IN MEMORIAM FLOR1AN CAJORI v-w of inverse squares applied to electric force, so nearly proved it as to lead one to think that more careful experimentation, were such possible, would demonstrate the exactness of the law. Assume, therefore, the law and trace the results. Let p be the electric density, or the amount of electricity per square centimetre of surface. On a sphere removed from other conductors the density is uniform, and the quantity on two surfaces varies as the area of the surfaces. The quantity on the surface AB (Fig. 98, " Thompson ") is p x Area AB pA. The quantity on CD is p x Area CD = pC. Assuming that electric force varies inversely as the square of the distance, the force exerted on a unit of elec- tricity at the point P inside the sphere by the quantity on ABis The force exerted on the same unit by the quantity on CD is 1 6 NOTES ON Cf Since the tangents drawn to the sphere are equally in- clined to JSC C x BP* A\C\\BP\ CP* , orA = = ' LP* substituting above/ = /'. If the sphere be cut up into small cones 2f = 2f ; or in other words, if electric force varies inversely as the square of the distance, there should be no resultant force on the inside of a closed conductor. If it follows any other law /cannot equal/'. The most careful experiments fail to detect any force existing, and in corroborating the result of the above demonstration, confirm the hypothesis made that electric force follows the law of inverse squares. 1 6. Capacity (3 246). The capacity of a conductor is by the definition given a fixed quantity, while "the amount of electricity the con- ductor can hold," the definition as generally given by be- ginners, is variable, depending on the potential as well as on the capacity. An illustration may make the distinction clearer. If a jar has a volume of one litre, its capacity is a litre, and it will hold a litre of air at atmospheric press- ure. If the pressure be doubled, however, the quantity of air in the jar is also doubled, although the capacity is the same. "The amount" the jar "will hold" is evi- dently determined by the capacity and pressure. If the pressure be unity, that of one atmosphere, the capacity is then the amount the jar actually holds, but quantity and capacity are under other conclitious different. With ref- erence to electricity, the capacity is similarly the charge ELECTRICITY AND MAGNETISM. I? the conductor will hold at unit potential, or the charge which will raise. the potential to unity, and the actual charge in a conductor is the product of the capacity and potential. 17. Unit of Capacity ( 247). In an electrified sphere, as the surface is an equipoten- tial surface, the charge may be considered as concentrated at the centre and the potential at the surface is - As the capacity is equal to the quantity divided by the poten- tial Capacity = - = r. A sphere of one centimetre radius is, therefore, of unit capacity. 18. Electric Force exerted by a Charged Plate ( 252). Let a be the radius of the plate and r the radius of the ring x, x,' x." p is the density, or the charge per unit of area. The quantity on any small circular element is p (27trdr). The force exerted by this quan- tity on a unit at O is p(2itrdr) and the force acting nor- mal to the plate is Fig. 5 1 8 NOTES ON 2-itrdr h \ h P [ vv- ., - = - ) , Since COS = * The total force exerted by the plate in a direction normal to its surface is, therefore * iitrdr h , C a 2 rdr ' = Integrating, = 27tph (h + r 2 )i I = 27tph h ' 2 7tp , 27tp (I COSO'). If O is very near the plate, or if the plate is very large, 6' = 90, and the electric force of a charged plate on a unit of electricity very near it is 2itp. Care must be taken not to confuse this with the force exerted by a sphere as deduced in g 251. If a plate is charged with positive electricity, and a positive unit is placed very near it on each side, the force will be one of repulsion in each case, but if one unit is repelled upward the other tends to move downward, and if one force is 2it p, the other must be 2rtp. The force changes, there- fore, by 47tp in passing through any charged surface. 19. Dimensions of Units ( 258). An important use of the dimensional equations is in the conversion of units based on one system of fundamental units of mass, length and time to others based on different fundamental units. The French use units based on the metrical system, and although the centimetre-gramme- second, or C. G. S. system, is now almost universally used ELECTRICITY AND MAGNETISM. 19 in electrical work, there are many observations made and recorded in which other units are used. In electrical work the English have heretofore used the foot-grain- second system, and it is still used in some government observatories. It is a matter of the highest importance, therefore, that the method of converting values expressed in one system to corresponding values in another should be thoroughly understood. As the ratio of the centi- metre to the foot is that of I to 30.48, it is evident that the ratio of the units of length in the C. G. S. and foot- grain-second systems is the same. The units of area in the two systems are the square centimetre and the square foot, and no one would think of saying that this ratio was the same as the preceding -, but rather - 1 . Similarly the ratio between the units of volume (3048)' i i 3 is not but . 5T7-. This is exactly the relation 30.48 (30.48)' shown by the dimensions of area and volume on page 211, they being respectively Z, 2 and D . In simple cases like the above the change is easily made, but in others, where the dimensions of the unit are more complex and the unit itself an unfamiliar one, the dimensions must be used to calculate the ratio. As an illustration, let it be required to express in units of potential based on the foot, grain and second, the difference of electrostatic potential expressed by 2.7 C. G. S. units of potential. The dimensions of electrostatic potential are M^ L T~ , The C G. S. unit /;;Ai//\i = \M) \L) then Foot-Grain-Sec, unit _ 30.48; = >7II6 20 NOTES ON .'. I C. G. S. unit = .7116 Foot-grain-second unit 2. 7 C. G. S, =. 1.92 Foot-grain sec. The ratios between the different units in the two systems are given in Note 40. 20. Attracted Disc Electrometers ( 261, p. 215). Let the difference of potential between two plates be V, and the distance apart be D. By Note 14 the average y electric force between the plates is . As proved in Note 1 8, the electric force changes by ^itp in passing through a surface, and being zero in the conductor is therefore y 47tp just outside. Equating, /o = - . The density on each plate being p, the attraction exerted by the lower plate on a unit of electricity on the upper one is 2jrp when the plates are near each other. The upper plate contains, however, Sp units and the total attraction is iitp x Sp = 21. Absolute Electrometer (p. 216). Sir William Thompson's absolute electrometer, so named from giving the potential in absolute units, is an attracted disc electrometer. The disc C (see Figure TOO, " Thompson") is held in place by springs, instead of a counterpoise as shown, and is in metallic connection with B. When no part of the appa- ratus is electrified, small weights are placed on C, to bring ELECTRICITY AND MAGNETISM. 21 it into a standard position such that a small hair attached to it is seen midway between two dots, as shown in the figure. The weights are then removed and B and C con- nected to one of the bodies whose difference of potential is required and A to the other. The electric force of attraction between the two plates will act to lower C, but as the accuracy of the instrument depends on its being in the plane of B, the plate A is moved up or down until the force of attraction is such as to bring the movable plate into this standard position, which is known by see- ing the hair again midway between the dots. It is now under the attraction of the electrical forces, in exactly the same position as when acted upon by the weights, and the two forces are therefore equal. Substituting, there- fore, for .Fin the formula of Note 20, 981 times the weight in grammes required to bring C into the standard position, and for D the distance in centimetres between A and C t all quantities in the equation are known and the difference of potential may be calculated. Another method is more common, dispensing with the use of weights. If the difference of potential between two bodies P and P' is required, one of the bodies is connected to A, and B and C are then electrified to a high potential. The plate A is then moved up or down until the plate C comes into the standard position, the hair showing mid- way between the dots, and the distance D of A from Cis noted. Then Potential of B - Potential of P = D y The plate A is next disconnected from the first body and connected with the second. As the difference of potential between A and Chas now been changed, the force acting between the plates is different and C is no 22 NOTES ON longer in the standard position. It is brought there by raising or lowering A, and when adjusted the distance D' between the plates is noted r Then Potential of B Potential of P' = D 'i/ 8 ^ V S. Subtracting this from the former, Difference of Potential between P and P' (DD') x constant of instrument. It is of course necessary that the potential of B should be the same in both cases. This is verified by a separate attracted disc, which is in each case electrified until its attraction for another disc at a fixed distance brings it into a standard position. The absolute potential of B is immaterial, the only requirement being that it should be the same in each case. If the absolute potential of P is wished, connect A first to P and then to earth. ELECTRICITY AND MAGNETISM. 23 III. THEORY OF MAGNETIC POTENTIAL. (Thompson's Electricity, pages 265-278.) 22. Magnetic Field. Any region throughout which forces act is called a " field," but the term is more frequently used in connec- tion with magnetic than with other forces. A magnetic field is, therefore, a region in which magnetic effects are produced. Any movement of a magnet pole can take place only in a magnetic field, and the term is of use, as it disregards all ideas of how the field is caused, and con- siders only the forces and the direction in which they act. If at any point a line is drawn indicating the direction of the force at that point, it is called a line of force. This direc- tion is that shown by a magnet placed at the point, and may therefore be easily determined by experiment ; but as any representation of a magnetic field must present the whole field at once, the determination of the position of the lines of force by this process would be tedious. The reasoning in \ 126 leads to an easier though less accurate method, but one of great utility in enabling clear concep- tions to be formed. If a magnet is covered by a sheet of paper and iron filings are sprinkled over the paper they will on being gently tapped arrange themselves in curves passing from pole to pole. From the definition of lines of force, these curves must be the lines of force in the plane of the paper, and the mind has only to conceive the space above and below the magnet to be similarly filled to gain a clear idea of the field. It is necessary, however, to 24 NOTES ON know not only the direction of the force at any point, but also its strength, and a correct plotting of the field must furnish this. Maxwell has shown that if in any part of their course, the number of lines of force passing through unit area of a perpendicular plane is proportional to the strength of the force there, the number passing through unit area in any other part of the field is in the same pro- portion to the strength in that part. The closeness of the lines of force is therefore a measure of the strength of the forces of the field, or, as more commonly expressed, of the intensity of the field. By drawing the lines of force therefore in this way, the strength and direction of the forces in all parts of the field are indicated. As a south pole moves always in the opposite direction to that in which a north pole moves, it is necessary in order to establish the direction of the force to consider the nature of the pole acted upon. All investigations in magnetism are made by considering a north pole free to move, and the positive direction of the lines of force is therefore that in which a free north pole moves. This definition is of great importance in many of the demonstrations given later. 23. Mapping a Field by Lines of Force. A magnetic field is of unit intensity when unit pole is acted upon by a force of one dyne. As by definition unit pole acts on an equal and similar pole at a distance of one centimetre with a force of one dyne, it follows that unit pole causes unit field at unit distance. As in- tensity of field is measured by the force acting on unit pole, unit field exists at a greater distance from a more powerful pole. It is, therefore, unnecessary to consider the question of distance from the pole producing the field, but simply bear in mind that the intensity of the field at ELECTRICITY AND MAGNETISM. 25 any point is measured by the force in dynes acting on a unit pole at that point. If the pole be of a strength m, the force with which it is attracted or repelled is ;// times that experienced by unit pole, or Force acting: on pole Intensity of field = = -^~ f - Strength of pole The value of H, or the strength of field, is given numer- ically. Thus the horizontal force of the earth's magnetism at London being .18, a pole of unit strength is impelled to move in a horizontal plane by a force of .18 dynes. A pole of strength 100 would be acted on by a force of 1 8 dynes in a horizontal direction, or by a force of 47 dynes in the line of dip. To represent the field graphic- ally, recourse is had to Maxwell's demonstration of the fact that the number of lines cutting unit area in different points of the field is proportional to the intensity at those points, and the numerical value of //"is interpreted as the number of lines per square centimetre of a surface per- pendicular to the direction of the lines. Thus at London the earth's horizontal field would be represented by draw- ing horizontal lines of force in the magnetic meridian, equidistant and so spaced that they cut a vertical east and west plane at the rate of .18 per square centimetre or of one line to every 5.56 + square centimetres. The positive direction is toward the north. The total field at London would be represented by lines of force in the di- rection of the dipping needle, equidistant and spaced so as to cut a perpendicular plane at the rate of .47 per square centimetre, or one to every 2.13 square centimeters. These lines projected intersect at the magnetic pole, and are, therefore, sensibly parallel within ordinary limits, in which case the field is said to be uniform. 26 NOTES ON 24. Equipotential Surfaces, Being surfaces in which no work is done in moving a unit pole, are necessarily perpendicular to the lines of force. If not, some component of the force would act, and work would be done in moving against it. Knowing the direction of the lines of force, the equipotential surfaces can be readily drawn by cutting all the lines at right angles. Like lines of force they may be drawn in any number required, but it is customary to have them rep- resent unit difference of potential, and this requires that they should be so far apart that an erg of work is done in moving a unit pole from one to the other. The dis- tance may be readily calculated. Work = Hx, but by definition work is unity I y . . L H ' or the distance of the equipotential surfaces is inversely as the intensity of the field. The field may, therefore, be represented in this way as accurately as by the lines of force. Taking the case already considered, the earth's horizontal field at London would be represented by verti- cal surfaces, sensibly planes, extending east and west and 5.56 + centimetres apart. 25. Lines of Force due to a Single Pole. In the case of a free pole of strength in, the number of lines of force is determined by the fact that there are ;;/ lines intersecting every square centimetre at unit dis- tance, or that there are m lines cutting every square centi- metre of a sphere of unit radius. The surface of this sphere being 471 there are in all ^itm lines of force radial- ELECTRICITY AND MAGNETISM. 27 ing equally in all directions. The equipotential surfaces are spheres, and the radii may be found from the for- mula for magnetic potential, V .which may be here assumed to be correct. By substituting values for Fdif- f . , . . f mm fenng by unity, r is found to be successively ;;/, , etc. As a free pole can never exist, but is always as- 4 sociated with another of equal but opposite polarity, any actual field is much more complex, but the cases given will illustrate the application of the principles traced, and will give clear ideas of the conventions made which underly further investigation. It is possible without changing the number of lines of force to change their distribution in the field, and as it is frequently desirable to intensify a certain part of the field, the method of doing so becomes of importance. A com- parison of Figures 52 and 53 " Thompson's Electricity," shows that the distribution may be greatly changed by an arrangement of magnet poles in the field, and as iron near a magnet becomes magnetized by induction, the field is similarly affected by the introduction of iron. In this case the iron seems to gather the lines of force in the vicinity, causing a great number to pass through its sub- stance. Iron placed near a magnet pole becomes mag- netized, so that dissimilar poles are adjacent, producing the state of affairs shown in Fig. 52. Jenkin compares this peculiarity of iron in concentrating the lines of force to that of a lens in converging rays of light. It is likewise possible to screen any part of a magnetic field from in- duction, by inclosing it in an iron shell. It may be easily demonstrated by experiment that if an iron ring be placed between the poles of a horse-shoe magnet, no lines of 28 NOTES ON force pass through the interior of the ring, but entering at one side pass through the metal of the ring issuing on the opposite side. A magnet inside an iron sphere is in- dependent of all outside influences. By the use of iron it is, therefore, possible either to concentrate the lines of force, or to divert them entirely from any desired part of the field. 26. Lines of Force due to a Current. As shown in g 191, the lines of force due to a current are circles perpendicular to the current and having it for a centre. If the conductor is straight, the circles are all in parallel planes, and the equipotential surfaces are planes radiating from the conductor and each containing it. The number of these planes is such that an erg is required to move unit pole from one to the other. As shown below, the intensity of the field at unit distance is 2(7. A pole moving in a circle of unit radius having the conductor for a centre passes over a distance of 2?r against a force of 2C, doing 4?r(7 ergs of work. The number of equipotential surfaces is, there- fore, 4-TtC. After the pole had made one revolution it would reach the equipotential surface from which it started, but hav- ing done 4/rC ergs in its revolu- tion the numerical value of the surface would now be 471- C more than before. It is impossible, A therefore, to give an absolute value to an equipotential sur- . 6- face due to a current. Let LL be a portion of an infinite rectilinear current, and let ELECTRICITY AND MAGNETISM. 29 BC be the force exerted by an element of length, dx> of this current = Cdx. Place a unit pole at A. The force exerted by the element dx at A is _ Cdx cos Q ^ Zg* If <9^ = i, ~A& = i + x 1 i Cdx. Vi + x- _ Cdx ar = t TT The total force = F C \ T~a = c T cos 6 ^ 9 - Loot 1 + *) J'i* Integrating, ^=Csin0 zC = //. zC = To find the intensity at a distance r from the conductor we have from Note 24 that the intensity is the reciprocal of the dis- tance between two equipotential surfaces. There being 4rtC surfaces cutting a circumference of 2rtr, the distance between them is - and hence 26? 27. Magnetic Potential. Magnetic potential has already been alluded to in Note 6. The conception is strictly analogous to that of elec- trostatic potential, and the demonstration given for the formula for electrostatic potential in Note n, is applica- ble to magnetic if for q a quantity of electricity, is substi- tuted m a strength of magnet pole. The same reasoning leads to the formula of V =. 2 r 3 NOTES ON Magnetic potential at a point equals the potential energy possessed by a unit north pole at that point, and is measured by the work in ergs done in bringing a unit north pole from infinity to that point. Zero of magnetic potential exists at an infinite distance from all mag-nets. dV Magnetic force is , or the rate of change of potential per unit of distance as in Note 14. The difference of magnetic potential between two points is measured by the work clone in moving a unit north pole from one to the other. Wherever work has to be done in moving a north pole, it would be done in re- sisting the motion of a south pole. In all investigations of magnetic potential, force or work, a unit north pole must always be considered. 28. Tubes of Force. The conception of tubes of force is frequently of utility. The lines of force radiating from a pole may be regarded as forming cones, and any section through a cone would cut all the lines of force. But as the number of lines of force is proportional to the intensity, the force on all cross sections of the cone is therefore the same. By conceiv- ing the magnetic force to be equal throughout the cone, existing between, as well as along the lines of force, a more accurate idea of the field is attained. It is easy to imagine a field of so slight intensity that a square centi- metre would not have any lines of force passing through it. The example of the earth's horizontal field at London, already referred to, is a case in ooint. One line of force passes through every five units of area, but the magnetic forces are felt just as strongly on the four units through which the line does not pass as on that which it cuts. By ELECTRICITY AND MAGNETISM. 31 thinking, therefore, of the lines of force as indicating only the direction and strength of the forces which act between them as well as in them, this difficulty is overcome. 29. Intensity of Magnetization. If a bar magnet be broken in half, instead of obtaining one piece of nortli and the other of south polarity, each is found to possess both and to be a perfect magnet. How- ever far the subdivision be carried, the result is the same, and the ordinary explanation is that the magnet is an aggregation of magnetized molecules, the magnetic axes of the molecules being to a greater or less extent parallel. If this were so the north pole of one molecule would be counteracted in its magnetic effects by the south pole of the next, and the only molecules capable of exert- ing external magnetic effects would be those on the sur- face, and the effect is exactly the same as would be pro- duced by a distribution of a magnetic matter or fluid, or avoiding the idea of a fluid, a distribution of magnetism over the surface of the magnet. The amount of magnetism per unit area is called the magnetic density. If the magnet- ism is regarded as being uniformly distributed throughout the mass of the magnet, the quotient of the magnetic cur- rent by the volume is called the intensity of magnetization. Let p be the magnetic density, a the cross section and / the length of the magnet. Then the strength of pole is m p a m I p a I ml ' = Tt _ Magnetic moment Volume The intensity of magnetization and magnetic density are therefore practically the same, the one presupposing a 32 NOTES ON uniform distribution of magnetism throughout the mass, the other a surface distribution. If the magnetism is due to the bar being situated in a magnetic field, the intensity of magnetization is equal to k H, k being what is called a " coefficient of magnetiza- tion." A few values of k are given in 340 (" Thomp- son "). Assuming Barlow's value for iron 32.8, the formula, intensity of magnetization k H indicates that the in- tensity of magnetization is dependent only on the intensity of the field ; but there is found to be a limiting value of magnetization which cannot be exceeded, however power- ful the field is. This is stated to be for iron 1390 (p. 269, " Thompson"), and the strongest field that could be util- ized in magnetizing iron is therefore ^9 = 42.4. The value of this coefficient is, however, uncertain, and appears to be much less at a high intensity of magnetization than at a low 30. Solenoidal Magnets. A filament of magnetic matter so magnetized that its strength is the same at every cross section is called a magnetic solenoid. A long thin bar magnet uniformly magnetized is called a solenoidal magnet, or simply a solenoid, in distinction to a magnetic shell. The name solenoid is also applied to a helix through which a current passes. (See Note 42.) As the magnet poles are points at which the magnetism of the magnet may be supposed to be concentrated, and from which magnetic forces act, the potential of any point near the magnet is determined by its distance from the two poles, or V 2 = m ( , j. The exact position of the poles is difficult to determine, but is stated in g 122 to be in long thin steel magnets ELECTRICITY AND MAGNETISM. 33 about rV of the distance from the end. If the poles are bent to meet, forming a ring, r = r' for all external points, and there is therefore no potential due to a magnetized ring. 31. Potential due to a Magnetic Shell. As defined in \ 107 (" Thompson") a magnetic shell is a thin sheet so magnetized that the two sides of the sheet have opposite kinds of magnetism. The demonstration and use of the expression for the potential due to a mag- netic shell requires a preliminary definition of a solid angle, and of the method of measuring it. The solid angle subtended at any point by a closed curve or surface is measured by the area of a sphere of unit radius de- scribed from the point as a centre, intercepted by lines drawn from all parts of the curve to the point. (See Fig. 64, " Thompson.") As the areas of similar surfaces on spheres are as the square of the radii the solid angle, &> = area on unit sphere, A _ area on sphere of radius r _ area on sphere of radius r^ ~^\~ To compute the solid angle. When the closed curve is circular and the point is in its axis it is necessary only to compute the area of a zone of one base on a sphere of unit radius. The area of the zone formed by the revolution of AD around AO as an axis is (see Chauvenet's Geometry, Book IX., Prop. X. Cor. Ill) Ad x 2 it OA. But Ad = A O dO r r cos .'. Area = zitr (r r cos 6) ; 3 Fig. 7. 34 NOTES ON but if r i, area is solid angle, .'. GO = 27t (i cos 6). To calculate the potential. Let r, and r 2 be the distances from the point Z> to the faces of the small element ds, ft be the angle between ds and its projection ,D Fig. 8. perpendicular to r 1 and p be the magnetic density. The strength of the shell i being the product of the density and thickness, - i = pt .-. p = -. The quantity of magnetism on the small element ds is dm = ds.p = ds. (I) The potential at D due to ds is - *=<*-). . . (a) ELECTRICITY AND MAGNETISM. 35 But cos ft = ; - 1 . Substituting this value and that of dm rfF=-cos/S (3) r~ But as ds is an infinitely small element, its plane projection ds cos ft perpendicular to r l is sensibly equal to the area on a sphere of radius r. The solid angle, therefore, subtended at D by the element ds is _ ds. cos ft f" Substituting in (3) dV = deal V ooi. 32. Equipotential Surfaces and Lines of Force of a Magnetic Shell. The computation of the solid angle is, as shown, simple when the point D is on the axis normal to the shell at its centre, but when D is oblique the area becomes an ellipse on a spherical surface of unit radius, and the calculation is extremely difficult. From the formula, however, for magnetic potential a few relations are readily deduced. As at all points where the shell subtends the same solid angle the potential is the same, any equipotential sur- face is evidently most remote from the shell on the axis normal to its centre. As the point of view becomes oblique it must approach the shell that the solid angle may be the same, and at all points in the plane of the shell the solid angle, and consequently the potential, are zero. The general form of the equipotential surfaces is, there- fore, that of deep bowls concave to the shell, and most remote from it on its perpendicular axis. At a point close to the shell the solid angle is a hemisphere or 2Tt and the 36 NOTES ON potential 27rz". On the opposite side the potential is 27T/, or 47T* ergs must be expended in moving unit pole from one side of the shell to the other. If the equipotential surfaces indicate unit difference of potential there are, therefore, ^Tti surfaces. From the equipotential surfaces the direction of the lines of force may be traced, as they start from the side of the shell having north polarity and curve so as to cut each surface at right angles, finally en- tering the south pole of the shell at right angles. 33. Work Done in Moving Pole near Shell. Potential being measured by the work done on unit pole in bringing it up to a point from an infinite distance, the work done on a pole of strength in is mcoi. It is pos- sible under the conventions made as to the number of lines of force to express this in another way. As already shown, the number of lines of force given off by a pole of strength ;;/ is ^nm, but as these radiate in all directions, they are given off throughout a solid angle 47? subtended at the centre of a sphere. Through any solid angle &? the number of lines is, therefore, moa. Calling this number N, the above expression becomes Ni, or the work done in bringing a pole up to a position near a magnet shell is measured by the product of the strength of the shell and the number of lines of force of the pole cut by the shell. This is evidently a measure of the work done either in bringing the pole up to the shell or the shell to the pole, and is, therefore, sometimes called the mutual potential of the pole and shell. The work done in bringing the pole from infinity to a point where it intercepts N l lines is Nil. If now it be moved to another in which it intercepts N* lines, the work done between the points is Work done = i (N., - ,V,). ELECTRICITY AND MAGNETISM. 37 The difference of potential between the points is i (Ni A T ,) ^ -' or magnetic potential being the quotient of work done in moving pole, by the strength of pole. The work done may be either positive or negative and the above expression may, therefore, have either sign. If A 7 ! > A 2 the work done in passing from N\ to A 2 is negative, or the shell tends to move in such a direction as to include a minimum number of lines of force. As these pass in the positive direction, exactly the same relation is expressed by saying that a magnetic shell in a field tends to place itself so as to enclose the maximum number of negative lines of force. If the north pole of a magnet shell is brought up to the north pole of a magnet, this relation is readily seen, as the shell will be repelled into a position in which it will enclose as few lines of force taken in the positive direction as is possible. If the same face be ap- proached to a south pole, it is attracted and moves into a position in which the maximum number of lines cut the shell in the negative direction. 34. Equivalent Magnetic Shells. The relations deduced for magnetic shells are of great service, as they are applicable to the case of a voltaic circuit in a magnetic field. If a wire carrying a current be looped into a circle, the lines of force which ordinarily encircle the conductor combine to act in the same direc- tion on a pole at a distance from the circuit. Thus in Fig. 86 (" Thompson "), it is seen that all the lines of force due to the current pass in the same direction through the plane of the circuit as do those of a magnetic shell. A closed voltaic circuit in a magnetic field, as may be readily 38 NOTES ON shown by experiment, is acted upon as a magnet would be. It is found that the magnetic effects of the north pole of a magnet are identical in nature with those of a circuit, in which the current flows in a direction opposite to that in which the hands of a watch move. This direction is known as the negative direction of the current, and the magnetic effects of a positive pole and of a negative cur- rent are, therefore, similar. Looking at the other side of the loop, the current would appear to pass in the direction in which the hands of a watch move, or in the positive direction ; but if a north pole be approached to the circuit from the side on which the current appears to have this direction it is attracted, showing that a positive current produces magnetic effects similar to those of a negative pole. As the direction in which a north pole moves shows the direction of the lines of force, it is seen from the above that the lines of force enter that face of the plane of the circuit in which the current appears to move " with the sun," or in the positive direction, and emerge from the other face. As it is a matter of great importance to be able to connect the direction of the lines of force with that of the current to which they are due, several rules have been given, one of the best of which is the comparison between the direction of rotation of a corkscrew and that of the motion of its point. If the wrist be rotated in the right-handed direction, the point advances ; and con- sidering the motion of the wrist to be that of the current, the movement of the point corresponds to that of a north pole, and indicates the direction of the lines of force. This relation is said to be that of " right-handed cyclical order," and the direction of the current and of the lines of force are spoken of as being thus related. The magnetic action of a voltaic circuit is found to depend upon the strength of the current, and on the area ELECTRICITY AND MAGNETISM. 39 of the enclosed surface. It is, therefore, evident that for every closed circuit, a magnetic shell whose edges coin- cided in position with the circuit could be substituted, if a certain relation were established between the units measur- ing the strength of the current and the strength of the shell. This relation is that expressed in the definition in g 195. The absolute electromagnetic unit of current is that current which in passing through a conductor one centimetre long, bent so as to be in all parts distant one centimetre from a unit pole, acts on the pole with a force of one dyne. By this definition unit current produces unit field at unit distance. But so does a shell or pole of unit strength. By expressing, therefore, the current in these units, the magnetic effect of the current is the same as that due to a magnetic shell whose edges coincide with the circuit, and whose strength is equal to that of the cur- rent. This is called an equivalent magnetic shell, and all relations hitherto traced for the shell are now applica- ble to the closed circuit. 35. Potential due to a Closed Voltaic Circuit. The potential due to a current at a point is therefore Coo, where Cis the current measured in absolute units, and GO is the solid angle subtended by the circuit at the point. As, however, a positive current produces the same magnetic effects as a negative pole, the sign of the potential is always the opposite of that of the current, or The difference of potential between two points is The magnetic force due to a current at a distance x is (Note 14) 40 NOTES ON dx The potential is iitC on one side of the circuit and 2.TfC on the other, changing by 4?rC. Hence there are 4;rCequipotential surfaces. 36. Work Done in Moving a Circuit Near a Pole. This is a problem of the greatest importance, as it under- lies the action of the dynamo machine. As already traced (Note 33), the work done in moving a magnetic shell near a pole, or conversely the pole near the shell, is Work done = moot = Ni. Similarly the work done in moving a closed circuit near a pole is Work dons = moaC = NC, N being the number of lines of force of the pole passing through the circuit. If the circuit be brought up from an infinite distance to a point where it intersects /V, lines due to the pole the work is JV } C. If now moved still farther so that it intersects a greater number, /V 2 , the work done between the points is V/ork = and Difference of potential = - C(A/2 ~ N $ = - C(oo, - GO,). If TVa > N\ the work is negative, and the circuit tends to move therefore in such a manner as to make the num- ber of lines enclosed a maximum. If a circuit be placed in a magnetic field so that the lines of the field while parallel to those of the current pass in the opposite direc- ELECTRICITY AND MAGNETISM. 41 tion, the circuit will, if free, first turn to bring the lines in the same direction, and will then move to make the num- ber enclosed a maximum (Fig. 87, "Thompson"). It may be useful to obtain an expression for the work done in this last case, as an understanding of the theory will assist in the comprehension of the working of electric- motors. Imagine a closed circuit placed in a uniform field. If the circuit be moved parallel to itself, the num- ber of lines enclosed is constant, and consequently no work is done in whatever direction the circuit be moved. If, however, the coil be rotated on an axis in its plane, it will enclose a varying number. If be the angle between the normal to the plane of the coil and the direction of the lines of force of the field, and the number of lines passing through the coil when its plane is perpendicular to them be N t the number enclosed when at an angle 6 is N cos 0. If the angle be now changed to 6' the number enclosed is JV cos V , and the work done in passing from one position to the other is Work = - C(Ncos 0' - NCOS 6). Suppose that the coil be rotated. The work done is easily calculated : In the first quadrant, work = - C (Ncos 90 - ^Ycoso ) = CN\ in the second quadrant, work = - C(A^cos 180 - N cos 90) = CN; in the third quadrant, work = C(A^cos 270 A^cos 180) = CN\ and in the fourth quadrant, work = - C(/Vcoso - A 7 cos 270) = - CN. In the first half of the revolution, therefore, work equal to 2 CN has to be done in order to more the coil, but in 42 NOTES ON the latter half the coil will do the same amount of work. The potential energy of the coil is, therefore, greatest when = 1 80, or when the lines of force of the field are parallel to those of the coil, but in the opposite direction, and if the coil be then left free to move, it will rotate to make 6 0, doing work equal to 2QV, and then requiring work to be done on it to cause further movement. On the supposition already stated that the field is uniform, N = HA, H being the intensity of the field and A the area of the coil. The work done by the coil in rotating through two quadrants may then be expressed as 2CHA, this also measuring the work done on the coil in the other two quadrants. As a resume of the above, we have the rule, that a mov- able circuit in a magnetic field tends to place itself so as to enclose the maximum number of lines of force in right-handed cyclical order. 37. To Calculate the Intensity of the Field due to a Voltaic Circuit. The force acting on unit pole, or the intensity of the field, is by Note 35 the rate of change of po- tential per unit of length. The in- tensity of field at a distance x is The difficulty of calculating the value of oo makes the general solu- tion extremely complicated. It is, however, easy to calculate the inten- sity of the field at any point on the P. axis of the circuit, as in that case GO 2it (i cos 6). Let x = the distance of the point A from the circuit. r r= radius of the coil. ELECTRICITY AND MAGNETISM. 43 Then dV = zrtC.d (L cos 0) cos = dx 2.x Cr* At the centre of the circle the force is a maximum, and is = as in 195. The sign shows that with a posi- tive current the force is one of attraction. 38. Position of Equilibrium of a Circuit and Magnet. Consider the magnet as composed of two poles of strength m and m connected rigidly. -. The formula for the force in the field may be written = sin 3 0. The forces acting on the two poles are and /I 1 1 \ ^v. e X & > / A B --- sin r Fig. 10. The resultant force is ~ (sin^ - sin' G'). This is zero when = 0', or when the centre of the magnet is at the centre of the coil. In any other position there will be a force acting and the equilibrium will be unstable. If, there- fore, either the coil or magnet is free to move, the coil will NOTES ON place itself so that the middle of the magnet is at its centre. (See Fig. 87, " Thompson.") 39. Mutual Potential of two Circuits ( 320). The work done in bringing one circuit up to another, or the " mutual potential " of the two circuits is, as given in , cos s | 320, cc . ss . This expression is one of great theoretical importance, but its derivation is difficult and out of place in an elementary treatise. The work done in moving a circuit near a pole or in a field has already been shown to be NC, and it is obviously immaterial whether the lines offeree TV are due to a pole, a magnetic shell or another circuit. Suppose two circuits A and B, carrying currents of strength C and C,' and let N^ be the number of lines of force due to A enclosed by B, and N* the number due to B enclosed by A. If B is moved out of the field caused by A the work done is A^ C If A is now moved so as to resume its former relative position to B the work done is N^ C'. The coils are now in the same relative position as at first and if there are no external magnetic forces, no work can have been done in moving the system. Hence Ni C-N n C = O. If the current in each is of unit strength .M = N v or each encloses the same number of the other's lines of force. Returning to the expression for the work done. CC ^- ss', and making Cand C' each unity, the number of lines enclosed by each is - ss', and this number may ELECTRICITY AND MAGNETISM. 45 be represented by the single symbol J/and is dependent only on the position and areas of the two coils. Let the planes of the two circuits be parallel, and the current flowing in the same direction in each, that in which the hands of a watch move. The negative sign of the formula shows that the circuits attract each other, and this is also evident from Maxwell's rule that a voltaic cir- cuit free to move always places itself so as to enclose the greatest possible number of lines of force. The nearer the circuits, the greater the value of M, and if they become coincident M would be , or infinite. As, however, r will o always have a finite value, the maximum value of M exists when the coils are touching, or r is a minimum. As the coils tend to approach or to diminish, r, the " coefficient of mutual potential " M, always tends to a maximum. This quantity is hereafter referred to (Note 63) as the " coefficient of mutual induction." 40. Conversion of Units ( 324). The use of the dimensions of units in passing from one system to another has been illustrated in Note 19. In electrical calculations, the most frequent change to be made is that from the C.G.S. system to the British units based on the foot, grain, and second. The ratios between these units are shown in the following table from Jenkin's " Electricity." 46 NOTES ON i a Number of C.G.S. 5 Units in j^J one British bco Unit (^4). ^ f Number of British Units in one C.G.S. Unit (B). g Mass o B T T,-.S | r Length . . "O Time.. . r 1.1' |] Force ._ p 1; ["Work * 60.198 1.7795820 2.2204179 0.01661185 ;*' Quantity ... ? 42.8346 1.631794912.3682051 0.0233456 3 Current. * or C 42.8346 1.6317949 2.3682051 0.0233456 ^ g Potential V 1.40536 0.14778741.8522125 o.7 T i=;6i i Resistance 0.03280899 2.5159929 i. 4040071^0.4704!; w Capacity r 30.47945 !l-4S4007i;2.^I!?QQ2Q 0.03280899 r Strength of Pole... m 42.8346 1.6317949 2.3682051 0.0233456 gl Magnetic Potential. V 1.40536 0.1477874 1.8522125 0.711561 g Intensity of Field... H 0.0461085 2.6637804 1.3362196 21.6880 * o c Current i or C 1.40536 0.1477874 1.8522125 0.711561 i Quantity ... j @ 1.40536 0.1477874 I.8522I25 0.711561 -Potential Electromotive F'ce f F E 42.8346 1.6317949 2.3682051 0-0,33456 o Resistance. . . E> Q Q o 5 Capacity c 30.47945 IJ^J^^/i 0.0328089912. 5159929 1.4840071 30.47945 The following table, showing the relations between the practical units in common use, may be convenient for reference : i metre = 39.37043 inches = 3.28087 feet. I kilogramme = 2.20462 avoirdupois pounds. i kilogrammetre, or i kilogramme raised one metre per second = 7.23307 foot-pounds per second. ELECTRICITY AND MAGNETISM. 47 i Force cle cheval or French H. P. = 75 kilogram- metres per second. I English H. P. = 33000 foot-pounds per minute. = 550 " " " second. := 76.04 kilogrammetres per second. = 1.014 force de cheval, I gramme 981 dynes (980.868 at Paris), i gramme centimetre, or I gramme raised one centi- metre in a second = 981 ergs. I pound avoirdupois = 4.45 x io 5 dynes nearly. I foot-pound per second = 1.356 x io 7 ergs nearly. i Volt = io 8 absolute electromagnetic units of potential, i Ohm io 9 " electromagnetic units of resistance. Practical Ohrn = .9895 x io 9 absolute units (Lord Ray- leigh). I Ampere = nj of an absolute electromagnetic unit of current. i Coulomb = iV of an absolute electromagnetic unit of quantity. = I ampere per second. i Farad = io~ 9 absolute electromagnetic units of ca- pacity. i Watt (see Note 55). = IO T ergs = .7373 foot-pounds, = TJB- English H. P. I thermal unit = i gramme of water raised i C. I Joule = io 7 ergs = .24 thermal unit (See Note 55). Mechanical equivalent of heat = 772 foot-pounds i F. = 1390- rc. Same in metric system = 424 kilogrammetres i C. = 42400 gramme-centimetres i C. = 4.16 x io 7 ergs. i Siemens unit = .9536 practical ohm. 48 NOTES ON i Jacobi = current evolving i Cm '. of mixed gas per minute at o and 760 mm. = .095 amperes. At any place the weight of the gramme is equal to g dynes. The value of g for any latitude may be found approximately from the formula g 980.6056 2.5028 cos 2/1 .000003^, A being the latitude and h the height above sea level. The limiting values of g are 978.1 at the equator and 983.1 at the poles. 41. Determination of the Horizontal Component of the Earth's Magnetism ( 325 a). The time of vibration of a particle acted on by a con- stant force is / = it , / being the time of a half or sim- ple vibration, and ju the acceleration. The latter is in any case equal to the moment of the impressed forces divided by the moment of inertia. When the arcs of vibration are small this may be applied to a magnet oscillating in a uniform field and the time of a complete or double oscilla- tion of a magnet is therefore K being the moment of inertia. M the magnetic mo- ment and //the horizontal intensity. A, To make the observation, a magnet is allowed to oscillate and the time of vibration is determined as ac- ELECTRICITY AND MAGNETISM. 49 curately as possible. This is best done by determining the approximate time of one oscillation, and allowing the magnet to oscillate a known time. Dividing this time by the approximate time of one oscillation gives the approxi- mate number of oscillations. Taking the nearest whole number to this, and dividing the whole time by it gives the exact time of one vibration. If the approximate num- ber of vibrations fell midway between two whole numbers, the observation would have to be repeated until it was known with certainty how many oscillations had been made in the observed time. The oscillations must be of small amplitude, and in very exact observations must be reduced to an infinitely small arc. If possible, the magnet should be supported by a single fibre to avoid torsion, but if a wire has to be used, the torsion must be allowed for. If the magnet is of simple form, either bar or cylindrical, the value of K may be determined from the formulas ing 325^, but if these do not apply, A' may be determined by observa- tion of the time of vibration of the magnet, and of the time, /,, when its moment of inertia is increased by the addition of a weiht of known moment K l . From (i) the value ofMffis - . . . (2) i When the weight is added, MH = ^ , + ^ or K = Knowing A" it is possible to compute Mfffrom (i). In the case of the oscillating magnet, the magnet and earth acted mutually on each other. In order to obtain the ratio T-^the force of the magnet must act against that NOTES ON of the earth, and this is done by measuring the deflection from the magnetic meridian that the magnet will cause in a small magnetic needle near it. A'S is the magnet the time of whose oscillation has been determined, and it is placed at right angles to the magnetic meridian, so that its centre is due north or south of the small needle at O. Let r be the distance of either A^or S from (9, and m -be the strength of N N 1 ^S and S. Then the force exerted by S on a south pole in 1 at O is Fig. ii. in the direction Oa, which may be taken to represent it in magnitude and direction. Similarly Oc would represent the force of attraction of the pole N. The re- sultant force is Ob: From similar triangles Oa : Ob : : So : NS ; or calling Ob, T, and NS t L, mm' mm' L t-:T::r:L .\T- -=. Let M' = ;'/', the moment of the small magnet, and M = mL the moment of the large one. Then the couples acting on the small needle when it has a perma- nent deflection are ; , a <*** mm LI' MM' to deflect, 77 cos = cos 6 = - 3 cos 6 ; to retain in the meridian, m'l'H sin = M' //sin 0. Equating these moments and reducing ELECTRICITY AND MAGNETISM. 51 ^ = r 3 tan0 ....... (3) By combining (2) and (3) J3, In the Kew Magnetometer the deflecting magnet is placed east or west of the small magnet instead of north or south, and the formula is different. Let r be the distance between the pole s and the centre ;* Fig. 12. of the deflecting magnet, L be the length of the deflecting magnet, m the strength of pole of the deflecting and m' that of the deflected magnet, and /' the length of the latter. The mm' force of repulsion exerted by 5 on j is -- - , r be- ing great in comparison with L. The force of attraction of TV on s is - . These forces may be considered as acting in the same line but in opposite directions, and the resultant force acting on s is = mm' z r _L _ \ L\* } r + - ) I a/ 52 NOTES ON (5) The moment of the deflecting couple on the small magnet is FT cos = 2AfM' . cos , r-?) 2 or since as above, L is small in comparison with r, its square and higher powers may be neglected, and the moment of the couple is *MM' .- !l_cosfl = 4 The moment of the couple tending to retain the small magnet in the meridian is M'Hsm 0. Equating and reducing ^=ir'tan0 (6) C. This derivation contains many assumptions. A more rigorously correct formula is that given by Kohl- rausch, M i r * tan 9 ~~ ri * tan 6 ' 7f ~ a ~ ""f^^C In which r and ri are the distances between the centres of the magnets in two successive positions, and and 0' the corresponding angles of deflection. The deflectir. B magnet is placed east or west of the deflected needle as in the last case. ELECTRICITY AND MAGNETISM. 53 The formula of (6) is the one generally used in work with the Kew Magnetometer, but is true only when r is large in comparison with /,. The more accurate formula can be readily derived. From equation (5) : r' The moment of the deflecting couple is zMM' Fl cos6 = T-5T- Li + - ' ~? )cos Equating this with M' H sin 0, the moment tending to retain the couple in the meridian, and reducing, By repeating the observation by placing the deflecting mag- net so that its centre is at a distance of r\ from the needle, a new deflection, 0', is obtained, and Multiplying (7) by r 5 and (8) by r, 8 , and subtracting the lat ter from the former, r* tan - r, 6 tan C' = 2 . ^ (r 1 - r 2 ) ; H or, 54 NOTES ON IV. MEASUREMENTS AND FORMULAE. 42. Solenoids ( 327). Ampere's Theory of Magnetism < 338). As stated in \ 327, a spiral coil of wire through which a current passes is called a solenoid. The definition has already been given as that of a magnetic filament uni- formly magnetized, and as a spiral coil carrying a current exerts the same forces, and is similarly acted upon in a magnetic field, it is called by the same name. Theo- retically the turns of the coil should be exactly parallel, and at right angles to the longitudinal axis, but as this is impossible the ends of the helix are brought in through the coil from each end to the centre as in Fig. 116. The current flowing in these branches exerts an effect equal and opposite to that due to the longitudinal component of the spiral, and the resultant effect of a solenoid thus con- structed is that of a number of parallel turns only, or "* the theoretical solenoid. If the helix be free to move it will, when a current is passed through it, move so as to include the maximum number of lines of force in the field. In the earth's field, it therefore assumes the same position as the dipping needle. The end pointing north is called the north pole of the sole- noid, as with the ordinary bar magnet. Let a solenoid be suspended so as to move freely, and a magnet be brought near it so that the north poles are nearest. The solenoid will be repelled, but if after repulsion, the south pole of the magnet is brought up, the north pole of the solenoid is attracted. Magnetic forces are evidently act- ing between the coil through which a current is flowing ELECTRICITY AND MAGNETISM. and the piece of steel which we call a magnet. Many other similar effects have been alluded to, and they suggested to Ampere a theory of magnetism, which ex- plains very many peculiar relations and is of great prac- tical utility. He conceived that a magnet was composed of a great number of molecules, around each of which flowed an electric current in a constant direction. In an ordinary unmagnetized bar of steel, these currents lie in all possible planes, so that their resultant magnetic effect is zero. If, however, the bar be magnetized, the energy expended in so doing operates to turn the molecules so that the currents are now parallel. In looking at the end of a magnet, the molecu- lar currents would, as shown in the figure, coun- teract each other in the substance of the magnet, but the current on the outer edge of the outer row of molecules being unbalanced would cause a resultant current on the surface in the direction opposite to the motion of the hands of a clock. In a solenoid a current flows in this direction when looking at the north pole of the solenoid, and the figure therefore shows the theoretical condition of the north pole of the magnet. Looking at the other face the currents appear to flow in the opposite direction. From these suppositions follows the rule given on p. 284. The theory explains many peculiar effects, such as those referred to in \ 112 and | 113, but is no more than a theory. It seems unques- tionable that the process of magnetization is attended by 56 NOTES ON molecular movements, but it is not proved that magnetism is due to molecular currents. Prof. Hughes has recently experimented on tempered steel, using an ingenious mod- ification of the induction balance, and asserts his belief that each molecule possesses magnetic polarity. In tem- pered steel the molecules are comparatively fixed, where- as in soft iron they possess considerable freedom of move- ment. He thinks the process of magnetization is merely one of molecular movement, by which the similar poles of the molecules are brought facing the same way. This final position is retained by the steel but lost by the iron. Ampere investigated the mutual action of magnets and currents by the theory of action at a distance between currents, but if in Fig. 13 the current in each molecule is supposed to have one line of force, the aggregation of molecules would produce the large number of lines all passing in the same direction that the magnet is found to possess, and the theory of lines of force due to Faraday and Maxwell explains all magnetic phenomena as well if not better than the method of Ampere. 43. Best Arrangement of Cells ( 351). From general formula, letting w be the total number of cells, w m n mE R E * . . (i) mr , r R mr R 4~ ./ t -i 4 n n m w m * Proof without calculus. In equation (i) + - = 2 , / ' Rr + ( / r.:r _ / R V \V ^~ V This is a minimum when the square it contains is zero, or when in-r mr In that case R = as before. w n ELECTRICITY AND MAGNETISM. 57 This is a maximum when the denominator is a mini- mum. Differentiating with regard to m and making first derivative equal zero. du r R r R m^r mr dm ~~ iv ~ m* ~ w ~ m' 1 O1 w ' ' n But is the internal resistance of the battery. Hence n the rule, that the best arrangement is secured when the internal resistance of the battery equals the external resistance in circuit. 44. Long and Short Coil Galvanometers ( 352). In the use of a galvanometer it is desirable that it should produce a readable deflection without greatly re- ducing the current. If the current is large a single turn of wire will cause a sufficiently strong field, but if the cur- rent is small, it is necessary to multiply its effect by pass- ing it through many turns in order to obtain a good deflection. By Note 37 the field at the centre of the coil is in a galvanometer having n> and - - in another having only one turn around its needle. The resistance of the first will be nearly n times that of the second. Let all the resistance in the circuit external to the galvanometer be r, and g be the resistance of the galvanometer, and E the E. M. F., supposed constant. Then C = ^ and C = r + g r + ng \{r is small C= nC' nearly, or the current is reduced by the high resistance galvanometer in almost the same 58 NOTES ON ratio that the field is increased. There is, therefore, no gain, and the great reduction of the current renders the use of such an instrument inadvisable. If r is large, the resistance is not increased n times by the introduction of ng instead of g, and C is and - - is less than r. The resistance of a divided circuit is, therefore, always less than that of any of the resistances entering into it. If there are three conductors, r, r', r", we have ELECTRICITY AND MAGNETISM. 59 = + ,+ = conductivity, R r r .r" and R = - , "T , . rr + rr' + r r 1 To find the current in each branch : Let C be the total current, B the battery resistance, O be the current in r', and C" that in r". Then The currents are inversely as the resistances through which they flow. Taking, therefore, the resistances be- tween A and B t Figure 129, r' r" r" also C: C" : : r" : r ' r " or C" -C -7-^-7, . . (2) r' + r" r' + r" 46. Shunts. These formulas are of great importance in the use ot shunts for galvanometers. If a current is so powerful that there is danger of its injuring the galvanometer coils, or if it produces a deflection too near 90, the galvanom- eter may be shunted by introducing a resistance in parallel circuit, so that less current will pass through the galvanometer. Letting C be the current when the galvanometer is B unshunted, C' the total current when Fiff- *4- the galvanometer is shunted, C g and C s the currents 60 NOTES ON through the galvanometer and shunt respectively, G the resistance of the galvanometer, S that of the shunt, and /? all other resistance, * E But as ~ is less than G, C' is greater than C, or Lr + O the introduction of the shunt has increased the total cur- rent in circuit. The current through the galvanometer is from (i), Note 45, c , 5^_ _ E S > (3) j.\. -r , G + ES ' R (G + S) + GS If the deflections of the galvanometer are proportional to the currents, .-. d : d' : : R (G + These formulas may be simplified if - ~ - be called u. This proportion is sometimes called the multiplying power of the shunt. By its use (3), the current through C' the galvanometer becomes . The current through the shunt is C' The resistance of the shunted gal- f~ *\ (^ variometer -^ - - becomes , and the resistance G + 5 u ELECTRICITY AND MAGNETISM. 6 1 necessary to add to the circuit to retain the same total current is G = G ( U ~ }. This ratio u is u \ u J that between the sensibility of the shunted and the un- shunted galvanometer. Thus, if the resistance of a shunt which will reduce the sensibility of a galvanometer of 1,000 ohms one hundred times is required, looo + S _ 1000 i ~ U = IOO = ~ .'. O = I.OI = (jr. S 99 99 If the current is kept the same by adding a resistance of G ( - ), C C, and will pass through the galva- \ u J u nometer. 47. Kirchhoff's Laws (353). The application of theae useful laws may be illustrated by the figure. The fol- B lowing equations are derivable. From the first law, that in any b- network of wires the algebraic sum of the currents meeting at a v point is zero. Fig. 15. I A At A c = Ci + c^ MB c^ + c^ c. From the second law, that in any closed circuit the electromotive force is equal to the sum of the separate resistances, each multiplied by the strength of current flowing through it. In the left hand circuit c (r + b] + c l r l = E. In the right hand circuit r, r l c-ir* = zero. 62 NOTES ON 48. Fall of Potential ( 357). Let A and B be the poles of a battery. They will have different potentials, numerically equal but of opposite signs, and the battery may be considered to preserve a constant difference of potential between them. Connect the points A and B through an external resistance. The p potential will fall along this resistance, and it is required to find the potential at any point. Draw A B to rep- resent in length the B value of the external resistance, and at A and B erect perpen- diculars, one positive and the other nega- tive, to represent pro- r- l6 - portionally the poten- tials at these points. From Kirchhoff 's second law, in any part of a circuit of resistance r, E C r. In another part of resistance r', E = C' r 1 , E and E' representing the difference of potentials between the ends of the portions of the circuit considered. If the resistances are both in the same cir- cuit C = C', hence E \ E' : : r : r', or the differences of potential in the same circuit are proportional to the resist- ances through which they act. Assuming one potential to be zero, the potentials at the other points vary directly as the resistances separating them from the point of zero potential, and as in the figure the horizontal line repre- sents resistance and the ordinate BH potential, it is evi- ELECTRICITY AND MAGNETISM. 63 dent that ordinates at other points cut off by the line CH will correctly represent the potential at these points, as they and they only will satisfy the above proportion. The line CH represents, therefore, the fall of potential in the resistance BC. If CH\s prolonged to K the triangles I~4C and CHB are similar, and since by hypothesis A K is equal to BH, AC = BC, or C, the point of zero potential, is midway between the poles, and the line HK represents the fall of potential along the resistance AB. It is to be noticed that in the figure the difference or potential between A and B is BH + AK \ or if Fis the potential at A and + V at B, the difference of potential is V - (- F) =2 V. The difference between A and D is v ( V] = V + v. If the point C is connected with the earth, as it is already at zero potential, the potential is not changed anywhere in the circuit. If, however, another point D t whose potential is + v t is connected to earth its potential is lowered by the amount v, and as the battery preserves a constant difference of potential be- tween A and B, the absolute potential of all points in the circuit is also lowered by the amount v. The fall of potential, is, therefore, the same, and is represented by drawing a line M N through D parallel to HK, and the potential at any point is the length of the ordinate at that point intercepted byMN. The difference of potential between A and B is now BN+ AM= ( F - -z/ ) + ( F + ?/ ) = 2 Fas before. If the negative pole A of the battery is connected to earth, the potential of all points of the circuit is raised by the amount V, and the line of potential assumes the position A P. The potential of every point of the circuit NOTES ON is now positive, but the differences of potential are the same as at first. 49. Wheatstone's Bridge ( 358). Connect the poles of a battery by two resistances, P Q 2 Fig. 17. and PQ', and at P erect a perpendicular to represent the difference of potential due to the battery. Then the lines ZQ and ZQ will represent the fall of potential in the resistances PQ and PQ'. The potential at any point on QQ' being the ordinate cut off by the lines ZQ or ZQ, if a galvanometer be joined to two points TV 7 and M at which the ordinates A^Tand MY are equal there will be no de- flection of the needle. But from the figure, since XY'is parallel to QQ, the triangles ZRX and XNQ are similar and NQ : RX : : A 7 X : RZ ; RY::MY:RZ\ but NX = MY RX: : MQ : RY, B -.::>, also or MQ .NQ A as in Fig. 130 and % 358. When, therefore, a galvanom- eter joined to the junctions of two pair of resistances ELECTRICITY AND MAGNETISM. through which a current is flowing shows no deflection, the resistances are proportional to each other. An important fact somewhat difficult to understand at first is apparent from Fig. 17. The resistances PNQ and PMQ' (see also Fig. 130) are unequal, while having the same electromotive force acting in each. The currents in the two branches are therefore unequal. Beginners are liable to regard the balance in Wheatstone's Bridge as due to an equality of currents ; but this is wrong, the equality of potentials at the galvanometer terminals being the condition of balance. This is equivalent to saying that there is no current through the galvanometer, offering another method of proof as follows : 50. Proof of Theory of Wheatstone's Bridge by Kirch- hoff's Laws. Let the currents in the different branches be represented by A Fig. 18. c z , etc., and the corresponding resistances by r, r lf 5 66 NOTES ON etc. Let j be the E. M. F. and b the resistance of the battery. From the first law At A f i / \ *** ** ^2 ^3 i C '5 . . . .. . . . (I) B c = fa + d (2) C* ^4 = c\ + c<> (3) " D c = c* + c, (4) By the tin ABC, c a r 3 - c t r t c*, G = . . . . (5) second j " ADC, c* r- 2 + c<, G d ri = . . . . (6) law, ( " BbDC, c (b -f r\ + Cl ^ + c t r t - E - O. (7) Adjust ri,r) CLARK'S METHOD. Clark's method requires three cells. E furnishes a current and has the highest E. M. F., E' is a \ C standard cell, gener- ally that of Clark ( 177), and v/ isthe cell to be tested. If E" has a higher E. M. F. than 1.457 two or more Clark's cells must be used at E' '. The similar poles of the cells are connected to A. In measuring, Fig. 19. E" is first disconnected and the needle of G' is brought to zero by adjusting P. E" is then connected at A and the slid- 68 NOTES ON ing piece C is moved along the resistance A B, shifting contact until G" also shows no deflection. From Kirchhoff's laws : At A c + c' + c" K o . . . . (i) " C K-c" - K' = o . . . . (2) " B K' - c' -c =o .... (3) In circuit G'AB c' (r> + G'} + Ka + A"b - E' - o . (4) A G"C c" (r" + G") + Ka - E" = o . (5) But when adjustment is secured c' and c" are each zero. Substituting these values from (\)c K from (2) K - K' from (3) c = K' from (4) Ka + K'b = ' ; or, K (a + b) = E' . . . (7) from (5) Ka - E" . (8) Combining (7) and (8) E' : E" : : a + b : a. (C) QUADRANT ELECTROMETER. The two poles of a standard cell are connected to the quad- rants, the same pole being in connection with opposite seg- ments, as I and 3, Fig. 101. The deflection of the needle is then noted. The standard cell is then disconnected and the one to be tested substituted in the same way. From the ratio of the deflections, the ratio of the electro-motive forces may be obtained. Care must be taken that the needle is electrified to the same potential in the two measurements. 52. Measurements of Internal Resistance ( 361). (a) Connect the battery in circuit with a galvanometer and a box of resistance coils, the resistances being B, G and R re- spectively. Note the deflection d in the galvanometer. In- crease R to R' and note the deflection d' . Then if a tangent galvanometer is used ELECTRICITY AND MAGNETISM. 69 tan d:ia.nd' Reducing, Jj = R' tan d' R tan d - : - r tan d tan ^ _ Cr. If G is of no resistance, and the first deflection was taken with no other appreciable resistance than that of the battery in circuit _ R' *an CR is the E. M. F. in the coil, while E is that originally due to the cell. The former is less than the latter by - , which must also be an E. M. F. and due to the at work done by the coil. As a result, therefore, of the motion of the coil in the field, the E. M. F. originally in dN circuit is diminished by the amount 7- . The E. M. F. remaining in the circuit is = *- f ...... The induced E. M. F. is therefore measured by the rate of change in the number of lines of force which pass through the circuit, and is opposite in direction to that originally existing, which caused the motion. By increas- ing this rate, by diminishing dt, or, what is the same thing, dN making the velocity of motion greater, - may be made to equal or exceed E, and the direction of the induced cur- rent would therefore be the same whether there was any E. M. F. to be overcome or not. If a battery current flows, the induced current diminishes it ; if there is no bat- tery current, the coil would have to be moved by external agency, and the induced current is in the opposite direc- 86 NOTES ON tion to a current which would cause the motion. This is seen directly from (2). If E O The fact that the induced current acts in an opposite direction to that causing the motion is easily deduced from the principle of conservation of energy, for, if a current flowing in a given direction caused motion of the circuit, and this motion induced a current in the same direction as the original, it would increase the motion and consequently the energy of the system. Lenz's law, given in $ 396, is therefore a direct result of the conservation of energy. On the supposition that the original E. M. F. is zero, or that there is no current flowing when the coil is at rest, O = C*RM +C. dN ..... (4) The original energy being zero, the energy of the current when the coil is moved can be obtained only from the work done in moving the coil, or Work done in moving coil = heating effect + work do^p by the coil. If the coil does no work, the total energy appears as heat in the circuit. dN The E. M. F. induced in the circuit being the dN current is equal to , in which R is the total resistance in circuit. 61. Practical Rule for Direction of Induced Current ( 395). From the rule given in $ 186, " Suppose a man swim- ming in the wire with the current, and that he turns so ELECTRICITY AND MAGNETISM. 87 as to look along the lines of force in their positive di- rection, then he and the conducting wire with him will be urged towards his left," combined with Lenz's Law (Note 62), the following rule for the direction of the in- duced current in a conductor is easily deduced. Suppose .a man swimming in any conductor to turn so as to look along the lines of force in their positive direction ; then if he and the conductor be moved toward his left hand he will be swimming against the current induced by this motion ; if he be moved toward his right hand the current will be with him. Through some error this rule is given incorrectly in $ 395, and differs there from the rule as given by Prof. Thompson in his Cantor lectures. 62. Lenz's Law ( 396). In Note 60 it was shown that in accordance with the principle of the conservation of energy, the induced cur- rent resulting from any motion of a conductor must be in the opposite direction to that of the current which would cause the motion; Lenz deduced this relation independently, and his statement that " in all cases of electromagnetic induction the induced currents have such a direction that their reaction tends to stop the motion which pro- duces them " is known as Lenz's Law. As an illustration of the use of the law, suppose a magnet to be inserted in a hollow coil. The induced currents must be in such a direction as to oppose the motion. As opposite currents repel each other, the current induced in the coil will be opposite in direction to the Amperian current of the magnet. If the magnet is withdrawn, the withdrawal would be opposed by a current in the same direction as the Amperian current, and by the law a current would therefore be induced in that direction. The same reason- ing applies to currents. 88 NOTES ON 63. Mutual Induction of Two Circuits ( 397). In \ 320 and Note 39 it was seen that two circuits tended to place themselves in such a position as to inclose as many of each other's lines of force as possible, and the number inclosed when each carried unit current was denoted by M. It has since been shown that any move- ment of either circuit induces a current in the other, and consequently changes the value of M. This quantity is therefore called the " coefficient of mutual induction." By a course of reasoning similar to that in Note 18 it may be shown that the force just outside the plane of a voltaic circuit of unit area is4?rC, and if 6" is the area inclosed by the circuit the total force is ^nCS, which becomes ^nS when C is of unit strength and tytnS when there are n turns each of area S. But as the number of lines of force is numerically equal to the strength of the field, this num- ber would therefore be 4?r5, and if all these lines passed through the other circuit, the maximum value of Mis $7tS when the two circuits are coincident. 64. Self-induction ( 404). The extra current is a current induced in the same conductor in which the original current flows. By Lenz's law, or by the table in Note 59, when a current begins in a conductor, a momentary current is induced in the opposite direction, and this phenomenon is noticeable as well in the original circuit as in another near it. The current in beginning is, therefore, opposed by an induced current in the opposite direction, and its increase is made more gradual, and more time is necessary for it to gain its full strength. The fact that the primary current is greatly reduced by the induced current accounts for the fact that when a circuit is closed, but little of a spark is seen. After ELECTRICITY AND MAGNETISM. 89 the current attains its normal strength it remains un- affected by induction, unless acted upon by external causes ; but if the circuit is broken, the law and table already re- ferred to indicate a current in the same direction as the primary, retarding its decrease. As this current prac- tically reinforces the primary, the spark at breaking the circuit is much brighter than at making. The extra current being strictly an induced current is subject to the general laws of induction. The induced E, dN M. F. is therefore -jr. If dN is constant, as it is with at any given current, the E. M. F. of the extra current varies inversely as dt, or the more quickly the circuit is closed or broken the greater the extra current. The value of L in 404 determines the other important relation, that the self-induction varies as the square of the number of turns, and as the resistance of the coil increases only as the number of turns, the extra current is much greater the more turns the coil possesses. This is of great importance in enabling extra currents of high E. M. F. to be obtained when wished, or avoided when they would be detrimental. 65. Helmholtz's Equations ( 405), The current is prevented by its self-induction from obtaining its full strength immediately. The electromotive force of the induced current is L . > L being the coefficient of self- dt induction, and the current acting in opposition to the primary T sJ r* current is . In any interval of time, *//, after the circuit is R at closed, the current has a strength of E L dC dC R . 90 NOTES ON and the current at a time / from the instant of closing the cir- cuit is 66. Induction Coil ( 398). It is important to remember that in the induction coil there are two circuits, not only independent of each other but carefully insulated. The primary coil, or the one im- mediately surrounding the iron core, is of comparatively few turns, and of low resistance, that a given E. M. F. may cause as powerful a current as possible and consequently induce as many lines of force as possible through the core. If the primary current is made and broken rapidly, this number of lines is alternately added and subtracted at very short intervals, and the E. M. F. induced in the second- ary coil, through the axis of which all the lines of the primary pass, is therefore very great. From the general dN formula E = j- > the induced E. M. F. in the secondary is evidently increased by using greater battery power in the primary or by making and breaking the primary cir- cuit with greater rapidity. The E. M. F. of the secondary circuit becomes so great that extreme care has to be taken with the insulation, and parts of the coil at widely dif- ferent potentials must not be brought near together. It is noticeable that the quantity of electricity passing ELECTRICITY AND MAGNETISM. in the secondary coil is extremely small. This is at once apparent when it is considered that the energy of the secondary coil, which may be expressed as C'E', is all derivable from that of the primary coil CE and cannot exceed it. If the energy in the two coils is assumed equal, C : C : : E' : E, and the enormous increase of E. M. F. in the secondary is, therefore, attended with a great reduction in the quantity of electricity passing. As all the energy of the primary current cannot be transferred to the secondary, this pro- portion is not strictly correct, but it illustrates the im- portant point that an induction coil which might kill a man could not heat a wire red hot, or perform other work where quantity was necessary. The condenser is made of sheets of tin foil insulated from each other by paper soaked in paraffine. Alternate sheets are connected throughout, so as to form two large coatings. The action may be understood from the figure. The current passes from the battery through Wto I the Fig. 23. interrupter, and thence through o back to the battery. The core, on being magnetized, attracts /, breaking the current at o. If there were no condenser the extra current would leap across from / to o in a bright spark, but when the condenser is used it darts into it, charging P 92 NOTES ON positively and N negatively, but immediately afterwards the two charges re-combine, the positive charge passing from P to L WBN t demagnetizing the core and making the " break " more rapid, and also opposing the current at the " make." In this way the time dt of the break is diminished while that of the make is increased, and the E. M. F. induced in the secondary coil at the former is, therefore, much the greater. By separating the poles of the secondary circuit, they may be placed so that the break or similar current can strike across them while the make or reverse current cannot. ELECTRICITY AND MAGNETISM. 93 VII. DYNAMO MACHINES. 67. General Principle of Dynamo Machines ( 407). The principle underlying all production of electricity by machines, is that of Note 59, that if a coil of wire is moved in a magnetic field a current is induced in the coil. The successive machines have simply been developments of this fact, improvements having been made either in the distribution of the lines of force in the field, or in the con- struction and movement of the coils. In the first machines, those of Pixii, Saxton and Clarke, permanent steel magnets were used, but only a portion of the lines due to these poles were cut by the coils, and the machines were, therefore, in- efficient. As a rule, the coils moved in front of the poles, in- tercepting the lines passing off from the poles in one direction only. The same general prin- ciple was followed in the Holmes and Alliance machines, there being a greater number of mag- nets and coils, with a poor dispo- sition of the different parts. The introduction of Siemens' armature Flg ' 24 A ' in 1857, was a great step in advance. N and 6* are the poles of several horseshoe magnets bolted together side by side, and between the opposite poles rotates a soft iron cyl- inder Con which the wire is coiled. This armature is thus N W 94 NOTES ON placed in the strongest part of the field, the greater number of lines of force passing directly from N to S through the core C, and the construction admits of the coils approach- ing the poles very closely. Very strong currents were ob- tained from machines in which this armature was used, but great difficulty was experienced from the heating of the armature. If a disk of any metal is rotated rapidly between the poles of a powerful magnet it becomes greatly heated by the currents which are induced in the metal. Tyndall melted fusible metal in a copper tube Fig. 24 B. rotated rapidly in a strong field. The induced currents are in this case in the metal and not in the coil and are generally known as " Foucault " currents. The heat evolved depends on the form and material of the armature and on its velocity, and heat from this cause has always been a serious objection to the early form of Siemens' armature. The next step \vas to have two armatures in the same machine ; one rotating between the poles of permanent magnets inducing a current which passed through the coils of a large electromagnet, between the poles of which the other armature was placed. The adoption of electro-magnets greatly intensified the field, and as the current causing them was also generated by the machine, a great gain of power and efficiency was secured. The machines of Ladd and Wilde were of this type. The above were called magneto-electric machines, as they all depended on permanent magnets to start them, but in 1867 the permanent magnets were suppressed, the current from the armature passing through the coils of the electro-magnet, the " field " coils being in series with the external circuit and armature. When the machine was stopped it was found that the cores of the electro-magnets ELECTRICITY AND MAGNETISM. 95 possessed sufficient residual magnetism to induce a cur- rent in the armature when it was started again, and this current once induced, strengthened the electro-magnets and in turn induced more current. Machines of this type were called dynamo-electric or simply dynamo machines. The advantages they possess over the magneto-electric are greater power, the field being stronger ; and greater econ- omy, the magnets being of wrought iron instead of steel. The distinction between magneto and dynamo machines is now so slight, by the introduction of a variety of new types, that it is hardly worth preserving. As has already been shown, the energy of the induced current is deriva- ble from the energy expended in moving the coil, and Prof. Thompson in his Cantor lectures has given a broad defi- nition of a dynamo machine as "a machine for converting energy in the form of dynamical power into energy in the form of electric currents by the operation of setting con- ductors (usually in the form of coils of copper wire) to rotate in a magnetic field." Accepting this definition, the theory of the dynamo is best understood by a reference to the laws of electro-magnetic induction already examined. The induced electromotive force in a conductor moving in dN a magnetic field is E . As an illustration, exam- ine the case of a coil spinning in a uniform field, and the application of the formula to dynamos may be considered later. Suppose a coil, as in Figure 25, rotating on a vertical axis, the lines of force passing from the reader down Through the paper perpendicularly. It incloses a maxi- mum number of lines offeree, and if rotated so that the right-hand edge comes to the front, while the left-hand goes behind the paper it will inclose a constantly decreas- ing number of lines, and a positive current will be in- duced. The E. M. F. will at first be small, as the rate of 96 NOTES ON change is small, the edges of the coil moving almost along the lines of force. The rate will gradually increase until the coil has moved through one quadrant and is edge on to the observer, when, as the motion of the edges is at right-angles to the lines, the rate, and consequently the E. M. F., is a maximum. In this position the coil incloses no lines of force, and during the second quad- rant it will move, inclosing an in- creasing number, and inducing, therefore, an inverse current. But the side of the coil now seen is the Fig. 25. opposite to that in view during the first quadrant, and the inverse current is, therefore, in the same absolute direction in the coil as the former direct current. During the second quadrant the rate and E. M. F. decrease, becoming a minimum when the coil has com- pleted a half revolution and is again in the plane of the paper. On entering the third quadrant, the number of lines inclosed decreases, and a direct current is induced ; but as the same side of the coil is presented to the observer as in the second, the direction of the current is reversed in the coil. In the fourth quadrant the number of inclosed lines increases, but the other side of the coil is toward the observer, so that the absolute direction of the current is the same as in the third. The general direction of the current is, therefore, downward in that part of the coil in front of the paper, and upward in the other half ; but as regards the coil itself, the direction of the current changes twice in every revolution, the point of change being where the circuit incloses the maximum number of lines of force. By the use of a commutator which shifts its connections at this point of the revolution, the current ELECTRICITY AND MAGNETISM. 97 may be made to flow in one direction in the exterior circuit. Considering this coil as the armature of a dynamo ma- chine, it is apparent that the current could be kept in one direction in the exterior circuit, but would be of varying strength. If another coil were fixed on the same axis but at right angles to the first, its E. M. F. would be a max- imum when that of the first was a minimum, making the current in the external circuit more nearly uniform. By increasing the number of coils a practically uniform current could be obtained, but at the expense of a very complicated commutator. 68. Electromotive Force. dN From the formula E = it is evident, I. That the at E. M. F. varies as the rate of change of the field. For a constant time dt, the rate varies as the number of lines taken out or introduced, and the field should therefore be intense. Mere intensity is not, however, enough, as a coil could be moved in the most intense uniform field without inducing any current. The field must be so arranged that the coil either passes from a maximum positive to maximum negative, or what amounts to the same thing, that it rotates in a constant field, the lines being alternately added and subtracted. In this case the number of lines should be a maximum or the intensity cf the field as great as possible. Perfect working in a dynamo requires a constant change of E. M. F., and consequently a constant rate. If a large coil revolves between the poles of two bar mag- nets, and no iron is present to modify the distribution of the lines of force in the field, the greater part pass direct" 7 NOTES ON between the poles, and are cut during a small part of the revolution of the coil, during which time the rate and in- duced E. M. F. are high, but in other parts of the revolu- tion the rate is very small. The available E. M. F. is induced suddenly, but the sudden creation of a current causes high self-induction and temporary strong extra currents, which in a dynamo are not only prejudicial but dangerous, on account of the high E. M. F. they may have. Idle wire in the armature also reduces the current. It is therefore desirable to prevent a concentration of the lines of force in a small part of the field. In a coil rotat- ing in a uniform field, the advantage of the constant rate is attained by the change of the number of lines inclosed in the ratio of the sine of the angle between the plane of the coil and the direction of the lines, and as a field tends to become uniform would this advantage be gained. To secure uniformity and prevent concentration large pieces of iron called pole pieces are fre- quently attached to the poles, partially encircling the arma- ture. By using long magnets and heavy pole pieces, the field may be made nearly uniform. An impor- tant modification of the field arises from the lines of force due to the current in the armature. Thus in Fig. 26, representing a cross section of a Siemens arma- ture, A being the end of the commutator and TT the Fig. 26. ELECTRICITY AND MAGNETISM. 99 commutator brushes ; the lines of force of the field ordi- narily pass from N to S in approximately straight lines. When the armature is in revolution, each coil in succession has its maximum current when it is in the position C in the figure, and the effect of the armature current is there- fore to induce two poles in the rim of the armature at A ' and S'. The poles A 7 and N' may be supposed to form a resultant pole at N" and 5 and S 1 at S", and the gen- eral direction of the lines of force of the field is therefore N" S". As shown in the discussion of the revolving coil, the commutator brushes should make contact at the neutral points, at right angles to the lines of force. If the lines of force of the machine in motion were in the same position as when at rest the brushes might remain at TT in a line perpendicular to the lines of force NS. If left in this position, however, it will be observed that there is a constant succession of sparks at the brushes, which evidently do not press at the neutral points. This sparking may generally be suppressed by rotating the brushes into positions T'T'. The explanation is simple : they have been brought into a line PP perpendicular to the changed direction N" S" of the lines of force of the field. The stronger the current, the stronger the induced pole of the armature N' ', and the nearer TV" is the resultant pole N". The stronger the current, therefore, the more the brushes must be advanced. The different types of dynamos vary principally in the way in which the field is formed. The principal methods are: (i.) The magneto in which the field is due to permanent magnets. (2.) The separately excited dynamo, a separate machine being used, the current of which passes through the field magnet coils of the generator. This possesses the great advantage of having a constant field, and when 100 NOTES ON several machines are used in one place, one may be used to actuate the field magnets of all the others. (3.) The series dynamo, the field coils being in the main cir- cuit. As the whole current of the machine passes around the magnets, an intense field is produced, but with the great disadvantage that any increase of resistance in the external circuit weakens the field, and consequently the E. M. F., just when a high E. M. F. is necessary to over- come the increased resistance. (4.) The shunt dynamo has the field magnet coils in a shunt of the main circuit. In this type an increased external resistance sends a greater current through the magnet coils, causing a more intense field and a higher E. M. F. By having the resist- ance in the magnet coils adjustable, a shunt dynamo may be made to give a practically constant E. M. F., whatever the external resistance may be. (5.) A mixture of the last two types has been used for special purposes, and is known as a series and shunt, or compound dynamo. It is a shunt dynamo having in addition a number of coils of wire on its field magnets in the main circuit. In a shunt dynamo, if the external resistance is suddenly increased, a greater part of the current flows around the field coils, inducing a higher E. M. F. To keep this constant the speed would have to be decreased. In a series dynamo, however, an increase of external resistance diminishes the E. M. F., and to keep it constant the speed must be in- creased. By making the magnet coils partly in series and partly in a shunt circuit, it is, therefore, possible to keep the E. M. F. practically constant at a certain speed within wide changes of the external resistance. (6.) Some machines have two coils on the armature, one of which sends a current through the field coils, while the other is in the external circuit. (7.) Alternate current machines are machines giving currents first in one direction and ELECTRICITY AND MAGNETISM. 1OI then in the opposite. Their field is caused generally by another machine, but some types send a part of their own current, rectified by a commutator, through the field coils. 2. The E. M. F. varies as the velocity. This relation is almost absolutely true in magneto machines and in others having a constant field, but in the series dynamo, where the field is itself a function of the current, the rate of increase of E. M. F. is much greater than that of increase of velocity up to the point of saturation of the magnets, beyond which, the field being constant, the above relation holds. Its correctness is as- sumed in practice. A great waste sometimes occurs from the commutator brushes not being adjustable. As already shown, the lines of force of the field are distorted by the current, and this distortion is greater as the velocity is increased. The brushes must, therefore, be advanced, or they will take off the current at the wrong time, involving a waste of energy and causing "sparking," injuring both commutator and brushes. 3. The E. M. F. varies as the number of turns of wire in the armature. dN The formula E -= is derived from a conception of the work done in moving a single coil in a magnetic field. If n coils were moved either one by one or all to- gether, n times the work would be done, and n times the E. M. F. induced. Increasing the number of turns in- creases the internal resistance in the same ratio, but if the external resistance is large there is a gain by taking more turns of wire on the armature. 4. The E. M. F. is greatest when the coil cuts the lines 102 NOTES ON of force at right angles. The rate of change is then greatest, and hence the electromotive force. 5. The E. M. F. varies as the area of the coil. If the field were uniform the gain would be directly as the area. In any case it varies directly as the number of lines of force inclosed, and there is, therefore, generally speaking, an advantage in having the coil of large area. In the five considerations on which the E. M. F. of in- duction depends, the only variable, after the machine is made, is the velocity. Several of the types referred to admit of a partial adjustment to meet changed circum- stances, but in general a machine should be adapted to the work expected of it, and should not be expected to be efficient under very different conditions. Although the velocity may be easily varied, it cannot be indefinitely in- creased without mechanical injury. 69. Efficiency. A dynamo has properly two efficiencies. As it is a vehicle for the transformation of mechanical into electrical energy its gross efficiency is the ratio of the current energy to the mechanical energy actually applied to the machine. If an engine developing 16 H. P., is working a dynamo, two H. P. being lost in transmission to the dynamo, in friction and in overcoming the inertia of the engine, only 14 H. P. are actually applied to turn the armature. If in this case the electrical energy developed by the dynamo, was 10 H. P. the gross efficiency is \\ t or 71 per cent. A good dynamo possesses an efficiency of from 90 to 95 per cent, when working under the most favorable conditions, and therefore far surpasses any other machine in its capacity for transforming energy. ELECTRICITY AND MAGNETISM. 103 The ordinary use of the dynamo is to produce light. Whatever its use may be, all the electrical energy not utilized in producing the desired result is practically wasted. The net efficiency is the ratio of the electrical energy in the external circuit to the mechanical energy applied to the armature. If in the above case only 6 H. P. existed in the external circuit, the net efficiency = -, & 4 -, or 44 per cent. The distribution of the energy in the circuit is one of the most important problems relating to dynamos. The total work in circuit is, by Note 55, C^Rt, R being the total resistance, consisting of internal r, and external /. The work is then CV + CY, and the ratio of the work done in the machine to that in the ex- C'r r ternal circuit is - = The fraction of the total C-l electrical energy in +he external circuit is similarly -^5 = That this proportion should be great, / must be nearly equal to R, or in other words, the resistance of the machine must be small compared with that of the circuit. From the above the important relation is evident, that the distribution of energy in an electrical circuit is determined by the distribution of the resistances in cir- cuit. The efficiencies may now be calculated. Let a resist- ance of armature, /that of the field coils, /of the external circuit, and R be the total resistance. E is the electro- motive force, and Cthe current in circuit. The gross efficiency of a series dynamo Work of current = H. P. applied ' 104 NOTES ON This by Note 55 is CV? _ _ H. P. applied ~ 746 x H. P. applied* This energy is given off in all parts of the circuit, that in the armature being C*a (in watts), that in the field coils Cy, and in the external circuit C 2 /. The net efficiency is Work in external circuit- H. P. applied and this is CW "746 cn H. P. applied 746 x H. P. applied The energy wasted as heat in the machine is C 2 (a + f) and the ratio of energy wasted is / / That this may be small the internal resistance (a + f) must be small in comparison with the external. The energy wasted takes the form of heat, and is thus not only wasted but directly harmful, as heating of the machine increases its resistance and thereby increases the ratio of wasted energy. In the shunt dynamo the relations are more complex, as the current in the various branches of the circuit is different. Let a = armature resistance and A armature current, f = field coil " " .F current in field coil, / = external " " L " " external circuit, R = Total resistance = a + -, s ELECTRICITY AND MAGNETISM. 105 Work in armature = A-a, in field coils F*f, and in ex- ternal circuit Ul. Total electrical energy = A*R = A 1 (a + ^ J . Gross efficiency = H. P. applied 746 x H. P. applied To find the net efficiency, the ratio of the electrical en- ergy utilized is '"- r. ( and by multiplying this into the value of the gross efficiency, previously obtained, the product is the net efficiency. The above expression contains only resistances. If L is measured the net efficiency is evidently Z,V 746 x H. P. applied 70. Electromotive Force in Circuit. In using CE to calculate the electrical energy in any case, E must not be taken as the difference of potential at the machine terminals. Calling this difference of potential E', and considering the case of the series dynamo as being more simple, we have from Kirchhoff 's second law, E' = Cl, I being the external resistance, or c = ?- , but from Ohm's Law C = j^ 106 NOTES ON Whenever E' is measured E must be calculated if the total electrical energy is to be computed. The product CE' is evidently the energy in the external circuit, and is less than the total energy by the quantity CV expended in the machine. The total energy is, therefore, CE' + C*r = C-(l -f r). If the resistances are all known the total energy may be calculated from the last formula with- out any risk of error. It has been shown by Sir William Thomson that in shunt dynamos the best results are obtained when the external resistance is a mean proportional between the resistance of the magnet coils and that of the armature, the latter being small in comparison with the resistance of the magnet coils. 71. Siemens' Machine ( 409, Fig. 151). This is a shunt dynamo. The armature is similar in shape to Siemens' armature already described, being a cylindrical drum, but having several coils coiled on it lengthwise instead of one. There are as many divisions of the commutator as there are coils, the divisions being longitudinal. An eight-coil machine has, therefore, its commutator ring divided into eight segments, to each of which connect the ends of two coils. The other ends of these coils are connected to other commutator divisions, so that the eight coils are all in a continuous circuit be- tween the commutator brushes, so wound that in all eight the current at any given instant flows in the same direc- tion. In some coils the E. M. F. is greater than in others, but as there are so many, the total E. M. F. of all in series varies but slightly from time to time, and the current is, therefore, practically constant. By placing the commu- tator brushes opposite each other, they are in contact with points of the circuit differing most widely in potential, and ELECTRICITY AND MAGNETISM. 107 a permanent difference of potential is therefore maintained between the terminals of the machine. The induction of the current in any one coil is analogous to that in the coil described in Note 67. 72. The Gramme Machine ( 410, Fig. 153). The Gramme is generally a series dynamo, although sometimes separately excited, and sometimes having its field coils excited by a separate armature coil. The armature is a ring of soft iron wire, widened till it might be consid- ered a short hollow cylinder. Around this ring are coiled a great number of armature coils, as shown in Fig. 152, the ends of the coils being brought to divisions of the commu- tator. The commutator consists of a number of plates radially arranged around the axis of the armature, and insulated from each other. The commutator divisions are seen on the right of the armature in Fig. 153, and cor- respond in number to the armature coils, which are con- nected through them in one continuous circuit. The action of the Gramme may be easily understood from the rules of Note 59. In Fig. 152 the positive direc- tion of the lines offeree is from A^to S, the lines entering the ring opposite N, and dividing, running through each half of the ring to that part opposite S, where they leave the ring and pass to S. The poles N and 6* cannot be considered as points, and the lines, therefore, enter the ring all along its lower portion (as shown in the figure) and emerge along the upper part. A coil in the position E" has, therefore, the maximum number passing through its plane. If, now, the armature is rotated, so that E" passes towards E, it continually incloses a decreasing number of lines of force, and a direct current viewed from N is induced. The E. M. F. varying as the rate of change, 108 NOTES ON is zero at E" and a maximum at a point opposite S, where the coil cuts all the lines at right angles. As the rotation of the armature continues, the coil after leaving E in- closes an increasing number of lines of force, and the current is therefore inverse as viewed from N. But from E to E' the side of the coil viewed is the oppo- site of that seen from E" to E, and the inverse current in the quadrant from E to E' is therefore in the same absolute direction in the coil as the direct from E" to E. Throughout the half revolution from E" to ', therefore, the induced current flows in the same direc- tion, being strongest when the coil is nearest the pole S. By connecting all the coils in series, the E. M. F. in the circuit becomes the sum of all in the individual coils, and as these occupy all possible positions at any instant, the total electromotive force is constant, the machine thus yielding an almost absolutely constant current. The action during the other half of the revolution may be traced in the same way. The coil in moving from E' to N incloses a decreasing number, inducing a direct current, which is opposite in direction to that in the quadrant from E to E'. If, therefore, in the latter the current had flowed away from the point E' towards E, it would in E'N flow away from E' towards N, and although the absolute direction of the currents is dif- ferent they combine to lower the potential of E'. During the quadrant between N and E" t the coil incloses an in- creasing number, and consequently has an inverse current induced, but this inverse current is in the same absolute direction as the direct in the preceding quadrant. If, therefore, throughout the upper half of the revolution the current flows away from E', it will in the lower half of the revolution flow away from E'also. Throughout the ELECTRICITY AND MAGNETISM. 109 whole revolution the effect is to raise the potential of E" and lower that of ', and if brushes touch the commu- tator at these points they will possess a difference of po- tential which may be utilized in the production of a cur- rent through an external circuit. The Gramme machine has been the subject of much investigation, and its action has been variously explained. The most general ex- planation in any case of electromagnetic induction is that obtained from a consideration of the lines of force, and this is the one adopted by Prof. Thompson, which has only been given here in slightly greater detail. The armature cylinder is made of soft iron wire, both to facilitate the rapid magnetization and demagnetization, and to prevent heating from the Foucault currents which would take place if solid metal were used. The change of direction of the lines of force of the field by those due to the current is frequently very marked in the Gramme machine, M. Breguet having found it necessary to ad- vance the commutator brushes 70 when working with a Gramme at 1770 revolutions. As the internal resistance of the Gramme is generally small, it is specially adapted for working a single powerful arc light, while the steadi- ness of its current renders it well adapted for incandescent lighting. 73. The Brush Machine ( 411). This machine has received its main development in the United States, but is now extensively used throughout the world. It contains many peculiar features, and is distinct- ly a separate type, although frequently alluded to, espe- cially by French authorities, as a modification of the Gramme. The general appearance of the machine is shown in Fig. 27. The first noticeable peculiarity is in the disposition of TIO NOTES ON ELECTRICITY AND MAGNETISM. m the four field magnets, which are placed so that the arma- ture coils pass between similar poles. The magnets are oval in cross section, and are furnished with large pole pieces, approaching very closely on each side to the arma- ture. The armature is a soft iron disc, with deep circular furrows cut in its sides to break the continuity of the sur- face and thus prevent the heating of the metal by the in- duction of Foucault currents. On the periphery of the armature of the small machine there are eight coils, the two coils diametrically opposite being in one, but coiled in opposite directions (See Fig. 30), so as to act in unison in the induction of currents. The coils project from the ar- mature as seen in Fig. 27, the reason assigned being, that the fanning of the air thus caused prevents overheating. The commutator consists of four rings each split into four segments. A cross sec- tion of one of the rings is as in Fig. 28, the two ends of one pair of coils being connected to the segments marked I, I, which are insulated from the segments 2, 2. When the brushes of the commutator touch the latter the coils are cut out of circuit. These cut- ting out segments in the dif- Fi - 2g - ferent rings of the commutator are so placed that at every instant one coil is cut out ; the connections being made so that a coil is not in circuit in that part of the rev- olution when no current is being induced in it. Each of the four brushes presses on the commutator rings of two coils not adjacent. Numbering the coils on the armature I, 2, 3 and 4 (Fig. 30) in order, the brushes Z? and B* are in circuit with coils I and 3 and * and B* with coils 2 and 4. 112 NOTES ON As the armature revolves each coil successively passes through all parts of the field. When a coil is midway between the dissimilar magnet poles, at the highest point of its revolution, the number of lines of force inclosed is a maximum, but changes so slowly that for this portion of the revolution the induced current is but small, and the coil is, therefore, cut out. As the coil approaches the large pole pieces and passes between them the rate changes rapidly. If a piece of soft iron be placed between two powerful similar magnet poles, the lines of force pass into it almost parallel on each side, and a coil moved along the bar cutting them perpen- dicularly, has a high rate of change in the number of lines inclosed, and conse- _ quently a high electro- motive force induced. Thus in Figure 29, as nearly all the lines of both poles pass through the soft iron between a and b, the coil A in moving with- in that region experiences but little change in the <, number inclosed, but as it approaches either end the rate of change is very great. The electro- Vl '** motive force is thus in- duced somewhat suddenly, but the efficacy of the pe- culiar arrangement of poles for the induction of a high electromotive force is evident. A comparison between Figs. 29 and 30 shows this to be nearly the condition ex- isting in the Brush machine. The latter figure is a plan ELECTRICITY AND MAGNETISM. 113 of the machine. As each pair of coils is connected to a separate commutator ring, the study of the connections is necessary to understand the complete working. In the figure, L represents a lamp in the external circuit. B l , B*, B z and B* are the commutator brushes, and the rings are Fig. 30. numbered I, 2, 3, 4, as illustrating the way in which the ends of the coils, taken in regular order around the arma- ture, are connected at the commutator. The currents in- duced in the several coils at any one instant will have the following circuits, coil 4 being supposed to be cut out : Coil i i, B\ L, X, 13*, 2, B\ M, B\ Coil 2 2, B\ , B l <*> B*,LX, B\ Coil 33, B\ L, X t B\ 2, B\ M, B\ ' These paths are the same except in the armature coils and the resultant current will have the path B l <^> B\ Z, X, B\ 2 B*, M, B\ An instant later coil 4 will be in circuit and I cut out. The resultant current then flows 3, B\ L, X, \ M, B\ 3. 114 NOTES ON The E. M. F. in circuit is evidently that due to two coils in series, and the internal resistance of the machine is diminished by the fact that there are always two of the four armature coils in parallel arc. The gross efficiency of the Brush machine is lower than that of some others, but it possesses the great advantage of yielding so high an electromotive force as to be able to burn forty arc lights in series, a feat which no other machine can accomplish. As there are only two coils in series at one time, the resultant electromotive force is far from constant, and the fluctuations are so great as to utterly unfit the machine for incandescent lighting or other purposes requiring a constant current. The E. M. F. of the largest Brush machine is 2000 volts and the current about 10 amperes. 74. Edison Machine. The Edison machine ($ 411) is a shunt dynamo. Its chief peculiarities are its long cylindrical magnets ending in remarkably heavy pole pieces almost encircling the armature, and the peculiar construction of the armature itself. Theoretical investigation and experiment both point to long cylindrical magnets as most efficient ; and in a shunt dynamo, in which there is a perpetual endeavor for a permanent adjustment of the strength of the field to the necessities of the case, it is advantageous to have mag- nets of considerable mass, as the change of field brought about by a variation in the strength of the field current is thus made more gradual. The large pole pieces tend to make the field more uniform, and thus act to secure a constant rate or a uniform change of electromotive force throughout the rotation. Edison calls his large machine a "steam dynamo," the engine and dynamo being on the same bed-plate. It is ELECTRICITY AND MAGNETISM. 115 specially designed for use at a central station to supply power or work incandescent lights throughout a district of a city. As established in New York, the whole weight of dynamo and engine is nearly thirty tons, sixteen of which are in the magnets and pole pieces. The core of the armature is made up of sheet iron discs, separated from each other by tissue paper and bolted together. This prevents heat currents. Instead of wire, the armature circuit is made of heavy copper bars, each bar being in- sulated from the next and from the iron core by an air space. The bars are connected together at each end of the armature by copper discs, there being half as many discs at each end as there are bars. Each disc has lugs formed on it on opposite edges, to which two bars are connected, and the whole being bolted together, the bars and discs form one continuous circuit of wonderfully low resistance, the total armature resistance of a machine sent to London being .0032 of an ohm. This very low resist- ance is necessary from the fact that the machine is in- tended to work 1300 incandescent lights, each of about 137 ohms, in parallel arc. The external resistance would, therefore, be only .095 ohms. As the number of lamps in circuit changes, the resistance in the magnet coils, which are in a shunt of the main circuit, is regulated so as to keep a practically constant electromotive force, and each lamp then burns with the same intensity under all condi- tions. Edison's large machine gives an E. M. F. of no volts and an ordinary current of 1000 amperes. 75. Alternate Current Machines. Alternate current machines have been used in Europe to a considerable extent for incandescent lamps and the Jablochkoff and other candles. Almost any machine yields alternate currents if used without a commutator, but most 1 1 6 NOTES ON alternate current machines have a large number of armature coils which pass between the poles of a system of opjx>,< a south, the lever moves and closes the local circuit. When no current passes in the line, or when it passes in the opposite direction, the local circuit is open. In sending two messages at the same time in the same direction, two keys are used, one reversing the current, send- ing positive or negative currents, the other sending weak or powerful. The strength of the current is, therefore, controlled by one key, its sign by the other. The method i 3 6 NOTES ON used by Edison for transmitting is shown in the figure. In the position shown the battery B has its terminals at N L1NE and P t the current passing from B through A" 2 to the spring 6* and thence to P, If the key K' is worked cur- rents of either polarity may be sent into the line, and passing through a polarized relay at the receiving station, a sounder in the local circuit is \vorked whenever a cur- rent in a given direction is transmitted by A". The strength of the current is immaterial, the polarized Fig. 3 3. relay answering only to cur- rents in one direction. As shown in the figure, the circuit of the battery B\ which is much larger than B, is open. If, however, the key /v" 2 is depressed the spring 6" comes in contact with the point m and breaks contact with #, and as it is separated from A" 2 by the insulating material /, the current of B now has to pass through B' m and 5 to P, and is, of course, reinforced by the the powerful current of B' in the same direction. Whenever A" 3 is depressed, therefore, the points N and P retain their polarity, but the current is of three or four times its original strength. In practice all contacts are made by springs, so that the cir- cuit is never broken at A%, but one current is followed di- rectly by the other. The message transmitted by K ' 8 is received by an ordinary relay in the same circuit with the polarized relay at the receiving station, the lever of which is controlled by a spring so adjusted that the weak cur- rent of B will not cause sufficient magnetism in the elec ELECTRICITY AND MAGNETISM. 137 tromagnets to attract the armature against the action of the spring, but when A% is worked the current due to B + B' easily overcomes it, whether the current be posi- tive or negative, and the relay, therefore, transmits all signals made by K *. The quadruplex is merely an extension of the duplex, using the diplex or double transmission. If in Fig. 163 (Thompson) the transmitting apparatus just described is used instead of the keys R and R, and if between A and B two relays are placed in series, one an ordinary relay and the other a polarized, the figure would represent the general arrangement of Edison's quadruplex system widely used in the United States. 86. Blake's Transmitter. In most telephone circuits, the receiving instrument is a Bell telephone, but the transmitting is a modification of Edison's telephone, known as Blake's Transmitter. The waves of sound impinge on a metallic diaphragm, caus- ing it to press with more or less force on a carbon button. A current from a battery passes through the button and the varying pressure of the diaphragm causes a varying resistance in the circuit, and produces in the current fluc- tuations, corresponding in number and time to the waves of sound. If this current is passed through a Bell telephone, the message could be heard. As now used, however, the battery circuit is entirely local. In this local circua is the primary coil of an induction coil, the secondary being in circuit with the line to the next station. Every fluctua- tion, therefore, in the strength of the local circuit, due to the change of pressure on the carbon button of the trans- mitter, induces a current in the secondary coil which works a Bell telephone at the distant station. The induc- tion coil is small, but it causes the electromotive force of I3 8 NOTES ON the line circuit to be much greater than that due to the battery and extends the use of the telephone to greater distances. 87. Telephone Exchanges. The use of the telephone has been greatly extended by the system of exchanges. A large number of persons have telephoae circuits to a central office, where any two circuits may be joined, thus enabling any two to converse. A great difficulty in all telephone circuits is due to induc- tion. The instrument is so extremely delicate that any in- constant current near it induces sufficiently powerful cur- rents in the telephone circuit to frequently obliterate a message entirely. Telegrams may be read in telephones if the telegraph and telephone circuits approach each other very closely, and telephone messages may also be heard in other circuits than that in which they are trans- mitted. Most of the disturbances commonly attributed to induction are, however, in all probability due to grounded telegraph circuits. ELECTRICITY AND MAGNETISM 139 REFERENCES TO PROF. THOMPSON'S ELEMENTARY LESSONS. 191, Note 26. 353, Note 45, 46, 47. ig2, 34- 357, " 48. 199, I. 358, 49, 50- 200, i. 36o, Si- 201, 2. 36i, " 52. 202, 3- 362, 53- 203, 4- i 364 ' 54- 204, 5- 367, 55- 237, 6, 7, 8, g, 10. 371, 44 56. 238, ii. 372, 57- 239, 12. 374, 44 58- 24O, 13- 375, " 76. 241, 14. 376, 77- 245, 377, 78, 79- 2 4 6, ii' 378, " 55- 247, 17. 380, " 80. 252, 18. 39i, 258, ig. 392, " 59- 26l, 20, 21. 393, 262, 51, c. 394, ' " 59, 60. 310, 22, 23, 24,25, 26, 395, " 61. 27. 396, 41 62. 3 11 , 28. 397, " 63. 312, 23. 398, " 66. 313, 2g. 404, " 64. 3M, 30. 8 405, 65. 315, 3i6, 3i, 32. 33- 407, i " 67, 68, 69, 317, 33- | 409, 71. 3i8, 34, 35, 36, 37, 38. i 4IO> 41 72. 319, 36. 4ii, 73, 74, 75. 320, 39- 415, 44 81. 324, 40. 425, 44 82. 325^, 41. ts 426, " 83. 327, 42. 427, ' 4 84. 338, 42. 428, 44 85. 43- 436, 44 86, 87. 352,' 44- 14 DAY USE RETURN TO DESK FROM WHICH BORROWED In* \y fl 1 ^1 Ib^ Lw 1 If This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. f iditt'a/Rfisr ' JAN 2 1957 ^.lillSSRK 66 u-uSaSttoSKmi. Berkeley Murdock, J. B. M8 Notes on sleotricity ana magnet 1 sm THE UNIVERSITY OF CALIFORNIA LIBRARY