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EXPERIMENTAL PHYSICS 
 
 A COURSE FOR FRESHMEN 
 
 Being a Revision of Alexander's Manual 
 
 BY 
 
 GEORGE K. BURGESS, S.B, 
 
 Docteur de /' Universite de Paris 
 
 Instructor in Physics, University 
 
 of California 
 
 BERKELEY, CAL- 
 1902 
 
Entered According to Act of Congress in the Year 1897, by 
 
 GEORGE K. BURGESS, 
 
 In the Office of the Librarian of Congress, at Washington. 
 
PREFACE. 
 
 This manual represents the latest step in the development of a 
 course in physics for Freshmen at the University of California under 
 the direction of Professor Slate; the modifications of previous texts 
 are not radical, but reflect the present instructor's views of what is 
 suitable for the freshman class at this time. There is no serious 
 claim to originality, either in subject matter or in method of pres- 
 entation, both of which are largely those of the late Professor 
 Whiting, and of Dr. A. C. Alexander, who, until recently, gave the 
 instruction in this course. 
 
 The course has been modified by decreasing the time of instruc- 
 tion in the laboratory from two three-hour periods a week to two 
 periods of two hours, and instead of one lecture there are now two 
 recitations a week. By this change it is hoped that the students will 
 get a better grasp of the principles involved in the experiments. 
 
 Among the main points by which this manual . differs from its 
 predecessors are the following: 
 
 Because the laboratory period has been reduced from three to 
 two hours, some of the exercises have been shortened. 
 
 The details of a considerable number of exercises differ from those 
 of previous texts, and many of the experiments have been entirely 
 rewritten, although treating in general of the same principles as 
 heretofore, v/ith a few exceptions. 
 
 More emphasis is given to the graphical representation of results. 
 
 Where possible, the principle of an experiment is summarized in 
 an equation by the student. 
 
 Optional experimental parts of an exercise have been removed, 
 due to the shorter laboratory period, and also because in practise 
 this has been found by the author to be of questionable benefit in 
 large elementary classes for which the ratio of the number of students 
 to the number of instructors is great. 
 
 In place of the optional portions are put questions or problems 
 that the student may solve outside of the laboratory, if he finishes 
 the experimental part only in the regular period. 
 
 Questions are occasionally appended to an exercise that require 
 a knowledge of principles developed in the class room, or reference 
 to some standard descriptive work. 
 
 Finally, the exercises have been arranged in four groups of eleven 
 
 (iii) 
 
 102273 
 
IV PREFACE. 
 
 each, the exercises of each group being so written that the student, 
 with the aid given in the recitations, may intelligently begin with 
 any one. Although in certain instances there is an apparent lack of 
 sequence, yet, on the whole, this system seems more efficient than 
 the one previously in vogue, in which the students were started by 
 eights in succession, when in a section of eighty students some were 
 six weeks late in starting. By the new arrangement two weeks are 
 gained in every eight, when all the students may devote their time to 
 back work. 
 
 The author is indebted to the members of the Physical Depart- 
 ment for helpful advice, and especially to Mr. C. A. Kraus, who has 
 aided in many ways the preparation of these notes. 
 
 GEORGE K. BURGESS. 
 
 Berkeley, July, 1902. 
 
CONTENTS. 
 
 GENERAL DIRECTIONS. 
 
 GROUP I. 
 
 PROPERTIES OF FLUIDS. 
 
 PAGE 
 
 1. Liquid Pressure and Density . 10 
 
 2. Vapor Pressure and Dalton's Law 12 
 
 3. Variation of Vapor Pressure with Temperature 15 
 
 4. Boyle's Law and Voluminometer 17 
 
 5. Pressure of Gas at Constant Volume 18 
 
 HEAT. 
 
 6. Expansion of Gas under Constant Pressure 19 
 
 7. Specific Heat 20 
 
 8. Latent Heat. 22 
 
 9. Mechanical Equivalent of Heat 24 
 
 SURFACE TENSION. 
 
 10. Surface Tension 25 
 
 MECHANICS. 
 
 11. Principle of Moments 27 
 
 GROUP II. 
 
 12. Composition of Forces 29 
 
 13. Elasticity; Laws of Stretching 31 
 
 14. Action of Gravity ... 32 
 
 15. The Pendulum, I ... 34 
 
 16. The Pendulum, II . . , 35 
 
 SOUND. 
 
 17. Resonance Tube 36 
 
 18. Velocity of Sound in Solids . . . . . ; . *. .* . . . . . . 38 
 
 19. Laws of a Vibrating String. ........'.. 39 
 
 (v) 
 
VI CONTENTS. 
 
 PAGE 
 
 LIGHT. 
 
 20. Photometry 40 
 
 21. Refraction 41 
 
 22. Refraction and Dispersion 43 
 
 GROUP III. 
 
 23. Images in a Spherical Mirror 44 
 
 24. Convex Lenses 47 
 
 25. Concave Lenses 48 
 
 26. Drawing Spectra 51 
 
 MAGNETISM. 
 
 27. Laws of Magnetic Action . . . 52 
 
 28. Magnetic Fields 54 
 
 29. Intensity of Earth's Magnetic Field, 1 56 
 
 30. Intensity of Earth's Magnetic Field, II 57 
 
 31. Comparison of Magnetic Fields 58 
 
 ELECTRO-MAGNETIC RELATIONS. 
 
 32. Electro-Magnetic Relations 59 
 
 33. Laws of Electro-Magnetic Action 62 
 
 GROUP IV. 
 
 34. Current Determination 64 
 
 ELECTRICITY. 
 
 35. Electrical Resistance 65 
 
 36. Electromotive Force. 67 
 
 37. Ohm's Law 69 
 
 38. Divided Circuits and Fall of Potential 70 
 
 39. Arrangement of Battery Cells; E. M. F. and Resistance . . 72 
 
 40. Comparison of Resistances by Wheatstone's Bridge .... 73 
 
 41. Heating Effect of an Electric Current 74 
 
 42. Laws of Electrolysis 76 
 
 ELECTRO-MAGNETIC INDUCTION. 
 
 43. Electro-Magnetic Induction 77 
 
 44. Earth Inductor 79 
 
GENERAL DIRECTIONS. 
 
 BEGINNING WORK. The class will be divided into four sec- 
 tions, each having two laboratory periods per week of two hours, 
 preceded by a recitation, for which each section will be divided 
 into halves. 
 
 The laboratory work includes forty-four exercises, divided into 
 sets of eleven. As soon as registered, each student will report 
 at the laboratory in East Hall and will be assigned to one of the 
 first eleven experiments. He will then perform in succession at 
 the following exercises the cycle of eleven experiments. Exam- 
 ple: A student assigned to the yth experiment will perform the 
 first eleven in the order 7, 8, 9, 10, n, i, 2, 3, 4, 5, 6. Two 
 weeks will be allowed at the close of this cycle for the correction 
 and completion of work. A new set of experiments will then be 
 mounted, and it will then be impossible to reperform any of the 
 first eleven experiments this year. 
 
 IN THK LABORATORY. The following directions are necessi- 
 tated largely by the size of the class. 
 
 Students will work in pairs and may choose their partners. 
 Each student however will be required to take a separate set of 
 observations for each experiment and to write up his notes inde- 
 pendently. All data must be recorded at the time of observation 
 in the note-book and not on scrap paper. 
 
 In general, at least three independent observations of each 
 quantity measured are to be taken and every observation recorded 
 when it is taken. Notes are to be neatly arranged (see sample 
 note-book) and observations recorded so as to be distinct from 
 descriptive or other written matter, and when practicable results 
 should be tabulated. 
 
 Concise but clear answers are wanted to questions asked; all 
 inferences should be in the words of the student, and demonstra- 
 tions should be complete. Fractions are to be expressed as 
 
 decimals, and calculations given in detail. 
 
 (vii) 
 
8 
 
 GENERAL DIRECTIONS. 
 
 For the heading of sheets, name, date, etc., consult sample 
 note-book; the arrangement there indicated must be exactly 
 followed. Separate sheets of a single exercise are to be fastened 
 securely together; turned over corners will not be accepted. 
 
 PLOTTING. In several exercises the results are to be expressed 
 graphically on plotting paper. When the data permits, such 
 scales for plotting should be chosen as will give a line extending 
 diagonally across the paper. Observed points on the curve 
 should be indicated by crosses and not by dots or circles. The 
 known quantity is to be plotted horizontally and the quantity to 
 be studied, vertically. Plots should be carefully drawn and 
 properly labeled. In general, a smooth line drawn among the 
 points corresponding to observations best represents these obser- 
 vations. For further details of construction of a plot, see sample 
 note-book. 
 
 PROBLEMS. A certain number of problems will be assigned 
 during the year. They are to be worked on laboratory paper 
 and the carbon prints are to be handed in. 
 
 TRIGONOMETRICAL RELATIONS. For those students who are 
 not familiar with the elements of trigonometry, the following 
 definitions will suffice. 
 A 
 
 Consider a right-angled triangle ABC of sides a, b, and c. 
 The various trigonometrical functions are most conveniently 
 defined in terms of the parts of such a triangle. 
 
 The sine (written sin) of an angle is the ratio of the opposite 
 side to the hypothenuse. 
 
 sin A = ~ and sin G=* 
 b b 
 
GENERAL DIRECTIONS. 9 
 
 The cosine (written cos) is the ratio of the adjacent side to the 
 hypothenuse. 
 
 . c , ~ a. 
 
 cos A = r and cos C = .- 
 
 b b 
 
 Evidently also 
 
 cos A=sin C and sin Ar=cos C. 
 a=b cos C=b sin A, c=b cos A=b sin C. 
 The tangent (written tan) is the ratio of the side opposite to 
 the side adjacent. 
 
 a c. 
 
 tan A=- and tan C - 
 c a 
 
 Also sin A sin C. 
 
 tan A= --- -and tan C~ 
 
 > 
 
 cos A cos C 
 
 For very small angles the sine and tangent may be replaced 
 by the angle itself. 
 
 UNFINISHED WORK. At the close of a laboratory period the 
 student will present the carbon print of his notes to the instructor, 
 -and if the exercise has not been finished, the records will be 
 stamped with the date, and the exercise may be completed later, 
 but is to be handed in complete within two weeks after the date 
 last stamped upon it, otherwise it must be repeated. All experi- 
 mental data taken out of the laboratory must be stamped. 
 
 In general, a student will have ample time to complete the 
 experimental part of any exercise in a laboratory period; but if 
 pressed for time, calculations, inferences, demonstrations, and 
 answering of questions may be performed outside of the labora- 
 tory, as above indicated. No experiments which are taken home 
 and for which the data have been changed will be accepted. 
 Corrections are to be made in the manner indicated in the sample 
 note-book. 
 
 GRADES. The following system of marking will be used: 
 
 i . Excellence. 
 
 2. Satisfactory. 
 
 3. Deficient in inferences, proofs, or answers to questions. 
 
 4. Repetition of part of experimental work required. 
 
 5. Repetition of whole exercise required. 
 
10 LIQUID PRESSURE AND DENSITY. [i 
 
 Unsatisfactory work will be returned for correction. All defi- 
 cient exercises are to be raised to grade 2, otherwise the grade 
 INCOMPLETE will be given for the term's work. 
 
 ORDER AND BREAKAGE. Those working at any exercise will 
 be held responsible for the apparatus used and will be expected 
 to leave it in good order when through. Breakages should be 
 reported to the instructor. 
 
 GROUP I. 
 
 In some of these experiments mercury is used. Care must be 
 taken not to spill it, and all metals should be kept away from it. 
 Refer to the sample note-book for suggestions as to arrangement 
 of data and writing of notes. In general, seek to finish the 
 experimental work in the time allowed, leaving computations and 
 answers to questions to be done outside of the laboratory if 
 pressed for time. 
 
 i. LIQUID PRESSURE AND DENSITY. 
 
 I. Clamp a U-tube in a vertical position to a burette stand, 
 with the bend of the tube resting on the table. Pour into this 
 tube enough mercury to stand about 5 cm. above the table in 
 each arm. Then pour into the longer arm enough water to 
 stand about 13.6 cm. above the end of the mercury column. 
 Work out all air bubbles with a fine wire, and mop up any 
 water resting on the mercury in the short arm with a bit of 
 blotting-paper tied to the end of the wire. Measure the 
 heights above the table of the ends of the mercury and water 
 columns, measuring as nearly as possible to the center of the 
 meniscus in each case. Are the liquids in the two branches 
 at the same level? If not, why? What differences are 
 there between the shapes of the free ends of the two columns? 
 Account for these differences. 
 
i] LIQUID PRESSURE AND DENSITY. II 
 
 Find the length of the mercury column that balances the 
 water column, and also the ratio of the two balancing columns 
 (water column to mercury column). 
 
 II. Fill the longer arm of the U-tube nearly full of water, 
 and measure the length of the water column, and also of the 
 mercury column that balances it. Find again the ratio of the 
 balancing columns. Is it the same as in I? This ratio will be 
 shown to be equal to the specific gravity of mercury. 
 
 III. Fill one of the two beakers, or jars, with water, and the 
 other with a saline solution. Place a leg of an inverted Y-tube 
 in each of the liquids Cautiously draw the liquids up in both 
 legs by suction, and close the stem of the Y air tight. Why 
 is the liquid higher in either branch than in the corresponding 
 open vessel? Measure the height of each column of liquid 
 above the level of the liquid in the open vessel. Is it the same 
 for both liquids, or not? Why? 
 
 Does it make any difference if the branches of the Y-tube 
 are not of the same diameter, or are not held vertically ? 
 Calculate the specific gravity of the saline solution. 
 
 IV. Fill the two branches of a W-tube, one with water and 
 the other with wood-alcohol. This should be done by pour- 
 ing the liquids into them alternately, a small quantity at a 
 time. Why is it necessary to observe this precaution in 
 filling? 
 
 Make the proper measurements and calculate the specific 
 gravity of the wood-alcohol. Draw diagram in illustration. 
 
 Why is it unnecessary to have the ends of the columns at the 
 same level? 
 
 V.' Answer the following questions: 
 
 1. To what class of liquids is the method of the U-tube 
 inapplicable? Why? 
 
 2. In the case of highly volatile liquids, what advantage has 
 the method of the W-tube over that of the Y-tube ? 
 
 3. Which of the three do you consider to be the most general 
 method? 
 
12 VAPOR PRESSURE AND DALTON'S LAW. [2 
 
 VI. Distinguish between specific gravity and density. 
 
 If pressure is defined as force per unit area, form an equation 
 expressing the equality of pressures of the two liquids in the 
 arms of the U-tube and show that the heights are inversely as the 
 densities; and that when water is used in one arm, the ratio of 
 the heights is the specific gravity of the other liquid. Show that 
 for a liquid, pressure is proportional to depth. 
 
 VII. Calculate the total outward pressure of a cube of mercury 
 20 cm. on a side. What is the weight of this mercury? 
 
 2. VAPOR PRESSURE AND DALTON'S LAW. 
 
 I. Take a closed tube, at least 80 cm. long, and wipe it clean 
 and dry with a swab tied to a long and stiff wire. Then fill it 
 with mercury by means of a small funnel.* Close the open end 
 with the thumb and invert the tube in a reservoir of mercury. 
 After removing the thumb, does the mercury in the tube fall to 
 the same level as the mercury in the reservoir? If not, why 
 What is meant by the barometric pressure? 
 
 * Observe the following directions in filling the tube and removing air 
 bubbles: 
 
 Fill to within a couple of cm. of the open end. Close with the thumb 
 and invert a number of times, gathering all the air bubbles adhering to 
 the sides into one large bubble. Then hold erect and fill completely, 
 pouring the mercury in slowly and working out all air bubbles with a 
 fine wire. Again invert in the reservoir. (The amount of air in the tube 
 can be observed by tilting it until the closed end is about 70 cm. above 
 the table.) To further remove the air, place the thumb tightly ov.er the 
 open end of the tube while in the reservoir, and then raise and carefully 
 invert it a number of times, letting the partial vacuum pass slowly from 
 one end of the tube to the other, and finally, holding it erect with the 
 open end up, take the thumb off and fill completely, as directed above. 
 This operation should be repeated until the air bubble seen when the 
 tube is tilted has been reduced to the smallest possible size. The height 
 of the mercury column ought now to agree, w r ithin one cm., with the 
 barometric reading for the day. If it does not so agree, repeat. 
 
2] VAPOR PRESSURE AND DAI/TON'S LAW. 13 
 
 Measure the height of the mercury in the tube above that in 
 the reservoir. Is it the same as the height of the barometer? If 
 it is not, explain why. 
 
 II. Having measured the height of the mercury column 
 above the level of the mercury in the reservoir, draw as much 
 ether as possible into a medicine dropper, and, inserting it 
 into the reservoir under the open end of the tube, introduce a 
 few drops into the tube, taking great care not to introduce any 
 air. Introduce enough so that some of the liquid will remain 
 unevaporated on top of the mercury column. Describe in detail 
 what takes place when the ether is introduced. Does the ether 
 all evaporate, or does it cease to evaporate after a certain amount 
 has been introduced? Explain why. When is a vapor said to 
 be saturated ? 
 
 After waiting 10 minutes for the ether vapor to come to the 
 temperature of the room, measure the height of the mercury 
 column. Why is it less than before the introduction of the ether? 
 What do you find to be the pressure of the ether vapor, in cm. of 
 mercury, at the temperature of the room ? (Record this tem- 
 perature.) 
 
 III. (.) Pour more mercury into the reservoir, leaving 
 enough space for the mercury in the tube when it is taken out. 
 With the tube resting on the bottom of the reservoir, measure 
 again the height of the mercury column, and also the length of 
 the tube occupied by the ether vapor. 
 
 (.) Raise the tube so that its lower end is just below the level 
 of the mercury inthe reservoir and after a few minutes repeat 
 the measurements of (a). 
 
 (V.) Answer the following questions: 
 
 1. Was the pressure of the ether vapor in (a) the same as 
 in ()? 
 
 2. Was its volume the same ? 
 
 3. The temperature being kept constant, do you find the 
 pressure of saturated ether vapor to depend on its volume, 
 or not? 
 
14 VAPOR PRESSURE AND DALTON'S LAW. \2 
 
 IV. Remove the ether from the mercury by wiping its surface 
 with a piece of clean blotting-paper and then passing it through 
 a pinhole at the point of a paper filter. Pour the mercury into a 
 I5o-cm. bottle with a rubber stopper, to a depth of 2 or 3 cm. 
 Be sure that the bottle is clean and dry and free of ether vapor. 
 (If there is any ether vapor in the bottle, it can be removed by 
 inserting a tube and blowing it out.) Insert the short arm of a 
 U-tube, at least 50 cm. long, through the rubber stopper. See 
 that the stopper fits closely into the mouth of the bottle and press 
 it in as tightly as possible. Invert the bottle, taking care not to 
 entrap any air in the mercury column. Resting the bend in the 
 tube on the table, measure the height of the mercury in the tube 
 above, or below, its level in the bottle. Four ether into the tube 
 so as to stand in an unbroken column 15 or 20 cm. deep, and 
 attach a rubber bulb to the open end of the tube. By pressing 
 the bulb, force a little of the ether into the bottle, taking care not 
 to force in any air. What is the effect of introducing the ether? 
 
 V. Force in about 15 cm. of the ether in the tube so that the 
 ether in the bottle is at the same level as the mercury had been 
 before, or a trifle above this level. The volume of the mixture of 
 air and ether vapor being approximately the same as the volume 
 of the air before the introduction of the ether, how does the 
 pressure within the bottle compare with the pressure when it 
 contained air alone ? Did the evaporation cease immediately 
 after the introduction of the gasoline, as in III? If it did not, 
 explain why. What do you find to be the effect of mixing ether 
 vapor with air, the volume being kept constant ? 
 
 Watch the mercury column and see that its height becomes 
 constant before taking the measurements in VI. The mercury 
 ought to become stationary in 15 minutes. 
 
 VI. Find by appropriate measurements the increase of pressure 
 within the bottle over the pressure before the introduction of the 
 ether. What does this increase of pressure represent? How 
 does it compare with the pressure of ether vapor when unmixed 
 with air as determined in II ? 
 
3] VAPOR PRESSURE AND TEMPERATURE. 15 
 
 Observe and record the temperature of the room. Is it the 
 same as when II was performed? How would any difference in 
 temperature affect the pressure of the ether vapor? 
 
 According to Dattori s law the pressure of any vapor, or gas, 
 in a gaseous mixture* is the same as it would be if it occupied 
 the space alone. Do the results obtained in VI and in II tend to 
 confirm the truth of this law ? 
 
 VII. Calculate in dynes per square centimeter the barometric 
 pressure, also the pressure of the ether vapor in II, in the same 
 unit. 
 
 Is evaporation a cooling or a warming process? Explain. 
 
 3. VARIATION OF VAPOR PRESSURE WITH 
 TEMPERATURE. 
 
 I. Fill a deep hydrometer jar with water at about 55. When 
 the water has cooled to 48 (not before) set in the jar a closed 
 U-tube with a few cm. of ether, free of air bubbles, f in the closed 
 end, and at least 50 cm. of mercury in the rest of the tube. 
 The mercury before inserting in the water should stand a few cm. 
 lower in the open arm than in the closed, and there should be 
 enough water to completely cover the ether. Describe what takes 
 place when ether is warmed in this way. 
 
 Suspend a thermometer in the jar on a level with the ether and 
 read the temperature of the water. J At the same time measure 
 
 *Dalton's law does nut apply to a mixture of gases, or vapors, that 
 act on each other chemically, or to a mixture of vapors from liquids that 
 are mutually soluble. 
 
 flf there is any air above the ether, ask to have it removed. 
 
 JTo read a thermometer accurately, the observer's eye should be 
 placed so that the first degree mark below the top of the mercury coin- 
 cides with its reflection in the mercury. The fraction of a division 
 above this mark should be carefully estimated and recorded in tenths of 
 a degree. 
 
16 VAPOR PRESSURE AND TEMPERATURE. [3 
 
 the difference in level between the mercury in the two arms of the 
 U-tube. Do this as accurately as you can by placing a metre rod 
 against the side of the jar and sighting across the top of each 
 mercury column. It will injure the rod to put it into the water. 
 Using this last measurement and the barometric pressure for the 
 day, find the pressure, in cm. of mercury and in dynes per square 
 centimetre, of the ether vapor within the closed arm of the tube. 
 Stir thoroughly when taking readings. 
 
 II. If necessary, siphon off a small quantity of the water and 
 replace it with enough cold water to lower the temperature about 
 3 or 4 degrees, not more. Repeat the measurements of the last 
 section. 
 
 In this way make a series of some ten observations of the tem- 
 perature and pressure of the ether vapor, cooling it down to the 
 temperature of the room or lower. 
 
 III. Plot the results of I and II on co-ordinate paper and draw 
 a smooth curve to show the relation between the pressure and 
 temperature of ether vapor. 
 
 Do you find the pressure of the ether vapor to vary uniformly 
 with the temperature or not ? 
 
 IV. Take some ether in a small test-tube and immerse it in 
 water at about 30, adding hot water gradually until the ether 
 begins to boil. A small, clean tack or other sharp-pointed object 
 placed in the ether will facilitate boiling. Record the tempera- 
 ture of the ether when it first begins to bubble as the boiling 
 point. 
 
 Find from the plot obtained in III the temperature of ether 
 vapor when its pressure is equal to the barometric reading for 
 the day. How does this agree with the boiling point of ether 
 just found ? What relation may one infer exists between the 
 temperature at which a liquid boils and that at which the pres- 
 sure of its vapor becomes equal to the atmospheric pressure? 
 Explain. 
 
 V. Write not less than one hundred words on the properties of 
 saturated vapors. Explain the phenomenon of boiling. 
 
. 
 4 ] BOYLE'S LAW AND VOLUMENOMETER. 17 
 
 4. BOYLE'S LAW AND VOLUMENOMETER. 
 
 I. With the Boyle's law apparatus take a set of five readings 
 of pressure and of corresponding volumes, covering the range of 
 the apparatus. To the difference in mercury levels what quantity 
 must be added to give the total pressure on the inclosed gas? 
 Assuming the tube to be of i cm. section, plot applied pressures, 
 i. e., the difference in mercury levels, in terms of reciprocals of 
 volumes. What does this plot show to be the relation between 
 the pressure and volume of a gas when the temperature is con- 
 stant? Produce the line drawn until it cuts the pressure axis 
 and compare the intercept on the pressure axis with the barom- 
 eter reading. 
 
 II. Unscrew the iron cap of the volumenometer and by raising 
 or lowering the open tube adjust the level of the mercury in the 
 other tube to some point between the middle and the upper 
 marks. See that the iron cap is empty and replace it, screwing 
 it down air-tight. 
 
 Test the apparatus to see that it is air-tight. (Describe how 
 you do this.) 
 
 Notice that there are three horizontal marks on the closed tube 
 and that the mass of mercury that fills this tube between each 
 pair of marks is recorded on the apparatus, so that the volumes 
 between the marks may be calculated. The density of mercury 
 is 13.6. Find the volume of the air enclosed above the middle 
 mark by noting the change in volume when the mercury is set at 
 the upper and at the middle marks and also the accompanying- 
 change in pressure. Two equations may thus be formed, one 
 giving the difference in volumes between the upper and lower 
 marks and the other the ratio of these two volumes (in terms of 
 the ratio of the pressures). 
 
 Write these equations and find volume called for. 
 
 III. Introduce a piece of iron into the iron cap and find as in 
 II the volume of the inclosed air to the middle mark. Calculate 
 
 2 
 
1 8 PRESSURE OF GAS AT CONSTANT VOLUME. [5 
 
 the density of the piece of iron, after finding its mass, explaining 
 the process you use and writing out the equations. 
 
 IV. Repeat II and III, using the middle and lower marks, 
 and compare results. 
 
 5. PRESSURE OF GAS AT CONSTANT VOLUME. 
 
 I. Set a metre rod in a vertical 'position alongside the open 
 tube of a simple constant-volume air thermometer with a fixed 
 bulb. Fill the space about and above the bulb with water at 
 about 5 or 10, and stir continuously. Allow ten minutes for 
 the, inclosed air to come to the temperature of the bath, and then 
 raise or lower the open tube so as to bring the mercury in the 
 stem of the bulb to the bottom of the tube through which the 
 stem is thrust. Read on the metre rod the heights of the two 
 mercury columns, and take the temperature of the bath, stirring 
 all the while. 
 
 II. Draw off some of the water and replace it with warmer 
 water so as to raise the temperature of the bath about 10.* 
 After waiting ten minutes, repeat the operations and measure- 
 ments of I. 
 
 In this way make a series of observations on the pressure and 
 temperature of the inclosed air, raising the temperature about to 
 at a time, and carrying it as high as can be conveniently done 
 with boiling water. Arrange the results in tabular form. How 
 did the pressure of the inclosed gas (air) alter as its temperature 
 increased? Was the rate of change uniform? 
 
 III. Calculate the average increase in pressure for a rise of one 
 degree in temperature. If no observation was made at o, calcu- 
 late from your results, using the atmospheric pressure for the day, 
 the pressure that the gas would have at o, if its volume was kept 
 
 *Do not try to obtain a rise of exactly 10 in temperature. Better 
 results can be obtained and time saved if the bath is raised a trifle over 
 10 and then stirred till the inclosed air has had time to come to the 
 same temperature as the surrounding water, whatever that may be. 
 
6] EXPANSION OF GAS UNDKR CONSTANT PRKSSURK. IQ 
 
 constant. Find the ratio of the average increase in pressure per 
 degree to the pressure at o. 
 
 Calling P the pressure at o, P t the pressure at /, and a the 
 ratio just found, write the equation connecting the pressure and 
 temperature of a gas when the volume is constant. 
 
 IV. Plot the results of II, plotting the temperatures as abscissae 
 and the pressures as ordinates. 
 
 Draw the straight line that agrees most nearly with the points 
 located on the plot. Find the rise of this line (/. e. , the increase 
 in pressure of the gas) for the change of 100 in temperature, and 
 also, from the plot, the pressure of the gas at o. From these 
 calculate the ratio of the increase in pressure per degree to the 
 pressure at o. How does this agree with the result found in 
 III? Why should this last be the more reliable of the two 
 results? 
 
 V '. What would be the pressure of a gas at 273 C. , suppos- 
 ing there was no change of state or volume ? If the pressure 
 of a gas depends on the motion of its molecules, would the mol- 
 ecules have any motion at 273 C.? Then, as heat is the energy 
 due to molecular motion, according to this reasoning could a gas 
 be cooled below 273 C.? 
 
 This temperature is called absolute zero. The temperature 
 measured in Centigrade degrees from absolute zero is called the 
 absolute temperature. 
 
 6. EXPANSION OF GAS UNDER CONSTANT 
 PRESSURE. 
 
 I. Fill the space about the closed tube, or bulb, of the air 
 thermometer with ice-cold water. Set the slider at the zero of 
 the vertical scale, and adjust the mercury columns so that the 
 mercury in both tubes is at the level of the lower end of the 
 stuffing box. (The mercury column can be set quite accurately 
 by sighting across the end of the brass tube surrounding the 
 glass. ) Read the volume of the inclosed gas (air) and take the 
 temperature of the water bath, stirring thoroughly. 
 
20 SPECIFIC HEAT. [7 
 
 II. Raise the temperature of the bath as in Exercise 5, II, 
 about 10 at a time, and repeat for each temperature the opera- 
 tions and measurements of I, waiting ten minutes between suc- 
 cessive temperatures to allow the air to take on the temperature 
 of the bath. What was the pressure of the inclosed air in each 
 case ? Was it the same ? Was the expansion of the air uniform ? 
 Arrange the results in tabular form. 
 
 III. Calculate the average expansion for a rise of one degree in 
 temperature. If no observation was made at o, calculate from 
 your results the volume that the gas would have had at o. Find 
 the ratio of the average expansion per degree to the volume at 
 o, in other words, the cubical coefficient of expansion between 
 o and i. 
 
 Calling V the volume of a gas at temperature o, V t the 
 volume at t, and a the coefficient just found, write the law of 
 expansion of a gas at constant pressure in the form of an equation. 
 This is called the law of Charles or Gay-Lussac. 
 
 IV. Plot the results of I and II on co-ordinate paper, plotting 
 the temperatures as abscissae and the volumes as ordinates. 
 
 Find from this plot, by the method of Exercise 5, IV, 
 the expansion for a change in temperature of 100 and the 
 volume of the gas at o. Calculate from these the coefficient of 
 expansion between o and i. Is the result the same as that 
 obtained in III ? 
 
 V. When experiments 4, 5 and 6 have been performed, hand in 
 a paper of at least two hundred words on the properties of gases. 
 
 7. SPECIFIC HEAT. 
 
 I. Weigh out about 300 gr. of lead shot and heat it in a 
 double boiler. After the water begins to boil, stir the shot thor- 
 oughly with a wooden paddle, continuing until the temperature 
 of the shot becomes constant. 
 
 Have ready about 75 gm. of water (weighed to 0.5 gm.) at a 
 temperature of 5 to 10, in a calorimeter of known mass. 
 
7] SPKCIFIC HEAT. 21 
 
 Note carefully the temperature of the shot (stirring) and of the 
 water (stirring), and as quickly as possible pour the shot into the 
 water, stirring vigorously all the while and note the rise in tem- 
 perature of the water. Read the temperature of the mixture 
 every half minute for five minutes, counting from the instant of 
 mixing. 
 
 From the results obtained calculate: 
 
 1. The number of heat units gained by the water, using as 
 heat unit the calorie or the heat required to raise the temperature 
 of one gramme of water one degree. 
 
 2. The number of heat units lost by the shot (in terms of s, 
 the specific heat of lead) or the ratio of the heat required to raise 
 i gm. of lead one degree to that required to raise i gm. water 
 i. 
 
 Assuming that the shot and water are alone concerned in the 
 transfer of heat, what relation exists between the heat lost and 
 gained by the shot and water respectively ? Write the equation 
 representing this relation and calculate the specific heat of lead. 
 
 II. Calculate from this result, using above equation, the mass 
 of water which would have brought the mixture to a temperature 
 two degrees higher than that of the room. 
 
 Repeat I, using this mass of water and the same amount of 
 shot as before and other conditions also the same as in I. 
 Why should the latter result be the better ? 
 
 III. The result found in II is to be corrected for the heat lost 
 to cup, assuming the specific heat of the cup to be 0.095; a d 
 also corrected for radiation as follows: 
 
 Construct a plot with times as abscissae and temperature of 
 water and mixture as ordinates; project the line (which should 
 be straight if the stirring has been thorough), representing the 
 temperatures of the mixture, back until it cuts the ordinate at the 
 instant of mixing. This ordinate will be approximately the true 
 temperature of the mixture. Why ? 
 
 Write the complete equation involving all of the above quanti- 
 ties and recalculate the specific heat of lead. 
 
22 LATENT HEAT. [8 
 
 IV. If the water at the start had a temperature higher than 
 that of the room, would the value of s found have been high or 
 ow ? Explain. 
 
 If one gramme of water were spilt in stirring, what would be 
 the effect on the value of s ? 
 
 8. LATENT HEAT. 
 
 I. Weigh out in a metal cup, which has been previously 
 weighed, at least 500 gm. of water at about 30. After record- 
 ing the exact temperature of the water, take a piece of ice (about 
 loo gm.) and place it in the cup, first wiping it carefully with 
 damp cotton. Stir the mixture thoroughly and take its tempera- 
 ture just as the ice disappears. 
 
 Having previously weighed the water, the mass of the dry ice 
 used can be found by weighing the mixture and subtracting the 
 mass of the water. 
 
 II. Calculate in order the following quantities, using the same 
 unit of heat as in Exercise 7 : 
 
 1. The heat lost by the water surrounding the ice. 
 
 2. The heat lost by the cup. (In calculating this quantity 
 it will be sufficiently accurate to take the specific heat of the metal 
 as 0.095.) 
 
 3. The heat required to raise the water from the melted ice 
 from o to the temperature of the mixture. 
 
 4. The total heat absorbed by the ice in melting. 
 
 5. The heat absorbed by each gramme in melting. 
 
 The latter quantity is called the latent heat of fusion of 
 water. 
 
 III. Fill a small copper boiler about two-thirds full of water 
 and insert through the cork stopper a safety-tube with an opening 
 about 2 cm. from its lower end. Connect to the boiler a rubber 
 tube with a trap for collecting the water condensed in the tube 
 and a delivery-tube 4 or 5 cm. long. Bring the water in the 
 boiler to a boil. (If at any time steam issues vigorously from the 
 
8] I.ATKNT HKAT. 23 
 
 safety-tube, it means that the water is low and the boiler needs 
 refilling.) 
 
 Weigh out about 500 gm. of ice-water in a metal cup of known 
 mass, and take its temperature. Empty the water out of the trap 
 and hold it so that the end of the delivery-tube is immersed 
 in the ice-water. Stir and observe the temperature as it rises. 
 When the temperature reaches a point two-thirds as much above 
 the temperature of the room as the original temperature of the 
 ice-water was below, remove the delivery-tube. Stir and take 
 the temperature again carefully. Replace the cup on the balance, 
 and find the increase in the mass of the water due to the steam 
 that has been condensed. 
 
 IV. If the temperature of the water was two-thirds as much 
 above the temperature of the room after the condensation of the 
 steam as it was below before the introduction of the steam, we 
 may safely neglect the effect of the air and surrounding bodies, for 
 the cup will lose to the room, by radiation and conduction, as 
 much heat in the latter part of the experiment as it gains from 
 it in the first part. Using the same unit of heat and the same 
 value for the specific heat of the metal cup as in II, calculate in 
 order the following quantities: 
 
 1. The total amount of heat imparted to the water and the 
 cup. 
 
 2. The heat given out by the water from the condensed steam 
 in cooling from 100 to the temperature of the mixture. 
 
 3. The total amount of heat given out by the steam or water 
 vapor in changing from the state of a vapor to that of a liquid. 
 
 4. The heat given out by each gramme of water vapor in 
 changing from the gaseous to the liquid state. 
 
 The latter quantity is called latent heat of vaporization of 
 water. 
 
 5. Write the equations representing this experiment. 
 
 V. Write at least one hundred words on the phenomena of 
 fusion and evaporation. 
 
24 MECHANICAL EQUIVALENT OF HEAT. [9 
 
 9. MECHANICAL EQUIVALENT OF HEAT. 
 
 I. Take two bottles and put in each of them a kilogramme of 
 lead shot. Place these bottles in a mixture of ice and water. 
 
 When the shot in one of the bottles has cooled about 3 below 
 the temperature of the room, shake it thoroughly, and pour it 
 into the tube provided, about one metre long, and close the end 
 of the tube securely after taking the temperature of the shot by 
 inserting a thermometer. Raise the end of the tube containing 
 the shot with sufficient velocity to keep the shot from falling, and 
 when it reaches a vertical position, let the shot fall vertically, like 
 a solid mass, through the length of the tube. Repeat this again 
 and again, keeping count of the number of times the shot falls.* 
 
 After the shot has fallen through the length of the tube a 
 hundred times, insert a thermometer through a side opening, and 
 take its temperature again. Why has the temperature of the 
 shot risen above that of the room ? 
 
 II. Replace the shot in the ice- water to cool, and while the 
 tube is still warm, repeat the operations and measurements of I, 
 using the shot from the other bottle, which should be about 3 
 below the temperature of the room. (Its temperature can be 
 raised by shaking the bottle, if it is too low.) Repeat the 
 experiment in this way, cooling one bottle of shot while using 
 
 * PRECAUTIONS, ETC. The shot should not be raised too suddenly, so 
 as to throw it violently against the side of the tube, nor should the tube 
 be raised or lowered so as to lengthen or shorten the distance fallen 
 through by the shot. 
 
 It is well, also, to hold the tube about a foot from each end, so that 
 there is no danger of any heat being imparted to the shot from . the 
 hands. The following method of raising the shot and reversing the tube 
 is recommended: Lay the tube on the table, and raise the end contain- 
 ing the shot, while the other end rests on the table. Let the shot fall, 
 and then lower the raised end. Raise the other end, which now con- 
 tains the shot, and let the shot fall again. Then lower this end, and 
 again raise the end which contains the shot; and so on. 
 
ID] SURFACE TENSION. 25 
 
 the other, making five determinations and using the average 
 result in what follows. 
 
 III. Remove the stopper and measure the distance from the 
 inner end of the stopper to the top of the shot. What is the 
 average distance fallen through by the shot in each reversal of 
 the tube? Explain. In one hundred reversals? How far would 
 the shot have to fall to raise its temperature one degree? How 
 for would one gramme have to fall to raise its temperature the 
 same amount (one degree) ? How much work, in ergs, would 
 be required to raise one gramme of shot one degree in tempera- 
 ture? The specific heat of lead is about 0.032. Using this, 
 calculate, in the ergs, the amount of work necessary to raise one 
 gramme of water one degree in temperature. This last quantity 
 is called the mechanical equivalent of the heat unit. 
 
 IV. Write the equation representing this exercise. 
 What are the chief sources of error in the experiment? 
 
 V. Power is the rate of doing work, and may be measured in 
 ergs per second, or in watts, which is io 7 ergs per second. 
 io 7 ergs is a joule. Calculate the work done in joules by the 
 shot falling 100 times the length of the tube, and if this operation 
 akes 3 minutes, calculate the power developed in watts. 
 
 io. SURFACE TENSION. 
 
 It is of capital importance that the rectangles and beakers used 
 in this exercise be clean. They should be thoroughly washed in 
 hot water before being used and for every change from one liquid 
 to another. 
 
 The Jolly balance should be read by bringing a definite point, 
 as the lower end of the spring, in a horizontal line with its image 
 in the mirror. The reading is facilitated by bringing a card 
 pierced with a small hole (3 mm. in diam.) close before the eye 
 and standing in front of the scale at such a distance that the 
 object and image are seen sharply focused at the same time. 
 
 I. Fill a beaker, about 7 cm. in diameter, with a solution of 
 
26 SURFACE TENSION. [lO 
 
 soap in water. Replace the pans of a Jolly balance by a wire 
 rectangle 2 cm. wide, hung vertically, and hold the beaker so 
 that the rectangle is immersed to a certain definite depth in the 
 soap solution. See that there is no soap film within the rectangle, 
 and read the balance. 
 
 Let the rectangle dip in the soap solution so that a film is 
 formed within it. Raise or lower the beaker so that the rect- 
 angle is immersed to the same depth as before and again read 
 the balance. What difference does the presence of the film make 
 in the reading of the balance ? To what force is the elongation of 
 the spring due? 
 
 Take four independent sets of readings. 
 
 II. Repeat the measurements of I, using rectangles about 4 
 and 6 cm. wide. How do you find the tension of the film to 
 vary with its width ? 
 
 III. Find the elongation of the spring produced by a small 
 known weight, some fraction of a gramme. How does the 
 elongation vary with the force producing it? Test this. 
 
 Calculate the tension in dynes (980 dynes=weight of one 
 gramme) of each of the three films in I and II. As a film has 
 two surfaces, the width of the surface in apparent tension, 
 neglecting that about the wires, will be equal to twice the width 
 of the rectangle. Using this, calculate in dynes the average 
 tension of the soap solution across each cm. of the surface. 
 
 The tension across a unit length of the surface of a liquid is 
 called the surface tension of that liquid. 
 
 IV. Clean the beaker and rectangle thoroughly, and repeat 
 the measurements of II with water fresh from the faucet. 
 
 As a film can not be formed with pure water, take the reading 
 of the balance when the upper side of the rectangle is just above 
 the surface of the water and again when it breaks away from this 
 surface. The force measured in this way may be regarded as due 
 entirely to surface tension, although this is not strictly true. 
 Take four sets of readings. 
 
 Calculate the surface tension of the water. How does it com- 
 pare with that of the soap solution? 
 
Il] PRINCIPLE OF MOMENTS. 2J 
 
 V. Using the same rectangle, find the surface tension of hot 
 water from the heater at the sink. Does the temperature affect 
 the surface tension appreciably, and how ? 
 
 VI. If the rectangle 4 cm. wide carries a soap film 2 cm. high, 
 what is the work done in forming this film? What is the energy 
 per square centimeter of this film ? How does this quantity 
 compare with the surface tension? 
 
 ii. PRINCIPLE OF MOMENTS. 
 
 I. (#.) Attach a light metal frame to the table so that it can 
 rotate freely about a pivot. ' Fasten two spring balances to the 
 frame with twine, at equal distances on opposite sides ol the 
 center, and draw them out so that they are parallel. Read the 
 balances. Does a force produce the same effect if transferred 
 along its line of action ? How test this? 
 
 Pull one of the balances out until the tension is doubled, 
 keeping them still parallel. What does the other balance regis- 
 ter? When a force tends to produce rotation about a pivot, 
 what is the effect of doubling this force upon the force opposing 
 the rotation ? 
 
 (<5.) Move one of the balances to a point twice the distance 
 from the center as in (a) and pull it (parallel to the other balance) 
 until it registers the same tension as before. Read both balances. 
 
 The perpendicular distance from the center of rotation to 
 the line of action of a force is called its lever arm. When a 
 force tends to produce rotation about a point, what do you find 
 to be the effect of doubling the lever arm upon the force oppos- 
 ing the rotation ? 
 
 (V.) The tendency of a force to produce rotation about a point, 
 according to (a) and (^), is proportional to. the product of what 
 two quantities ? This product is called the moment of the force 
 about the point considered, and is usually taken positive in sign 
 when the force tends to produce rotation in a counter-clockwise 
 direction, and negative when it tends to produce rotation in the 
 opposite direction. 
 
28 PRINCIPLE OF MOMENTS. [ll 
 
 II. (a.) Take a beam suspended so as not to rub the surface of 
 the table, and connect its middle point to a nail in the table by 
 means of a spring balance. Attach two balances to two screw- 
 eyes, one metre apart, on the opposite side of the beam at 
 unequal distances from its middle point and to corresponding 
 nails in the table. Tighten the cord attached to the first balance. 
 Read all three balances, and measure the distances between their 
 points of attachment to the beam. 
 
 (<.) Loosen, or tighten, the cords a little and read the balances 
 again. 
 
 (V.) Calculate the moment of each of the forces in (a) about 
 some point of the beam. Give these moments their proper 
 signs, and find their algebraic sum. Do the same for the 
 forces in (6). What is your conclusion as to the value of 
 the sum of their moments when a number of parallel forces in 
 the same plane act on a rigid body so that it is held in 
 equilibrium ? 
 
 III. Attach three balances at random to the frame used in I, 
 and to nails in the table. Tighten the cords and read the 
 balances. Draw, on a sheet of paper laid underneath the frame, 
 a line parallel to the line of action of each of the forces measured 
 by the balances. Remove the frame and measure carefully the 
 lever arm of each force about the pivot as a center. Calculate 
 the moments of the forces about the pivot and find their algebraic 
 sum. In addition to finding the sum of the moments about the 
 pivot, find also the sum of the moments of the forces about some 
 point outside the pivot. Do you find the sum of the moments to 
 be approximately the same wherever the center of moments is 
 taken, or not? Explain. 
 
 IV. Repeat III, removing the pivot, so that the frame is free to 
 move in any horizontal direction. Make the proper measure- 
 ments and calculate the sum of the moments of the forces about 
 some point on the table taken at random. Do the same for some 
 other point on the table. Do you find the sum of the moments 
 to be approximately the same wherever the center of moments is 
 taken ? 
 
12 
 
 :] COMPOSITION OF FORCES. 2Q 
 
 V. If any number of forces in the same plane act upon a rigid 
 body so that it is held in equilibrium, what do you conclude from 
 the results of this exercise must be the algebraic sum of their 
 moments about any point in that plane? The correct answer to 
 this question is called the principle of moments. 
 
 VI. Two equal, parallel forces in opposite directions constitute 
 a couple. The perpendicular distance between them is called the 
 arm of the couple. 
 
 Let a be the arm, and F one of the component forces of a 
 couple. Find the moment of this couple about any point. Is it 
 the same for all points ? Demonstrate this for any point within 
 and one without the lines of action of the forces. 
 
 GROUP II. 
 
 In general, students will begin with the exercise corresponding 
 to that they began with in Group I. Thus he who started with 
 the 5th exercise will now take the i6th, and so on. 
 
 12. COMPOSITION OF FORCES. 
 
 I. Take a stout beam, over a metre long, and find its weight 
 (in Ibs. ) by means of a spring balance. 
 
 Attach cords of equal length to screw-eyes near the ends of the 
 beam, and suspend it by these cords from two 3O-lb. spring 
 balances hung from nails in the wall, at the same distance apart 
 as the screw-eyes in the beam. Read the balances. What 
 relation exists between the combined readings of the balances 
 and the weight of the beam ? 
 
 II. Suspend a mass of metal, weighing over 30 Ibs., from the 
 middle of the beam and read the balances again. Do the 
 balances read alike ? Why? How can you find the weight of 
 the metal from the readings of the balances? What is the 
 weight as thus found? 
 
30 COMPOSITION OF FORCES. [l2 
 
 III. Hang the mass of metal from a point to one side of the 
 middle of the beam and read the balances again. Why do they 
 not read alike now ? Does the relation found in I between the 
 total suspended weight and the combined readings of the balances 
 still hold true ? Measure the horizontal distances from the cord 
 by which the weight is hung to the cords to which the balances 
 are attached. How do the products formed by multiplying each 
 distance by the reading of the corresponding balance (less one- 
 half the weight of the beam) compare? 
 
 In general, what is the resultant of two parallel forces in the 
 same direction equal to: what is its direction: and how is its line 
 of action situated with reference to the component forces? 
 
 IV. Hang two 3o-lb. spring balances from two nails above the 
 blackboard, at least one metre apart, and connect the balances by 
 a cord somewhat over a metre long. From the middle point of 
 this cord suspend the mass of metal used in II and III. Draw 
 on the blackboard lines parallel to the two parts of the cord and 
 lay ,orT on these lines, from their intersection, length? proportional 
 to the tension in each part of the cord as registered by the proper 
 balance. Construct a parallelogram with these lines as sides and 
 draw the vertical diagonal. Measure the length of this diagonal 
 in Ibs., using the same scale as was used for the sides of the 
 parallelogram. How does this diagonal compare in direction 
 and length with the downward force (weight) of the mass sus- 
 pended from the cord? What is the value of the weight as found 
 by this method ? 
 
 V. Hang the mass of metal to one side of the middle of the 
 cord, and construct another similar parallelogram of forces. Is 
 the relation between the diagonal and the weight of the suspended 
 mass the same as in IV? What is the value of the weight as 
 found from this parallelogram? 
 
 VI. Hang the mass of metal by a single cord from one of the 
 nails. Attach a spring balance to the cord, near the bottom of 
 the blackboard, and pull it horizontally one foot from the vertical. 
 Note the reading of the balance, and measure the vertical distance 
 from the nail to the line of action of the horizontal force. 
 
^ 
 
 13] ELASTICITY: LAWS OF STRETCHI 
 
 By what two forces was the cord acted upon, and in what 
 direction was their resultant ? Which one of these two forces was 
 measured directly? Find the value in Ibs. of the other force. 
 (As the two forces are at right angles, this may be done either 
 graphically by constructing a triangle of forces, or by calculation 
 from similar triangles.) Find also the tension in the inclined part 
 of the cord. 
 
 VII. Repeat VI, drawing the cord two feet to one side instead 
 of one foot and find again the value of the weight. 
 , VIII. If three forces are in equilibrium about a point, show that 
 they may be represented in magnitude and direction by the three 
 sides of a triangle taken in order. 
 
 13. ELASTICITY: LAWS OF STRETCHING. 
 
 I. (a.) Attach a spring balance to the finer of the wires hang- 
 ing freely from the ceiling. Set the scale immediately behind 
 this wire and adjust the index on the wire, if necessary, until this 
 index is opposite the upper part of the scale, and read its position 
 on the scale by means of a lens, taking care that the index, lens, 
 and eye are in a horizontal line. Read the spring balance also. 
 
 (b.) Hang on a weight putting it gently into place and re- 
 peat (a). Add successively three other weights, noting the 
 index and balance readings in each case. 
 
 (c.') Remove the weights one by one, taking the same readings 
 as in (d) and (a). Average the corresponding results of (a), (6), 
 and (). 
 
 (d.) Find the length of wire used, measure its diameter in four 
 places and compute its mean cross-section. 
 
 II. Clamp the wire used in I at about midway its length and 
 repeat I, taking care to use the Weights in the same order as 
 before. 
 
 III. With a wire of greater diameter but same material, repeat 
 I, noting the precautions of I and II. 
 
 IV. Make a plot of the results, in I with weights in dynes' 
 
32 ACTION OF GRAVITY. [14 
 
 (453.6 grammes are equivalent to a pound Avoirdupois) as 
 abscissae and elongation of the wire as ordinates. What relation 
 do you find to exist between the stretching force and the resulting 
 elongation ? The statement of this relation is known as Hooke* s 
 Law. 
 
 V. (a.) Show from I and II the relation between the length 
 and elongation. 
 
 (.) From land III, show the relation between the diameter 
 and elongation; between the cross-section area and elongation. 
 
 (c.) Form an equation giving the elongation in terms of length, 
 cross-section, force applied, and a constant K. 
 
 VI. Stress is defined as force per unit area; strain is the 
 elongation per unit length; and the measure or modulus of 
 elasticity is the ratio of the stress to the strain. Find the ex- 
 pression for the modulus of elasticity in terms of the quantities in 
 V and calculate its value in C. G. S. units for the substance used. 
 What relation exists between the modulus M (called Young's 
 Modulus) and the constant K in V (c) ? Is the value of M the 
 same for all substances? (Compare results with your neighbors' 
 who used wires of different material. ) 
 
 VII. If a very considerable weight were hung on a wire, would 
 the conclusions of IV, V, and VI hold? Explain. Why does 
 no correction have to be made for the position of the spring- 
 balance ? 
 
 14. ACTION OF GRAVITY. 
 
 I. (a.) Find the time of a quarter- vibration by counting and 
 timing 100 complete vibrations of a rod pendulum freed from any 
 weights that may have been attached to it. 
 
 (.) Fasten a strip of impression paper, dark side out, and on 
 it a piece of white paper, to the lower end of the pendulum. 
 Suspend a metal ball by a thread passing over two nails above the 
 pendulum, another at the base, and attach to the lower end of the 
 pendulum, pulling the latter aside. The weight of the ball and 
 
14] ACTION OF GRAVITY. 33 
 
 friction will be sufficient to hold the pendulum aside. Burn the 
 string near the ball after it is at rest and find by trial to what 
 height the ball must be raised so as to hit the paper when falling. 
 Make three determinations of this distance, measuring from the 
 center of the ball above to the corresponding mark on the paper 
 in each case. 
 
 II. Clamp the weight provided to the pendulum near the lower 
 end at the place marked and repeat I (a) and (). 
 
 III. Repeat with the weight clamped near the top of the 
 pendulum. 
 
 IV. (.) Find in each of the above cases, for the time of a 
 quarter vibration, the average velocity of the ball, and its final 
 velocity. Show that the final velocity is twice the average 
 velocity if the ball starts from rest and increases its velocity at a 
 constant rate. 
 
 (.) Deduce from the results of I, II, and III the relation 
 between the space passed over by the ball and the time, indicat- 
 ing clearly the process you use. 
 
 (/-.) Calculate the distance the ball would have passed over in 
 one second, averaging the results of I, II, and III. 
 
 (d. ) Show to what power of the time the acquired velocity is 
 proportional; see (a) and (c). 
 
 (<?.) Calculate the velocity acquired in one second, i. e., the 
 acceleration (usually denoted by the letter g) , averaging results as 
 before. 
 
 V. (.) Express IV () in the form S=Kt x and calculate the 
 value of the constant K. What relation does it bear to the value 
 of g ? What then is the equation for the space passed over in 
 terms of the time and acceleration ? 
 
 (b. ) Similarly find the value of the constant in the relation found 
 in IV (d) and write the corresponding equation. 
 
 (c.) Deduce the expression for the velocity of a body, starting 
 from rest and moving under the action of gravity, in terms of the 
 acceleration and space. 
 3 
 
34 THE PENDULUM. [15 
 
 VI. If a ball of greater mass had been used, would the same 
 results have been obtained for the final velocities and for the 
 acceleration? Explain. 
 
 Distinguish between mass and weight. 
 
 15. THE PENDULUM. I. 
 
 I. (a.) With a metal ball attached to the longest wire that the 
 apparatus allows, pulling aside the bob not more than iocm., 
 find the period of the pendulum to o.oi second by the following 
 method: 
 
 First find the approximate period* by timing about twenty 
 vibrations. (Be careful to count "one" when the bob passes the 
 middle of the swing at the end of the first vibration.) Next note 
 the time that the bob passes to the right (say) through the center 
 of swing, the eye being in line with this position; wait about three 
 minutes and note again the time of transit in the same direction; 
 repeat this timing two or three times. Between each pair of 
 observations there was a whole number of vibrations. Divide 
 the first interval by the approximate period found above; if this 
 period were the true one the quotient would be an integer. 
 Divide the interval by the nearest integer to the quotient last 
 found and the result will be a closer approximation to the true 
 period. Repeat this operation for the other observed intervals 
 and take the mean as the best value for the period. (Note: 
 The computations may be done after the whole experiment has 
 been performed.) Measure the length of the pendulum, i. e., 
 from point of suspension to center of ball. 
 
 (.) Repeat (a), pulling aside the bob not more than 5 cm. 
 
 (<:.) Repeat (a), pulling aside the bob some 50 cm. 
 
 (d.) What is the effect of increasing the amplitude on the 
 period of a pendulum ? Which is the best value to take for the 
 period, that given by (a), (), or (<;)? Why? 
 
 *The period is the time of a complete vibration, or the time between 
 two successive transits in the same direction. 
 
16] THE PENDULUM. 35 
 
 II. (<2.) Repeat I (a) or (b), using- two shorter pendulum 
 lengths. 
 
 (.) Substitute a wooden ball of same size as the metal one 
 and repeat, using any length of pendulum, but measuring it. 
 
 III. (a.} From I and II (a) find the relation existing between 
 the length and period of the pendulum. Indicate clearly your 
 method. 
 
 (<.) Show whether or not the relation III (a) applies to II (). 
 What is the effect on the period of changing the mass of the 
 bob? 
 
 IV. From the results of I and II and the relation of III (a) 
 calculate the length of the seconds pendulum at Berkeley. (A 
 seconds pendulum is one whose half-period is one second.) 
 
 16. THE PENDULUM. II. 
 
 I. (a.) Find the period of the pendulum, to a hundredth of a 
 second, when set so that it vibrates in a vertical plane. (See 
 Ex. 15.) 
 
 (<.) Find the period when the plane of vibration makes an 
 angle of 60. 35 with the vertical, (cos. 60. 35=0. 49.) 
 
 (c.) Find the period when the plane of vibration makes an 
 angle of 75. 5 with the vertical, (cos. 75. 5=0.25.) 
 
 (d,} What is the vertical force acting on the pendulum bob? 
 What is the vertical force acting on unit mass of the bob? 
 Suppose this vertical force acting on unit mass to be resolved into 
 two components, one perpendicular to the plane of vibration of 
 the pendulum, and the other in the direction of its length when 
 at rest. If a pendulum is constrained to vibrate in a particular 
 plane, as in this case, would a force perpendicular to its plane of 
 vibration affect its period or not? Why? Draw a diagram 
 showing forces acting on bob. 
 
 What is the ratio between the force per unit mass in the 
 direction of the length of the pendulum in (#) to that in (<); in 
 (#) to that in (^)? (Express these ratios as reciprocals.) What 
 
36 RESONANCE TUBE. [17 
 
 is the ratio of the period in (a) to that in (); in (a) to that in 
 (r)? (Calculate these ratios in decimals.) By comparing these 
 results, find the law connecting the period of a pendulum with the 
 force on unit mass, or the acceleration, in the direction of its 
 length when at rest, assuming that the period varies as some 
 integral root, or power, of the acceleration. 
 
 II. (To be done when Ex. 15 has been completed.) (a.) From 
 the law deduced in III (a) of Ex. 15 find the length of the 
 simple pendulum equivalent to the physical pendulum of I (a), 
 Ex. 1 6, using the seconds pendulum as comparison. 
 
 (.) Express the period P of a simple pendulum in terms of a 
 constant k, its length, L, and acceleration in the direction of its 
 length. The latter quantity is g (see Ex. 14) if the pendulum 
 vibrates in a vertical plane. 
 
 (f.) Calculate the values of g and of k as follows: The equation 
 of II (&) applies to the case of the seconds pendulum of Ex. 15 
 and to the simple pendulum of II (a}. Form two simultaneous 
 equations for the values of k and g in terms of the known quanti- 
 ties L and P, for each pendulum, and solve fork and g. 
 
 III. Is the length of the seconds pendulum the same over the 
 surface of the earth? Why ? Write not less than one hundred 
 words on the uses of the pendulum. 
 
 17. RESONANCE TUBE. 
 
 The resonance tube to be used consists of a long vertical glass 
 tube connected at its lower end by a rubber tube and siphon with 
 a jar of water, so that when the jar is raised and lowered, the 
 water flows in and out of the tube. The siphon can be started 
 by setting the jar on the floor and pouring water into the tube 
 until it flows into the jar. The water in the tube may be kept at 
 any desired level by turning the cock at the base. 
 
 Tuning forks are to be set in vibration by striking with rubber 
 hammer, and in no other way. 
 
 I. Hold a vibrating A-fork over the nearly full tube, and mark 
 
17] RESONANCE TUBE. 37 
 
 with a rubber band as the jar is lowered the level of the water 
 when the air in the tube vibrates in unison with the fork and 
 causes a marked increase in the intensity of the sound. 
 
 Raise the jar, and as the water rises readjust the rubber band 
 to the level of the water when the sound swells out again. Let 
 the water rise and fall past this point a number of times and 
 determine the level when the air in the tube vibrates in unison 
 with the fork, as accurately as you can, recording each measure- 
 ment. 
 
 As the air has no freedom of motion in a vertical direction at 
 the surface of the water, this plane where the column of air may 
 be cut off without prejudice to its rate of vibration must be one 
 of minimum vibration, i. e., a nodal plane. 
 
 What is the condition of the air at the open end of the tube ? 
 
 Find all the prominent nodal planes you can. Measure the 
 distances between them and between the highest one and the 
 open end of the tube. Is the latter the same as the distance 
 between two consecutive nodal planes ? Can you account for the 
 fact that the ratio of these distances is not exactly 1:2? How 
 are these distances related to the wave-length in air of the 
 particular note sounded ? Explain and draw diagram in illustra- 
 tion. 
 
 II. Repeat I with the two C-forks, and also with a G- or D- 
 fork. 
 
 Find the ratio of the distance between the nodes when the 
 A-fork was used to that when the large C-fork was used. This 
 gives the ratio between the wave-lengths. How is this ratio 
 related to the ratio between the vibration frequencies of the two 
 notes? The latter ratio measures the musical interval between 
 the notes. 
 
 Calculate from your results the musical intervals between the 
 lower C-fork and each of the others. 
 
 III. If the larger C-fork makes 256 complete vibrations per 
 second, calculate the velocity of sound in air in the tube used. 
 Find the vibration number of each of the other forks. 
 
38 VELOCITY OF SOUND IN SOLIDS. [l8 
 
 IV. Explain why the column of air emits a note. 
 Is more or less energy used by the fork and air column sound- 
 ing together than when the fork is sounding alone? Explain. 
 
 18. VELOCITY OF SOUND IN SOLIDS. 
 
 I. Clamp a long brass rod exactly at its middle point in a vice. 
 Take a long glass tube, provided with a piston at one end and 
 containing powdered cork, and set its rubber-covered end 
 against one end of the rod. A cardboard disc of some -2 cm. 
 diameter should be glued to this end of the rod. 
 
 Set the rod in vibration by stroking it from the center out 
 slowly and with but slight pressure, with a cloth wet with wood 
 alcohol or rubbed with resin. The piston should be adjusted by 
 trial until the cork-dust takes up its characteristic arrangement. 
 Describe the behavior of the cork dust. Where are the nodes ? 
 The loops? Which is found at either end of the tube ? Explain. 
 Measure the length of the tube and find the wave-length of the 
 sound in air. What is the wave-length of the sound in the rod? 
 Explain. Make three determinations. 
 
 Calculate the ratio of the wave-length in brass to the wave- 
 length in air for the same note. How is this ratio related to the 
 relative velocity of sound in brass and air? Calculate the velocity 
 of sound in brass,* and write out the equations involved. 
 
 II. Repeat the measurements of I with a glass rod and find the 
 velocity of sound in glass. 
 
 III. Repeat with an iron rod. 
 
 IV. How with this apparatus might the velocity of sound in 
 gases be found? Write the equations involved. 
 
 V. Write at least one hundred words on the subject of station- 
 . ary waves. 
 
 *The velocity of sound in air is: 
 
 V=33i i/ 1+0.0041 metres per second, where t is the temperature in 
 centigrade, 
 
19] LAWS OF A VIBRATING STRING. 39 
 
 19. LAWS OF A VIBRATING STRING. 
 
 I. (#.) Attach two piano steel wires (No. 27 and No. 22 
 B. & S. gauge) to a sonometer and stretch the lighter wire over 
 the sounding-board with a weight of 4 Ibs. Move the sliding 
 bridge until the note given out by the wire when plucked is in 
 unison with the tuning-fork provided. (The note of the fork can 
 be made more audible by holding the end of its handle on the 
 sonometer board. ) Measure the length of the vibrating part of 
 the wire. 
 
 (&.) Move the sliding bridge so that the note given out by the 
 wire is in unison with the note an octave * below that of the 
 tuning-fork, and again measure the vibrating part of the wire. 
 
 (<:.) What is the relation between the vibration frequency of 
 two notes separated by an interval of an octave? What, by 
 comparing the results of () and ($), do you find to be the law 
 connecting the length of the vibrating wire (or string) with its 
 vibration frequency? 
 
 II. With additional weights increase the tension of the wire to 
 four times its tension in I, and adjust the sliding bridge, if neces- 
 sary, so that the note given out is in unison with that of the tun- 
 ing-fork. How does the length of the vibrating part of the wire 
 compare in this case with its length in I (b) ? What do you con- 
 clude to be the law connecting the vibration frequency of a 
 stretched wire with its tension ? 
 
 III. (a.) Repeat the experiment of I with the heavier wire, and 
 by comparison with the result of I (a} find the ratio between the 
 lengths of the two wires when their vibration frequencies are 
 equal. From this find the ratio between the vibration frequencies 
 of the two wires when their lengths are made equal ? (See law 
 found in I.) 
 
 (b.) Measure the length of a piece of each kind of wire and find 
 the ratio of their masses per unit length, using a Jolly balance for 
 weighing. 
 
 * Two notes have an interval of an octave where one has twice the 
 pitch of the other. 
 
40 PHOTOMETRY. [20 
 
 What do you find to be the relation between the vibration 
 frequency and linear density of a stretched wire, the length and 
 tension being constant? 
 
 IV. Form an expression for the pitch of a wire in terms of its 
 length, tension and linear density. What does this equation 
 assume regarding the elastic properties of the wire and the nature 
 of its motion? 
 
 20. PHOTOMETRY. 
 
 I. Set a diffusion photometer, two rectangular blocks of paraf- 
 fine separated by a sheet of tin-foil, so that the two blocks of 
 paraffine are equally illuminated by the diffused light of the room. 
 Light a set of four simple gas jets and a single separate jet of the 
 same form, and regulate the flow of gas so that the jets are all of 
 the same height and brightness. Place the single jet at a 
 distance of 50 cm. on one side of the photometer, so as to 
 illuminate one block of the paraffme, and the set of four jets on 
 the other side at such a distance that the two blocks of paraffine 
 will be equally illuminated. Measure the distance from the 
 photometer to the four jets. 
 
 How does the illumination of the paraffine due to the single jet 
 compare with that due to the four jets? How does the intensity 
 of the illumination due to a single jet at 50 cm. compare with 
 that due to a single jet at the distance of the four jets ? The 
 intensity of the illumination is proportional to an integral power 
 of the distance; what, from your results, do you conclude the 
 power in question to be? Is it direct or inverse? 
 
 II. Place the four jets at 50 cm. from the photometer and the 
 single jet on the opposite side at such a distance that the blocks 
 of paraffine are again equally illuminated. Are the conclusions 
 drawn from the results of I corroborated, or not, by the results 
 thus obtained ? Record measurements. 
 
 III. Light a candle and place it at a certain distance from the 
 photometer. Light a coal-oil lamp and place it on the opposite 
 side of the photometer, so that the candle and lamp illuminate 
 
21] REFRACTION. 41 
 
 the blocks of paraffine equally. (The height of the lamp wick 
 should not be altered during the course of this experiment, and 
 the lamp should burn at least five minutes before taking read- 
 ings.) How can you find the ratio between the illuminating 
 power of the candle and that of the lamp ? (Take four readings 
 with varying distances of the candle.) What is this ratio as 
 derived from your measurements? 
 
 Weigh the lamp and the candle. Let them burn for 30 
 minutes. Reweigh and find the mass of the coal-oil and of the 
 paraffine, or candle substance, consumed. For one gramme of 
 matter consumed by the candle, how many grammes were con- 
 sumed by the lamp ? Calculate the relative illuminating power of 
 coal-oil and paraffine for equal masses consumed, assuming that 
 the illuminating power varies directly as the amount of matter 
 consumed. 
 
 IV. Alter the height of the lamp flame and repeat III. 
 Calculate again, from the result obtained, the relative illuminating 
 power of coal-oil and paraffine for equal masses consumed. How 
 does the value found compare with that found in III? Is the 
 assumption made above, that the illuminating power varies 
 directly as the amount of matter consumed, corroborated by the 
 results of III and IV, or not? 
 
 V. Form an equation expressing the candle power of any 
 source of light in terms of the proper variables. Distinguish 
 between intensity of illumination and illuminating power. 
 
 21. REFRACTION. 
 
 I. Take a rectangular cell, having one side of plate glass and 
 containing a mirror revolving on a vertical axis, and fill it about 
 half full of water. Set this cell so that the axis on which the 
 mirror revolves is over the center of a large circle drawn on the 
 table. Adjust the cell so that its glass side is perpendicular to 2t 
 radius of the circle drawn parallel to the end of the table. This 
 may be done by stretching a white string along this radius, and 
 
42 REFRACTION. [21 
 
 moving the cell until the image of the string in the plate glass 
 coincides in direction with the string itself. (A piece of blackened 
 tin held back of the plate glass will help in locating the image of 
 the string.) 
 
 Move your eye along the edge of the table until you see the 
 image of the string in the mirror above the water. With another 
 white string locate the direction of this image, and stick a pin in 
 line with it on the circle drawn on the table. In the same way 
 look for the image of the string seen through the water, and mark 
 with another pin on the circle the direction of this image. 
 
 Measure the perpendicular distance from each of these pins to 
 the radius represented by the first string. 
 
 Answer the following questions: 
 
 1. Does the light from the first string undergo any change in 
 direction on entering the cell ? 
 
 2. Will it, therefore, strike the mirror at the same angle within 
 the liquid as without, i. e. , above the liquid ? 
 
 3. Will it be reflected at the same angle within as without the 
 liquid ? 
 
 4. Will the reflected light, passing through the liquid, have the 
 same direction after leaving the liquid as that which does not 
 pass through the liquid? What do you find by experiment ? 
 
 Remembering that the first string is perpendicular to the sur- 
 face of the water at which the light is refracted, how are the 
 sines of the angles of incidence and refraction related to the 
 distances measured above? 
 
 The ratio of the sine of the angle of incidence to that of refrac- 
 tion when the light is incident in air, or, more properly, in a vac- 
 uum, is called the index of refraction of the substance. (If the 
 light is incident in the substance and refracted in air, the index of 
 refraction, on the contrary, is equal to the ratio of the sine of the 
 angle of refraction to that of the angle of incidence.) Calculate 
 from your results the index of refraction of water. 
 
 II. Repeat I with the mirror at a slightly different angle and 
 calculate again the index of refraction of water. 
 
22] REFRACTION AND DISPERSION. 43 
 
 III. Rotate the mirror a little more and repeat I, calculating 
 again the index of refraction. 
 
 Do you find the index of refraction to vary with the angle of 
 incidence, or not ? 
 
 IV. Take a cubical block of glass and lay it on a sheet of 
 brown paper. Mark on the paper the position of two of its 
 opposite edges, and continue the lines with a ruler held against 
 the face of the cube. Stick a pin in the table about 30 cm. from 
 the cube, and as far to one side as it can be placed without 
 becoming invisible when looked at diagonally through the oppo- 
 site faces of the cube. Looking at this pin through the cube, 
 place three pins in line with it, one on the same side close to the 
 cube, and two on the side of the observer. Remove the cube 
 and draw lines on the paper to show the direction of the light 
 from the first pin before entering the glass, after passing through 
 the glass, and within the glass. How did the direction of the 
 light before entering the glass cube compare with its direction 
 after passing through the cube ? 
 
 Draw a perpendicular to the face of the cube through the point 
 where the light entered, make the proper measurements, and cal- 
 culate the index of refraction of the glass. Turn the cube through 
 1 80 and repeat. 
 
 V. How with the rectangular cell, if the dimensions of the 
 apparatus permitted, might the critical angle for water be found ? 
 Explain, giving a diagram. 
 
 22. REFRACTION AND DISPERSION. 
 
 I. Set a mirror in a small rectangular cell, and fill it about half 
 full of water. Place the cell with its glass side perpendicular to 
 the line formed by a linear source of light (an electric lamp with 
 ' ' hofseshoe ' ' filament) and a narrow slit and at a distance of 
 100 cm. from the scale of a metre rod set at right angles to the 
 line of the light and slit. Move your eye along the metre rod 
 until the image of the light in the mirror above the water becomes 
 
44 IMAGES IN A SPHERICAL MIRROR. [23 
 
 plainly visible and read the scale. Look in the same way for the 
 image of the light in the mirror as seen through the water. Is 
 this image similar in appearance to that seen above the water ? 
 Describe and explain the difference. Can you locate its direction, 
 as was done for the image seen above the water ? 
 
 Locate the direction of the extreme red of the spectrum seen 
 through the water. As the distance of the scale from the cell is 
 one metre (100 cm.), the respective readings of the scale in 
 metres will be equal to the tangents of the angles of incidence 
 and refraction. The source of light may be considered as at the 
 surface of the mirror, hence the light is incident in the dense 
 medium. Show by diagram the angles of incidence and refrac- 
 tion. Using a table of natural sines and tangents, find the sines 
 corresponding to these tangents, and calculate the index of 
 refraction of water for red light. (See Expt. 21 for definition of 
 index of refraction.) 
 
 Find in the same way the index of refraction for blue light, 
 using the extreme blue of the spectrum; and also for yellow 
 light. 
 
 II. Repeat I with a saline solution instead of water. What 
 effect do you find salt in solution to have upon the index of 
 refraction of water? 
 
 The angle between the rays of red and blue light after refraction 
 is called the dispersion for red and blue light. What is the 
 dispersion for these rays for water and for the salt solution ? 
 
 III. Show how from this experiment the velocity of light in the 
 salt solution may be computed, if the velocity in water is known, 
 and demonstrate the relation between the index of refraction and 
 velocity. 
 
 GROUP III. 
 
 23. IMAGES IN A SPHERICAL MIRROR. 
 
 I. Place a concave spherical mirror so as to form as clear an 
 image as possible of the window-sash on a screen, and measure 
 the distance from the mirror to the screen. 
 
23] IMAGES IN A SPHERICAL MIRROR 45 
 
 Repeat, using- some distant object, as the tops of the trees 
 across the road, instead of the window-sash, and measure again 
 the distance from the mirror to the screen. Was this distance 
 greater or less than when the window-sash was focused on the 
 screen ? The principal focus is the point through which all 
 parallel rays are reflected. Its distance from the mirror is called 
 the principal focal length of the mirror. Which of the meas- 
 urements above may be taken as the principal focal length of the 
 mirror ? 
 
 II. Place an upright rod at a distance in front of the mirror 
 equal to twice its principal focal length. Adjust the position of 
 the rod by the method of parallax, so that some definite point 
 on it will coincide in position with its image in the mirror. Do 
 this by adjusting the rod first so as to coincide with its own 
 image*, and then sliding a piece of paper up or down the rod 
 until it meets its image. This will give the required point on the 
 rod. (Do not confound the image formed by the front, plane 
 surface of the glass with that formed by the spherical mirror on 
 the back.) What measurement will now give the radius of 
 curvature of the mirror ? Why ? How does this compare with 
 the principal focal length? 
 
 III. (a.) Place the screen at as great a distance from the mirror 
 as the table will allow, and place two gas jets so that their images 
 formed on the screen will be as distinct as possible. To obtain 
 images beyond the center of curvature of the mirror, where did 
 the gas jets have to be placed, between the mirror and the prin- 
 cipal focus, between the principal focus and the center of curva- 
 ture, or beyond the center of curvature? 
 
 (^.) Measure the distance from the mirror to the gas jets and 
 the distance from the mirror to the screen; also the distance 
 between the gas jets and the distance between their images. 
 
 *This can he done by changing the position of the observer's eye and 
 adjusting and readjusting the position of the rod until it will always 
 coincide in direction with its own image from every point of view. 
 
46 IMAGES IN A SPHERICAL MIRROR. [23 
 
 How does the ratio between the first two distances compare with 
 the ratio between the last two? Find the ratio between the 
 distance of the object and that of its image from the center of 
 curvature of the mirror instead of from its surface. How does 
 this ratio compare with the other two? 
 
 (Y.) Reduce to decimals the reciprocals of (i) the distance 
 from the mirror to the gas jets; (2) the distance from the 
 mirror to their images; (3) the principal focal length; (4) the 
 radius of curvature. Of these four reciprocals find two whose 
 sum is equal to a third, and also equal to a simple multiple of 
 the fourth. 
 
 IV. Interchange the positions of the gas jets and the screen. 
 (In the new positions they will, of necessity, have to be placed on 
 opposite sides of a line normal to the mirror.) Adjust the screen 
 so as to obtain as definite images as possible, and repeat the 
 measurements of III (<). Does the proportion found in III ($) 
 still hold true? Does the relation between the reciprocals in III 
 (Y) still hold true? When the gas jets are beyond the center of 
 curvature, are the images formed between the mirror and the 
 principal focus, between the principal focus and the center of 
 curvature, or beyond the center of curvature? 
 
 V. Place a vertical rod between the mirror and its principal 
 focus, within 8 or 10 cm. of the mirror, and locate its image by 
 means of another rod, using the method of parallax. Measure 
 the distance from the mirror to the object and its image respec- 
 tively. In order that the relation between the reciprocals found 
 in III (Y) shall still hold true, what change in sign is necessary? 
 Write the equations representing the conditions in III and V. 
 
 VI. Suppose an object at an infinite distance from the mirror; 
 where would its image be found, and how would it change in 
 position as the object approached the mirror, supposing the 
 object to approach until it touched the surface of the mirror ? 
 State whether the image would be real, or virtual; erect, or 
 inverted; larger than the object, or smaller. 
 
24] CONVEX LENSES. 47 
 
 24. CONVEX LENSES. 
 
 I. (0.) With a convex lens form an image of the window-sash 
 on a screen and measure the distance from the lens to the screen. 
 
 (<.) With the same lens form an image of some distant object 
 on the screen, and measure again the distance from the lens to 
 the screen. Is this distance the same as in (a)? Which of 
 these distances may be taken as the principal focal length of the 
 lens? Why? 
 
 -II. Light two gas jets and place them at a distance from the 
 lens equal to twice its principal focal length, and place the screen 
 so as to form as distinct images of the jets as possible. Measure 
 the distances respectively from the lens to the screen, and from 
 the lens to the gas jets. How do these distances compare? 
 Measure the distances between the gas jets and between their 
 images. How do these distances compare? 
 
 III. Set the gas jets at a distance from the lens equal to about 
 five times its principal focal length, and place the screen so as to 
 form as distinct images as possible of the jets. 
 
 Measure the distances: (i) From the lens to the screen; (2) 
 from the lens to the gas jets; (3) between the gas jets; (4) 
 between their images. Find a relation existing between these 
 quantities and express it in the form of a proportion. 
 
 Reduce to decimals the reciprocal (i) of the principal focal 
 length; (2) of the distance of either gas jet from the lens; (3) of 
 its image from the lens. The sum of what two of these recipro- 
 cals is approximately equal to the third? 
 
 IV. Interchange the position of the gas jets and the screen 
 and adjust the lens, if necessary, so as to make the images as 
 distinct as possible. Repeat the measurements of III. 
 
 Form a proportion, if you can, similar to that formed in III, 
 and find, if you can, a similar equation connecting certain 
 reciprocals. Indicate any difference in the two cases. 
 
 V. Set an upright rod between the lens and the principal focus. 
 On which side of the lens is the image of the rod ? Is the image 
 
48 CONCAVE IvENSKS. [25 
 
 real, or virtual; erect, or inverted? Locate this image by means 
 of another upright rod, by the method of parallax, (Exercise 23, 
 II.) In order that the relation between the reciprocals previously 
 found should still hold true, what change in sign is necessary ? 
 
 VI. Answer the following questions as applied to a convex or 
 converging lens: 
 
 1 . Where should an object be placed in order that its image 
 may be real? In order that its image may be virtual? 
 
 2. When will the image be erect, and when inverted ? 
 
 3 Where should the object be placed in order to form an 
 enlarged image? In order to form a diminished image? 
 
 4. Where should the object be placed in order to use a con- 
 verging lens as a magnifying glass? 
 
 25. CONCAVE LENSES. 
 
 I Locate with an upright rod the image formed by a concave 
 lens of some vertical part of the window-sash, using the method 
 of parallax. (Exercise 23, part II.) (The rod used in locating 
 the image should be looked at over, not through, the lens.) 
 Measure the distance from the lens to the image. 
 
 Locate in the same way the image of some vertical object in 
 the distance, as the corner of a house, or a telegraph pole, and 
 find the principal focal length of the lens. Explain. 
 
 II. (.) Place the vertical rod at a distance from the lens equal 
 to about twice its principal focal length, and locate its image by 
 means of another vertical rod. Measure the distance from the 
 lens to the image. 
 
 (.) Repeat with the stationary rod at the principal focus. 
 
 (c.) Repeat with the stationary rod between the principal focus 
 and the lens. 
 
 Reduce to decimals the reciprocal: (i) of the distance from the 
 lens to the image in either (a), (), or (c); (2) of the corre- 
 sponding distance from the lens to the object; (3) of the princi- 
 pal focal length. Which one of these distances should be made 
 
25] CONCAVE LENSES. 49 
 
 negative in order that the sum of the first two reciprocals should 
 be equal to the third ? 
 
 III. Set two vertical rods attached to the same support at a 
 suitable distance from the lens (to be determined by the student), 
 and locate their images by means of two other separate rods. 
 
 Measure (i) the distance of the fixed pair of rods from the lens, 
 (2) the distance of their images from the lens, (3) the distance 
 between the rods, and (4) the distance between their images. 
 Find the relation existing between these four quantities and ex- 
 press it in the form of a proportion, and state it in words. 
 
 IV. Answer the following questions: 
 
 1 . Can a real image be formed by a concave lens ? 
 
 2. Can a concave lens be used as a magnifying glass? 
 
 3. Suppose an object at an infinite distance from a concave 
 lens; where would its image be located, and how would it change 
 in position as the object approached the lens, supposing the 
 object to approach until it touched the lens ? 
 
 4. Can there be, when a single lens or mirror is used, such a 
 thing as a real and erect image, or a virtual and inverted image? 
 
 V. * When Exercises 23-25 have been done, copy and fill out 
 the following table. 
 
 VI. Also construct geometrically the following: 
 
 (i.) The image of an object within the focus of a concave 
 mirror. 
 
 (2.) The image of an object at the center of curvature of a 
 diverging lens. 
 
 (3.) The image of an object between the focus and center of 
 curvature of a converging lens. 
 
 Demonstrate that to produce a real image with a converging 
 lens the object and image must be separated by a distance of 
 at least 4/~. 
 
 VII. Write what you can of the analogies between a concave 
 mirror and a converging lens, between a convex mirror and a 
 diverging lens. 
 
 * Parts V and VI and VII may be handed in as a separate exercise. 
 4 
 
CONCAVE LENSES. 
 
 
 
 CONCAVE 
 MIRROR. 
 
 CONVEX 
 LENS. 
 
 CONCAVE 
 LENS. 
 
 
 
 
 
 
 Dco 
 
 
 
 
 
 
 Magnified or diminished 
 
 
 - 
 
 
 
 
 
 
 
 
 Location of Image 
 
 
 
 
 00 > D> 2f 
 
 Real or virtual 
 
 
 
 
 
 Magnified or diminished 
 
 
 
 
 
 
 
 
 
 
 Location of Image 
 
 
 
 
 Dyf 
 
 
 
 
 
 
 Magnified or diminished 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2f> D > f 
 
 Real or virtual 
 
 
 
 
 
 Magnified or diminished 
 
 
 
 
 
 
 
 
 
 
 Location of Image 
 
 
 
 
 f> D~> o 
 
 Real or virtual 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 /}=Distance of object from mirror. 
 /=Principal focal length. 
 
26] DRAWING SPECTRA. 51 
 
 26. DRAWING SPECTRA. 
 
 The spectroscope should be examined and its construction 
 understood before proceeding. The instrument should be set so 
 that the slit in the collimator does not point toward any outside 
 source of light, as a window. The instrument may be adjusted 
 for use as follows: Place a colorless Bunsen flame, in which is 
 held asbestos soaked with salt solution, directly before the slit and 
 narrow the latter, focusing upon it with the telescope, the 
 prism being in place, until the slit appears as a sharp, bright line. 
 Light the gas illuminating the scale in the third arm of the 
 instrument, and focus the scale by moving it in and out until the 
 figures upon it can be distinctly read. (The eye-piece should not 
 be touched during this last operation.) Bring the 5 (or 50) 
 mark of the scale into coincidence with the yellow line due to 
 the sodium. If now the adjustment has been carefully done, by 
 moving the eye slightly back and forth before the eye-piece the 
 sodium line and the mark 5 will not appear to move with respect 
 to each other. If there is such motion repeat the adjustment. 
 
 I. The spectra of the salts provided are to be examined and 
 drawn upon plotting paper, the spectroscope scale being plotted 
 as abscissae and each spectrum on a separate horizontal line. 
 (See sample note-book.) State in each case the general color 
 of the flame and the colors of the various lines and bands. 
 
 To observe successfully the potassium spectrum it will be nec- 
 essary to open the slit somewhat and insert a piece of cobalt 
 glass between the flame and the slit. The sodium spectrum will 
 probably be ever present, but is readily distinguished from that of 
 the salt under examination. 
 
 II. Draw the spectrum of a luminous flame, and also of the 
 same flame seen through red, green, yellow, and blue glass. Is 
 the light transmitted by any of these glasses monochromatic? 
 
 Distinguish between absorption spectra and emission spectra. 
 
 III. The wave-lengths corresponding to certain spectral lines 
 are furnished; draw a smooth curve in terms of their position on 
 
52 LAWS OF MAGNETIC ACTION. [27 
 
 the scale and from this curve determine the wave-lengths corre- 
 sponding to the calcium lines and the strontium lines. 
 
 IV. Draw a diagram representing the optical principles in- 
 volved in the construction and use of the spectroscope you used. 
 
 V. Write not less than one hundred words on the uses of the 
 spectroscope. 
 
 27. LAWS OF MAGNETIC ACTION. 
 
 Prove that if a compass-needle is deflected by a horizontal force 
 acting in an east and west direction, the magnitude of the force 
 will be proportional to the tangent of the angle of deflection. 
 
 I. Place two pocket compasses side by side. Do the like poles 
 attract or repel each other? Do the unlike? 
 
 II. Lay a compass on a large sheet of brown paper, draw a 
 circle around it, and mark on the paper the center of the circle, 
 i. e., the position of the center of the compass. Draw a line east 
 and west through this point and mark off on this line points in 
 both directions at distances of 10, 15, 20, 30, and 40 cm., re- 
 spectively, from the center of the compass. Remove all magnetic 
 substances from the neighborhood, replace the compass, and 
 adjust it so that its needle reads zero degrees. (The compass 
 should be tapped very lightly as the needle comes to rest, with 
 the finger or with a rubber pencil-tip.) 
 
 Hold a long magnetized steel strip in a vertical position with 
 its lower end on the table at 10 cm. either east or west of the 
 compass, and read the deflection of the compass-needle. (Tap 
 the compass as before, and read both ends of the needle, averag- 
 ing the readings.) Repeat with the end of the long magnet at 
 10 cm. on the other side and average the two deflections of the 
 compass-needle, recording each observation. To what function 
 of the angle of deflection is the force exerted by the lower pole of 
 the long magnet proportional, assuming that the needle is com- 
 paratively short ? (See proposition above.) 
 
 III. Repeat the last part of II with the end of the long magnet 
 
27] LAWS OF MAGNETIC ACTION. 53 
 
 at 15, 20, 30, and 40 cm., respectively, from the center of the 
 compass, changing sides and averaging as before. Calculate from 
 your results (using a table of natural tangents) the ratio of the 
 horizontal force due to the lower pole of the magnet at 10 cm. to 
 that at 20 cm. ; at 15 cm. to that at 30 cm.; at 20 cm. to that at 
 40 cm. ; etc. Does the force vary directly, or inversely, with the 
 distance? Assuming that it varies (directly or inversely) as 
 some integral power of the distance, what do you find to be the 
 power in question? Arrange results in tabular form. 
 
 IV. Take a comparatively short magnet and lay it on the table 
 on a line drawn east and west through the center of a compass- 
 needle, at such a distance as to deflect the needle about 40. 
 Read the deflection and measure the distance from the center of 
 the magnet to that of the compass-needle. Place the magnet at 
 double this distance, and read the deflection again. Do you find 
 the horizontal force to vary with the distance in this case accord- 
 ing to the law found in III, or not? Was the needle in II and 
 III acted on in a horizontal direction by both poles of the magnet, 
 or practically by one alone? Was it in IV? 
 
 When a magnet is comparatively short, how do you find the 
 force exerted by it at any point to vary with the distance of the 
 point from the center of the magnet, assuming that it varies as 
 some exact integral power of this distance? 
 
 V. A unit magnetic pole is a magnetic pole of such strength 
 that it will exert a force of one dyne on a similar pole at the dis- 
 tance of one cm. 
 
 The pole strength of a magnetic pole is defined as the force 
 exerted by it on a unit magnetic pole at the distance of one cm. 
 
 What is the force between two magnetic poles at the 
 distance d apart, the strength of the poles being m v , and m^ 
 respectively ? 
 
 VI. If in IV the magnet were in the E and W line and the 
 compass on a line perpendicular to the middle point of the mag- 
 net, in the same horizontal plane, find geometrically the expres- 
 sion for the force between a compass pole and the magnet. 
 
54 MAGNETIC FIELDS. [28 
 
 28. MAGNETIC FIELDS. 
 
 I. Take a magnet 16.5 cm. long, and locate approximately 
 the mean distance of either pole from the end, by the following 
 method : 
 
 Lay the magnet on a sheet of paper, and trace its outline with 
 a pencil. Place a compass on the paper so that the compass box 
 is about one cm. from the magnet. Commencing near the end 
 of the magnet, move the compass, one or two cm. at a time, 
 parallel to the magnet, drawing, for each position of the compass, 
 lines to indicate the direction of its needle. Remove the magnet, 
 draw a line through the position of its axis, and extend the above 
 lines until they intersect this line. Find a medium point and 
 measure its distance from the end of the magnet. 
 
 II. Lay the magnet used in I lengthwise on a large sheet 
 of brown paper. Draw the outline of the magnet with a pencil, 
 and sprinkle iron filings on the paper around it. Trace the 
 lines in which the iron filings set themselves when the paper is 
 tapped. 
 
 Brush the iron filings off the magnet, and return them to the 
 sprinkler, taking care not to scatter and waste them. (In re- 
 moving iron filings from a magnet, brush them towards the 
 center, and not towards the ends.) 
 
 Replace the magnet, and place a small compass at different 
 points of the tracing. How does the direction of the compass- 
 needle at any point coincide with that of the lines of iron 
 filings ? 
 
 III. Take a sheet of cardboard and place it with its sides 
 parallel to the edges of the table. To the most northerly or 
 southerly corner of the cardboard fasten a small compass with 
 wax*, and, after removing all magnetic substances from the 
 neighborhood, draw a pencil line to correspond with the 
 magnetic meridian through the compass. On this line place 
 
 * Attach t.ie wax to the edge of the compass, and do not put it under- 
 neath. 
 
28] MAGNETIC FIELDS. 55 
 
 a short magnet with its north pole directed toward the south, 
 and adjust the distance between it and the compass so that 
 the compass-needle is in neutral equilibrium (i. e., will point 
 indifferently in any direction). Fasten the magnet in this 
 position to the cardboard with wax. The compass-needle will 
 not be affected now by the earth's magnetic field, while the 
 sides of the cardboard are parallel to the edges of the table. 
 Why? 
 
 IV. Take the drawing made in II. Mark the position of the 
 poles of the magnet, and draw a circle, about 2 or 3 cm. in 
 diameter, around each. Divide these circles into 12 or more 
 equal parts, and through each division draw a line, following the 
 directions in which the iron filings set themselves, as far as these 
 directions can be determined. 
 
 Replace the magnet on the paper, and place the compass- 
 needle, protected as in III from the influence of the earth's 
 magnetic field, at the end of one of these lines. Extend this 
 line an inch or so in the direction indicated by the needle. 
 Prolong all the lines through the divisions of the circle in this 
 way, an inch or so at a time, as far as the limits of the paper 
 will allow. 
 
 V. Take a point on one of these lines about 9 or 10 cm. from 
 one of the poles of the magnet, and 12 or 15 cm. from the other 
 pole. Suppose a north or south magnetic pole to be placed at 
 this point. Draw lines in the directions that this pole would be 
 urged by each pole of the magnet, and lay off on these lines 
 distances proportional to the forces in these directions due to the 
 poles taken separately. (Force varies inversely as- the square of 
 the distance. ) Construct on these lines a parallelogram of forces, 
 and find the direction of the resultant force due to both poles of 
 the magnet. How does the direction of this resultant compare 
 with that of the magnetic line of force at point considered ? 
 
 If it were possible to produce an isolated north magnetic pole 
 and place it in a magnetic field, how would the path along which 
 it would move be related to the magnetic lines of force ? Deduce 
 
56 INTENSITY OF KARTH's MAGNETIC FIELD. [29 
 
 from this a definition of a magnetic line of force. How is the 
 strength of the magnetic field due to the magnet indicated by the 
 distribution of the lines of force at any region in the preceding 
 diagram ? 
 
 The sheet of brown paper used in II, IV, and V is to be signed 
 and handed in with the other notes. Each student, however, 
 should make in his note-book a reduced copy of the diagram 
 before handing it in. 
 
 VI. Lay two short magnets on a sheet of white paper with 
 impression paper and another sheet of white paper underneath 
 (or they may be laid directly on a page of the note-book). Lay 
 them parallel, side by side, about 1.5 or 2 cm. apart, with their 
 unlike poles opposite. Sprinkle iron filings about them, and 
 trace the lines along which the filings set themselves. 
 
 29. INTENSITY OF EARTH'S MAGNETIC FIELD. I. 
 
 Caution. Keep the magnet used in this exercise away from 
 other magnets or magnetic bodies. 
 
 I. (a.) Place a magnet in the east and west line east or west 
 of a compass-needle, at such a distance as to deflect the needle 
 through an angle of 45. Measure the length of the magnet and 
 the distance of its nearer end from the center of the compass. 
 
 (b.) Reverse the magnet and repeat the measurements of (a), 
 (c. ) Repeat (a) and (b} with the magnet on the other side of 
 the compass-needle. 
 
 II. (a.) Suspend a carriage for the magnet by two fine parallel 
 wires of equal length, adjustable from above, so that they are 
 east and west of each other. Place a brass rod of about the same 
 size as the magnet in the carriage and carefully draw a line par- 
 allel to the rod on a piece of paper placed underneath it. 
 Remove the brass rod and place the magnet in the carriage. 
 Does the magnet lie, as the rod did, east and west, or not? 
 Explain why. Mark on the paper the position of the magnet. 
 
 (.) Reverse the magnet and mark its position again. 
 
30] INTENSITY OF EARTH *S MAGNETIC FIELD. 57 
 
 (V.) Measure the distance between the two wires of the bifilar 
 suspension, and mark their position carefully on the paper in the 
 three cases above. Find, by measurement from the drawing, the 
 average distance that the lower end of either wire is pulled out 
 from the vertical when the magnet is hung in its carriage. 
 
 What forces cause the magnet to be deflected ? What is the 
 direction of these forces, and where do they act on the magnet, 
 assuming that the poles of the magnet are at its extremities ? 
 Measure on the paper the arm of the couple (see Exercise 1 1, VI) 
 formed by these forces. 
 
 Measure the length of the bifilar suspension and also find the 
 weight of the magnet and carriage. Express these weights in 
 dynes. 
 
 Repeat Part II, using another part of the same paper. 
 Average the two sets of results. 
 
 Preserve the paper diagram for reference. 
 
 30. INTENSITY OF EARTH'S MAGNETIC FIELD. II. 
 
 This exercise need not be performed in the laboratory, and is 
 to be done only when the other exercises on magnetism have 
 been performed. 
 
 I. In Exercise 29, I, how did the horizontal force at the center 
 of the compass due to the magnet compare in each case with that 
 due to the earth's magnetic field? 
 
 Calculate the average force on a unit magnetic pole at the 
 center of the compass due to the nearer pole of the magnet, 
 calling the pole-strength of the magnet P (see Exercise 27, V) 
 and assuming that its poles are situated at its extremities. Do 
 the same for the farther pole of the magnet. How did these 
 forces compare in direction? Find their resultant. How does 
 this resultant compare with the horizontal force (usually denoted 
 by the letter H) on a unit magnetic pole due to the earth's 
 field? (See question above.) 
 
 Form an equation from these results and find from it the 
 numerical value of the quotient H/ P. 
 
5& COMPARISON OF MAGNETIC FIELDS. [31 
 
 II. Assuming that the weight in Exercise 29, II, was evenly 
 divided between the two wires of the bifilar suspension, calculate 
 the horizontal force on the lower end of each wire tending to pull 
 it back into a vertical position. Do this by means of a triangle 
 of forces as in Exercise 12, VI, using the length of the wire and 
 the deflection from the vertical, as measured in Exercise 29. In 
 what direction did these forces act, and what was the arm of the 
 couple (see Exercise 11, VI) formed by them? Calculate the 
 moment of the couple formed by these forces. Draw diagrams 
 of forces. 
 
 What two forces tended to deflect the magnet ? To what was 
 each of these forces equal in terms of //and />? Calculate the 
 moment of the couple formed by these forces. 
 
 What relation exists between the moments of the two couples 
 just calculated? Express this relationship in the form of an 
 equation, and calculate the numerical value of the product //x P. 
 
 III. Combine the results found in I and II so as to eliminate 
 the unknown quantity P and . find the value of H in dynes. 
 Calculate also the pole-strength; in what unit is it expressed? 
 
 IV. Write not less than two hundred words on the subject: 
 Terrestrial Magnetism. 
 
 31. COMPARISON OF MAGNETIC FIELDS. 
 
 I. Suspend a magnet in a horizontal position by a long thread 
 (a torsionless thread, if possible), and protect it from air currents 
 by hanging it in a box. When the suspended magnet has been 
 brought to rest, set it vibrating about a vertical axis by bringing 
 an open knife blade near it, and determine its period of vibration 
 within a few hundredths of a second.* Be careful not to touch 
 the magnet with magnetic substances, and also keep all movable 
 magnetic bodies away from the neighborhood of the vibrating 
 magnet. 
 
 Find the period by the method of Exercise 15. 
 
32] ELECTRO-MAGNETIC RELATIONS. 59 
 
 II. Mark in some way the position of one end of the magnet, 
 remove it, and place a compass with a short needle at this point. 
 Place a long magnet at right angles to a line drawn east and west 
 through the thread with its center on this line and its south pole 
 towards the south. Move this magnet parallel to itself until the 
 earth's horizontal field at the center of the compass is as nearly 
 neutralized as possible. Then turn the magnet through 180, 
 i. e., end for end. Will the intensity of the horizontal magnetic 
 field at the compass-needle now be greater or less than the 
 earth's horizontal field, //? How much greater or less? 
 
 Remove the compass, replace the suspended magnet, and 
 determine its period of vibration again as in I. Calculate the 
 ratio of the periods in the two cases. How does this compare 
 with the intensity of the horizontal magnetic fields in the two 
 cases? 
 
 Assuming that the period of a vibrating magnet varies as 
 some integral root, or power, of the intensity of the magnetic 
 field parallel to the magnet, what do your results indicate this 
 root, or power, to be? Is it direct or inverse? 
 
 III. Suspend your magnet at two designated places in the 
 aboratory, determining its period of vibration at each place and 
 also at a place where //is known. From your results and the 
 law just found, calculate the value of H at each of the places 
 where the magnet was vibrated. 
 
 IV. If the suspending string were not torsionless, would the 
 calculated values of H be too high or too low ? Explain. 
 
 What analogies exist between this magnetic pendulum and the 
 simple gravity pendulum ? 
 
 32. ELECTRO-MAGNETIC RELATIONS. 
 
 I. Connect the plates of a Daniell cell by a flexible wire cord. 
 Stretch a portion of this cord out straight and hold it near a 
 compass-needle placed on the edge of a wooden block. The 
 electric current is supposed to flow through the external circuit 
 
60 ELECTRO-MAGNETIC RELATIONS. [32 
 
 from the copper plate of the cell to the zinc plate. In what 
 direction is the north pole of the compass-needle deflected, or is 
 it deflected at all, when the current and the needle are in the 
 following relative positions: 
 
 1. Current flowing north, needle below? 
 
 2. Current flowing north, needle above ? 
 
 3. Current flowing north, needle east or west? 
 
 4. Current flowing south, needle below ? 
 
 5. Current flowing south, needle above? 
 
 6. Current flowing south, needle east or west? 
 
 7. Current flowing upward, needle north? 
 
 8. Current flowing upward, needle south? 
 
 9. Current flowing downward, needle north ? 
 
 10. Current flowing downward, needle south? 
 
 1 1 . Current flowing east or west, needle above or below ? 
 
 12. Current flowing east or west, needle north or south? 
 
 II. Answer the following questions: 
 
 1. How is the direction in which the compass-needle is de- 
 flected affected by reversing the direction of the current? 
 
 2. How is it affected when its position is changed from one 
 side of the current to the other, i. e., from above to below and 
 from east to west ? 
 
 3. Is the force exerted by an electric current on a magnetic 
 pole parallel to the direction of the current or not ? What do the 
 results of I, i and 2, indicate? 
 
 4. What is the direction of this force, with reference to the 
 plane containing the current and the magnetic pole, as indicated 
 by the results of I, 3 and 6 ? 
 
 5. If the needle was not deflected in I, n, explain why. 
 
 6. Suppose the current is represented in position and direction 
 by the fingers of the right hand and the palm to be turned 
 towards the compass-needle, which pole was deflected in the 
 direction indicated by the thumb in I, i; in I, 2; in I, 3, etc.? 
 Frame a rule including all the above cases. 
 
32] ELECTRO-MAGNETIC RELATIONS. 6l 
 
 III. Connect the plates of the Daniell cell to a rectangular coil 
 suspended with its terminals in mercury cups so as to turn freely 
 about a vertical axis. Set the coil with its plane north and south. 
 Follow the path of the electric current from the copper plate of 
 the cell through the coil to the zinc plate, and find in what part 
 of the coil the current flows in a northerly direction, in what in a 
 southerly direction, in what part upward, and in what part 
 downward. 
 
 Take a magnet and hold its north pole in the following 
 positions relative to the current, observing in each case the 
 direction in which the wire carrying the current tends to 
 move: 
 
 1. Current flowing north, north pole below. 
 
 2. Current flowing north, north pole above. 
 
 3. Current flowing south, north pole below. 
 
 4. Current flowing south, north pole above. 
 
 5. Current flowing upward, north pole north. 
 
 6. Current flowing upward, north pole south. 
 
 7. Current flowing downward, north pole north. 
 
 8. Current flowing downward, north pole south. 
 
 How does the force exerted by a magnetic pole upon an electric 
 current compare in direction with that exerted by the current 
 upon the pole? (Compare the results of I and III.) 
 
 IV. Trace by means of iron filings the magnetic field due to a 
 helical coil carrying a current. To the field of what shape mag- 
 net does this resemble ? 
 
 Test the coil with a compass-needle and determine which end 
 attracts the north pole and which the south pole of the needle. 
 Could the position of its poles be determined beforehand? How ? 
 
 V. What do you conclude from I, II, and III to be the form 
 of the magnetic field about a wire carrying a current? 
 
 How does an electric circuit tend to set itself with respect to 
 the number and direction of the magnetic lines of force in its 
 neighborhood? (A magnetic line of force proceeds from the 
 north pole to the south pole outside the magnet.) 
 
62 LAWS OF ELECTROMAGNETIC ACTION. [33 
 
 Does the coil used in III act as if it were itself a magnet? If 
 so, of what form ? 
 
 33. LAWS OF ELECTRO-MAGNETIC ACTION. 
 
 I. Take an upright wooden circle about 30 cm. in diameter, 
 having apiece of insulated copper wire wound once around it, 
 with two free ends of about equal length twisted together so that 
 the effect of an electric current in one will be neutralized by that 
 of an equal and opposite current in the other. Place a compass- 
 needle at the center of the coil, and set the coil so that its plane 
 is parallel to the magnetic meridian. 
 
 Connect this galvanometer with some source furnishing a 
 constant electric current. Read the angle of deflection of the 
 compass-needle. Reverse the direction of the current and read 
 the angle again. Average the two results. 
 
 In what direction is the force tending to deflect the needle? 
 (See Exercise 32.) To what function of the angle of deflection 
 is this force proportional? (See Exercise 27, Proposition.) 
 
 II. Repeat I with a coil of the same diameter, but having 
 twice the length of wire as in I, i. e., having twice as many turns 
 of wire. 
 
 III. Take another wire and wind it once around a wooden 
 circle concentric with and of half the diameter of that used in I. 
 Connect these two coils so that the same current will flow through 
 them in opposite directions. Increase the number of turns of the 
 larger coil until the effect of the smaller coil on the compass- 
 needle is neutralized. How many turns of wire were necessary 
 to do this ? How many times did the length of the wire have to 
 be increased from that of the single turn on the inner coil in order 
 to neutralize the effect due to the decrease in the diameter of the 
 coil? 
 
 IV. Set up three such galvanometers having coils of the same 
 diameter and length, placing them as far apart as the table will 
 allow, and connect them so that the whole current passes through 
 
33] LAWS OF ELECTRO-MAGNETIC ACTION. 63 
 
 one coil and half of the current through each of the other coils. 
 Read the angle of deflection of each compass-needle. Reverse 
 the direction of the current and average the east and west deflec- 
 tions of each galvanometer. 
 
 V. Answer the following questions, showing in each case the 
 numerical process by which you arrived at your conclusion: 
 
 1. How does the force at the center of a circular coil carry- 
 ing an electric current vary with the length of wire in the coil, 
 according to the results of I and II, assuming that it varies with 
 some integral power (direct or inverse) of the length ? 
 
 2. How with the diameter or radius of the coil, according to 
 the results of II and III? 
 
 3. How with the current, according to the results of IV? 
 Assuming that the force F on a unit magnetic pole at the 
 
 center of a circular coil depends only on the length, L=2.nRN, 
 of wire in the coil, its radius, R, and the current, C, express this 
 force in terms of these three quantities and a constant K. 
 
 VI. Draw diagrams of all electrical connections. 
 Represent graphically the magnetic field at the needle when the 
 
 latter is deflected 45. 
 
 GROUP IV. 
 
 In all the electrical experiments, diagrams of electrical connec- 
 tions are to be made. 
 
 Instruments of the tangent galvanometer type a loop of wire 
 about a magnetized needle should be set with the plane of the 
 coil in the magnetic meridian and leveled so that the needle 
 swings freely. Wires leading to such an instrument, or near it, 
 should be twisted or laid side by side so that the magnetic fields 
 of currents in opposite directions neutralize each other. Two 
 instruments should never be nearer each other than one metre. 
 
 To take an observation, read both ends of the pointer, reverse 
 the direction of the current, read both ends of the pointer again, 
 and average the four readings. Reading both ends of the 
 
64 CURRENT DETERMINATION. [34 
 
 pointer eliminates eccentricity of mounting of the needle with 
 respect to the scale. Reversing the current corrects for the 
 imperfect orientation of the coil in the magnetic meridian. Note 
 that the pointer is usually mounted at right angles to the needle. 
 In the case of needles mounted on pivots slight tapping of the 
 instrument may be necessary to insure a correct reading. The 
 influence of one instrument upon "another may be tested by 
 reversing the current through one of them. 
 
 It is important in all electrical work that the connections be 
 tight. Always disconnect from batteries when through. 
 
 In all cases the above methods are to be used and the indicated 
 precautions taken. 
 
 34. CURRENT DETERMINATION. 
 
 I. Connect a tangent galvanometer, such as was used in 
 Exercise 33, in series with an ammeter and a source of constant 
 current, following the preceding directions for setting up and 
 reading. Take five sets of readings on different parts of the 
 scale of both instruments simultaneously, varying the current by 
 introducing into the circuit various lengths of German silver 
 wire. Record all readings and take the proper averages. 
 
 II. From the laws of electro-magnetic action studied in 
 Exercise 33 we may calculate the value of the current for the 
 various readings of the galvanometer, and comparing these values 
 with the ammeter readings, both expressed in the same unit, we 
 may calibrate or test the ammeter. 
 
 The C. G. S. unit of current in the electro-magnetic system is 
 the current that will act with a force of one dyne on a unit mag- 
 netic pole at the center of an arc i cm. long of i cm. radius. 
 If C in the equation of Exercise 33, V, was measured in terms of 
 this unit, F in dynes, and R and L in cm., the constant K may 
 be eliminated. How? Solve the resulting equation for C. 
 
 Also F=H tan 6 where H is the horizontal component of the 
 earth's magnetic field and the angle of deflection of the needle. 
 
35] ELECTRICAL RESISTANCE. 65 
 
 Now F is the same quantity in the above two equations. The 
 two values of F may therefore be equated and an expression for 
 the current through the galvanometer found in C. G. S. units in 
 terms of the four measurable quantities: the radius R of the coil, 
 the length L=27rRN of wire in fhe coil (where N is the 
 number of turn,s), the horizontal component H of the earth's 
 field, and the tangent of the angle of deflection 6. (The value 
 of H will be given.) 
 
 III. Measure the radius of the galvanometer coil and find its 
 length. Calculate the values of the current, in C. G. S. units, for 
 the readings of the galvanometer taken in I. 
 
 How do the calculated values agree with the ammeter readings 
 of the current ? What, then, is the ratio between the C. G. S. 
 unit of current and the ampere the practical unit indicated by 
 the ammeter? 
 
 IV. Make a table of corrections to the readings of the ammeter 
 in terms of the current as calculated. 
 
 V. Explain what reversing the current in a tangent galvano- 
 meter eliminates? 
 
 35. ELECTRICAL RESISTANCE. 
 
 I. (a.) Connect an ammeter directly with the battery terminals 
 and read the current. Disconnect as soon as possible. 
 
 (. ) Introduce 50 cm. of No. 25 German silver wire into the 
 circuit in series with the ammeter. Read the current. How was 
 its value altered, by introducing this wire into the circuit? 
 
 (c.) Repeat with 100 cm. of No. 25 German silver wire, at 
 the same time introducing into the circuit a wire equal in size and 
 length to the wires leading to the battery. What is the effect on 
 the current of doubling the length of the wire in the circuit? 
 
 If we consider that the wire offers resistance to an electric 
 current, and assume that the resistance varies as some integral 
 power of its length, what do the results of ($) and (<:) show this 
 power to be ? Is it direct, or inverse ? 
 
66 ELECTRICAL RESISTANCE. [35 
 
 II. (a.) Repeat I (<:) with two No. 25 German silver wires, 
 each 100 cm. long, connected in parallel, instead of the single 
 wire. What is the effect upon the current of paralleling the 
 resistance wire with another wire of the same material and of 
 equal diameter and length ? 
 
 (.) Remove the extra wire inserted in the circuit in i (V), and 
 adjust the length of the two wires, so that the current through 
 the ammeter is the same as in I (). How does the resistance of 
 the two wires in parallel, after this adjustment, compare with the 
 resistance of the single wire in I (b) ? How do their lengths 
 compare? What do you find to be the ratio of the resistance of 
 a single wire to that of two wires of the same material, length, and 
 diameter connected in parallel? 
 
 III. (0.) Connect a No. 25 German silver wire 20 cm. long in 
 series with the ammeter and read the current. 
 
 (.) Replace the No. 25 German silver wire by a No. 20 
 German silver wire of the same length, and measure the current 
 again. What do you find to be the effect of increasing the cross- 
 section of a wire upon the current ? 
 
 (V.) Adjust the length of the No. 20 German silver wire so that 
 the current through the ammeter is the same as in III (a). How 
 does the length of the No. 20 wire compare with that of a No. 
 25 wire having the same electrical resistance ? 
 
 (d.) With a screw gauge meaure the diameter of the No. 25 
 and also of the No. 20 wire. What is the ratio of the diameters 
 of the two wires? What is the ratio of the resistance of a No. 
 25 wire to that of the same length of No. 20 wire ? Explain. 
 Assuming that the electrical resistance varies as some integral 
 power of the diameter of a wire, what do you find the power in 
 question to be ? Is it direct, or inverse ? How must the resist- 
 ance vary, then, with the cross-section of the wire ? How do the 
 results of II () confirm your answer to this last question ? 
 
 IV. (a.) Introduce 50 cm. of No. 25 nickel wire into the cir- 
 cuit, instead of the German silver wire, and measure the current. 
 
36] ELECTROMOTIVE FORCE. 67 
 
 What is the ratio of the resistance of the German silver and nickel 
 wires of the same length and diameter? 
 
 (.) Replace the brass wire by the No. 20 German silver 
 wire and adjust its length so that the current through the 
 ammeter is the same as in IV (a). 
 
 Having found a certain length of No. 20 German silver wire 
 equal in resistance to 50 cm. of No. 25 nickel wire, and knowing 
 the diameters of these wires, calculate the relative resistance of 
 nickel and German silver wires of the same diameter and length. 
 
 V. The resistance of a cubic centimeter is called the specific 
 resistance of a substance. If the specific resistance of German 
 silver is known, show how that for nickel may be calculated from 
 your results. Write the equation representing this. 
 
 36. ELECTROMOTIVE FORCE. 
 
 I. (a. ) Connect a low-resistance galvanometer (an ammeter) 
 directly to a Daniell cell and note the reading. Introduce 
 another Daniell cell into the circuit in series with the first cell, 
 connecting the copper plate of one cell to the zinc plate of the 
 other, so that the currents due to both flow in the same direction 
 through the ammeter. What change did the second cell produce 
 in the reading of the ammeter, if any ? 
 
 (.) Repeat (a) with a high-resistance galvanometer, con- 
 structed so that the effect on the deflection due to diminishing the 
 current is offset by having a great number of turns in the coil. 
 How did the change in the reading produced by introducing an 
 additional cell into the circuit compare with that produced by the 
 additional cell when an ammeter was used ? Should a galva- 
 nometer of high or low resistance be used to show the effect of 
 connecting two battery cells in series? 
 
 The effect of connecting two cells in series is to double the 
 electromotive force* tending to produce an electric current in 
 
 *The practical unit of electromotive force is called a volt, and a high- 
 resistance galvanometer graduated to give the electromotive force 
 
68 ELECTROMOTIVE FORCE. [36 
 
 the circuit. What sort of a galvanometer (high or low resistance) 
 do your results indicate should be used to measure the electro- 
 motive force due to any source of electric currents, or between two 
 points of a circuit carrying a current ? What is the objection to 
 using a high-resistance galvanometer to measure the current in a 
 circuit? 
 
 II. With a voltmeter measure the electromotive force of the 
 following cells and combinations of cells, anal answer the questions 
 asked. (The directions for using a galvanometer apply also to a 
 voltmeter.) 
 
 1. A Daniell cell. 
 
 2. Two Daniell cells in series, connected copper to zinc. 
 
 3. Two Daniell cells in series, connected copper to copper. 
 
 4. Two Daniell cells in parallel. 
 
 Are the electromotive forces of the individual Daniell cells 
 equal? (Compare i, 2, and 3.) How does the electromotive 
 force of two Daniell cells in parallel compare with that of a single 
 cell? With that of two cells in series? (Compare 1,2, and 4.) 
 
 5. A Leclanche cell. (Zinc and carbon plates in a solution of 
 sal ammoniac, ammonium chloride.) 
 
 6. A Leclanche and a Daniell cell in series, connected carbon 
 to zinc and copper to zinc. 
 
 7. The same cells in series, connected carbon to copper and 
 zinc to zinc. 
 
 Is the electromotive force of a battery cell altered in any way 
 when it is connected to another cell of different construction ? 
 (Compare 1,5, 6, and 7.) 
 
 8. Any other cells or sources of electromotive force provided. 
 
 III. Measure the electromotive force of a Daniell cell, and of a 
 Leclanche cell, after being short-circuited for fifteen or twenty 
 minutes. Was the electromotive force of the Daniell cell the 
 
 between its terminals in volts is called a vo'tmeter. The instruments of 
 this class used in (b) were designed for use as voltmeters and will be 
 designated as such hereafter. The electromotive force of a Daniell cell 
 is 1.07 volt. 
 
37] OHM'S LAW. 69 
 
 same as that found in II? Was that of Leclanche the same? If 
 not, why? Which cell do you conclude is unsuitable for use 
 where a constant current is required, as in telegraphing ? Give a 
 reason why the other cell would be unsuitable for use where the 
 circuit would only be closed for a moment at a time and at long 
 intervals, as on a bell circuit. Disconnect all wires from cells. 
 Is electromotive force a force? Explain. 
 
 37. OHM'S LAW. 
 
 I. (.) Connect a single Daniell cell in series with a rheostat 
 and an ammeter. Take out enough plugs from the rheostat to 
 introduce a resistance of 5 ohms* into the circuit. Read the 
 ammeter carefully, reversing as usual. (Do not be surprised if 
 the current is small.) 
 
 (3.) Introduce another Daniell cell into the circuit in series with 
 the first cell. Read the ammeter again. 
 
 (The electromotive force in. (6) is twice that in (a). (See 
 Exercise 36, II.) What relation do you find to exist between 
 the electromotive force and the current when the resistance is 
 constant ? 
 
 II.. (a.) With the connections as in I () take out enough 
 plugs from the rheostat to increase the introduced resistance to 7 
 ohms. Read the ammeter. 
 
 (<.) Repeat II () with all the rheostat plugs out. (Resist- 
 ance=io ohms.) 
 
 How do the currents through the ammeter in I (), II (a), 
 and II ($), compare? How do the resistances of the circuits 
 compare, neglecting the comparatively small resistance of the 
 battery cells ? What relation do you find to exist between the 
 resistance and the current when the electromotive force is 
 constant ? 
 
 *The ohm is equal t the resistance at o C. of a column of mercury 
 106.3 cm. long and i sq. mm. in cross section. 
 
7O DIVIDED CIRCUITS. [38 
 
 What, from the results of I and II, is the relation between 
 the current in a circuit (or part of a circuit), the electromotive 
 force acting through the circuit (or between its terminals, if it is 
 not a complete circuit), and the resistance of the circuit (or part 
 of a circuit)? This relation, when written correctly in the form 
 of an equation (assuming the units of current, electromotive force, 
 and resistance to be so related that the constant factor is unity), 
 is called Ohm' s Law. 
 
 III. Connect the ammeter in series with a rheostat and two 
 Daniell cells in parallel. Vary the resistance by steps of one 
 ohm over the range of the rheostat. Read the corresponding 
 currents. Make a plot with values of resistance as abscissae and 
 products of current by resistance as ordinates. Also plot resist- 
 ances and currents on the same paper. Explain by Ohm's law 
 the forms of the lines drawn. 
 
 Show how the resistance varies with the electromotive force 
 when the current is constant. 
 
 38. DIVIDED CIRCUITS AND FALL OF POTENTIAL 
 ALONG A CONDUCTOR. 
 
 I. (a.) Join two rheostats in parallel and connect them in 
 series with the battery provided and an ammeter. Cut out the 
 resistances in the rheostats, leaving but one ohm in one branch of 
 the circuit, and two ohms in the other. Read the current through 
 the ammeter. 
 
 (.) Place the ammeter in the branch circuit of one ohm's 
 resistance, and measure the current in this branch. 
 
 (<:.) Measure in the same way the current in the branch circuit 
 of two ohms' resistance. 
 
 (d.^) Answer the following questions: 
 
 1. How does the current in the main circuit compare with the 
 sum of the currents in the two branch circuits? 
 
 2. Does the greater current flow through the circuit of greater 
 or less resistance? 
 
38] DIVIDED CIRCUITS. 71 
 
 3. The currents in a divided circuit are proportional to an 
 integral power of the resistances of the branches. What do your 
 results indicate this power to be? Is it direct, or inverse? 
 
 II. Connect the two rheostats with a third so as to form 'three 
 parallel circuits of one, two, and three ohms' resistance, respect- 
 ively. Measure with an ammeter, as was done in I, the current 
 in the main circuit and in each of the branch circuits. Measure 
 with a voltmeter the electromotive force between the two junctions 
 of the parallel circuits. 
 
 Is the relation found in I between the currents in the 
 branch circuits and the resistances of the circuits confirmed by the 
 results of II ? Explain. 
 
 III. Connect an external resistance of 10 ohms having steps of 
 2 ohms, in series with the battery. 
 
 Connect one of the voltmeter terminals to one plate of the 
 battery and the other terminal to points on the rheostat separated 
 from this plate by resistances of 2, 4, 6, 8, 10 ohms, respectively. 
 Plot resistances as abscissae and voltmeter readings as ordinates. 
 What does the plot indicate to be the relation between the fall of 
 potential along a conductor and the corresponding resistances ? 
 Is the fall of potential over 2 ohms' resistance the same in all parts 
 of the circuit? 
 
 IV. Calculate from the readings of the ammeter and voltmeter, 
 by means of Ohm's law,* the combined resistance of the three 
 circuits in II when joined in parallel. 
 
 What relation exists between this resistance and the resistances 
 of the separate branches? 
 
 The reciprocal of the resistance of a conductor of electricity is 
 called its conductivity. Calculate the conductivity of each of the 
 parallel circuits in II separately, and also the conductivity of the 
 three in parallel. What relation exists between the conductivity* 
 of the whole, and the sum of the conductivities of the separate 
 branches, of the circuit ? 
 
 For statement of Ohm's law, see text-book or Exercise 37. 
 
72 ARRANGEMENT OF BATTERY CELLS. [39 
 
 V. Deduce algebraically from Ohm's law the relations found 
 experimentally in I and II, finding the equations for the resistance 
 of circuits of two and of three branches in parallel. 
 
 39. ARRANGEMENT OF BATTERY CELLS; THEIR 
 
 ELECTROMOTIVE FORCE AND INTERNAL 
 
 RESISTANCE. 
 
 I. Connect three Daniell cells in series with each other (zinc to 
 copper) and in series with an ammeter and a rheostat. Also 
 connect a voltmeter to the terminals of the battery. Read the 
 voltmeter and ammeter simultaneously, varying the external 
 resistance from o by steps to the limit of the rheostat. Discon- 
 nect the ammeter and rheostat and read the voltmeter. 
 
 II. Repeat I with the three cells in parallel (coppers together 
 and zincs together). 
 
 III. What from I and from II is the value of the electromotive 
 force of a single Daniell cell? Measure this quantity directly 
 with the voltmeter. Also read the ammeter connected to a single 
 Daniell cell. 
 
 By Ohm's Law find the internal resistance of a single Daniell 
 cell, of three in parallel and of three in series. What are the 
 corresponding electromotive forces ? 
 
 IV. Construct a plot, from the results of II, with external 
 resistances (R) as abscissae and terminal potential differences 
 (E'=CR where C=current) as ordinates. The electromotive 
 force E of the battery is given by the voltmeter reading on open 
 circuit. Indicate this quantity on the plot also. For what resist- 
 ance in the external circuit does the terminal potential difference 
 become zero ? On what does the terminal potential difference 
 depend? Deduce an equation (based on Ohm's law) giving the 
 relation between the electromotive force of a battery in terms of 
 the internal and external resistances and the current. Modify 
 this to include the terminal potential difference. 
 
40] COMPARISON OF RESISTANCES. 73 
 
 V. In I and II, which arrangement of cells gave the greatest 
 current when there was no external resistance in the circuit? 
 Which when the highest resistance used was introduced? Ex- 
 plain why in each case. 
 
 In general, how should a number of cells be connected in order 
 to obtain the greatest possible current ? 
 
 (i.) When the resistance in the external circuit is very small. 
 
 (2.) When comparatively large. 
 
 Explain these results algebraically by Ohm's law. 
 
 40. COMPARISON OF RESISTANCES BY WHEAT- 
 STONE'S BRIDGE. 
 
 I. Connect a Leclauche cell to the bridge-wire of a Wheat- 
 stone's bridge, and connect a sensitive galvanoscope, by one 
 terminal, to the sliding contact. (As the galvanoscope is simply 
 used to show the presence or absence of an electric current, the 
 motion of its needle is restricted to a few degrees.) Connect also 
 two rheostats in series with each other and in parallel with the 
 bridge-wire, and join the free terminal of the galvanoscope to the 
 junction of the two rheostats. A circuit of six branches is thus 
 formed, with the galvanoscope in one branch, the battery cell in 
 another, the rheostats in two branches, and two branches formed 
 by portions of the bridge-wire. 
 
 With a resistance of five ohms in each rheostat set the sliding 
 contact so that there is no current through the galvanoscope. 
 Interchange the rheostats and repeat. 
 
 Measure the lengths of the two portions into which the bridge- 
 wire is divided in each case. What is the mean ratio of these 
 two lengths? How does this ratio compare with the ratio between 
 the two resistances in the rheostats ? 
 
 II. Repeat I, with resistances of 5 and 10 ohms, respectively, 
 in the rheostats; with resistances of 7 and 10 ohms. What 
 proportion do you find can always be formed between the resist- 
 ances in the rheostat branches and the two lengths into which 
 
74 HEATING EFFECT OF AN ELECTRIC CURRENT. [41 
 
 the bridge-wire is divided when there is no current through the 
 galvanoscope ? Indicate clearly. 
 
 III. What must be the difference of potential between the two 
 points where the galvanoscope is connected when there is no 
 current indicated? Why? Show, by applying Ohm's law to 
 the four branches formed by the two parts of the bridge-wire 
 and the two resistances, that, when this is the case, the proportion 
 found in II must hold true. 
 
 IV. Replace one of the rheostats by 100 cm. of No. 25 German 
 silver wire. Adjust the sliding contact so that there is no current 
 through the galvanoscope, and measure the lengths into which 
 the bridge-wire is divided. Using the rheostat resistance as a 
 standard, calculate, by means of the proportion found in II, the 
 resistance of 100 cm. of No. 25 German silver wire. 
 
 V. Repeat IV with various coils of wire on the table, instead 
 of the German silver wire, and find the respective resistances of 
 these coils. Record the numbers on the coils. 
 
 VI. Repeat IV with a coil of fine copper wire immersed in 
 cold water, and then in hot water, taking the temperature of the 
 water in each case after stirring. From your "results calculate: 
 (i) The resistance of the coil at each temperature; (2) the 
 change in resistance per degree rise in temperature; (3) the 
 resistance at o; (4) the change in resistance per degree rise in 
 temperature of each ohm at o. The last result will be the 
 temperature coefficient of the electrical resistance of copper. 
 
 41. HEATING EFFECT OF AN ELECTRIC 
 CURRENT. 
 
 I. Fill a small calorimeter, that has been weighed with its 
 stirrer, two-thirds full of icercold water and weigh. Adjust in 
 place the heating-coil provided having the higher resistance, and 
 insert a thermometer in the water through the opening in the 
 cover to which the coil is attached. Stir thoroughly, taking 
 care not to splash the water and keep stirring throughout the 
 exercise. 
 
41] HEATING EFFECT OF AN ELECTRIC CURRENT. 75 
 
 Connect the heating coil in series with an ammeter and with 
 the terminals of the power circuit marked ' ' large current, 
 making the final connection at a noted minute and taking the 
 temperature at the same instant. Read the temperature and the 
 current each every minute on alternate half minutes, until the 
 temperature is half as high above that of the room as it was below 
 at the start. 
 
 II. Repeat I connecting to the terminals marked "small 
 current." 
 
 III. Repeat I (with the same terminals as in I) using the coil 
 of lower resistance. 
 
 IV. Calculate the heating effect of the current in each of the 
 three cases, in degrees per second. 
 
 Show from the results of I and II to what power of the current 
 the heating is proportional. 
 
 From the results of I and III show how the heating varies 
 with the resistance when the current is constant. 
 
 V. What becomes of the energy expended in maintaining an 
 electric current through a conductor? 
 
 Form an equation representing the relation of the heating to 
 the current, resistance and time. 
 
 Calculate the heat imparted by the coil in I on the assumption 
 that all goes to the water, calorimeter and stirrer. (The neces- 
 sary specific heats are given.) 
 
 The energy expended electrically is given in joules when ex- 
 pressed in terms of the units: the ampere, ohm, and second. 
 From the relation found above calculate in joules the energy ex- 
 pended in I, using the average value of the current. Deduce the 
 ratio of the calorie to the joule. What is this quantity? 
 
 Calculate in watts the power required in I. 
 
 What is it now necessary to know to calculate the value of the 
 watt in ergs per second? Explain. 
 
 VI. By means of Ohm's Law and the relation of V, find an 
 expression for the heating effect in terms of the electromotive 
 force and the current. 
 
76 LAWS OF ELECTROLYSIS. [42 
 
 42. LAWS OF ELECTROLYSIS. 
 
 1. Scour with emery cloth the six plates of the three copper 
 voltameters, then wash and dry them, taking care not to touch 
 the polished surfaces 'and not lay them on anything other than 
 clean white paper. Weigh these plates carefully on a sensitive 
 Jolly balance, recording the numbers on the plates in order to 
 identify them later. Place the plates in the dilute, slightly acid 
 copper sulphate solution which fills each of the voltameters, two 
 plates in each voltameter, adjusting so that the plates are parallel 
 in each cell. Connect the voltameters so that the whole current 
 will go through one of them and half through each of the other 
 two, arranging so that the current will go from the thick to the 
 thin plate in each cell. Diagram. State how you determine the 
 direction of the current. The circuit is completed by connecting 
 in series an ammeter and storage battery. The final connection 
 completing the circuit is to be made at a noted instant of time. 
 Leave the circuit closed for fifty minutes exactly and read the 
 current every two minutes. At the close of the run wash, dry, 
 and weigh the plates with the same precautions as before. 
 Record on the diagram the gain or loss of each plate. 
 
 II. i. Was the copper carried with or against the current? 
 Which, then, are the gain plates, those by which the current 
 enters or leaves the cells? How does the electrolytic cell com- 
 pare in this respect with the voltaic cell? 
 
 2. In each voltameter how did the gain of mass in one plate 
 compare with the loss of mass in the other? 
 
 3. What relation exists between the gain in mass of the. gain 
 plate in the voltameter through which the whole current passed 
 and the corresponding quantities for the other two voltameters? 
 Does the same relation hold for the loss plates ? 
 
 Find how the mass of copper deposited varies with the current. 
 
 III. Using the average value of the current in I, calculate for 
 each cell the mass of copper that would be deposited, from a 
 copper sulphate solution by a current of one ampere in one 
 
43] ELECTROMAGNETIC INDUCTION. 77 
 
 second, and find the average. This quantity is known as the 
 electro-chemical equivalent of copper. 
 
 IV. If zinc electrodes in a zinc sulphate solution were used, 
 would you expect the same quantity of zinc to be deposited in 
 the same time by the same current, as above ? 
 
 Look this up and state the remaining law of electrolysis. 
 
 Express in the form of an equation the laws of electrolysis. 
 
 43. ELECTROMAGNETIC INDUCTION. 
 
 I. (a.) Connect a coil of wire to a sensitive galvanometer, after 
 testing with a Leclanche cell what is the direction of the 
 current corresponding to a deflection to the right and left. (Be 
 careful not to disturb the galvanometer and accessory apparatus.) 
 Connect an electro-magnet to the storage battery terminals. 
 
 Hold the coil in the field of the electromagnet perpendicular to 
 the direction of the field, opposite the north pole of the magnet. 
 Then turn the coil quickly through 90, so that it becomes 
 parallel to the direction of the field. Note the deflection of the 
 galvanometer, and whether a clockwise or counter-clockwise 
 current is induced in the coil, looking along the lines of force. 
 
 (^.) With the coil held as in (a) remove the electromagnet 
 from before the coil, noting deflection and direction of induced 
 current as before. 
 
 (c.*} With the coil and magnet as in (a), break the circuit of 
 the electromagnet. Record as before. 
 
 In (d), (), and (c) how do the currents compare in magnitude 
 and direction? Viewing the coil in the direction of the lines of 
 force, was the number of lines through the coil diminished or 
 increased in each case? Does then diminishing the number of 
 lines of force through a closed circuit induce a clockwise or a 
 counter-clockwise current in the circuit? 
 
 II. Repeat I (a) with the same coil but inserting a resistance 
 in the circuit equal to the previous total resistance of the circuit. 
 Compare the current induced with that in I. How did it vary 
 
7 8 ELECTROMAGNETIC INDUCTION. [43 
 
 with the resistance in the circuit ? Which do you conclude is the 
 quantity that remained constant, the induced current or the 
 induced electromotive force? Is it better then to speak of induc- 
 ing an electromotive force or a current by moving a closed circuit 
 in a magnetic field? 
 
 III. (#.) Remove the extra resistance, and rotate the coil from 
 its final position in I through another 90. Apply the rule de- 
 duced in III, for the direction of the induced current (electro- 
 motive force). Does it still hold true? 
 
 (.) Rotate the coil through 180 more by steps of 90. What 
 is the effect of increasing the number of force-lines through the 
 coil on the direction of the induced current (electromotive force) ? 
 
 (r.) Repeat I (<) with the electromagnetic turned end for end. 
 Is the result of III () confirmed? 
 
 IV. Repeat III () with a coil having twice as many turns of 
 wire. How do you find the induction to vary with the number 
 of turns of wire in the coil? If you consider each turn as enclos- 
 ing a certain number of force-lines, how then does the induction 
 vary with the total change in the number of the force-lines 
 threading through the coil ? 
 
 V. (a.'} Hold the coil stationary, as in I (<:), and remove the 
 core only of the electromagnet. If the galvanometer is deflected, 
 read the deflection. Replace the core and read the deflection, if 
 any, again. Explain the effect in each case. 
 
 (<5.) Remove the core very slowly and read the deflection of 
 the galvanometer. Does this experiment indicate that the in- 
 duced current (electromotive force) varies with the rate at which 
 the change in the magnetic field is produced? How does the 
 rate of change affect the induced electromotive force ? 
 
 VI. The general laws of electromagnetic induction may be 
 stated thus: When the magnetic field is altered in any way with 
 respect to an electric conductor, an electromotive force is induced 
 in the conductor. This induced electromotive force is propor- 
 tional to the rate of change in the magnetic field, and its direc- 
 
44] EARTH INDUCTOR. 79 
 
 tion is such as to produce a current that will oppose the change 
 in the field. 
 
 Show how the results obtained in I-V may be explained by 
 means of this law. 
 
 44. EARTH INDUCTOR. 
 
 I. (a.) Set up a sensitive galvanometer and connect it with an 
 earth-inductor, placing them as far apart as the table will allow. 
 Place the earth -inductor so that the two stationary, upright sup- 
 ports are in an east and west line, and set the circle so that its 
 axis- of rotation is horizontal. 
 
 Turn the circle slowly into a horizontal position, let the 
 galvanometer-needle come to rest, and then turn the circle sud- 
 denly through 1 80, noting the effect on the galvanometer. 
 Explain the cause of the current produced. 
 
 (b. ) Turn the circle in the same direction through another 
 1 80, and compare the induced current with that in I (). Was 
 its direction the same ? What would its direction have been if 
 there had been no commutator? 
 
 (V.) Rotate the coil continuously and uniformly, recording 
 the number of turns per minute and the deflection of the 
 galvanometer. 
 
 II. Set the coil so that its axis of rotation is approximately in 
 the direction of the earth's magnetic field (at an angle of about 
 62 with the horizontal). Rotate it continuously as was done in 
 I (V), recording again the number of turns per minute and the 
 deflection of the galvanometer, if any. How does the current 
 induced compare with that in I (V)? Explain the difference, if 
 there is any. 
 
 III. Set the coil as in I, and rotate it continuously at a rate 
 either one-half or twice as great as in I (V). What effect do you 
 find a change in the rate of rotation to have upon the value of 
 the induced current ? 
 
 IV. Repeat I (c) with the axis of rotation vertical, rotating 
 
80 EARTH INDUCTOR. [44 
 
 the coil as nearly as possible at the same rate. To what com- 
 ponent of the earth's magnetic field is the induced current pro- 
 portional in this case? To what component was it proportional 
 in I (V)? How might the angle of dip be calculated from the 
 observations made in this section and in I (V) ? Using a table of 
 natural tangents, calculate thus the angle of dip at Berkeley. 
 
 V. By varying the angle of inclination of the coil, find a posi- 
 tion for which there will be no current induced when the coil is 
 rotated. Read the angle of inclination, if the earth-inductor has 
 a graduated circle. What is the relation between this angle and 
 the angle of dip? How does the value of the angle of dip found 
 in this way compare with that found in IV? 
 
 VI. Turn the base of the earth-inductor through 90 and 
 rotate the coil continuously about a vertical axis, as in IV, at the 
 same rate. How do you find the induced current to compare 
 with that in IV ? Explain the difference, if there is any. 
 
 VII. Answer the following questions and give reasons for your 
 answers: 
 
 1. Would there have been any current induced if the coil had 
 been moved parallel to itself? 
 
 2. Would there have been any current induced if the coil had 
 been moved parallel to itself with a strong magnet in its 
 neighborhood ? 
 
 3. What would be the effect on the induced current if a soft 
 iron core were placed within the coil of the earth inductor? 
 
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