right angles to the plane of the page, and represented by the dots in the line sm, and the other in the plane of incidence and represented by the straight lines drawn across the path of the ray sm. Now when the ray sm strikes the mirror it is clear that the general law of reflection, namely angle of incidence equals angle of reflection, requires that there be some' angle of incidence such FIG. 224 318 ELECTRICITY, SOUND, AND LIGHT that the refracted ray mr and the reflected ray mm' are at right angles to each other. But when this is the case that component vibration of the refracted ray which lies in the plane of incidence coincides with what should be the direction of the reflected ray, namely the line mm'-, hence this component vibration obviously has no component which is perpendicular to mm'. But if light vibrations are always perpendicular to the direction of propaga- tion of the light, this means that there should be one angle of incidence for which no part of the component vibration of the original light which is in the plane of incidence can be reflected. Considerations of symmetry require, however, that that compo- nent vibration of the ray sm which is perpendicular to the plane of incidence, and represented by the dots in the figure, should be reflected at all angles of incidence. Now this is precisely what experiment shows to be the case. At the angle of incidence of the ray sm for wliich the angle rmm' is a right angle, the reflected ray mm' consists only of vibrations which are perpendicular to the plane of incidence. The ray mm' is said to be a ray of plane polarized light, and the angle of incidence at which the ray sm must fall upon the mirror in order that the reflected ray mm' may consist only of vibrations in this one plane is called the polariz- ing angle. That this angle is always the angle for which the reflected and refracted are at right angles was discovered in 1815 by Sir David Brewster (1781-1868), and is known as Brewster's law. It may easily be shown that another form of statement of the same law is the following: the angle of complete polariza- tion is the angle the tangent of which is the index of refraction of the reflecting substance. This is the form in which Brewster announced Ms law.* That the two forms of statement repre- sent one and the same physical relation may be seen from the following : If the angle cod (Fig. 225) is 90, then we have 90 ^+90 - r = 90, or i + r = 90; hence sin r = cos i ; hence the index n(= sin fc'/sin r) may be written in the form sin* . . _! n = tan ^ ) or ^ = tan n. (1) COS2 POLAEIZED LIGHT 319 FIG. 225 The reason that we originally set the mirror m so as to make an angle of 33 with the vertical was that the index of refraction of crown glass is about 1.55, and the angle the tangent of which is 1.55 is 57. In order that the angle of incidence might be 57 it was necessary to make the angle between the plane of the mirror and the vertical 33. It will now be obvious why we obtained no re- flected light at all from m f when we had rotated it from position 1 to po- sition 2 (Fig. 223). For in this latter position m r bore precisely the same relation to the vibration of the ray mm' as did the mirror m to the component of sm which was vibrating in the plane of incidence smm'. Experiment 2. Kow set m and m' again so that there is no light from/ reflected at mf and then rotate mf about a horizontal axis, observing the middle of in' all the while. You will find that there is always some of the light ray mm' reflected from m' except when m' is set exactly at the polar- izing angle. The amount of the light thus reflected will be found to increase rapidly as the position of the mirror departs in either direction from the polar- izing angle. Experiment 3. Replace the glass mirror m' by a pile of about fifteen thin glass plates set at the polar- izing angle (see Fig. 226), and then observe not only, as above, the reflected ray e, but also the ray e' trans- mitted by the plates as the pile is turned about a vertical axis. You will find that when the pile of plates is in position 2 (Fig. 223), that is, in the posi- tion such that the reflected ray disappears, the trans- mitted ray is of maximum brightness, and when the plates are rotated into position 1 (Fig. 223), that is, into a position such that the reflected ray is of maximum brightness, the transmitted ray has almost entirely disappeared. In explanation of these effects consider that an incident beam sm (Fig. 227) of ordinary light is resolved into two components, FIG. 226 320 ELECTRICITY, SOUND, AND LIGHT FIG. 227 one vibrating in, and one normal to, the plane of incidence. Let the intensity of each of these components be represented by 50 (see Fig. 227). At the polarizing angle none of the 50 parts which vibrate in the plane of incidence are reflected, while photometric measurements show that about 16 per cent of the light which is vibrating perpendicular to the plane of incidence is reflected; that is, 8 per cent of the incident beam is reflected at the polarizing angle. Hence, after the first refraction, the transmitted light consists of 50 parts vibrating in the plane of incidence and 42 parts vibrating in the plane perpendicular to the plane of incidence. After the second refraction these numbers have become 50 and 35.3; after the third refraction, 50 and 29.7; after the fourth, 50 and 25, and so on-. After passage through twelve or thirteen plates the transmitted light has become nearly plane polarized by this process, the plane of its vibrations obviously being at right angles to the plane of vibration of the reflected light. A pile of plates of this sort furnishes a very inexpensive means of obtaining plane polar- ized light, but it suffers from the disadvantage that the polarization is not quite complete. If no light whatever were absorbed or scattered by the glass, the transmitted ray would be- come more and more nearly plane polarized the larger the number of plates, but in prac- tice there is found to be no advantage in increasing the number of plates beyond thir- teen or fourteen. Experiment 4. Replace the upper mirror mf by a FIG. 228 Nicol prism (Fig. 228), the construction of which will be considered later, and looking down through this Nicol at the image of f reflected in m, rotate the Nicol about a vertical axis. You will find POLAEIZED LIGHT 321 that the ray mm' is cut off completely by the Nicol when the latter is in a certain position, but that the light from the flame is transmitted with maximum brightness when the Xicol has been rotated through an angle of 90 from this position. From a knowledge of the plane of vibration of the ray mm' (Fig. 223) decide what must be the plane of vibration of a ray with respect to the face abed of the Nicol, in order that it may be wholly transmitted by the latter, and mark the direction of this transmit- ting plane of the Xicol by an arrow drawn on the face of mounting contain- ing the Xicol. Henceforth you can use the Xicol as a detector of the plane of vibration of any polarized light which you may observe. 198. Polarization by double refraction. The phenomena which will be presented in the following experiments were discovered in 1670 by the Danish physicist Erasmus Bartholinus (16251698), who first noticed the fact of double refraction in Iceland spar, and by Huygens (1629-1695) in 1690, who first noticed the polarization of the doubly refracted beams produced by the Ice- land spar, and first offered an explanation, of double refraction from the standpoint of the wave theory. Experiment 5. Make a pinhole in a piece of black cardboard, and lay the cardboard on a piece of plane glass on the frame h (Fig. 223). Some inches beneath this, for example on the plate m turned into the horizontal position, lay a piece of white paper and illuminate it well. Then lay a crystal of Iceland spar (Fig. 229) over the hole in the cardboard. Remove m' and look vertically down upon the crystal. You will see two pin holes instead of one. Rotate the crystal about a vertical axis. One image will remain stationary, while the other will rotate about it. FIG. 229 That the image which remains stationary is produced by light which has followed the usual laws of refraction is evident from the fact that it behaves in all respects as it would if viewed through a glass plate. The image which rotates, however, must be produced by light which has followed some extraordinary law of refraction ; for although it has passed into the crystal in a direction normal to the bottom face, and out of it in a direction normal to the top face, it must have suffered bending inside the crystal, since it emerges from the crystal at a point different from 322 ELECTRICITY, SOUND, AND LIGHT that at which the other ray emerges. We must conclude, then, that a ray of light which is incident upon the lower face of such a crystal of Iceland spar is split into two rays by the spar, and that these two rays travel in different directions through the crystal. The ray which follows the ordinary law of refraction is called the ordinary ray, the other the extraordinary ray. Experiment 6. To find the direction in which the extraordinary ray travels, rotate the crystal about a vertical axis above the pin hole and note that the extraordinary image always lies in the line connecting the ordi- nary image and the solid ob- tuse angle of the face which is being viewed, and, further, that the extraordinary image is always on that side of the ordinary which is away from this solid obtuse angle. It will be evident, then, from these experiments that if Figure 230 rep- resents a section of the crystal made by passing a plane normal to the top and bottom faces and through the two solid ob- tuse angles of the rhomb, the line io will represent the path of the ordinary ray through the rhomb, while the broken line imne will represent the path of the extraordinary ray. Experiment 7. In order to determine whether the ordinary or the extraor- dinary ray travels the faster through the rhomb, observe again the two pin holes, or, better, observe at close range a dot on a piece of white paper upon which the crystal lies, and note which image, the ordinary or the extraordinary, appears to be the nearer to the upper face. This will evidently correspond to the ray which has suffered the largest change of velocity in emerging into the air (see p. 272); that is, it will correspond to the ray which travels more slowly in the crystal. This will be found to be the ordinary ray. FIG. 230 POLARIZED LIGHT 323 Experiment 8. If you can obtain a crystal which has been cut so that its top and bottom faces are planes which are at right angles to the line ab (Fig. 229) which connects the two obtuse angles of a perfect rhomb, that is, a rhombohedroii having all of its faces equal, view the pin hole normally through this crystal. You will observe that there is now but one ray, and that this ray does not change position upon rotation ; that is, that it behaves in the ordinary way. The direction of the line connecting the two obtuse solid angles of a crystal all of whose sides are equal is the optic axis of the crystal. This axis is not a line, but rather a direction. Any ray of light which passes through the crystal in a direction parallel to the line ab (Fig. 229), that is, parallel to the optic axis, does not suffer double refraction. Experiment 9. Place the Iceland spar again over the pin hole in the manner indicated in Experiment 5, and view the two images through the Nicol prism as the latter is rotated about a vertical axis. You will find that both the ordinary and the extraordinary images consist of plane polarized light, but that the planes of vibration of the waves which pro- duce the two images are at right angles to one another. Hence we may conclude that the Iceland spar has in some way separated the incident light into two sets of vibrations, one of which consists of all the components of the initial vibrations which were parallel to a particular plane in the crystal, while the other con- sists of all of the components of the initial vibrations which were perpendicular to this plane. Experiment 10. With the aid of the Nicol, the transmitting plane of which you determined in Experiment 4, find whether the ordinary or the extraordinary ray consists of vibrations which are parallel to the plane which includes the optic axis of the crystal. You will find that it is the extraordinary ray the vibrations of which are in this plane, while the vibrations of the ordinary ray are perpendicular to this plane (see Fig. 230). 199. Theory of double refraction. The elementary theory of double refraction is as follows : For the sake of simplicity we shall confine attention to the wave form in a single plane in the crystal, namely the plane which is perpendicular to the upper and lower faces of the crystal 324 ELECTRICITY, SOUND, AND LIGHT FIG. 231 and includes the optic axis. This is called the principal plane. It is the plane of the paper in Figures 231 and 232. As we have already seen, any incident beam of light which passes normally into the crystal through the hole in the cardboard (see Figs. 231 and 232) may be thought of as consisting of equal vi- brations in two planes, one perpendicular to the plane of the paper and the other parallel to this plane. Let us consider these two vi- brations as separated, so that we may treat of one in Figure 231 and the other in Fig- ure 232. Let us suppose, further, that any vibrations which are parallel to the direction of the optic axis ab pass through the crystal with greater facility, that is, with greater speed, than do vibrations which are perpendicular to this direction. The compo- nent in the plane of the paper (see Fig. 231) of the incident vibrations will give rise at the boundary mn of the crystal to transverse disturbances which will travel outward in all directions through the crystal. The portion of the wave front, however, which travels at right angles to the axis ab, that is, in the direction md (Fig. 231), will have its vibrations parallel to FIG. 232 POLAEIZED LIGHT 325 the optic axis, while the portion of the wave front which travels in the direction mg will have its vibrations perpendicular to this axis. If, then, vibrations parallel to ab travel faster than do such as are perpendicular to ab, the wave which originates at any point on mn will travel faster in the direction md than in the direction mg, and will consequently have an elliptical rather than a spher- ical form, the longer axis of the ellipse being in the direction at right angles to the optic axis ab. The envelope of all the ellipses which originate in the points on mn will be the line m'n f : The beam will therefore travel through the crystal in a direction other than that of the normal to its wave front ; that is, in the direction mm'. For the reason given in section 154 (p. 242) there w r ill be destruc- tive interference at all points outside of the parallels mm', nn f . On the other hand, the weaves which start out from each point on mn because of the propagation into the crystal of the vibrations which were perpendicular to the plane of the paper (see Fig. 232) will be everywhere perpendicular to the optic axis, and hence will travel with equal speeds in all directions. The beam will there- fore follow the usual law of refraction and will travel in a direc- tion at right angles to its wave front, the waves from each point being now spheres instead of ellipses. 200. Construction of the Nicol prism. In order that the light which is transmitted by a crystal of Iceland spar may consist of vibrations in one plane only, it is necessary to dispose in some way either of the ordinary or extraordinary beam so as to prevent it from passing through the crystal. This was first accomplished in 1828 by the German physicist Nicol in the following way. If the beam be (Fig. 233) is made to enter the face of the crystal at a certain oblique angle, the ordi- nary ray, being refracted more than the extraordinary (see Exp. 7), will travel in the crystal in the direction co, for example, while the extraordinary ray will take the direction ce. Now Nicol cut the crystal into two parts along the plane aa, and then cemented the parts together again with Canada balsam. This balsam has an index of refrac- tion which is smaller than that of the ordinary ray, but larger 326 ELECTRICITY, SOUND, AND LIGHT than that of the extraordinary ray ; hence it was possible, by using a long crystal like that shown in the figure, to choose the plane aa so that the ordinary ray would be totally reflected and absorbed in the blackened walls of the crystal, while the extraordi- nary ray would pass through. Several modifications of this device are now in use, but they are all essentially the same in principle. 201. Effects produced by the passage of polarized light through thin crystals. Experiment 11. Arrange the polarizing apparatus precisely as in Fig- ure 223, save that the Nicol prism replaces the mirror m'. Rotate the Nicol until the flame is completely extinguished. Then obtain from the instructor a so-called half -wave plate (for sodium light) of mica or selenite and place it on the slide holder Ji. You will find that, in general, the inser- tion of the mica causes the light to reappear. Rotate the mica about a vertical axis and note that in one revolution there are four positions, just 00 apart, at which there is extinction. These are the positions in which the plane of vibration of the light which is incident upon the mica is either parallel to or perpendicular to the plane containing the optic axis of the mica. Rotate the mica in a horizontal plane until it is just 45 from one of these positions of extinction. Then rotate the Nicol. The image of the flame will be found to disappear when the Nicol has been rotated through 90. We may explain this phenomenon as follows. In Figure 234 let a represent the plane of vibration of the light which is inci- dent upon the sheet of mica. Let l> represent the two vibrations a ^ into which the incident vibration is decomposed as soon as it enters the crystal, the one, e, in the plane of e the optic axis, and the other, 0, perpendicular to this plane. Since one of these two waves travels faster through the mica than does the other, a sheet may be chosen of just such thickness that the two beams will emerge from the crystal with o just one-half wave length behind e. Since, moreover, the beams are broad and the mica very thin, o and e are not separated from one another as they were in the case of the two nar- IT IP 9*34. row beams used in Experiment 5. The same portion of the ether must therefore transmit simultaneously the two beams after emergence from the mica. The fact that one of them has lost one-half wave length with respect to the other has been indicated X. X t POLARIZED LIGHT 327 at c (Fig. 234), by changing the position of the arrowhead on the line which represents this vibration from one end to the other. These two vibrations will recombine, upon emergence into air, into one single vibration, which is represented by Figure 234, c?. The light which emerges from the crystal will therefore be plane polarized, but the direction of its vibration will be at right angles to the direction of vibration of the beam when it entered the crystal. Since the Nicol was originally set so as to cut out vibrations in the direction shown in Q Figure 234, a, it will, of course, transmit with maxi- ^ /^ mum intensity vibrations in the direction shown in Figure 234,c In order to extinguish this vibration \7 it should be necessary to rotate the Nicol through 90, as we found in our experiment was the case. If the mica plate had been just one half as thick as it was, the two components would have emerged from the crystal one-fourth wave length instead of one-half wave length apart. They would have then recombined into a circular vibration (see Fig. 235), since two equal simple harmonic forces 90 apart, acting simultaneously upon the same particle, must cause it to describe a circular path. The analyzing Nicol should obviously transmit the same fraction of this circular vibration, no matter into what plane it is turned. Experiment 12. Replace the half -wave plate by one half as thick, that is, by a quarter-wave plate. Set it at first so that it is 45 from the point of extinction, the Xicol being set in the position corresponding- to extinction when no plate is interposed ; then rotate the Xicol and note that no change takes place in the intensity of the transmitted light because of this rotation. Light of this sort produced by passing plane polarized light through a quarter-wave plate is said to be circularly polarized. If the plate is of such thickness as to produce a retardation (of one component with respect to the other) of one wave length, two wave lengths, three wave lengths, and so on, the light will obvi- ously emerge from the crystal as plane polarized light vibrating in the same plane as that in which it entered the crystal. Hence it will be completely cut out by the Nicol when the latter is set in the position of extinction for the case in which no crystal is interposed. 328 ELECTRICITY, SOUND, AND LIGHT x; X FIG. 236 If the plate is of such thickness as to cause retardation inter- mediate between a quarter wave and a half wave, or a half wave and a whole wave, or a whole wave and a wave and a quarter, and so on, the light which emerges from the crystal is elliptically polarized, that is, the vibrations of the. ether particles take place in the form of an ellipse (see Fig. 236). The major and minor axes of this ellipse may be easily found by observing in what direction the analyzing Nicol must be turned in order to obtain a maximum or a minimum of transmitted light. 202. Colors produced by thin crystals in polar- ized light. Experiment 13. Set the polariscope in a window in the position shown in Figure 237, the black paper being removed from the mirror n. If m has precisely the same inclination which "was given it in Experiment 1 (Fig. 223) , then, when the polariscope is so turned that the prolongation of the line ma meets the clear sky, the white light from the sky will strike the lower side of m at the polarizing angle, be reflected to the mercury mirror w, and return with little loss as a plane polarized beam to the Nicol N. Set N so that this beam is extinguished. Place a sheet of mica about twice as thick as a half-wave plate for sodium light upon h and turn it until it is just 45 from a position of extinction. When viewed through the Nicol it will be found to be brilliantly colored. Rotate the Nicol slowly and notice that a rotation of 45 causes the color to disappear, but that a rotation of 90 causes a color which is the complement of the first color to appear. Further rotation through 90 will cause the first color to return, and so on. When sheets of mica of different thicknesses are used different colors will be produced, but a rotation of the Nicol through 90 will always cause the color to change to that of the complement of the original color. -^ 007 In order to understand the cause of this phenomenon suppose, for simplicity, that the mica is just thick enough to produce a retardation of one-half wave length of the longest red wave. POLARIZED LIGHT 329 Since the shortest violet waves have about one half the wave length of the longest red, this same plate will produce a retarda- tion in the violet of one whole wave length. The violet wave will therefore emerge from the crystal with both of its components in the same phase, and these components will recombine into a plane vibration precisely like that which entered the crystal, namely the vibration represented in Figure 234, a. The red ray, however, will emerge from the crystal with one of its components one-half wave length behind the other, and these two components will recombine into a vibration at right angles to that of the entering ray, namely one of the form shown in Figure 234, d. If, then, the Nicol is in the position for extinction when no crystal is interposed, it will cut out all of the violet in the incident white light and transmit all of the red, so that if these red and violet waves fell alone upon the crystal, a rotation of the Nicol would cause red and violet to appear alternately. As a matter of fact, however, if the incident light is white, all of the colors between the red and the violet will be present, and the vibrations of the transmitted light which correspond to them will be ellipses of some form. However, the wave lengths which are close to the red, namely orange and yellow, will be largely transmitted by the Nicol along with the red, and will have but small components to be transmitted with the violet, while the w r ave lengths near the violet, namely the blues and the shorter greens, will be mainly transmitted with the violet. Hence the light which passes through the Nicol in its first position will be some shade of red, because it will have most of the shorter wave lengths subtracted from it ; while, when the Nicol is turned through 90, all of the wave lengths which were before cut out will be transmitted. The color will therefore be exactly the complement of the first color, that is, it will be some shade of blue. A crystal which is too thin to produce one-half wave length retardation of the shortest visible rays, namely the violet, cannot show any marked color effects in polarized light, since no wave length can be entirely cut out for any position of the Nicol. On the other hand, a crystal which is so thick as to produce retarda- tion of very many wave lengths of any one color will produce also 330 ELECTRICITY, SOUND, AND LIGHT a retardation of an exact number of wave lengths for each of many other colors scattered throughout the spectrum. These colors will all be cut out by the Nicol, and the transmitted light will like- wise consist of wave lengths which are taken from all parts of the spectrum, and will therefore reproduce the effect of white light. Hence these color phenomena in polarized light can be observed only with crystals which produce a small number of ivave lengths of retardation. By scraping crystals down to proper thicknesses in different parts, color patterns of much beauty are often produced when the crystals so treated are viewed in the polarized light. All the colors, of course, change to the comple- ments upon rotation of the Nicol through 90. Experiment 14. Place a iiiimber of these designs in selenite or mica upon the slide holder h, and observe the appearance of the complementary colors in different portions of the design as the Nicol is rotated. Experiment 15. Observe in convergent polarized light a crystal of Iceland spar, say 1 mm. thick, the upper and lower faces of which are planes per- pendicular to the optic axes. The beam of convergent light is most easily obtained by placing the crystal very close to the Nicol in the ar- rangement of Figure 237, so that the observer looks down through the crystal upon a field of con- siderable width, from all parts of which polarized light is ap- F,o. 238 F.o. 239 Pcl>ing the eye If the Nicol was originally set tor extinction, you will observe a dark center surrounded by a series of brilliantly colored rings upon which is superposed a black cross (see Fig. 238). Rotating the Nicol will cause the black cross to change to a white one, and all of the colors to change to their complements (see Fig. 239). These effects may be explained as follows. The central rays pass through the crystal in a direction parallel to the optic axis. They therefore suffer no resolution into ordinary and extraordi- nary components, and hence no change in the character of their vibration. They are cut out by the analyzing Nicol, hence the black center. The rays, however, which converge upon the eye after passing through the outer edges of the crystal have trav- eled in directions slightly oblique to the axis, and have therefore POLARIZED LIGHT 331 suffered decomposition into ordinary and extraordinary rays, which have undergone different retardations. A given retardation of one ray with respect to the other corresponds to a given color pre- cisely as explained above. A given color must, of course, be sym- metrically distributed about the axis of the converging beam, since the thickness of the crystal is so distributed ; hence the concentric rings of color. The black cross is superposed upon these rings be- cause in two particular planes, namely those for which the inci- dent vibration is respectively in and perpendicular to the plane containing the axis and the ray, even these oblique rays are not split up into components, but pass through vibrating in their original direction, and are therefore cut out by the Mcol. Upon rotating the Nicol through 90 all of these extinguished rays are, of course, transmitted ; hence the white cross. 203. Rotary polarization. Experiment 16. Arrange the polarizing apparatus as in Figure 223, save that mf is replaced by a Xicol, and place upon h a crystal of quartz, say 5 mm. thick, the upper and lower faces of which are made by planes which are at right angles to the optic axis of the quartz. When the Xicol is set for extinction the introduction of the quartz into the path of the beam will be found, in general, to cause the extinguished image of the flame to reappear. Rotate the Xicol, and measure the amount of rotation required to cause the yellow flame to disappear again. According to accepted results this rotation for sodium light should be 21.7 per millimeter of thickness of the quartz. Replace the sodium flame by the ordinary violet flame of the Bunsen burner and repeat, rotating this time until all trace of the violet color in the flame has disappeared. The rotation will be found to be nearly double that found for sodium light. The rotation in the case of light filtered through red glass will be found to be less than that in the case of yellow light. These experiments show, first, that plane polarized light which passes through quartz in the direction of its optic axis remains plane polarized after transmission, and, second, that the plane of polarization of the light is rotated by the quartz, the amount of the rotation being greater for the short wave lengths than for the long. The discovery that quartz is able to produce these effects was made by Arago in 1811. From the difference in the amount of rotation of different colors it follows that if plane polarized white light is incident upon the 332 ELECTRICITY, SOUND, AND LIGHT lower face of the crystal of quartz, the light which is transmitted by the analyzing Nicol will be colored, since this Nicol will extin- guish completely only those vibrations which are perpendicular to its transmitting plane. It follows, further, that if the Nicol is rotated through 90, all of the components of the white light which were before extinguished will be now transmitted and vice versa, and hence that the rotation through 90 will cause the color to change to the complement of the first color. Experiment 17. Verify the above predictions by setting the polariscope so that light from the sky falls upon it in the manner indicated in Fig- ure 237. To show that the lights transmitted by the analyzer in planes 90 apart are complementary, it is best to replace the Nicol by a thick crystal of Iceland spar, or some other form of double-image prism, so that both of the lights to be compared may be transmitted at once. As the analyzer is rotated, the overlapping portions of the images will be found to maintain the color of white light, while the opposite non-overlapping portions will be of complementary colors. It has been found that there are two kinds of quartz crystals, one of which produces rotation to the right, the other to the left. This difference in optical behavior corresponds also to a difference in crystalline structure which makes it easy to distinguish the so-called right-handed from the left-handed quartz crystals with- out actually making the optical test. Furthermore, it was discovered by Biot, in 1815, that there are certain liquids which possess the same property shown in the above experiments by quartz. Of these, solutions of cane sugar have received most attention, for the reason that the amount of rotation produced by a column of sugar solution of fixed length is taken as the commercial test of the strength of the solution. As in the case of quartz, there are found to be two kinds of cane sugar of precisely the same chemical constitution, but of slightly different crystalline form, which rotate the plane of polarization in opposite directions. The form which rotates to the right is called dextrose, the other levulose. It is possible to convert dextrose to levulose sugar by acting upon it with hydrochloric acid. This conversion is actually made in sugar testing, the rotation due to the conversion being the quantity directly measured. CHAPTEK XXVIII RADIO-ACTIVITY 204. Types of radiation. Up to about 1895 the only types of radiation which were generally recognized were those comprised under the two heads, sound waves and ether waves. Ether waves included light waves, of wave length .00038 mm. to .00076 mm.; ultraviolet waves, studied by means of their photographic effects and having wave lengths extending from .00038 down to an unknown limit, but definitely explored down to .0001 ; ultrared rays, studied by means of their heating effects, and having wave lengths extending from .00078 mm. up to an unknown limit, but explored up to .06 mm.; and electrical waves studied by means of electrical resonators, and having observed wave lengths extend- ing from 3 mm. up to infinity. All of these forms of ether waves have been definitely proved to travel with the same velocity in a vacuum and to be essentially alike except in the matter of wave length. Since 1895 three new types of radiation have gained recognition. They are cathode rays, X rays, and the radiations produced by radio-active substances. Cathode rays were observed as early as 1859 and named in 1876, but their character was not understood until 1897. 205. Cathode rays. When an electrical discharge is forced through a glass tube from which the air is beiog exhausted, the appearance of the discharge is found to vary continuously as the exhaustion progresses. When the pressure within the tube has been reduced to say .01 mm. of mercury, a greenish fluorescence is found to have made its appearance on the walls of the tube about the cathode. As the exhaustion is carried still farther all appear- ance of a discharge through the gas within the tube vanishes and the walls become suffused with a greenish fluorescence. 334 ELECTKICITY, SOUND, AND LIGHT This fluorescence seems to be due to some sort of radiation which is emitted by the cathode in a direction normal to its surface ; for whenever an object is placed within the tube a sharp shadow of it is found to be cast upon the walls at a point directly op- posite the cathode, no matter where the anode may be placed. This is well illustrated by an experiment devised by Sir William Crookes in 1879, and repre- sented in Figure 240, a shadow of the Mal- tese cross M being evident upon the wall of the tube opposite the cathode C. 206. Theory of cathode rays. The nature of these so-called cathode rays was a subject of much dispute between the years 1880 and 1895. Some thought them to be some type of ether waves emitted by the cathode, while others, following the lead of Crookes, were convinced, that they consisted of material par- ticles projected with large veloc- ities from the surface of the cathode. The most convincing evidence in favor of the latter view was found in the fact, brought out clearly by Crookes in 1879, that the cathode rays are deflected by a magnet in F IG . 241 precisely the way in which neg- atively charged particles projected normally from the surface of the cathode ought to be deflected. The experiment illustrating this deflection is shown in Figure 241. The cathode beam from EADIO-ACTIVITY 335 c, the direction of which, after it passes through the slit e, is indicated by the fluorescence which it produces in the zinc sul- phide screen ad, along which it grazes as it progresses through the tube, is deflected by the magnet ns, in the position shown, precisely as the motor rule indicates that it should be deflected if the cathode beam consisted of a negative current flowing from the bottom toward the top of the tube, that is, of a positive current flowing from top to bottom. The strongest objection to the projected-particle theory was made on the ground of experiments conducted by Lenard, in Bonn, Germany, in 1893. These experiments show that the cathode rays can pass out of the vacuum tube through a thin aluminum win- dow and travel several centimeters in air at atmospheric pressure before being absorbed. Figure 242 shows the tube commonly used for repeating this experiment. In this figure c rep- resents the cath- ode, A the anode, in this case a cyl- inder behind the cathode, w a very minute and very thin piece of aluminum foil, and E the cathode rays diffusing in air at atmospheric pressure in the manner indi- cated. The presence of these rays at any point is detected by the fluorescence which they produce in an exploring zinc sulphide screen. This experiment shows that whatever cathode rays may be, they are able to pass through a sheet of aluminum which wiU not per- mit the passage of the molecules of gas, since otherwise the vacuum in the tube could not be maintained ; and that they are also able to pass through the million or so of molecules which any straight line which is drawn through several centimeters of air at atmos- pheric pressure must encounter. It is evident that this experiment is not easily reconciled with the projected-particle theory if the projected particles are assumed to be atoms or molecules of ordi- nary matter ; but all difficulty vanishes if we assume that the pro- jected particles are exceedingly small in comparison with ordinary FIG. 242 336 ELECTRICITY, SOUND, AND LIGHT atoms, and if we assume further that the atoms of ordinary matter form some sort of infinitesimal stellar systems, the distances be- tween whose members are large in comparison with the size of the members themselves. In 1897 it was definitely shown, both by J. J. Thomson and Lenard, that the cathode rays are deflected, not only by a magnetic field, but also by an electrostatic field, in precisely the way in which they should be deflected if they consist of negatively charged particles. Furthermore, by comparing, with the aid of mathemat- ical analysis, the amounts of deflection produced by magnetic and electric fields of known strengths it was found that the mass of the projected particles of the cathode rays, if this theory were adopted, came out but 1/1800 of the mass of the smallest known atom, namely that of hydrogen. This epoch-making discovery removed all the serious difficulties of the projected-particle theory. The ether theory, on the other hand, has now been entirely abandoned, since it does not seem possible to reconcile it in any way with the magnetic and electrostatic deflectibility of the rays. Furthermore, the cathode rays were definitely proved in 1896, by Perrin, of France, to impart negative charges to any objects upon which they fall. This accords with the projected-particle hypothesis, but it is wholly unlike any property possessed by known forms of ether waves. 207. Hypothesis as to the nature of the atom. The acceptance of the projected-particle theory as to the nature of cathode rays makes it necessary to assume that these minute negatively charged corpuscles, or electrons as they have been named, are constituents of the atoms of all kinds of matter, since cathode rays of the same magnetic and electric deflectibility can be produced with all sorts of electrodes and all sorts of residual gases in the discharge tube. This assumption met with very striking support in a discovery announced by Zeeman, of Amsterdam, in 1897. This observer found that the spectral lines produced by incandescent vapors become doubled, and otherwise modified, when the incandescent vapor which is being examined is placed between the poles of a powerful electro-magnet. From the character and magnitude of this effect it was found that the radiations emitted by these incan- descent vapors must be produced by the vibrations, or rotations RADIO-ACTIVITY 337 about a center, of negatively rather than positively charged par- ticles, and that, within the limits of experimental error, the mass of these rotating negative particles is precisely that of the cathode- ray particles. In view, then, of cathode-ray phenomena, the Zeeman effect, and radio-activity (sect. 210), the atoms of all substances are now regarded as containing negatively charged particles, or electrons, of about 1/1800 the mass of a hydrogen atom. These electrons, escaping from the material of the cathode under the influence of the powerful electric fields produced in cathode-ray tubes, stream away in straight lines from the surface of the cathode with veloci- ties which have been definitely measured and found to lie between one tenth and one third the velocity of light (3 x 10 10 cm. per second), the velocity being determined solely by the strength of the field which is employed to produce the discharge. In order to account for the fact that the absorbing power of different kinds of matter for the cathode rays is nearly directly proportional to density, it is sometimes assumed that the atoms of all substances are merely aggregates of electrons, so that the total weight of a cubic centimeter of any substance is determined solely by the number of electrons contained in it, and the total ab- sorbing power of this cubic centimeter for cathode rays is deter- mined by the number of electrons which a given cathode-ray particle would encounter in passing through this cubic centimeter. This view would be in accordance with Lenard's experiment on the absorbing powers of different substances for cathode rays, but there is not enough evidence to substantiate it fully, and it must be held at present, therefore, merely as a speculation. Since, however, the atoms certainly contain electrons, and since these electrons are negatively charged, in order to account for neu- tral atoms we must assume a total positive charge in each atom equal to the sum of the negative charges carried by its electrons. Since no definite evidence has as yet been brought forward that positive bodies of the minute mass of the negative electrons are ever set free from atoms, it has been suggested by J. J. Thomson that the atom may consist of a nucleus of positive electricity about which, or within which, the negative electrons are rotating. 338 ELECTKICITY, SOUND, AND LIGHT 208. The ionization of gases. One of the most notable proper- ties possessed by cathode rays is the property of rendering any gas through wliich they pass a conductor of electricity. Thus if a charged electroscope is placed anywhere within a foot or so of the window w (Fig. 242), it is found to lose its charge rapidly. The way in wliich this discharge takes place has been very convinc- ingly shown to be as follows. The cathode rays in passing through the gas produce in it, in some way, both positively and negatively charged particles. If, then, the electroscope is positively charged, for example, the negative particles produced in the gas by the cathode rays are drawn to the positively charged electroscope, give up their charges to it as soon as they touch it, and thus reduce its positive charge. If the electroscope is negatively charged, it is the positive particles of the gas which discharge it. The origin of these positive and negative particles is supposed to be as follows. The neutral molecules of the gas, each of which contains negative electrons and an amount of positive electricity equal to the sum of the negative charges on its electrons, have one or more of these electrons knocked off by the bombardment of the rapidly moving cathode-ray particles. The loss of an electron by a neutral molecule of the gas leaves it positively charged. This positively charged molecule, with a few neutral molecules which it probably attaches to itself, constitutes one of the positive ions of the gas. On the other hand, the electrons, which have been knocked off from the neutral molecules, probably soon attach themselves to other neutral molecules and thus impart negative charges to them. These negatively charged molecules, with neutral molecules which become attached to them, constitute the negative ions of the gas. When two of these oppositely charged ions approach near enough to each other, they doubtless are drawn together, their opposite charges neutralized, and the molecules restored to their original condition. It is in this way at least that we attempt to account for the fact that the gas rapidly loses its conducting power when the cathode rays are removed. If, however, the ions are formed in a sufficiently strong electrical field, they may be drawn to the elec- trodes before they have any opportunity to unite with any other oppositely charged ions. In this case the rate at which either of BADIO-ACTIVITY 339 the electrodes loses its charge is a measure of the total number of ions formed per second. The ionization of a gas then differs from the ionization of liquids, studied in Chapter XVI, in the following respects. While the ionization of liquids occurs spontaneously, the ionization of gases occurs only through the action of an outside ionizing agent. The ionization of liquids occurs only in the case of a mixture of differ- ent substances, that is, in the case of solutions, but the ionization of a gas occurs as well with simple, homogeneous gases as with mixed ones. 209. X rays. It was in December, 1895, that Professor Eont- gen announced the discovery of a new type of radiation which he named X rays. These rays have their origin in cathode-ray tubes, and are like cathode rays in the fluorescent, photographic, and ion- izing effects which they produce, but differ from cathode rays, first, in that they possess a very much higher penetrating power ; second, in that they are not deflected in the slightest degree either by a magnetic or an electrostatic field ; third, in that they do not impart negative charges to bodies upon which they fall ; and fourth, in that different bodies do not absorb them in amounts proportional to the absorption coefficients of these same bodies for cathode rays. That the X rays have a much greater penetrating power than the cathode rays is shown by the fact that while the latter can only be obtained outside a cathode-ray tube by passing them through a very thin aluminum window, as Lenard did, the former pass with considerable ease through the glass walls of the ordinary X-ray tube. It is because of their relatively great penetrating power, and because of the fact that, in general, they are absorbed in dif- ferent amounts by substances which have different densities, that it is possible to take with them shadow pictures of the bones of the body, or of any objects of different density inclosed in cover- ings opaque to ordinary light. It was doubtless on account of this property that the rays attracted so immediate and so widespread attention. Subsequent study of X rays has proved conclusively that they originate in the surface of any body upon which the cathode rays 340 ELECTRICITY, SOUND, AND LIGHT fall. In the ordinary X-ray tube a platinum plate P (Fig. 243) is used as a target for the cathode rays which are emitted at right angles to the surface of the concave cathode C and come to a focus at a point on this plate. This is the point at which the X rays are generated and from which they radiate in all direc- tions. This statement can be readily verified by turning about an X-ray tube in action, placed at a distance of a few feet from a charged electroscope, and noticing that the electroscope loses its charge rapidly when it is in any position such that rays radiating from the middle of the front face of P could fall upon it, but that the loss of charge ceases as soon as the tube is turned into such a po- sition that rays coming from this point could not fall upon FIG. 243 ^ i the electroscope. The nature of the X rays is not yet definitely settled, but it is very generally believed that they are very violent ether pulses set up by the sudden stoppage of the cathode rays as they strike upon the target P. If this theory is correct, we must regard the ioniza- tion which the X rays produce as brought about by the shaking loose of electrons from the atoms of air because of the suddenness of the ether pulses which strike them. This theory is in accord with the fact that the rays show no regular reflection, refraction, or polarization. 210. Discovery of radio-activity. At the time of the discovery of X rays, and before their origin was known, it was surmised that they might be in some way connected with the fluorescence of the walls of the cathode-ray tube, and since physicists were look- ing for other sources of the rays, it occurred to Henri Becquerel, of Paris, that the mineral uranium, which under the influence of sunlight fluoresces in a manner not unlike the fluorescence of the glass tube under the influence of the cathode rays, might also give rise to X rays when exposed to sunlight. He therefore wrapped up a photographic plate in black paper, placed a uranium com- pound above it, and exposed the uranium to sunlight. He found EADIO-ACTI VIT Y 341 that after an exposure of some days the photographic plate was darkened, thus showing that the uranium emitted rays which were capable of passing through opaque paper. He soon found, however, that the exposure of the uranium to sunlight was quite unnecessary. The effects could be produced just as well in a dark closet as in sunlight. Becquerel therefore drew the conclusion that uranium compounds spontaneously emit radiations which are akin to X rays in their photographic and penetrating effects. He soon found also that these rays were like X rays in their ionizing properties, but unlike them, and like the cathode rays, in that they were deflected both by magnetic and electrostatic fields, and in that they imparted negative charges to bodies upon which they fell. In a word, the rays which produce the photographic effects observed by Becquerel were found to be in every respect identical with cathode rays except that the electrons which are thus shot off sponta- neously from uranium atoms have speeds of about one half the velocity of light. This is considerably larger than the velocities of ordinary cathode rays. 211. Complexity of the radiations from uranium. In 1899 Rutherford proved clearly that the rays from uranium are of two kinds, the so-called alpha and beta rays. The beta rays are simply projected electrons, that is, cathode rays, but the alpha rays have a much less penetrating power and produce much stronger ioniz- ing effects. They have since been shown to consist of positively charged bodies of atomic size, their mass being either that of the hydrogen molecule or the helium atom. These alpha rays are com- pletely absorbed by a layer of air three or four centimeters thick. The name radio-activity has been given to the property, found to be possessed by quite a number of substances, of spontaneously projecting bodies of atomic or subatomic size with such velocity as to ionize a gas or produce an impression upon a photographic plate. The method which has been most commonly used for studying the radiations of uranium, and of other substances which have been found to possess similar properties, is the following. The uranium, or other substance to be tested, is placed in a shallow metal vessel between two plates, A and B (Fig. 244), to one of 342 ELECTRICITY, SOUND, AND LIGHT which is attached a gold leaf or other indicator of an electrical charge. The rays from the radio-active substance ionize the gas between the plates, and if the plate to which the gold leaf is attached is originally charged, it loses this charge at a rate which is proportional to the number of ions formed per second in the gas between the plates because of the pres- ence of the radio-active substance. The relative radio-activities of two substances are then compared by measuring the rel- ative rates of discharge of the electro- scope when equal weights of the two substances are spread over equal areas. When tested by this means the activity of uranium is found to be exceedingly constant; hence uranium is commonly taken as a standard, and the activities of all substances are rated in terms of the activity of uranium. The proof first given by Rutherford of the complexity of the radiations from uranium was as follows. He laid layers of alu- minum foil successively over the uranium and observed the diminu- tion in the rate of discharge of the electroscope produced thereby. The following table represents one set of his observations. FIG. 244 THICKNESS OF ALUMINUM FOIL .005 MM. Number of layers Discharge per minute in scale divisions Ratio of successive rates of discharge 182 1 77 42 2 33 43 3 14.6 44 4 9.4 65 12 7 It will be seen that at first each succeeding layer cuts down the rate of discharge of the electrometer by approximately the same fraction of the value which it had before the introduction EADIO-ACTIVITY 343 of that layer; in other words, the ratio of each two successive values of the leak is constant. This is, of course, as it should be if there is but one type of radiation which is being absorbed by suc- cessive layers. But it will be seen that after the third layer this relation breaks down completely, and the insertion of eight new layers after the fourth reduces the leak only from 9.4 to 7. This is interpreted as meaning that the alpha rays have been practically all absorbed by the first four layers, while the beta rays are but very slightly absorbed by the introduction of eight more layers. In order to cut the remaining leak down to one-half value, i.e. to 3.5, it was found necessary to introduce a thickness of aluminum equal to 100 more sheets. Since one layer cuts down the alpha radia- tion to less than one-half value, the beta radiation must be more than one hundred times as penetrating as the alpha radiation. The above example illustrates one method of separating the two types of rays. Another method is to pass the radiations through a strong magnetic field. The beta rays are found to be easily deflected, while the alpha rays are deflected exceedingly little and in the opposite direction. It is this experiment which shows that the two types of rays are oppositely charged. 212. Other radio-active substances. Immediately after Bec- querel's discovery of uranium rays M. and Mme. Curie, of Paris, made a careful study of all the then known elements by the elec- trical method, with a view of determining whether or not the property of radio-activity was possessed by other substances than uranium. They found that thorium and all the thorium compounds had about the same degree of activity as uranium and its compounds, but that no other of the then known elements behaved in this way. However, in investigating the ores of uranium they found that pitchblende, which is composed largely of uranium oxide, had about four times the activity of chemically prepared uranium oxide. Since, in general, the chemically prepared compounds of uranium were found to have activities which were proportional to the amounts of uranium present, it appeared clear that the activity of uranium was a property of the uranium atom and was not affected by the sort of combination in which the atom was found. From these facts the Curies concluded that the large activity of 344 ELECTRICITY, SOUND, AND LIGHT pitchblende must be due to the presence of some hitherto unknown element of very great activity. They at once applied themselves to the task of separating this element. Their methods were those commonly used in chemical analysis, except that, after each separa- tion by precipitation, both the filtrate and the precipitate were evaporated to dryness and tested for their activities by the elec- trical method. It was thus possible to find whether the active substance had been precipitated out of the solution, or whether it had remained in the filtrate. The result of this search was the defi- nite separation from a ton of pitchblende of a few centigrams of a new element which was named radium because it had the tremen- dous radio-activity of about two million as compared with uranium. In chemical properties radium is closely allied to barium, from which it was found very difficult to separate it. Another radio- active substance was later extracted from pitchblende and named actinium. This substance is chemically allied to thorium. The radiations from radium and actinium, like those from uranium, have been found to consist of alpha and beta rays. A third type of radiation, however, called the gamma rays, was also found to be emitted by radium and actinium. This is character- ized by a penetrating power a hundred times greater than that of the beta rays and by its entire freedom from deflectability by elec- trostatic or magnetic fields. These gamma rays are commonly supposed to be X rays, which in any case, according to the theory of section 209, would of necessity accompany the beta rays as they do in a cathode-ray tube. In the case of uranium also these gamma rays are doubtless present, but are so weak as to escape detection under the ordinary conditions of experiment. 213. The spinthariscope. The most striking, though perhaps not the most conclusive, evidence that radio-active substances are continually projecting particles from their atoms with enormous velocity is furnished by an experiment devised by Crookes in 1903. When a bit of radium is placed near a screen of zinc sulphide it produces fluorescence in the screen precisely as do cathode rays or X rays. In order to study more carefully this fluorescence Crookes placed a tiny speck of radium R (Fig. 245) about a millimeter above a zinc sulphide screen $ and then viewed the latter through R RADIO-ACTIVITY 345 a lens L which produced from 10 to 20 diameters magnification. The continuous soft glow of the screen, which is all that one sees with the naked eye, is, under these circumstances, resolved by the microscope into a thousand tiny flashes of light. The appearance is as though the screen were being vigor- ously bombarded by a continuous rain of projectiles, each impact being marked by a flash of light. The flashes are probably due indirectly to the impacts of the alpha particles. Since these alpha particles have been shown by measurements upon their electrostatic and magnetic deflectability to have about the mass of a helium atom, and to have velocities of about one tenth that of light, it will be obvious that they would strike an object with tre- mendous energy. 214. Radio-activity an atomic property. Whatever the cause of this ceaseless emission of particles by radio-active substances, it is now fairly well established that the activity of all radio-active substances is proportional to the amount of the radio-active ele- ment present, and has nothing whatever to do with the nature of the chemical compound in which the active element is found. Thus uranium, for example, may be changed from a nitrate to a chloride, or from a chloride to a sulphide or an oxide, without pro- ducing any change whatever in its activity. Furthermore, radio- activity has been found to be equally independent of all changes in physical as well as chemical condition. The activity of uranium, for example, is not altered at all by reducing its temperature to that of liquid air. This is strong evidence in favor of the view that radio-active change, that is, the change, whatever it be, which is responsible for the emission of the alpha and beta particles, involves a change in the nature of the atom itself. 215. Radio-active transformations. The first direct evidence that the atoms of radio-active substances are continually trans- muting themselves into atoms of new physical and chemical prop- erties was contained in some experiments made in 1900 by Sir William Crookes, but first interpreted by Eutherf ord. Crookes found that when he precipitated an aqueous solution of uranium nitrate with ammonium carbonate and then redissolved the precipitated 346 ELECTEICITY, SOUND, AND LIGHT uranium by an excess of the reagent, there remained behind an undissolved precipitate which contained a large part of the original activity possessed by the uranium nitrate. He called this undis- solved precipitate (or better, the portion of it which was responsi- ble for the activity, for when chemically tested it showed nothing but iron, aluminum, and other impurities) uranium X. He soon afterwards discovered that the uranium nitrate, which had lost a large part of its activity through the separation from it of uranium X, had, in the course of a few months, completely regained its original activity, while the uranium X had lost its power of radi- ating. This result pointed very strongly to the conclusion that a radio-active substance is being continually produced by uranium. Not long after this Eutherford found that a product analogous to uranium X could be separated from thorium by a similar method. He called this thorium X. The essential difference between ura- nium X and thorium X lies in the fact that the activity of the first falls to half value in twenty-two days, while that of the second falls to half value in about four days. The examination of radium revealed a behavior exactly similar to that of uranium, for it, too, was found to be continually produc- ing a radio-active substance which, when separated from the radium, slowly lost its activity, while the radium from which it was sepa- rated regained, at a like rate, its radiating power. In the case of radium this new substance, unlike uranium X and thorium X, could be distinguished by other physical properties besides its activity. Thus Rutherford found it to be of the nature of a gas which could be separated from a radium salt by heating the latter or by dissolving it in water. The radium which had been so treated lost, for the time being, all but one fourth of its original radiating power, the other three fourths being found in the gas, or emanation, as Rutherford called it. This gas could be set away in bottles and the change in its activity watched from day to day. It could be condensed by passage through tubes immersed in liquid air, its presence being, in general, detected by the ionization which it imparted to the air with which it was mixed. It has recently been obtained in sufficient quantities to show a charac- teristic spectrum and other qualities common to gases. This gas, RADIO-ACTIVITY 347 however, like uranium X and thorium X, has but a transitory existence, for the fact that it gradually loses its activity shows that it passes on into something else. Nor did physicists have long to look in order to discover this substance into which the emanation from radium is transformed. They found that when the gas comes into contact with a solid object, this object, especially if it is negatively charged, becomes coated with a film of radio-active matter which can be dissolved with hydrochloric or sulphuric acid, and which is left in the dish when the acid is evaporated, or which may be rubbed off with leather and found, by means of the property of activity which it possesses, in the ash of the leather after the leather has been burned. This active deposit, generally obtained by placing a nega- tively charged wire in a vessel containing radium, is so infinitesi- mal in amount that it has never been detected in any other way than through its activity. At first it might look as though it were nothing but the active gas itself condensed on the surface of the solid object, but since the rate at which it loses its activity is alto- gether different from the rate at which the emanation decays ; since, furthermore, the emanation atom is not positively charged, and therefore does not tend to collect on a negative wire; and since the active deposit is found to emit both alpha and beta rays, while the emanation emits only alpha rays, it seems necessary to con- clude that this film of active matter is the product of the emana- tion rather than the emanation itself. In fact, it appears to bear in all respects the same relation to the emanation which the emana- tion bears to radium; that is, 'it -is the result of the disintegration of the atom of the emanation, just as the emanation is the result of the disintegration of the atom of radium. Thorium is found to be precisely like radium in that it gives rise to a gaseous emanation, and in that this emanation disinte- grates into something else which collects upon a negatively charged wire. It has been definitely shown, however, that the thorium emanation is not the direct product of thorium, but rather of thorium X. In order to collect the active deposit from the thorium emana- tion it is only necessary to thrust a wire through a cork which 348 ELECTRICITY, SOUND, AND LIGHT Hi m THORIUM OXIDE iiii~ llll + FIG. 246 closes a small tin can in which some thorium oxide is placed, and make the wire the negative terminal of a battery of 100 volts or more, the can being made the positive terminal (see Fig. 246).* In the course of an hour or more the wire will have become appre- ciably active. The fact that this active mate- rial is deposited on a negatively charged wire throws some light on the probable nature of the dis- integration of the emanation atom, for this means, of course, that the atoms which are thus deposited are positively charged. In order to account for this we must assume, that although the emanation atom produces only alpha rays, yet it must lose one or more electrons in passing over into the atom of this active deposit. The reason that these electrons do not produce beta-ray effects may be that they are projected with insufficient velocities to ionize the air. If a wire is exposed to the thorium emanation for several days in the manner indicated above, upon removal its activity is found to decay fairly regularly, falling to half value in about 1 1 hours, but if it is exposed to a large amount of the emanation for only a few minutes, its activity is found to increase regularly for about 3J- hours and then to 2 decrease in the manner indi- cated in Figure 247. Jhe in- itial increase in activity shows that an active substance must p IG- 247 be forming on the wire for some time after its removal from the vessel containing thorium, and the fact that the initial activity of the wire is practically zero shows that the material from which this active substance is 100 ^ ^ ~~^~~ ACTIV/T) * 3 J / / / I o 1 4-0 8O 120 160 2.OO 24-O 26O 32 TIME IN MINUTES * Figures 246 and 247 are taken from Rutherford's "Radio-active Trans- formations." RADIO-ACTIVITY 349 Uranium Ur. X GxlOByr. 22 dv. formed is not itself active. The curve is completely explained if we assume that the atom of the thorium emanation disinte- grates into the atom of some nonactive substance called by Rutherford thorium A, which deposits upon the wire, while this nonactive substance then disintegrates into an active substance, fc. thorium B. Enough has been said to show how, one after another, the successive products of radio- active change have been discovered and studied. The following chart (Fig. 248), taken from Rutherford's " Radio- active Transformations," shows the various trans- formations which the radio-active elements are supposed to un- dergo, the time given beneath each product representing the time required for this sub- stance to lose half its Act. A Act. B 36 min. 2 min. activity. It IS nOW ""acdveTi^Tr' pretty well established, " .* ^ o-o-d-o- Thorium Th. X 24 x 109 yr< 4 dy. Eman. 54 sec. Th. A Th. B 11 lir. 55 min. active deposit . A Actinium Act. X 10 dy. Eman. 4 sec. Radium Eman. 1300 yr. 4 dy. Rad. A. Rad. B. 3 mill. . 21 min. Rad. C. Rad. D. Rad. E. Rad. F. active deposit, rapid change EIG. 248 28 min. 40 yr. Radio-Lead 6 dy. 143 dy. Polonium active deposit, slow change however, that radium is itself only a transformation product of uranium, being produced not by the immediate disintegration of uranium X, but by some rayless change in an intermediate product called ionium into which the uranium X immediately dis- integrates. It seems probable, too, that actinium is also a disinte- gration product of uranium. If this should turn out to be the case, 350 ELECTRICITY, SOUND, AND LIGHT there would be but two strongly radio-active families, namely the uranium family and the thorium family. It is worthy of remark that the atoms of uranium and thorium are the heaviest atoms known, their atomic weights being 239 and 232 respectively. These atoms, then, seem to be continually disintegrating into atoms of lower atomic weight, the process of disintegration apparently consisting in the expulsion of an alpha or a beta particle, or of both together. The residue of the atom after each such expulsion appears to have new physical and chemical properties, that is, to be a new chemical substance. This new substance is itself in geii- o eral unstable, and after a time passes over into something else, with the expulsion of other particles. What are the ultimate products of this series of radio-active changes is. not yet definitely known. It is certain that helium is one of them, for the spectrum of helium has been found to grow out of the emanation of radium. Rutherford -regards the alpha particles as themselves helium atoms. It is not impossible that lead and some other common elements are products of this disintegration. At any rate, many of these elements are regularly found in uranium ores. J. J. Thomson has recently (1907) shown that the same alpha particle which is pro- jected from radio-active substances is thrown off from all sorts of substances in a highly exhausted tube under the influence of very intense electrical fields. If any of these common substances are spontaneously emitting these alpha particles at a very slow rate, and with velocities too small to ionize the surrounding air, we should have no means of detecting the fact. Whether, then, the continuous transmutation of one element into another is confined to a few radio-active substances, or whether it is a general phe- nomenon of nature, is a question which must be left for the future to decide. EXPERIMENT 28 (A) Object. To compare the radio-activities of black uranium oxide, pitchblende, thorium nitrate, and uranium nitrate. Directions. With a mortar and pestle reduce to a fluffy powder each of the substances to be compared. Weigh out, say, 4 g. of each, and spread uniformly over a metal surface of about 25 sq. cm. This may be done either by pressing down the powder with some flat object in a shallow RADIO-ACTIVITY 351 vessel, of, say, 3 mm. depth, until the surface is smooth and the thickness uniform, or by making an emulsion, in a small test tube, of the powder in chloroform, alcohol, or some other liquid in which the powder will not dissolve, and then pouring the emulsion very quickly into the shallow vessel and setting away until the liquid has evaporated. Charge the electroscope (Fig. 249) by drawing a charged rod of sealing wax over the projecting end e of the metal rod which supports the gold leaf, and which is insulated from the metal frame E of the electroscope by means of an amber plug p. When a deflection of from 30 to 45 has been produced, focus the telescope, or microscope, upon the gold leaf and set the eyepiece so that the leaf in discharging moves as nearly as possible at right angles to the scale in the eyepiece. Insert the uranium oxide into the electroscope through one of the glass sides, and earth the metal case E by running a small wire from it to a gas or water pipe. Place the cylindrical metal tube b over e so as to cut off all outside inductive effects (see p. 12), and make two or three observations with a stop watch of the time required for the leaf to pass over a given five divisions of the scale in the eyepiece, recharging the leaf with the sealing wax between each set of readings. Replace the uranium by the other sub- stances to be compared, being careful to place them all in just the same position, and make similar sets of readings for each. Express the activities of all the substances in terms of that of the black uranium oxide. (B) Object. To determine the number of ions produced per second in a gas by the radiations from a uranium oxide film of a given weight per square centimeter. Directions. Replace the uranium oxide film in the electroscope, but this time screw upon e the solid brass cylinder a, and set over this, as coaxially as possible, the brass tube b. This operation places a condenser in parallel with the electroscope, so that the capacity of the system is greatly increased. Since the uranium produces the same number of ions per second as before, the same charge will be taken per second from the electroscope, but its capacity being now much increased, the rate of fall of the leaves will be FIG. 249 352 ELECTRICITY, SOUND, AND LIGHT much reduced. Let t l and / 2 represent the two times of fall of the gold leaf between the given limits before and after the condenser is added ; let PD represent the fall in potential corresponding to the observed fall of the leaf ; let c be the capacity of the electroscope alone, and C the capacity of the condenser ; let Q t and Q 2 be the charges removed from the electroscope in the times t 1 and t 2 respectively. Then = 1 But by Chapter VIII, Q 1 = PD x c, and Q 2 = PD x (c+C). Hence This gives the capacity of the electroscope in terms of t v and 2 , and the capacity C of the condenser. This last quantity may be computed from the mean area of the inner and outer cylindrical surfaces and their distance apart* (see eq. 17, p. 109). If now we can ob- tain the P.D. corresponding to the observed change in deflection of the gold leaf, we have at once the quantity of electricity passing per second to the gold-leaf system. Calling this quantity i, we have . = C 1 = P5xo ' To obtain PD in volts, attach the electro- scope in parallel with a Braun electrometer (Fig. 250), or other electrostatic voltmeter, and charging both by means of the sealing wax, find by direct comparison the P.D. cor- responding to the observed change in deflec- tion of the gold leaf. Divide this by 300 (i.e. j , to reduce it to absolute elec- FIG. 250 trostatic units. Express this current also in electro-magnetic units by dividing by 3 x 10 10 . The charge on each ion of an ionized gas is known to be the same as the charge on the hydrogen ion * The rigorous formula for a cylindrical condenser of this kind is C = > in which I is the length of the inner cylinder, a the external diameter of this cylinder, and b the internal diameter of the outer tube. This is not appreciably different from the above unless a and 6 differ considerably. EADIO-ACTIVITY 353 in electrolysis, namely about 4 x 1Q- 10 absolute electrostatic units. Hence the number of positive ions produced per second by a square centimeter of the uranium film is simply - J 4 xlO- 10 EXAMPLE (A) When 4 g. each of uranium oxide, pitchblende, uranium nitrate, and thorium nitrate were spread uniformly over circular vessels of 5.5 cm. diameter, and having rims 3 mm. in height, the observations on the times required for the leaf to fall from division 5 to division 10 were : 1st 2d 3d Mean Uranium oxide .... 19.6 19.8 19.4 19.6 Pitchblende ..... 6.4 6.4 6.6 6.5 Uranium nitrate .... 54.6 54.8 54.6 54.7 Thorium nitrate .... 16.4 16.6 16.8 16.6 In terms of the uranium oxide the activity of the pitchblende was therefore 3.17, that of the uranium nitrate .36, and that of the thorium nitrate 1.18. (B) When a cylindrical .condenser 5.89 cm. long, the diameter of whose inner and outer cylindrical surfaces were 1.594 cm. and 1.90 cm. respectively, was added, the mean time of fall of the leaf through 5 divisions when the uranium oxide was in the electroscope was 123.4 seconds. The capacity C was found to be 16.8 absolute electrostatic units. Hence c = 3.19 electro- static units. The loss of PD due to fall of the gold leaf through the given 5 divisions was 68 volts. The uranium oxide was spread over a circular area . Q PDxc 68 x 3.19 ft having a diameter ot o.o cm. Hence i - - -- = .Ooo9 * < 300 x 19,6 electrostatic units* and i per sq. cm. = - - = .00155 electrostatic units AO *Q = 5. 17 x 10~ 14 electro-magnetic units. Therefore the number of ions formed per second by the radiation from 1 sq. cm. = 3,900,000. PROBLEMS CHAPTERS I AND II 1. Find the intensity of the magnetic field due to an isolated south pole of 560 units strength at a point 20 cm. from the pole. If the value of //, the horizontal component of the earth's field is 0.24 in this neighborhood, what will be the resultant field intensity and its direction if the point con- sidered is due magnetic north of the pole ? 2. A magnet has a length of 10 cm. and a pole strength of 450 units. Calculate the direction of the field with respect to the axis of the magnet, and the intensity at a point 20 cm. from one pole and 15 cm. from the other. 3. What is the field intensity due to a magnet of moment M at a point distant r cm. along a perpendicular to the magnet at its middle point? 4. If a magnet vibrates with a period of 5 sec. where the horizontal intensity is 0.18, what is its period where the intensity is 0.24? 5. A very short magnetic needle is suspended 17.32 cm. below the cen- ter of a bar magnet of pole strength 200 units. The length of the bar magnet is 20 cm. between poles. Its axis (S to N) makes an angle of 90 with the direction of the horizontal component (H = 0.24) of the earth's field. What angle does the axis of the magnetic needle make with H when in equilibrium ? 6. If the period of oscillation of the magnetic needle in Problem 5 is 3 sec., what is it when the bar magnet has been rotated through 90? 7. A magnetic needle is suspended above and parallel to a horizontal bar magnet lying in the magnetic meridian. When the north end of the bar points northward the period of the magnet is 7 sec., but when the bar magnet is reversed the period is 5 sec. What would be the period in the earth's field alone ? 8. A bar magnet suspended to rotate about a vertical axis has a period of 4 sec. in the earth's field. Find the period of the system if a bar of aluminum is attached to the magnet so as to rotate with it about the same axis, the aluminum bar having one half the length and one half the width of the magnet, but the same vertical thickness. The density of steel is 7.7 and of aluminum 2.7. Find also the value of the moment of inertia of the bar if the magnetic moment is 1200 units and H is 0.22. 354 PROBLEMS 355 CHAPTER III 9. What current must be passed through a tangent galvanometer coil 45 cm. in diameter and consisting of 5 turns, in order that the needle may be deflected 30 in a neighborhood where H= 0.18? What current for 45 and for 60? 10. What must be the radius of a single coil which, when carrying a given current, would produce at its center a field intensity equal to that caused by the same current in passing through two concentric coaxial coils of 15 cm. and 45 cm. diameters ? 11. A circular wire coil of 15 turns and 50 cm. diameter must carry how much current to produce at its center a field of 0.60? 12. A current of 0.331 ampere is passed through two concentric, coaxial coils of 12 and 24 turns respectively, first when connected so that their magnetic fields are in opposite directions, and second so that they are in the same direction. The fields thus established are 0.20 and 0.30 respectively. Find the radii of the two coils. 13. Draw the curve representing the values of the tangent for all angles between and 90. Decide from a study of the curve for what value of the deflection in a tangent galvanometer the smallest error is introduced into the final result by a constant observational error in reading the deflection. 14. The reading of a tangent galvanometer was 46 unconnected for tor- sion. When the torsion head was twisted through that angle, the needle turned 1.1 against the restoring force of the earth's field. Find the per cent of error introduced into the final result by using the uncorrected instead of the corrected value of the deflection. 15. A current which gives a reading of 0.27 ampere on a milliammeter deposits 0.2008 g. of silver in 10 min. 42 sec. What is the error in the ammeter reading ? 16. How much water should be decomposed by a current of 0.80 ampere in one hour? 17. Given the electro-chemical equivalent of zinc as 0.03367 g. per cou- lomb, find how much zinc is consumed in a Daniell cell in generating a current of 0.5 ampere for 150 hr. How much copper is deposited under the conditions of the problem ? 18. A current is sent through three cells, one containing acidulated water, the second copper sulphate, and the third silver nitrate. How much copper will have been deposited in the second cell while 2 g. of silver are deposited in the third cell? What will be the volume, at 76 cm. of mercury pressure and 20 C., of the mixed gases liberated in the first cell? 356 ELECTRICITY, SOUND, AND LIGHT 19. How many ounces of aluminum are deposited from a suitable solu- tion of bauxite, a natural aluminum oxide, by the passage of 150 amperes for one hour? (Aluminum oxide = A1 2 O 3 : atomic weight of Al = 27.4.) 20. Hydrogen given off at an electrode of a hydrogen voltameter dis- places entirely the volume of 40 cc. of water contained by the vessel placed over the electrode. The atmospheric pressure is 747 mm. of mercury at the room temperature of 23 C. Find the weight of the liberated gas. 21. The atomic weights of three elements are in, n, and p. They all form sulphates of the form M 2 SO 4 . What is the ratio of their electro- chemical equivalents ? 22. An element which forms oxides of the form E 2 O 3 and EO has an electro-chemical equivalent of e for the compound of higher valency. What is the electro-chemical equivalent for the lower valency ? CHAPTERS IV AND V 23. When electrical energy costs 8 cents per kilowatt hour, how much does it cost to operate an incandescent lamp that takes 0.5 ampere from 110 volt mains ? 24. A vessel of 30 g. water equivalent contains 1600 g. of water. Its radiation constant is such that its temperature falls at the rate of 6 C. per minute at 90 C. How much current must be passed through a wire of 10 ohms' resistance immersed in the water that the temperature shall be maintained at 90 C. ? 25. From Experiment 4 it is seen that the rate at which heat is devel- oped in a wire is (PD)L Show that this is also equal to RI Z . 26. A wire immersed in water generates heat at the rate of 2 calories per second when carrying 0.5 ampere of current. Find the power in watts and in horse power expended in the wire and its resistance. 27. Show from Ohm's law that in two wires of the same circuit the ratio of the P.D.'s between the terminals of the wires is the same as the ratio of their resistances. 28. Neglecting the loss due to radiation, what will be the rise in tem- perature in 10 sec. of a No. 20 copper wire of mass 45.5 g. and resistance 0.3293 ohm, connected across constant potential mains of 25 volts P.D.? 29. If a wire of the same mass as that in Problem 28, but of half the diameter, is supplied with current from the same mains, what will be its rise in temperature in 10 sec.? 30. If the wire is of the same diameter as that in Problem 28, but of half the length, and hence half the mass, what will be its rise in 10 sec.? PROBLEMS 357 FIG. 1 31. If equal lengths of wires of different diameters but of the same material are placed across constant potential mains, which will burn out the sooner, the small wire or the large one? In answering this question consider radiation. 32. If the ammeter in Figure 1 reads 0.5 am- pere, the voltmeter 50 volts, and if the resist- ance of the voltmeter is 300 ohms, what is the resistance of the coil R ? 33. Three coils of re- sistance 2, 3, and 5 ohms are connected in series across 110 volt mains. Find the current through the circuit, the P.D. across the terminals of each coil, and the power expended in each coil. 34. A 2- and a 7 -ohm coil are part of a circuit in which 18 amperes are flowing. Calculate the P.D. across the terminals of this parallel circuit, the combined resistance of the two coils, and the current in each coil. 35. A room is lighted by a chandelier containing 11 incandescent lamps in parallel between wires of resistance negligible as compared to the lamp resistances. Each lamp has a resistance of 220 ohms. What is the com- bined resistance which they offer '? 36. A P.D. of 1.06 volts is maintained at the terminals of a battery E in Fig- ure 2. If resistances of 10,000, 1100, 900, and 1 ohms are connected at a, ft, c, and d respectively, what is the current through the resistance c ? 37. Given the resistance of the volt- meter used in Experiment 4 as 16,000 ohms, and the resistance of the coil of platinum wire as 13.4 ohms, find the per cent of error introduced in assuming that all the current indicated by an ammeter reading of 4.08 amperes passes through the coil. FIG. 2 CHAPTERS VI AND VII 38. One mil is a thousandth of an inch. One circular mil is the area of a circle one mil in diameter. The area of a circle d mils in diameter is therefore d 2 circular mils. The resistance of pure soft copper such as is used in making insulated wires is given as 10.38 ohms per mil-foot at 75 F. What in C.G.S. units is the specific resistance of such copper? 358 ELECTRICITY, SOUND, AND LIGHT 39. What is the resistance at 75 F. of 1000 feet of wire 31.96 mils in diameter ? 40. The resistance of aluminum wire is given as 17.03 ohms per mil- foot at 75 F. What is the ratio of the diameters of two wires, one cop- per and the other aluminum, such that for the same length the resistances are the same ? 41. The specific resistance of copper at 0C. is 1650 absolute C.G.S. units. Find the resistance of a trolley wire one kilometer long and one centimeter in diameter. 42. If in Figure 62, page 82, J R 1 = 10,000, R z = 100, R 3 = 100, G = 100, and V 1.06, what current was flowing through (7? 43. The mean temperature coefficient of copper is not exactly 0.0042. The mean rate of increase of resistance for Matthiessen's pure copper between and lC. is best expressed as a = (a + bt + c/ 2 ) where t is the temperature entering into the relation R t -K (l -f af). The constants a and b have the values 0.004019 and 0.00000214 respectively, for values of t between and 100. Find the value of a to use at 25 C. and at 50 C. 44. Compute the error introduced in finding the value of a resistance at 50 from an observation of its resistance at 25 by using the average value a 0.0042 instead of the value of a as given in Problem 43. 45. What is the rise in temperature of the field coil of a dynamo which at the beginning of a run had a temperature of 25 C. and a resistance of 290 ohms if its final resistance is 346 ohms ? Use a = 0.0042. 46. The P.D. between the two wires of the circuit shown in Figure 3 is 2 volts greater at a than at b. A current of 10 amperes is flowing in the circuit. What is the resistance of the line ? FIG. 3 47. If it is desired to transmit power so that 30 amperes may be deliv- ered at the end of a line at a P.D. of 110 volts, what must be the resist- ance of the line in order that the "line drop" shall be 2 per cent of the voltage at the transmitting end ? What power is expended in the transmission ? 48. A group of incandescent lamps takes 15 amperes. The line loss is not to exceed 2 volts. What must be the size of the copper wire to be used if the lamps are 2000 feet from the transmitting end of the line ? (Given No. 5 wire, 182 mils diameter, and No. 6 wire, 162 mils diameter.) How many watts are lost in the line ? PROBLEMS 359 49. No. 20 German silver wire is to be used in constructing a coil of resistance 5 ohms at 20 C. If the diameter of the wire is 31.96 mils and the specific resistance of the specimen is 0.00002076 per centimeter cube at C., how many centimeters of the wire will be necessary ? (a = 0.0004.) 50. The average temperature coefficient of platinum is 0.00366. What is the temperature of a furnace in which the coil of a platinum thermom- eter has a resistance of 1020 ohms? The resistance at 0C. is 300 ohms. 51. The resistance of the platinum thermometer coil of Problem 50 when placed in a bath of liquid air is 98- ohms. What is the temperature of the bath ? 52. What would be the resistance of the coil at the limiting tempera- ture of the absolute ;:ero ? 53. A standard resistance of 0.0001 ohm is connected in a circuit, and a millivoltmeter connected to the terminals of the standard resistance indicates a P.D. of 0.137 volt. Neglecting the resistance of the milli- voltmeter, what is the current flowing in the circuit? 54. A universal galvanometer shunt has the form shown in Figure 4. The conductor k attached to the galvanometer G may be connected to the points b, c, or d. What must be the ratio of .the resist- ances ab,bc, and cd in order that the ratio -. -^ of the deflections of 1 h the galvanometer for connections at 6, c, and d respectively shall be as 1:10: 100? (Consider that the resistance ad is small compared to the gal- vanometer resistance G.) 55. A galvanom- eter having a resist- ance of 225 ohms is shunted by a resistance of 25 ohms. Compare the values of the current through the galvanometer with and without the shunt. 56. A milliamrneter reading to 0.100 ampere has a resistance of 5 ohms. It is desired to use this instrument to measure current of from to 10 am- peres. What must be the resistance of a shunt such that the instrument may be read directly in amperes by applying a multiplier of 100 to its reading? FIG. 360 ELECTRICITY, SOUND, AND LIGHT 57. Two voltmeters of resistance 20,000 and 16,000 ohms respectively are connected in series across 220 volt mains. What current flows through the system, and what is the reading of each voltmeter? 58. If the voltmeters of Problem 57 are connected in parallel across 220 volt mains, what current flows through each, and what are the readings? 59. A millivoltmeter has a resistance of 20 ohms. What resistance must be connected in series with it in order that it may be used to read volts directly by multiplying its readings by 100? CHAPTERS VIII AND IX 60. A condenser is made of two flat metal plates separated by air. If the area of the plates is 500 sq. cm., and the average distance between them is 0.01 cm., what is the capacity of the condenser in electrostatic and in electro-magnetic units ? 61. If a plate condenser similar to that described in Problem -60 has a capacity of 0.002 microfarad, and another condenser with glass for a dielectric has twice the area of plates and four times the distance of sep- aration, and has a capacity of 0.0046 microfarad, what is the dielectric constant of the glass used ? 62. What quantity of electricity would charge each condenser to 110 volts? 63. What is the ratio of the resistance of a conductor as measured in the E.S. and the E.M. systems respectively? 64. Show from a consideration of the equation C = - that when there is a given charge upon the plates of a condenser, the value of the field be- tween the plates is inversely proportional to the dielectric constant of the medium which fills the space between the plates. 65. The charge carried by the hydrogen ion in electrolysis is about 4 x 10~ 10 absolute electrostatic units. Express this charge in electro- magnetic units. 66. An insulated gold-leaf electroscope shows a charge which corresponds to a potential of 450 volts. When an insulated sphere, the capacity of which is 15 electrostatic units, is connected with the gold leaf, the poten- tial falls to the point corresponding to 350 volts. Find the capacity of the gold-leaf electroscope. 67. When the terminals of a dynamo are applied directly to the termi- nals of an electrostatic voltmeter, the P.D. indicated is 500 volts. An oil condenser consisting of plates 25 sq. cm. in area and 1 mm. apart is placed in series with the electroscope. When the terminals of the same dynamo PKOBLEMS 361 are applied to the extremities of this condenser combination, the voltmeter indicates 50 volts. If the dielectric constant of the oil is 4, what is the capacity of the electrostatic voltmeter in microfarads? in electrostatic units ? 68. If a i microfarad condenser, charged by means of a 2-volt cell, pro- duces a throw of 10 cm. in a ballistic galvanometer, what number of microcoulombs passed through the galvanometer when it was thrown 6 cm. by the earth inductor? CHAPTERS X AND XI 69. A battery of E.M.F. 1.07 volts and internal resistance 1.8 ohms is connected to a coil of resistance 6 ohms. What is the P.D. at the battery terminals, and what current flows in the circuit? 70. A tangent galvanometer of resistance 1.5 ohms is deflected 54 when connected to a battery. The deflection due to the same battery is only 42 when an additional resistance of 2 ohms is connected in the circuit. What is the resistance of the battery ? 71. A galvanometer of 20 ohms resistance gives a deflection of 5 when connected to a battery of E.M.F. 1.08 volts and 2 ohms internal resist- ance. When connected to a second battery it gives a deflection of 8. When shunted by a wire of 20 ohms resistance and connected to this second battery it gives a deflection of 7.5. Find the E.M.F. and internal resistance of the second battery. 72. Six Daniell cells, each having an E.M.F. of 1.08 volts and an internal resistance of 2 ohms, are to be connected to a circuit of 6 ohms resistance by wires of negligible resistance. What is the current through the external circuit (a) when all the cells are in series ? (b) when all the cells are in parallel ? (c) when they are arranged in three groups in parallel, each group containing two cells in series ? (d) when arranged in two groups in parallel, each containing three cells in series? What is the P.D. at the terminals of the battery thus formed for each of the above arrangements ? 73. If battery cells of E.M.F. 2.16 volts and internal resistance 1 ohm each are connected as in Problem 72, for which arrangement is the current in the external circuit a maximum, and what is its value? 74. A 15 per cent solution of copper sulphate has a conductivity of 39 x 10~ 12 in C.G.S. units. Calculate the current due to a P.D. of 2 volts across the terminals of an electrolytic cell of this solution if the electrodes are flat plates 20 x 30 cm. and 3 cm. apart. 75. The resistance of an electrolytic cell similar to that of Problem 74 is 1.94 ohms at 18 C. and 0.80 at 100 C. Find the per cent increase in conductivity per degree. 362 ELECTRICITY, SOUND, AND LIGHT 76. Three Daniell cells of E.M.F. 1.1 volts and internal resistance 1.4 ohms each are connected in series with a storage battery of unknown internal resistance by leads of 0.3 ohm. The current observed is 1.17 amperes. The storage battery terminals are then reversed and a current of 0.26 ampere is observed to now. What is the E.M.F. of the storage battery, and what P.D. would a voltmeter across its terminals indicate for each connection ? CHAPTERS XII AND XIII 77. The field intensity in the gap space between the poles and the armature of a dynamo is 5000 units. Wires 25.5 cm. in length and 11 cm. from the axis of the armature are imbedded in the iron of the armature core. The armature makes 1400 revolutions per minute. 'Find the E.M.F. in volts induced in each armature conductor as it passes under a pole face. 78. A copper disk 20 cm. in diameter rotates 30 times per second in a uniform field of intensity 700 units. What E.M.F. is induced between the center and the circumference of the disk ? 79. A coil of wire of 30 turns and average area 15 sq. cm., carrying a current of 2 amperes, is suspended from the beam of a sensitive balance so that the plane of the coil is horizontal. If the intensity of the earth's magnetic field is 0.57 and the angle of dip is 63, what is the difference in weight of the coil produced by reversing the direction of the current passing through it ? 80. A horizontal wire 90 cm. long lies in a northeast direction. How many lines of force does it cut if moved vertically upward for 200 cm.? The value of H is 0.26 and the value of V is 0.51. CHAPTERS XIV AND XV 81. The core of an induction coil carries 100,000 lines of magnetic flux when a given current flows in the primary of the coil. Allowing 0.004 sec. for the flux to decrease to 10,000 lines after the primary current has been interrupted, how many turns of wire are necessary on the second- ary of the coil in order that an average E.M.F. of 20,000 volts may be induced ? 82. A coil of inductance 0.025 henry and 1600 turns of wire has how many lines of force per average turn for a current of 35 amperes ? 83. An iron ring of 2.5 sq.cm. cross section and 20 cm. circumference has a primary of 80 turns and a secondary of 4600 turns. If the permea- bility of the iron is 1600 for a current of 1 ampere, find the E.M.F. induced in the secondary by reversing the primary current in 0.08 sec. PROBLEMS 363 84. An iron ring of 0.75 cm. radius has a test coil wound on it of 50 turns of wire, which is connected to a ballistic galvanometer. The ballistic galvanometer circuit has a resistance of 300 ohms, and the galvanometer throws 9 scale divisions when 0.000072 coulomb passes through it. What throw is caused when the induction in the iron ring changes suddenly from 2400 lines in one direction to 3600 in the opposite direction? CHAPTERS XVII AND XVIII 85. Amagat found that a volume of a cubic centimeter of alcohol at 14 C. was decreased by .000101 cc. for each atmosphere increase in the pressure to which it was subjected. If the density of alcohol is 0.79, what is the velocity of sound in this medium? 86. Assuming that the aural impression of a sound persists for 0.1 sec., calculate the distance of a person speaking 4 syllables per second from a reflecting surface in order that the echo may be distinct. Temperature 20 C . 87. Find the change in pitch observed by a person standing on a rail- way platform from which a locomotive with whistle blowing is receding at the uniform rate of 30 miles per hour. 88. Notes of 225 and 336 vibrations per second are sounded simulta- neously. If the even overtones only are present in the first note, and both odd and even overtones in the second note, how many beats per second will occur, and to what overtones will they be due ? 89. Two musicians stationed some distance apart are playing slightly out of tune, so that 4 beats per second are noticeable. How' fast must a person travel from one toward the other in order that no beats are notice- able? Calculate for notes of 256 and 384 vibrations per second at a tem- perature of C. 90. If the velocity of a compressional wave in a gas is 320 m. per second at 20 C., what will be the velocity at 50 and twice the pressure? 91. How much must an organ pipe be heated from 0C. in order that the note may be changed by a semitone ? 92. A whistle blown normally with air is blown with hydrogen of den- sity 0.0692 as compared to air at the same temperature and pressure. What change is thus produced, and what is the interval between the notes ? CHAPTERS XIX AND XX 93. A brass rod 2 m. long stroked longitudinally is in tune with a 25-cm. length of a given sonometer wire. A steel rod 3 m. long stroked longitudinally is in tune with a 26-cm. length of the same sonometer wire. Find the relative velocities of sound in steel and brass. 364 ELECTKICITY, SOUND, AND LIGHT 94. The e string of a violin has a mass of 0.125 g. and a length of 33 cm. What is the tension in kilograms weight which it exerts when tuned to 640 vibrations? If the g, d, and a strings were to be made out of the same material and stretched by the same force, what would be the ratio of their diameters to that of the e string? CHAPTERS XXI AND XXII 95. The theoretical limit of resolution of a lens with a circular aperture is 1.22 times the wave length of light divided by the diameter of the aperture. Mizar, the larger of the two stars at the bend of the handle of the Great Dipper, is a double star. Its two components are separated by 14.5" of arc. What is the smallest aperture of telescope that can resolve this doublet? 96. If the distance between the images of the third order produced by a transmission grating placed 3 in. from a sodium flame is 40cm., what is the number of lines per centimeter in the grating? 97. The two D lines (sodium) are separated by an angle of 53" of arc in the first-order spectrum of a plane diffraction grating. What is the grating space ? CHAPTERS XXIII AND XXIV 98. If the absolute index of refraction of glass is 1.55 and of water 1.33, what is the minimum deviation produced in a beam of light by a prism of glass immersed in water? The angle of the prism is 60. 99. The deviation produced by a prism with a small angle is commonly written D = (/JL 1)^4. Justify this equation. 100. When a layer of liquid 5 cm. deep is placed over a dot on a glass plate, the position of the dot as found by changing the focus of a micro- scope is 1.45 cm. above the plate. What is the index of refraction of the liquid? 101. From the law "angle of incidence equals angle of reflection" deduce the fact that a rotating mirror turns through one half the angle through which the reflected ray is rotated. 102. What is the focal length of a lens which has conjugate focal lengths of 25 cm. and 35 cm.? 103. What is the length of the real image of a line 4 cm. long and distant 40 cm. from a lens of focal length 25 cm.? 104. If a concave mirror has a curvature of 0.05, what is its focal length? How far from the mirror will be formed the real image of an object distant 35 cm. from the mirror? What will be its size relative to that of the object ? PROBLEMS 365 105. The lens L forms at p' a real image of p (Fig. 5). When L is moved to L' it again forms a real image of /rat//. If LL' is 60 cm. and pp' is 110 cm., what is the focal P I\ L IV PI length of lens used ? FIG. 106. Parallel rays from the sun passing through the open- ings a and b (Fig. 6) in a screen placed before a concave lens L illuminate a second screen at c and d. The distance of this second screen from the lens is 15 cm. The distances ab and cd are 3 cm. and 7.5 cm. respectively. Find the focal length of the lens. 107. A combination of a doable concave lens and a more powerful convex lens of focal length 15 cm. has a focal length of 80 cm. Find the focal length of the con- cave lens. 108. An object A is placed 50 cm. from a concave lens L of unknown focal length YIG. (Fig. 7). A concave mirror M placed upon the opposite side of the lens forms a real image of the object at B, at a distance of 45 cm. from the mirror. The focal length of the mirror is 30 cm., and the distance from the mirror to the lens is 65 cm. Find the focal length of the lens. FIG. FIG. 8 109. Figure 8 shows the relative position of an object A, nearer to the lens L than its principal focus F, and its virtual image B. If A is one third the focal length from F, what are the relative sizes of A and B? CHAPTERS XXV AND XXVI 110. At what distance from a photometer must a Hefner lamp be placed in order that it may produce the same intensity of illumination as & stand- ard candle distant 85 cm. from the photometer? 111. Neglecting the motion of the earth, find an expression for the ap- parent wave length X' of light of wave length A from a distant star moving toward the earth with a velocity of s. Hence write an expression for the 366 ELECTRICITY, SOUND, AND LIGHT velocity of the star in the line of sight. From this equation find the velocity in the line of sight of a mass of hydrogen in the neighborhood of a sun spot that gave in the spectrometer an apparent wave length of 6566 x 10~ 8 instead of 6563 x 10~ 8 , which corresponded for a stationary source to the dark band observed in the solar spectrum. (See Young, " Manual of Astronomy," pp. 231-233.) 112. If to an observer on the earth, who is by the earth's motion moving through space at the rate of 18.5 miles per second, light from a distant star seems to come in a direction making an angle of 20.5" with the true direction of the star, find the velocity of light in miles per hour. Sin 20. 5" = 0.005842, cos 20. 5" = 0.999987. (See Young, "Manual of Astronomy," p. 150.) CHAPTERS XXVII AND XXVIII 113. At what angle must a beam of light be incident upon a smooth water surface in order that the reflected beam shall be plane polarized ? 114. How thick should be a piece of quartz with its parallel faces cut perpendicular in order that plane polarized light incident upon it shall, after transmission, have the same plane of vibration as before ? 115. The index of refraction of the ordinary ray in Iceland spar is 1.6543. The index of Canada balsam is 1.536. At what angle must the ordinary ray in a Nicol prism meet the interface in order that it may just suffer total reflection? 116. If an a particle projected from uranium produces on the average 100,000 positive ions, how many a particles are required to produce the ionization current given in the example at the end of Experiment 28? TABLES Table 1 SATURATED WATER VAPOR Showing pressure P (in mm. of mercury) and density D of aqueous vapor saturated at temperature t; or showing boiling point t of water and density D of steam cor- responding to an outside pressure P. t P D t P D t p D 10 2.2 2.3X10- 6 30 31.5 30.1X10- 6 88.5 496.2 9 2.3 2.5 " 35 41.8 39.3 " 89 505.8 8 2.5 2.7 " 40 54.9 50.9 " 89.5 515.5 7 2.7 2.9 " 45 71:4 65.3 " 90 525.4 428.4X10- 6 6 2.9 3.2 " 50 92.0 83.0 " 90.5 535.5 5 3.2 3.4 " 55 117.5 104.6 " 91 545.7 - 4 3.4 3.7 " 60 ' 148.8 130.7 " 91.5 556.1 3 3.7 4.0 " 65 187.0 162.1 " 92 566.7 2 3.9 4.2 " 70 233.1 199.5 " 92.5 577.4 1 4.2 4.5 " 71 243.6 93 588.3 4.6 4.9 " 72 254.3 93.5 599.6 1 4.9 5.2 " 73 265.4 94 610.6 2 5.3 5.6 " 74 276.9 94.5 622.0 3 5.7 6.0 " 75 288.8 243.7 " 95 633.6 511.1 " 4 6.1 6.4 " 75.5 294.9 95.5 645.4 5 6.5 6.8 " 76 301.1 96 657.4 6 7.0 7.3 " 76.5 307.4 96.5 669.5 7 7.5 7.7 " 77 313.8 97 681.8 8 8.0 8.2 " 77.5 320.4 97.5 694.2 9 8.5 8.7 " 78 327.1 98 707.1 10 9.1 9.3 " 78.5 333.8 98.2 712.3 11 9.8 10.0 " 79 340.7 98.4 717.4 12 10.4 10.6 " 79.5 347.7 98.6 722.6 13 11.1 11.2 " 80 354.9 295.9 " 98.8 727.9 14 11.9 12.0 " 80.5 362.1 99 733.2 15 12.7 12.8 " 81 369.5 99.2 738.5 16 13.5 13.5 " 81.5 377.0 99.4 743.8 17 14.4 14.4 " 82 384.6 99.6 749.2 48 15.3 15.2 " 82.5 392.4 99.8 754.7 19 16.3 16.2 " 83 400.3 100 760.0 606.2 " 20 17.4 17.2 " 83.5 408.3 100.2 765.5 21 18.5 18.2 " 84 416.5 100.4 771.0 22 19.6 19.3 " 84.5 424.7 100.6 776.5 23 20.9 20.4 " 85 433.2 357.1 " 100.8 782.1 24 22.2 21.6 " 85.5 441.7 101 787.7 25 23.5 22.9 " 86 450.5 102 816.0 26 25.0 24.2 " 86.5 459.3 103 845.3 27 26.5 25.6 " 87 468.3 105 906.4 715.4 " 28 28.1 27.0 " 87.5 477.4 107 971.1 29 29.7 28.5 " 88 486.8 110 1075.4 840.1 " 368 ELECTBICITY, SOUND, AND LIGHT Table 2 DENSITY OF DRY AIR AT TEMPERATURE t AND PRESSURE If mm. OF MERCURY t "=720 730 740 750 760 770 DIFFERENCE PER mm. 10 .001181 .001198 .001214 .001231 .001247 .001263 16 11 1177 1194 1210 1226 1243 1259 1 2 12 1173 1189 1206 1222 1238 1255 2 3 13 1169 1185 1202 1218 1234 1250 3 5 14 1165 1181 1197 1214 1230 1246 4 6 15 .001161 .001177 .001193 .001209 .001225 .001242 5 8 16 1157 1173 1189 1205 1221 1237 6 10 17 1153 1169 1185 1201 1217 1233 7 11 18 1149 1165 1181 1197 1213 1229 8 13 19 1145 1161 1177 1193 1209 1224 9 14 20 .001141 .001157 .001173 .001189 .001204 .001220 15 21 1137 1153 1169 1185 1200 1216 1 2 22 1133 1149 1165 1181 1196 1212 2 3 23 1130 1145 1161 1177 1192 1208 3 4 24 1126 1141 1157 1173 1188 1204 4 6 25 .001122 .001138 .001153 .001169 .001184 .001200 5 7 26 1118 1134 1149 1165 1180 1196 6 9 27 1114 1130 1145 1161 1176 1192 7 10 28 1110 1126 1142 1157 1172 1188 8 12 29 1107 1122 1138 1153 1169 1184 9 13 30 .001103 .001119 .001134 .001149 .001165 .001180 Correction for Moisture in Above Table Dew-point Subtract Dew-point Subtract Dew-point Subtract Dew-point Subtract -10 8 6 4 2 .000001 .000002 .000002 .000002 .000003 +2 +4 +6 -f8 .000003 .000003 .000004 .000004 .000005 +10 + 12 +14 + 16 +18 .000006 .000006 .000007 .000008 .000009 +20 +22 +24 +26 +28 .000010 .000012 .000013 .000015 .000016 Table 3 DENSITY OF WATER Table 4 DENSITY OF MERCURY TEMP. C DENSITY TEMP. C DENSITY TEMP. C DENSITY 0.999884 13 0.999443 13.596 1 0.999941 14 0.999312 10 13.572 2 0.999982 15 0.999173 12 13.567 3 1.000004 16 0.999015 14 13.562 3.95 1.000000 17 0.998854 16 13.557 4 1.000013 18 0.998667 18 13.552 5 1.000003 19 0.998473 20 13.547 6 0.999983 20 0.998272 22 13.542 7 0.999946 22 0.997839 24 13.537 8 0.999899 24 0.997380 26 13.532 9 0.999837 26 0.996879 28 13.528 10 0.999760 28 0.996344 30 13.523 11 0-999668 30 0.995778 32 13.518 12 0.999562 100 0.958860 34 13.513 TABLES 369 Table 5 DENSITIES AND ELECTRICAL CONDUCTIVITIES AT 18 C (FROM KOHLRAUSCH) C. OF NORMAL SOLUTIONS A = equivalent weight (O= 16.00) or concentration in gram equivalents per liter. p = number of grams of substance to 100 grams of the solution. d = density in grams per cubic centimeter at 18 C. 10 HC1 HN0 3 1H 2 S0 4 C 2 H 4 O 2 36.46 63.05 49.04 60 03 3.587 6. 107 4.758 5 959 1.0165 1.0325 1.0307 1 0074 300.0 299.0 197.0 .0159 .0150 .0120 Suffar 342 2 30 30 1 1294 Table 6 VISCOSITIES OF AQUEOUS SOLUTIONS OF VARYING CONCENTRATIONS If z is the viscosity of a dilute solution in terms of the viscosity of water, and x the concentration in gram molecules per liter, then, according to Arrhenius, A, therefore, represents the viscosity of a normal solution divided by the viscosity of pure water. The values in the following table refer to viscosities at 25 C. SUBSTANCE A SUBSTANCE A Alcohol {C 2 H 5 OH} Cane Sugar {Ci 2 H 22 O n } Copper Nitrate {Cu(NO 3 ) 2 } Copper Sulphate {CuSO 4 } Ether {(C 2 H 5 ) 2 0| Hydrochloric Acid {HC1} Nitric Acid |HNO 3 } Potassium Nitrate |KNO 3 } 1.030 1.046 1.1729 1.3533 1.026 1 0699 1.0233 0.9664 Potassium Sulphate {K 2 SO 4 } Sodium Acetate {Na 2 C 2 H 3 () 2 } Sodium Chloride {NaCl} Sodium Nitrate {NaNO 3 } Sodium Sulphate *Na 2 S0 4 } Zinc Nitrate {Zn(NO 3 ) 2 } Zinc Sulphate {ZnS0 4 | 1.0982 1.3998 1.0986 1.0522 1.2253 1.1666 1.3613 370 ELECTRICITY, SOUND, AND LIGHT Table 7 ATOMIC WEIGHTS (0 = 16.00) (International 1904) Aluminum . . . . . Al 27.1 Antimony .... Sb 120.2 Argon . . A 39.9 Arsenic . . As 75.0 Barium . . Ba 137.4 Bismuth .... . . Bi 208.5 Boron . . B 11. Bromine .... . . Br 79.96. Cadmium .... . . Cd 112.4 Caesium . . Cs 132.9 Calcium .... Ca 40.1 Carbon . . C 12.00 Cerium Ce 140.25 Chlorine .... . . Cl 35.45 Chromium . . . Cr 52.1 Cobalt Co 59.0 Columbium . . . . . Cb 94. Copper . . Cu 63.6 Erbium . . Er 166. Fluorine .... . . F 19. Gadolinium . . . . . Gd 156. Gallium .... . . Ga 70. Germanium . . . . . Ge 72.5 Glucinum .... . . Gl 9.1 Gold ...... . . Au 197.2 Helium . . He 4. Hydrogen .... H 1.008 Indium . In 114. Iodine . . I 126.85 Iridium Ir 193.0 Iron . . Fe 55.9 Krypton .... Kr 81.8 Lanthanum . . . . . La 138.9 Lead Pb 206.9 Lithium . . . . Li 7.03 Magnesium . . . . . Mg 24.36 Manganese . . . . . Mn 55.0 Mercury .... . . Hg 200.0 Molybdenum . . . . Mo 96.0 Neodymium ..... Nd 143.6 Neon Ne 20. Nickel . Ni 58.7 Nitrogen N 14.04 Osmium Os 191. Oxygen 16.00 Palladium Pd 106.5 Phosphorus P 31.0 Platinum Pt 194.8 Potassium K 39.15 Praseodymium . . . . Pr 140.5 Radium Rd 225. Rhodium Rh 103.0 Rubidium Rb 85.4 Ruthenium Ru 101.7 Samarium Sm 150. Scandium So 44.1 Selenium Se 79.2 Silicon Si 28.4 Silver Ag 107.93 Sodium Na 23.05 Strontium Sr 87.6 Sulphur S 32.06 Tantalum Ta 183. Tellurium ...... Te 127.6 Terbium Tb 160. Thallium Tl 204.1 Thorium Th 232.5 Thulium Tm 171. Tin Sn 119.0 Titanium Ti 48.1 Tungsten W 184.0 Uranium U 238.5 Vanadium V 51.2 Xenon Xe 128. Ytterbium Yb 173.0 Yttrium Yt 89.0 Zinc Zn 65.4 Zirconium . . Zr 90.6 TABLES 371 Table 8 ELECTRO-CHEMICAL EQUIVALENTS SUBSTANCE VALENCY GRAMS PER COULOMB SUBSTANCE VALENCY GRAMS PER COULOMB Aluminum . 3 .0009357 Magnesium . 2 .0001261 Bromine . . 1 .0008276 Manganese . 2 .0002849* Cadmium . 2 .000582* Mercury . . 1 .002075 Chlorine . . 1 .0003672 Mercury . . 2 .001037 Cobalt . . 2 .0003056* Nickel . . . 2 .0003042 Copper . . 1 .0006588 Nitrogen . . 3 .0000485* Copper . . 2 .0003294 Oxygen . . 2 .0000829 Fluorine 1 .0001968* Platinum . . 4 .000500* Gold . . . 3 .000681 Potassium . 1 .0004054 Hydrogen . 1 .0000104 Silver . . . 1 .001118 Iodine . . 1 .001314 Sodium . . 1 .0002387 Iron . . . 2 .0002895 Tin .... 2 .0006163 Iron . . . 3 .0001930 Tin .... 4 .0003081 Lead . . . 2 .001072 Zinc .... 2 .0003387 * Starred values are calculated. Table 9 SPECIFIC RESISTANCES AND TEMPERATURE COEFFICIENTS Metals SUBSTANCE RESISTANCE AT C. IN OHMS PER CENTIMETER CUBE MEAN TEMPERA- TURE COEFFICIENT 0-100C. Aluminum Antimony 2.906 x 10- 6 35 42 x 10~ 6 .00435 Bismuth . . 130 9 x 10- 6 Copper (annealed) 1 584 x 10- 6 0042 Copper (hard drawn) Iron (annealed) Iron (hard drawn) Gold . . 1.619 x lO- 6 9.693 x lO- 6 15. x 10- 6 2 088 x 10- 6 .00625 00377 Mercury Nickel Platinum Silver 94.34 x 10- 6 12.35 x 10- 6 9.035 x lO- 6 1 561 x 10~ G .0009 .00622 .00367 00400 Tin 10 5 x 10- 6 .00440 Zinc . 5 75 x 10- 6 .00406 German Silver Platinum Silver (Pt 33%, Ag 66%) . . Platinum Iridium (Pt 80%, Ir 20%) . . Platinum Rhodium (Pt 90%, Rh 10%) . Manganin (Cu 84%, Mn 12%, Ni 4%) . 20.89 x 10- 31.6 x 10- 6 30.9 x 10- 6 21.1 x 10- 6 46.7 x lO- 6 ; .00027 .00082 .00143 .0000 372 ELECTRICITY, SOUND, AND LIGHT Table 9 (continued) Electrolytes R = resistance in ohms per centimeter cube; =per cent decrease in resistance per degree Centigrade ; solutions given in per cent by weight of salts free from water of crys- tallization 5 fc 10 % 1 5% R a R a R KC1 14.5 2 7.34 1.9 4.95 1.9 NaCl .... 14.9 2.2 8.27 2.1 6.10 2.1 NH 4 C1 10.9 2. 5.62 1.9 3.86 1.7 CuSO 4 52.6 2.2 31.2 2.2 23.8 2.3 ZnS0 4 MgS0 4 AgN0 3 KOH . . 52.6 38.4 38.5 5.81 2.2 2.3 2.2 1.9 31.3 24.4 20.8 3.17 2.2 2.4 2.2 1.9 23.8 20.8 14.7 2 35 2.2 2.5 2.2 1 9 HC1 2.53 1.58 1.59 1.56 1 34 1.55 HNO 8 H 2 S0 4 3.88 4.78 1.5 1.21 2.17 2.55 1.45 1.28 1.63 1.84 1.4 1.36 Table 10 DIELECTRIC CONSTANTS Air Alcohol . . 1.00 26.5 2 -3 3-5 . 2.2-2.7 2. -8. . 1.9-2. . 2.1-2.2 Tabl YOUNG'S Quartz 4 5 3.7 4. 2.3 1.015 Shellac . 27 Ebonite Common glass .... India rubber .... Mica Sulphur . . . Benzol . 2. 22 Turpentine . . Water . . . . 2.2 . 81 Paraffin Petroleum oil .... Gases . . . Vacuum . . . .9998 - 0.9985 e 11 MODULUS Approximate Values in Dynes per Square Centimeter Cast iron Brass wire Copper wire Steel wire Aluminum Nickel Silver Glass Oak (cut longitudinally) Pine (cut longitudinally) 12 x 10H 10 x 10 11 12 x ion 19-20 x ion 6.5 x ion 20 x ion 7.3 x ion 6.5 x ion 0.9 x 10 n 0.5-0.6 x 10 11 TABLES 373 (about) Aluminum . Bismuth . . Brass . . . Brick . . . Copper Cork Diamond Glass (common crown) " (flint) .... Gold Ice at C Iron (cast) .... Table 12 DENSITIES Solids 2.58 9.80 8.5 2.1 8.92 0.24 3.52 2.6 3.0-6.3 19.3 0.9167 7.4 Iron (wrought) Lead. . . . Nickel . . . Oak .... Pine .... Platinum . . Quartz . . . Silver . . . Sugar . . . Sulphur . . Tin .... Zinc . 7.86 11.3 8.9 0.8 0.5 21.50 2.65 10.53 1.6 2.07 7.29 7.15 Alcohol at 20 C. . Carbon bisulphide Ethyl ether at C. Glycerin . . . Turpentine . . . Benzol . Liquids 0.789 Gasoline . . 79 1.29 0.735 1.26 0.87 Mercury Sulphuric acid . . . Hydrochloric acid . . Nitric acid . . . . . 13.596 . . 1.85 . . . 1.27 1 56 .88 Olive oil . 0.91 Gases at C. 76 cm. of Mercury Pressure Acetylene (C 2 Ho) .... 0.001185 Ammonia (NH 3 ) 0.000770 Carbon monoxide (CO) . . 0.001252 Carbon dioxide (C0 2 ) . . . 0.001974 Hydrogen (H 2 ) . Marsh gas (CH 4 ) Nitrogen (N 2 ) Oxygen (O 2 ) . . 0.0000895 0.000715 0.001257 0.001430 Table 13 AVERAGE SPECIFIC HEATS Alcohol .... Aluminum . Bismuth Brass .... Copper .... Carbon bisulphide Ebonite .... German silver . Glass Gold Graphite . . . Ice Iron . at 40 0.648 Lead . QO _ 100 0.2185 Magnesium 0.0303 Mercury . 0.094 Nickel QO _ IQQO 0.095 Paraffin . 40 0.2429 Petroleum 0.33 Platinum 0.0946 Quartz . 0.20 Silver . . 0.0316 Steel . . 11 0.160 Tin . . 0.504 Turpentine - 100 0.1130 Zinc . 25 10 21 20 100 0.0315 0.251 50 0.0333 100 0.1128 0.683 58 0.511 100 0.0323 100 0.19 100 0.0568 0.118 100 0.0559 0.467 0.0935 374 ELECTRICITY, SOUND, AND LIGHT Table 14 WAVE LENGTH IN MICRONS (^u) OR THOUSANDTHS MILLIMETERS FRAUN- HOFEK LINE SOURCE WAVE LENGTH IN fJi FRAUN- HOFER LINE SOURCE WAVE LENGTH IN /* Potassium .7699 61 Magnesium .5184 Potassium .7665 6 2 Magnesium .5173 A Sun .7604 63 Iron .5169 a Sun .7185 &4 Magnesium, iron .5168 B Oxygen .6870 Cadmium .5086 Lithium .6708 Barium .4934 C Hydrogen .6563 Calcium .4878 Strontium .6550 F Hydrogen .4861 Calcium .6499 Strontium .4607 Cadmium .6438 Barium .4554 Strontium .6408 Mercury .4358 a Oxygen .6278 f Hydrogen .4340 Mercury .6152 G' Iron .4326 Lithium .6104 G Iron, calcium .4308 Di Sodium .5896 Calcium .4227 Dz Sodium .5890 h Hydrogen .4102 Ds Helium .5876 Mercur^ .4078 Mercury .5790 Mercury .4047 Mercury .5769 Potassium, iron .4046 Mercury .5461 H Hydrogen, calcium .3968 Cadmium .5379 Cadmium .5338 TT Iron, calcium .5270 Jjj Iron .5270 Table 15 REFRACTIVE INDICES FOR DIFFERENT COLORS RED(C) YELLOW(Z>) BLUE (F) Water .... 1 3317 1 3335 1 3377 Alcohol . . 1 3606 1 3624 1 3667 Carbon bisulphide ... 1 6198 1 6293 1 6541 /light Crown glass H . I heavy . . 1.5127 1.6126 1.5153 1 6152 1.5214 1 6213 f lisrht 1 6038 1 6085 1 6200 Flint glass \ * 1 heavy 1 7434 1 7515 1 7723 r ordinary ray . Iceland spar 4 i. extraordinary ray . Quart z .f 0rdinal l ra7 i. extraordinary ray . . . 1.6545 1.4846 1.5418 1.5509 1.6585 1.4864 1.5442 1.5533 1.6679 1.4908 1.5496 1.5589 TABLES 375 Table 16 REDUCTION OP BAROMETRIC HEIGHT TO 0" C. (The table corrections represent the number of millimeters to be subtracted from the observed height h. They are obtained from the formula (.000181 .000019)^, the first number being the cubical expansion coefficient of mercury, the second the linear coefficient of brass.) t OBSERVED HEIGHT IN mm. 680 690 700 710 720 730 740 750 760 770 mm. mm. mm. mm. mm. mm. mm. mm. mm. mm. 10 1.10 1.12 1.13 1.15 1.17 1.18 1.20 1.22 1.23 1.25 11 1.21 1.23 1.25 1.27 1.28 1.30 1.32 1.34 1.35 1.37 12 1.32 1.34 1.36 1.38 1.40 1.42 1.44 1.46 1.48 1.50 13 1.43 1.45 1.47 1.50 1.52 1.54 1.56 1.58 1.60 1.62 14 1.54 1.56 1.59 1.61 1.63 1.66 1.68 1.70 1.72 1.75 15 1.65 1.68 1.70 1.73 1.75 1.77 1.80 1.82 1.85 1.87 16 1.76 1.79 1.81 1.84 1.87 1.89 1.92 1.94 1.97 2.00 17 1.87 1.90 1.93 1.96 1.98 2.01 2.04 2.07 2.09 2.12 18 1.98 2.01 2.04 2.07 2.10 2.13 2.16 2.19 2.22 2.25 19 2.09 2.12 2.15 2.19 2.22 2.25 2.28 2.31 2.34 2.37 20 2.20 2.24 2.27 2.30 2.33 2.37 2.40 2.43 2.46 2.49 21 2.31 2.35 2.38 2 42 2.45 2.48 2.52 2.55 2.59 2.62 22 2.42 2.46 2.49 2'. 53 2.57 2.60 2.64 2.67 2.71 2.74 23 2.53 2.57 2.61 2.65 2.68 2.72 2.76 2.79 2.83 2.87 24 2.64 2.68 2.72 2.76 2.80 2.84 2.88 2.92 2.95 2.99 25 2.75 2.79 2.84 2.88 2.92 2.96 3.00 3.04 3.08 3.12 Table 17 USEFUL CONSTANTS AND RELATIONS TT = 3.1416. 7T 2 = 9.8696, - = 0.31831. logarithm TT = .49715. 7T Naperianbasee = 2.7183. Iog 10 e = . 43429. = 2.30261og 10 #. 1 inch = 25.4 millimeters. 1 meter = 39.37 inches. 1 mile = 1.609 kilometers. 1 kilogram = 2.2 pounds. 1 pound = 453.59 grams. 1 ounce = 28.35 grams. 1 grain = 64.8 milligrams. Mechanical equivalent of lcalorie(15) =4.19 x 10 7 ergs. 1 horse power = 746 watts = 33000 foot pounds per minute. 1 watt = 1 joule per second. 1 joule = 10 7 ergs = 0.7373 foot pounds. 1 radian = 57.3 degrees. 180 = TT radians. Half diameter of the earth, equatorial, 6378.2 kilometers; polar, 6356.5 kilo- meters; mean, 6367.4 kilometers. Mean density of the earth, 5.5270. 376 ELECTRICITY, SOUND, AND LIGHT Table 18 RESISTANCES OF COPPER AND OF GERMAN SILVER WIRE Brown and Sharp Gauge PURE COPPER 18% GERMAN SILVER Number Diameter in Mils data in -) Ohms per 1000 ft. Ohms per 1000 ft. 000 409.64 .064 00 354.80 .081 o 324.95 .102 \ 289.30 .129 2 257.63 .163 ooq 40 4 .~-' . t 204.31 .259 , mQ4 Q0 . 7: 162.02 . O\J 411 7 144.28 * _L 1. FL1Q i 128.49 O L U .654 9 114^43 ]824 11.832 10 101.89 1.040 18.720 11 90.74 1.311 23.598 12 80.81 1.653 29.754 13 71.96 2.084 37.512 14 64.08 2.628 47.304 15 57.07 3.314 59.652 16 50.82 4.179 75.222 17 45.26 5.269 94.842 18 40.30 6.645 119.610 19 35.89 8.617 155.106 20 31.96 10.566 190.188 21 28.46 13.323 239.814 22 25.35 16.799 302.382 23 22.57 21.185 381.330 24 20.10 26.713 480.834 25 17.90 33.684 606.312 26 15.94 42.477 764.586 27 14.20 53.563 964.134 28 12.64 67.542 1215.756 29 11.26 85.170 1533.060 30 10.03 107.391 1933.038 31 8.93 135.402 2437.236 32 7.95 170.765 3073.770 33 7.08 215.312 3875.616 34 6.30 271.583 4888.494 35 5.61 342.443 6163.974 36 5.00 431.712 7770.816 37 4.45 544.287 9797.166 38 3.97 686.511 12357.198 39 3.53 865.046 15570.828 40 3:14 1091.865 19653.570 TABLES 377 Table 19 RESUME OF DEFINING EQUATIONS AND UNITS MAGNITUDE DEFINING EQUA- TION SYMBOL FOR IN C.G.S. UNITS (E. M. SYSTEM) RATIO E. M. UNIT TO E. S. UNIT RATIO PRAC- TICAL UNIT TO C.G.S. UNIT NAME Magnet pole .... F-Z*. r -2 g.2 cm. 2 sec. Magnetic field strength F fif * # s 1 gauss m cm. 2 sec. Magnetic flux . . . $ = Ma t g.* cm. 5 maxwell sec. Magnetic moment . . M - ml g.s cmj sec. Magnetization . . . M m g.- 3 V a cm.^ sec. Magnetic induction . $ = -=#, + 47Tj{ a g> cm. 2 sec. Current F = IIH g.^ cm. ^ 10-1 ampere sec. Quantity Q = It g.' cm. ^ V 10-1 coulomb Potential . . . '-? gj cm. 5 -. i 108 volt sec. 2 Resistance I- PD "IT cm. sec. v --2 109 ohm Capacity c Q sec. 2 V 1 10-9 farad ~ PD cm. Self-induction . . . $> = LI cm. 10 9 henry * For air. t For air, otherwise 4> = cBa. c& = cK for air, since cf 0. v = 3 x 10k 378 ELECTRICITY, SOUND, AND LIGHT NATURAL SINES A TIO'lP o 1 2 3 4 5 Q g g Complement XXllglC Difference 0.0000 0017 0035 0052 0070 008" 0105 0122 0140 0157 0175 89 1 0175 0192 0209 0227 0244 0262 0279 0297 0314 0332 0349 88 2 0349 0366 0384 0401 0419 0436 0454 0471 0488 0506 0523 87 8 0523 0541 0558 0576 0593 0610 0628 0645 0663 0680 0698 86 4 0698 0715 0732 0750 0767 0785 0802 0819 0837 0854 0872 85 5 0.0872 0889 0906 0924 0941 0958 0976 0993 1011 1028 1045 84 6 1045 1063 1080 1097 1115 1132 1149 116" 1184 1201 1219 83 7 1219 1236 1253 1271 1288 1305 1323 1340 1357 1374 1392 82 8 1392 1409 1426 1444 1461 1478 1495 1513 1530 1547 1564 81 9 1564 1582 1599 1616 1633 1650 1668 1685 1702 1719 1736 80 10 0.1736 1754 1771 1788 1805 1822 1840 1857 1874 1891 1908 79 11 1908 1925 1942 1959 1977 1994 2011 2028 2045 .2062 2079 78 12 2079 2096 2113 2130 2147 2164 2181 2198 2215 2233 2250 77 " 13 14 2250 2419 2267 2436 2284 2453 2300 2470 2317 2487 2334 2504 2351 2521 2368 2538 2385 2554 2403 25J1 ^2419 SJfcS 76 75 15 0.2588 2605 2622 2639 2656 2672 2689 2706 2723 2740 2756 74 16 2756 2773 2790 2807 2823 2840 2857 2874 2890 2907 2924 73 17 2924 2940 2957 2974 2990 3007 3024 3040 3057 3074 3090 72 18 3090 3107 3123 3140 3156 3173 3190 3206 3223 3239 3256 71 19 3256 3272 3289 3305 3322 3338 3355 3371 3387 3404 3420 70 20 0.3420 3437 3453 3469 3486 3502 3518 3535 3551 3567 3584 69 21 3584 3600 3616 3633 3649 3665 3681 3697 3714 3730 3746 68 22 3746 3762 3778 3795 3811 3827 3843 3859 3875 3891 3907 67 23 3907 3923 3939 3955 3971 3987 4003 4019 4035 4051 4067 60 18 24 4067 4083 4099 4115 4131 4147 4163 4179 4195 4210 4226 65 25 0.4226 4242 4258 4274 4289 4305 4321 4337 4352 4368 4384 64 26 4384 4399 4415 4431 4446 4462 4478 4493 4509 4524 4540 63 27 4540 4555 4571 4586 4602 4617 4633 4648 4664 4679 4695 62 28 4695 4710 4726 4741 4756 4772 4787 4802 4818 4833 4848 61 29 4848 4863 4879 4894 4909 4924 4939 4955 4970 4985 5000 60 30 0.5000 5015 5030 5045 5060 5075 5090 5105 5120 5135 5150 59 15 31 5150 5165 5180 5195 5210 5225 5240 5255 5270 5284 5299 58 32 5299 5314 5329 5344 5358 5373 5388 5402 5417 5432 5446 57 33 5446 5461 5476 5490 5505 5519 5534 5548 5563 5577 5592 56 34 5592 5606 5621 5635 5650 5664 5678 5693 5707 5721 5736 55 35 0.5736 5750 5764 5779 5793 5807 5821 5835 5850 5864 5878 54 36 5878 5892 5906 5920 5934 5948 5962 5976 5990 6004 6018 53 i* 37 6018 6032 6046 6060 6074 6088 6101 6115 6129 6143 6157 52 38 6157 6170 6184 6198 6211 6225 6239 6252 6266 6280 6293 51 39 6293 6307 6320 6334 6347 6361 6374 6388 6401 6414 6428 50 40 0.6428 6441 6455 6468 6481 6494 6508 6521 6534 6547 6561 49 41 6561 6574 6587 6600 6613 6626 6639 6652 6665 6678 6691 4 8 13 42 6691 6704 6717 6730 6743 6756 6769 6782 6794 6807 6820 47 43 6820 6833 6845 6858 6871 6884 6896 6909 6921 6934 6947 4<> 44 6947 6959 6972 6984 6997 7009 7022 7034 7046 7059 7071 45 Complement .9 .8 .7 .6 .5 .4 .3 .2 .1 .0 Angle NATURAL COSINES TABLES 379 NATURAL SINES Complement Angle .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 Difference 45 0.7071 7083 7096 7108 7120 7133 7145 7157 71G9 7181 7193 41 40 7193 7206 7218 7230 7242 7254 7266 7278 7290 7302 7314 43 19 47 7314 7325 7337 7349 7361 7373 7385 7396 7408 7420 7431 42 48 7431 7443 7455 7466 7478 7490 7501 7513 7524 7536 7547 41 49 7547 7559 7570 7581 7593 7604 7615 7627 7638 7649 7660 40 50 0.7660 7672 7683 7694 7705 7716 7727 7738 7749 7760 7771 C9 51 7771 7782 7793 7804 7815 7826 7837 7848 7859 7869 7880 38 " 52 7880 7891 7902 7912 7923 7934 7944 7955 7965 7976 7986 37 53 7986 7997 8007 8018 8028 8039 8049 8059 8070 8080 8090 36 54 8090 8100 8111 8121 8131 8141 8151 8161 8171 8181 8192 35 55 0.8192 8202 8211 8221 8231 8241 8251 8261 8271 8281 8290 3410 50 8290 8300 8310 8320 8329 8339 8348 8358 8368 8377 8387 33 57 8387 8396 8406 8415 8425 8434 8443 8453 8462 8471 .480 32 58 8480 8490 8499 8508 8517 8526 8536 8545 8554 8563 8572 31 59 8572 8581 8590 8599 8607 8616 8625 8634 8643 8652 8660 30 9 60 0.8660 8669 8678 8686 8695 8704 8712 8721 8729 8738 8746 29 61 8746 8755 8763 8771 8780 8788 8796 8805 8813 8821 8829 28 62 8829 8838 8846 8854 8862 8870 8878 8886 8894 8902 8910 27 s 63 8910 8918 8926 8934 8942 8949 8957 8965 8973 8980 8988 26 64 8988 8996 9003 9011 9018 9026 9033 9041 9048 9056 9063 25 65 0.9063 9070 9078 9085 9092 9100 9107 9114 9121 9128 9135 24 66 9135 9143 9150 9157 9164 9171 9178 9184 9191 9198 9205 23 ' 67 9205 9212 9219 9225 9232 9239 9245 9252 9259 9265 9272 22 68 9272 9278 9285 9291 9298 9304 9311 9317 9323 9330 9336 21 69 9336 9342 9348 9354 9361 9367 9373 9379 9385 9391 9397 20 6 70 0.9397 9403 9409 9415 9421 9426 9432 9438 9444 9449 9455 19 71 9455 9461 9466 9472 9478 9483 9489 9494 9500 9505 9511 18 72 9511 9516 9521 9527 9532 9537 9542 9548 9553 9558 9563 17 73 9563 9568 9573 9578 9583 9588 9593 9598 9603 9608 9613 16 s 74 9613 9617 9622 9627 9632 9636 9641 9646 9650 9655 9659 15 75 0.9659 9664 9668 9673 9677 9681 9686 9690 9694 9699 9703 14 76 9703 9707 9711 9715 9720 9724 9728 9732 9736 9740 9744 13 77 9744 9748 9751 9755 9759 9763 9767 9770 9774 9778 9781 12 78 9781 9785 9789 9792 9796 9799 9803 9806 9810 9813 9816 11 79 9816 9820 9823 9826 9829 9833 9836 9839 9842 9845 9848 10 80 0.9848 9851 9854 9857 9860 9863 9866 9869 9871 9874 9877 9 3 81 9877 9880 9882 9885 9888 9890 9893 9895 9898 9900 9903 8 82 9903 9905 9907 9910 9912 9914 9917 9919 9921 9923 9925 7 83 9925 9928 9930 9932 9934 9936 9938 9940 9942 9943 9945 6 2 84 9945 9947 9949 9951 9952 9954 9956 9957 9959 9960 9962 5 85 0.9962 9963 9965 9966 9968 9969 9971 9972 9973 9974 9976 4 86 9976 9977 9978 9979 9980 9981 9982 9983 9984 9985 9986 3 * 87 9986 9987 9988 9989 9990 9990 9991 9992 9993 9993 9994 2 88 9994 9995 9995 9996 9996 9997 9997 9997 9998 9998 9998 1 89 9998 9999 9999 9999 9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0<> Complement .9 .8 .7 .6 .5 .4 .3 .2 .1 .0 Angle NATURAL COSINES 380 ELECTRICITY, SOUND, AND LIGHT NATURAL TANGENTS 1 Complement Angle .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 Difference 0.0000 0017 0035 0052 0070 0087 0105 0122 0140 0157 0175 89 1 0175 0192 0209 0227 0244 0262 0279 0297 0314 0332 0349 88 2 0349 0367 0384 0402 0419 0437 0454 0472 0489 0507 0524 87 3 0524 0542 0559 0577 0594 0612 0629 0647 0664 0682 0699 86 4 0699 0717 0734 0752 0769 0787 0805 0822 0840 0857 0875 85 5 0.0875 0892 0910 0928 0945 0963 0981 0998 1016 1033 1051 84 6 1051 1069 1086 1104 1122 1139 1157 1175 1192 1210 1228 83 7 1228 1246 1263 1281 1299 1317 1334 1352 1370 1388 1405 82 8 1405 1423 1441 1459 1477 1495 1512 1530 1548 1566 1584 81 9 1584 1602 1620 1638 1655 1673 1691 1709 1727 1745 1763 80 10 0.1763 1781 1799 1817 1835 1853 1871 1890 1908 1926 1944 79 18 11 1944 1962 1980 1998 2016 2035 2053 2071 2089 2107 2126 78 12 2126 2144 2162 2180 2199 2217 2235 2254 2272 2290 2309 77 13 2309 2327 2345 2364 2382 2401 2419 2438 2456 2475 2493 76 14 2493 2512 2530 2549 2568 2586 2605 2623 2642 2661 2679 75 15 0.2679 2698 2717 2736 2754 2774 2792 2811 2830 2849 2867 74 16 2867 2886 2905 2924 2943 2962 2981 3000 3019 3038 3057 73 19 17 3057 3076 3096 3115 3134 3153 3172 3191 3211 3230 3249 72 18 3249 3269 3288 3307 3327 3346 3365 3385 3404 3424 3443 71 19 3443 3463 3482 3502 3522 3541 3561 3581 3600 3620 3640 70 20 0.3640 3659 3679 3699 3719 3739 3759 3779 3799 3819 3839 69 21 3839 3859 3879 3899 3919 3939 3959 3979 4000 4020 4040 68 20 22 4040 4061 4081 4101 4122 4142 4163 4183 42C4 4224 4245 67 23 4245 4265 4286 4307 4327 4348 4369 4390 4411 4431 4452 66 24 4452 4473 4494 4515 4536 4557 4578 4599 4621 4642 4663 65 21 25 0.4663 4684 4706 4727 4748 4770 4791 4813 4834 4856 4877 64 26 4877 4899 4921 4942 4964 4986 5008 5029 5051 5073 5095 63 27 5095 5117 5139 5161 5184 5206 5228 5250 5272 5295 5317 62 22 28 5317 5340 5362 5384 5407 5430 5452 5475 5498 5520 5543 61 29 5543 5566 5589 5612 5635 5658 5681 5704 5727 5750 5774 60 23 30 0.5774 5797 5820 5844 5867 5890 5914 5938 5961 5985 6099 59 31 6009 6032 6056 6080 6104 6128 6152 6176 6200 6224 6249 58 2 * 32 6249 6273 6297 6322 6346 6371 6395 6420 6445 6469 6494 57 33 6494 6519 6544 6569 6594 6619 6644 6669 6694 6720 6745 56 25 34 6745 6771 6796 6822 6847 6873 6899 6924 6950 6976 7002 55 35 0.7002 7028 7054 7080 7107 7133 7159 7186 7212 7239 7265 54 2S 36 7265 7292 7319 7346 7373 7400 7427 7454 7481 7508 7536 53^ 37 7536 7563 7590 7618 7646 7673 7701 7729 7757 7785 7813 52 23 38 7813 7841 7869 7898 7926 7954 7983 8012 8040 8069 8098 61 28 39 8098 8127 8156 8185 8214 8243 8273 8302 8332 8361 8391 50 29 40 0.8391 8421 8451 8481 8511 8541 8571 8601 8632 8662 8693 49 30 41 8693 8724 8754 8785 8816 8847 8878 8910 8941 8972 9004 48 31 42 9004 9036 9067 9099 9131 9163 9195 9228 9260 9293 9325 47 32 43 9325 9358 9391 9424 9557 9490 9523 9556 9590 9623 9657 46 33 44 9657 9691 9725 9759 9793 9827 9861 9896 9930 9965 1.0000 45034 Complement .9 .8 .7 .6 .5 .4 .3 .2 .1 .0 Angle NATURAL COTANGENTS TABLES 381 NATURAL TANGENTS Angle .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 Dii. 45 1.0000 1.0035 1.0070 1.0105 1.0141 1.0176 1.0212 1.0247 1.0283 1.0319 36 46 1.0355 1.0392 1.0428 1.0464 1.0501 1.0538 1.0575 1.0612 1.0649 1.0686 37 47 1.0724 1.0761 1.0799 1.0837 1.0875 1.0913 1.0951 1.0990 1.1028 1.1067 38 48 1.1106 1.1145 1.1184 1.1224 1.1263 1.1303 1.1343 1.1383 1. 1423 1.1463 40 49 1.1504 1.1544 1.1585 1.1626 1.1667 1.1708 1.1750 1.1792 1.1833 1.1875 41 50 1.1918 1.1960 1.2002 1.2045 1.2088 1.2131 1.2174 1.2218 1.2261 1.2305 43 51 1.2349 1.2393 1.2437 1.2482 1.2527 1.2572 1.2617 1.2662 1.2708 1.2753 45 52 1.2799 1.2846 1.2892 1.2938 1.2985 1.3032 1.3079 1.3127 1.3175 1.3222 47 53 1.3270 1.3319 1.3367 1.3416 1.3465 1.3514 1.3564 1.3613 1.3663 1.3713 49 54 1.3764 1.3814 1.3865 1.3916 1.3968 1.4019 1.4071 1.4124 1.4176 1.4229 52 55 1.4281 1.4335 1.4388 1.4442 1.4496 1.4550 1.4605 1.4659 1.4715 1.4770 54 56 1.4826 1.4882 1.4938 1.4994 1.5051 1.5108 1.5166 1.5224 1.5282 1.5340 07 57 1.5399 1.5458 1.5517 1.5577 1.5637 1.5697 1.5757 1.5818 1.5880 1.5941 60 58 1.6003 1.6066 1.6128 1.6191 1.6255 1.6319 1.6383 1.6447 1.6512 1.6577 64 59 1.6643 1.6709 1.6775 1.6842 1.6909 1.6977 1.7045 1.7113 1.7182 1.7251 68 60 1.7321 1.7391 1.7461 1.7532 1.7603 1.7675 1.7747 1.7820 1.7893 1.7966 72 61 1.8040 1.8115 1.8190 1.8265 1.8341 1.8418 1.8495 1.8572 1.8650 1.8728 77 62 1.8807 1.8887 1.8967 1.9047 1.9128 1.9210 1.9292 1.9375 1.9458 1.9542 82 63 1.9626 1.9711 1.9797 1.9883 1.99702.0057 2.0145 2.0233 2.0323 2.0413 88 64 2.0503 2.0594 2.0686 2.0778 2.0872 2.0965 2.1060 2.1155 2.1251 2.134S 94 65 2.145 2.154 2.164 2.174 2.184 2.194 2.204 2.215 2.225 2.236 10 66 2.246 2.257 2.267 2.278 2.289 2.300 2.311 2.322 2.333 2.344 11 67 2.356 2.367 2.379 2.391 2.402 2.414 2.426 2.438 2.450 2.463 12 68 2.475 2.488 2.500 2.513 2.526 2.539 2.552 2.565 2.578 2.592 13 69 2.605 2.619 2.633 2.646 2.660 2.675 2.689- 2.703 2.718 2.733 14 70 2.747 2.762 2.778 2.793 2.808 2.824 2.840 2.856 2.872 2.888 16 71 2.904 2.921 2.937 2.954 2.971 2.989 3.006 3.024 3.042 3.060 17 72 3.078 3.096 3.115 3.133 3.152 3.172 3.191 3.211 3.230 3.250 19 73 3.271 3.291 3.312 3.333 3.354 3.376 3.398 3.420 3.442 3.465 22 74 3.487 3.511 3.534 3.558 3.582 3.606 3.G30 3.655 3.681 3.700 25 75 3.732 3.758 3.785 3.812 3.839 3.867 3.895 3.923 3.952 3.981 28 76 4.011 4.041 4.071 4.102 4.134 4.165 4.198 4.230 4.264 4.297 32 77 4.331 4.366 4.402 4.437 4.474 4.511 4.548 4.586 4.625 4.665 37 78 4.705 4.745 4.787 4.829 4.872 4.915 4.959 5.005 5.050 5.097 44 79 5.145 5.193 5.242 5.292 5.343 5.396 5.449 5.503 5.558 5.614 52 80 5.67 5.73 5.79 5.85 5.91 5.98 6.04 6.11 6.17 6.24 7 81 6.31 6.39 6.46 6.54 6.61 6.G9 6.77 6.85 6.94 7.03 8 82 7.12 7.21 7.30 7.40 7.49 7.60 7.70 7.81 7.92 8.03 10 83 8.14 8.26 8.39 8.51 8.64 8.78 8.92 9.06 9.21 9.36 li 84 9.51 9.68 9.84 10.0 10.2 10.4 10.6 10.8 11.0 11.2 85 11.4 11.7 11.9 12.2 12.4 12.7 13.0 13.3 13.6 14.0 3 86 14.3 14.7 15.1 15.5 15.9 16.3 16.8 17.3 17.9 18.5 6 87 19.1 19.7 20.4 21.2 22.0 22.9 23.9 24.9 26.0 27.3 88 28.6 30.1 31.8 33.7 35.8 38.2 40.9 44.1 47.7 52.1 89 57. 64. 72. 82. 95. 115. 143. 191. 286. 573. Angle .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 NATURAL TANGENTS 382 ELECTRICITY, SOUND, AND LIGHT LOGARITHMS 10 0000 1 0043 2 0086 3 4 0170 5 6 7 0294 8 9 1 2 3 450 789 0128 0212 0253 0334 0374 4 8 12 17 21 25 29 33 37 11 12 13 0414 0792 1139 0453 0828 1173 0492 0864 1206 0531 0899 1239 0569 0934 1271 0607 0969 1303 0645 1004 1335 0682 1038 1367 0719 1072 1399 0755 1106 1430 4 8 11 3 7 10 3 6 10 15 19 23 14 17 21 13 16 19 26 30 34 24 28 31 23 26 29 14 15 16 1461 1761 2041 1492 1790 2068 1523 1818 2095 1553 1847 2122 1584 1875 2148 1614 1903 2175 1644 1931 2201 1673 1959 2227 1703 1987 2253 1732 2014 2279 369 368 358 12 15 18 11 14 17 11 13 16 21 24 27 20 22 25 18 21 24 17 18 19 2304 2353 2788 2330 2577 2810 2355 2601 2833 2380 2625 2856 2405 2648 2878 2430 2672 2900 2455 2695 2923 2480 2718 2945 2504 2742 2967 2529 2765 2989 257 257 247 10 12 15 9 12 14 9 11 13 17 20 22 16 19 21 16 18 20 20 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 246 8 11 13 15 17 19 21 22 23 3222 3424 3617 3243 3444 3636 3263 3464 3655 3284 3483 3674 3304 3502 3692 3324 3522 3711 3345 3541 3729 3365 3560 3747 3385 3579 3766 3404 3598 3784 246 246 246 8 10 12 8 10 12 7 9 11 14 16 18 14 15 17 13 15 17 24 25 26 3802 3979 4150 3820 3997 4166 3838 4014 4183 3856 4031 4200 3874 4048 4216 3892 4065 4232 3909 4082 4249 3927 4099 4265 3945 4116 4281 8962 4133 4298 245 235 235 7 9 11 7 9 10 7 8 10 12 14 16 12 14 15 11 13 15 27 28 29 4314 4472 4624 4330 4487 4639 4346 4502 4654 4362 4518 4669 4378 4533 4683 4393 4548 4698 4409 4564 4713 4425 4579 4728 4440 4594 4742 4456 4609 4757 235 235 1 3 4 689 689 679 11 13 14 11 12 14 10 12 13 30 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 1 3 4 679 10 11 13 31 32 33 4914 5051 5185 4928 5065 5198 4942 5079 5211 4955 5092 5224 4969 5X05 5237 4983 5119 5250 4997 5132 5263 5011 5145 5276 5024 5159 5289 5038 5172 5302 1 3 4 1 3 4 1 3 4 678 578 568 10 11 12 9 11 12 9 10 12 34 35 36 5315 5441 5563 5328 5453 5575 5340 5465 5587 5353 5478 5599 5366 5490 5611 5378 5502 5623 5391 5514 5635 5403 5527 5647 5416 5539 5658 5428 5551 5670 1 3 4 1 2 4 1 2 4 568 567 567 9 10 11 9 10 11 8 10 11 37 38 39 5682 5798 5911 5694 5809 5922 5705 5821 5933 5717 5832 5944 5729 5843 5955 5740 5855 5966 5752 5866 5977 5763 5877 5988 5775 5888 5999 5786 5899 6010 1 2 3 1 2 3 1 2 3 567 567 457 8 9 10 8 9 10 8 9 10 40 6021 6031 6042 6053 6064 6075 6085 6096 6107 6117 1 2 3 456 8 9 10 41 42 43 6128 6232 6335 6138 6243 6345 6149 6253 6355 6160 6263 6365 6170 6274 6375 6180 6284 6385 6191 6294 6395 6201 6304 6405 6212 6314 6415 6222 6325 6425 123 1 2 3 1 2 3 456 456 4 5 G 789 789 789 44 45 46 6435 6532 6628 6444 6542 6637 6454 6551 6646 6464 6561 6656 6474 6571 6665 6484 6580 6675 6493 6590 6684 6503 6599 6693 6513 6609 6702 6522 6618 6712 123 123 1 2 3 456 456 456 789 789 778 47 48 49 6721 6812 6902 6730 6821 6911 6739 6830 6920 6749 6839 6928 6758 6848 6937 6767 6857 6946 6776 6866 6955 6785 6875 6964 6794 6884 6972 6803 6893 6981 1 2 3 1 2 3 1 2 3 455 445 445 678 678 6 7, 8 50 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 1 2 3 345 678 51 52 53 7076 7160 7243 7084 7168 7251 7093 7177 7259 7101 7185 7267 7110 7193 7275 7118 7202 7284 7126 7210 7292 7135 7218 7300 7143 7226 7308 7152 7235 7316 1 2 3 1 2 2 1 2 2 345 345 345 678 677 667 54 7324 7332 7340 7348 7356 7364 7372 7380 7388 7396 1 2 2 345 667 TABLES 383 LOGARITHMS 55 l 2 3 4 5 6 7 8 9 1 2 3 456 789 7404 7412 7419 7427 7435 7443 7451 7459 7466 7474 122 345 567 56 57 58 7482 7559 7634 7490 7566 7642 7497 7574 7649 7505 7582 7657 7513 7589 7664 7520 7597 7672 7528 7604 7679 7536 7612 7686 7543 7619 7694 7551 7627 7701 122 122 1 1 2 345 345 344 567 567 567 59 60 61 7709 7782 7853 7716 7789 7860 7723 7796 7868 7731 7803 7875 7738 7810 7882 7745 7818 7889 7752 7825 7896 7760 7832 7903 7767 7839 7910 7774 7846 7917 112 112 112 344 344 344 567 566 566 62 63 64 7924 7993 8062 7931 8000 8069 7938 8007 8075 7945 8014 8082 7952 8021 8089 7959 8028 8096 7966 8035 8102 7973 8041 8109 7980 8048 8116 7987 8055 8122 112 112 112 334 334 334 566 556 556 65 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 112 334 556 66 67 68 8195 8261 8325 8202 8267 8331 8209 8274 8338 8215 8280 8344 8222 8287 8351 8228 8293 8357 8235 8299 8363 8241 8306 8370 8248 8312 8376 8254 8319 8382 1 1 1 1 1 1 334 334 334 556 556 456 69 70 71 8388 8451 8513 8395 8457 8519 8401 8463 8525 8407 8470 8531 8414 8476 8537 8420 8482 8543 8426 8488 8549 8432 8494 8555 8439 8500 8561 8445 8506 8567 1 1 1 1 1 1 234 234 234 456 456 455 72 73 74 8573 8633 8692 8579 8639 8698 8585 8645 8704 8591 8651 8710 8597 8657 8716 8603 8663 8722 8609 8669 8727 8615 8675 8733 8621 8681 8739 8627 8686 8745 1 1 1 1 1 1 2 3 2 3 2 3 455 455 455 75 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 1 1 2 3 455 76 77 78 8808 8865 8921 8814 8871 8927 8820 8876 8932 8825 8882 8938 8831 8887 8943 8837 8893 8949 8842 8899 8954 8848 8904 8960 8854 8910 8965 8859 8915 8971 1 1 1 1 1 1 2 3 2 3 2 3 455 445 445 79 80 81 8976 9031 9085 8982 9036 9090 8987 9042 9096 8993 9047 9101 8998 9053 9106 9004 9058 9112 9009 90G3 9117 9015 9069 9122 9020 9074 9128 9025 9079 9133 1 1 2 1 1 2 112 2 3 2 3 2 3 445 445 445 82 83 84 9138 9191 9243 9143 9196 9248 9149 9201 9253 9154 9206 9258 9159 9212 9263 9165 9217 9269 9170 9222 9274 9175 9227 9279 9180 9232 9284 9186 9238 9289 1 1 2 1 1 2 112 2 3 2 3 2 3 445 445 445 85 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 1 1 2 2 3 445 86 87 88 9345 9395 9445 9350 9400 9450 9355 9405 9455 9360 9410 9460 9365 9415 9465 9370 9420 9469 9375 9425 9474 9380 9430 9479 9385 9435 9484 9390 9440 9489 1 1 2 1 1 1 1 2 3 2 2 2 2 445 344 344 89 90 91 9494 9542 9590 9499 9547 9595 9504 9552 9600 9509 9557 9605 9513 9562 9609 9518 9566 9614 9523 9571 9619 9528 9576 9624 9533 9581 9628 9538 9586 9633 1 1 1 1 1 1 2 2 2 2 2 2 344 344 344 92 93 94 9638 9685 9731 9643 9689 9736 9647 9694 9741 9652 9699 9745 9657 9703 9750 9661 9708 9754 9666 9713 9759 9671 9717 9763 9675 9722 9768 9680 9727 9773 1 1 1 1 1 1 2 2 2 2 2 2 344 344 344 95 9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 1 1 2 2 344 96 97 98 9823 9868 9912 9827 9872 9917 9832 9877 9921 9836 9881 9926 9841 9886 9930 9845 9890 9934 9850 9894 9939 9854 9899 9943 9859 9903 9948 9863 9908 9952 1 1 1 1 1 1 2 2 2 2 2 2 344 344 344 99 9956 9961 9965 9969 9974 9978 9983 9987 9991 9996 1 1 2 2 334 INDEX (Numbers refer to pages) Aberration, spherical, 288 chromatic, 305 Achromatic lens, 305 Adiabatic compression, 193 Alpha rays, 341 Amagat, 190 Ammeter, 39, 84 Ampere, 35 Ampere, unit, 35 Amplitude, defined, 223 Arago, 331 Arc-light efficiency, 299 Arrhenius, 181 Atom, nature of, 336, 345 Atomic weights, table of, 370 cB = magnetic induction, 160 o/3 and SK curve, 161, 169 Ballistic galvanometer, 97 Ballistic method, for magnetization, 168 for comparing E.M.F.'s, 122 Barometric redactions, table of, 375 Bartholinus, 321 Battery, Daniell, 117 E.M.F. of, 113, 117 resistance of, 114, 117 polarization of, 121 Leclanche', 124 Clark standard, 121 Beats, 234 Becquerel, 340 Beta rays, 341 Biot, 332 Boyle's law, 43, 192, 194 Brewster's law, 318 Bright-line spectra, 307 Candle, British standard, 294 Capacity, denned, 90 units of, 97 measurement of, 93 laws of combination, 103 comparison of, 100 standard, 105 calculation of, 107 Carcel lamp, 294 Cathode rays, 333, 334 Caustics, 286 Charles's law, 43 Chromatic aberration, 305 Circular polarization, 327 Clausius, 177 Clausius's dissociation theory, 177 Coercive force, 165 Colladon, 190 Colors produced by polarized light in crystals, 328 Compressional waves, in fluids, 187 in rods, 213 Condensation, reflection of, 195 Condenser, capacity of, 91 formula for, 107 mica, 109 Conduction, electrical, 3, 24 electrolytic, 176 Conductivity, molecular, 180 specific, 183 tables of, see Resistance Continuous spectrum, 306 Cooper-Hewitt lamp, 299 Corpuscular theory of light, 238 Cosines, tables of natural, 378, 379 Coulomb, 35 Coulomb, unit, 35 Critical angle of total reflection, 288 Crookes, 334, 344 Curie, M. and Mme., 343 Current, electrical, defined, 28 magnetic effect of, 29 direction of, 31 unit of, 33, 35, 77 measuring instruments, 86 Curvature, defined, 271 Damping of galvanometer, 65 damping factor, 95 Daniell battery, 117 Dark-line spectra, 307 D'Arsonval galvanometer, 63, 80, 97 Defining equations, re'sume' of, 377 Demagnetization, method of, 170 385 386 ELECTRICITY, SOUND, AND LIGHT Density, tables of: air, 368 gases, 373 mercury, 368 solids, 373 liquids, 369, 373 water, 368 Dextrose, 332 Diamagnetic substances, 159 Diatonic scale, 208 Dielectric absorption, 103 Dielectric constant, 102-110 tables of, 372 Diffraction, 238 through apertures, 242, 266 Diffraction grating, 258 et seq. Direct-vision spectroscope, 305 Discontinuous spectra, 306 Dispersion, Chapter XXVI Dispersive power of grating, 263 Displacement law of radiation, 298 Dissociation, electrolytic, 177 Distribution of magnetism, 36, 137 Doppler's principle, 309 Double refraction, 323 Drude, 6 Dynamo, the ideal, 131 construction of, 134 rule, 128 Dynamometer, 88 E.M.F. = electromotive force, 113 Earth inductor, 142 Earth, magnetic constants of, 141 Electrical conduction, 3, 24 Electricity, quantity of, 2, 35 theory of, 2 electro-chemical equivalents, 38 table of, 371 Electrolysis, 37 Electrolytic conduction, 176 Electro-magnetic, induction, 126 system of units, 34 units, table of, 377 Electromotive force, denned, 113 induced, 127 comparison of, 122, 123 Electron, 6, 337 Electrostatic, induction, 4 system of units, 33 field intensity, 107 Elliptical polarization, 328 Emanation, radium, 346 Equipotential surfaces, 11 determination of, 13 Even temperament, 208 Extraordinary ray, 322 Farad, 97 Faraday, 37, 79, 97, 176 laws of electrolysis, 37 concept of lines of force, 79 Fields of force, 8 Figure of merit of galvanometer, 82 Fleming's method for measuring self- induction, 156 Fluorescence due to radiations, 333, 344 Focal lengths, 273 Franklin, 6 Fraunhofer lines, 307 Fundamental note, 201 Galvanic cell, 30, 117 et seq. Galvanometer, D' Arson val, 63, 80 tangent, 35 ballistic, 97 Thomson, 63, 86 direct-reading, 84 resistance of, 84 constant of, 77 damping of, 95 Gamma rays, 344 Gases, ionization of, 338 Gauss, unit, 9 eyepiece, 278 Gay-Lussac's law, 194 Gouy, 311 Gram-molecule, 179 Graphical transformation of cB and &C curve, 167 Grotthus chain, 177 H = horizontal component of earth's field, by magnetometer, Chapter II by earth inductor, 142 3C = magnetic field intensity, 8, 128 Heating effect of a current, 49 Hefner lamp, 294 Helmholtz, 112 Henry, 127, 149 Henry unit, 149 Buy gens, 238, 321 Hysteresis, 104 loop, 165 3 = magnetization, 161 3 and &C curve, 169 Iceland spar, 321 Illumination, laws of, 292 Images, formation of, 245 et seq. Incandescent lamp efficiency, 299 Index of refraction, 274 tables of, 374 by minimum deviation, 277 by total reflection, 289 INDEX 387 Induction, coil, 152 electro-magnetic, 126 electrostatic, 4 magnetic, 159 Intensity, of current, 33, 35, 77 of electrostatic field, 107 of magnetic field, 17, 31, 150 Interference of two wave trains, 240 Internal resistance of battery, 114, 117 Ionic speeds, 182 lonization, of gases, 338 of liquids, Chapter XVI Irrationality of dispersion, 304 Isothermal compression, 193 Just temperament, 210 Kelvin, 1 Kohlrausch, 180 Kundt's tube, 217 La Place, 193 Lenard, 335, 336 Lens, formulas, 273 images formed by, 245 Lenz's law, 127, 130 Levulose, 332 Light, two theories of, 238 velocity, 35, 105 Light waves, 238 et seq. Liquids, ionization of, Chapter XVI Logarithms, tables of, 382, 383 Longitudinal waves, 187 et seq., 213 et seq. Loops, 215 Luminescence, 298 Lummer-Brodhun photometer, 294 Major chord, 208 Magnet, 1 pole strength, 1, 17 Magnetic creeping, 171 Magnetic dip, 141 Magnetic field about current, 31 Magnetic field intensity, 17 et seq., 159 Magnetic moment, 18 Magnetic variation, 141 Magnetism, quantity of, 1, 17, 136 distribution of, 136 Magnetization, defined, 161 three stages of, 161 curve of, 168-1 70 Magnetometer, Experiment 2 Mains, 316 Melde's experiment, 229 Michelson, 265 Minimum deviation, angle of, 276 measurement of, 280 Mirror formulas, 273 images formed by, 245 et seq. , 287 Molecular conductivity, 180 measurement of, 183 Molecular theory of magnetism, 134 Moment of inertia of bar magnet, 22 Motor rule, 78 Musical scale, 208 Natural period, of pipes, 203 of rods, 214 Nernst lamp, 299 Newton's deduction of velocity of sound, 192 experiment on dispersion, 300 corpuscular theory of light, 238 Nicol prism, 320-325 Nodes, 215 Normal solution, 179 densities of, 369 Norrenberg polariscope, 316 Oersted's experiment, 31 Ohm, 58 Ohm, unit, 59 Ohm's law, 58 in electrolytes, 178 One-fluid theory, 6 Optical efficiency of light sources, 298 Ordinary ray, 322 Organ pipes, 205 Osmotic phenomena, 179 Overtones, 201 P.D. = potential difference, 46-51 Parallel resistances, 60 capacities, 103 Paramagnetic substances, 159 Permeability, 171 Perrin, 336 Phase, 191 Photometric standards, 204 Photometry, Chapter XXV Pipes, open, 201 closed, 201 notes to which p-'pes respond, 201 natural periods of, 203 Pitchblende, 343 Platinum black, 185 Polarization, by reflection, 316 by double refracting crystals, 321 Polarization, electrolytic, 116 Polarizfd light, Chapter XXVII Po-st-office box, G7-70 388 ELECTEICITY, SOUND, AND LIGHT Potential, electrical, 11 gravitational, 9 magnetic, 10 of conductor in electrical equilib- rium, 12 Potential difference, 46-51 Potentiometer method of comparing E.M.F.'s, 122 Practical units, 35, 51, 59, 97, 149 table of, 377 Prevost's law of exchanges, 297 Prism, measurement of angle of, 278 Quarter-wave plate, 326 Quartz, right- and left-handed, 332 Radiation, types of, 333 Radio-activity, Chapter XXVIII discovery of, 340 an atomic property, 345 theory of, 345 Radium, 344 Radium A, B, C, D, E, F, 349 Rarefaction, reflection of, 196 Ratio of E.M. and E.S. units, 35, 105 of specific heats, 193 Rayleigh, 192, 198, 311 Real image, 246 Reflection, of wave trains, 195 law of, 247 total, 88, 285 Reflection grating, 267 Refraction, of light, 270 through prism, 275 Residual magnetism, 165 Resistance, 58 laws of, 60 specific, 71 tables of, 371, 372 Resolving power, of grating, 265 of lens, 251 Resonance of air columns, 196 Retentivity, 165 Reversal of spectral lines, 309 Riecke, 6 Right-hand screw rule, 32 Rods, waves in, 213 et seq., 233 Rotary polarization, 330 Rowland, 29, 267 Saturation, magnetic, 135 Scale, musical, 208 Self-induction, nature of, 146, 151 coefficient of, 148 measurement of, 149 unit of, 148 illustrations of, 150 Sensibility of galvanometer, 83 Sensitive flame, 252 Series, resistances in, 60 capacities in, 104 Sines, tables of, 378, 379 Sodium lines, reversal of, 309 Solar spectrum, 307 Solenoid, field strength within, 166 Solid angle, 108 Sound, 187 et seq. velocity of, in air, Chapter XVII velocity of, in water, 190 Specific conductivity, 183 Specific resistance, 71 Specific resistance, table of, 371, 372 Specific heats, table of, 373 Spectrometer, 277 Spectro-photometry, 297 Spectroscopic analysis, 311 Spectrum, Chapter XXV pure, 303 grating, 262 normal, 262 prismatic, 304 solar, 307 Spinthariscope, 344 Stationary waves, 226 Stefan's law of radiation, 297 Strings, waves in, 222 et seq. Sturm, 190 Sugar, rotation of plane of polarization by, 332 Susceptibility, 171 Tangent galvanometer, 35 Tangents, table of natural, 380, 381 Temperament, scale of even, 210 scale of just, 209 Temperature coefficient of resistance, 71 tables of, 371 Test coil, 142, 158 Thermo-electro motive forces, 113 Thomson, 7, 336, 343, 350 Thorium, 346 Total reflection, Chapter XXIV prism, 289 Transformer, alternating current, 153 Transverse wave motion, 224 et seq., Chapter XXVII Tube of force, 9 Tuning fork, 234 Two-fluid theory, 5 Units, table of, 377 Uranium, radio-activity of, 341 IXDEX 389 Uranium X, 346 Useful numbers, table of, 375 V= vertical component of earth's field, 141 measurement of, by means of an earth inductor, 142 Van ' Hoff, 179 Vapor pressures, table of, 367 Variation, magnetic, 141 Velocity of compressional waves, 187, 213 of transverse waves, 230 Velocity of light, 35 in two media compared, 271 of ordinary and extraordinary ray compared, 323 et seq. Velocity of sound, 190, 194 effect of temperature on, 194 independent of pressure, 194 in two solids compared, 215 in two gases, 219 Virtual image, 246 Viscosity, table of, 369 Volt, 51 Voltameter, 39 Voltmeter, 51 electrostatic, 352 Water-vapor pressures, table of, 367 Wave, front, 239 construction of, 240 length, 224 lengths of light in air (tables), 374 motion, 222 et seq. length by grating, 261 length by Kundt's tube, 217 Waves, train of, 190 reflection of, 194 Welsbach lamp, 299 Wheatstone bridge, 61 et seq. measurement of self-induction by, 149 measurement of molecular conduc- tivity by, 184 Wind instruments, musical, 206 Wire, sizes and resistances, table of, 376 Wollaston, 307 X rays, 333, 339 Young, 238 Young's modulus, tables of, 372 Zeeman, 336 Zeeman effect, 336, 337 ANNOUNCEMENTS A FIRST COURSE IN PHYSICS By ROBERT A. MILLIKAN, Assistant Professor of Physics in the University of Chicago, and HENRY G. GALE, Instructor in Physics in the University of Chicago izmo. Cloth. 488 pages. Illustrated. List price, $1.25 ; mailing price, $1.40 A LIST OF LABORATORY EXPERIMENTS IN PHYSICS FOR SECOND ART SCHOOLS By ROBERT A. MILLIKAN and HENRY G. GALE List price, 40 cents ; mailing price, 45 cents THIS one-year course in physics has grown out of the experience of the authors in developing the work in physics at the School of Education of the University of Chicago, and in dealing with the physics instruction in affiliated high schools and academies. The book is a simple, objective presentation of the subject as opposed to a formal and mathematical one. It is intended for the third-year high-school pupils and is therefore adapted in style and method of treatment to the needs of students between the ages of fifteen and eighteen. It especially emphasizes the his- torical and practical aspects of the subject and connects the study very intimately with facts of daily observation and experience. 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