ADVANCED PRACTICAL PHYSICS FOR STUDENTS B.L.WORSNOP AND H.T. FLINT fiBl r.SSs'iV.i' w.ii*s I ^^H .^ikiatettMsfitti*. 1 ^H = WiSii^-'i'^r - 'r'".:'.'.r" ^^^B^l ^ii*---. H .tJI & .-l^lVr^t METHUEN iff, JbdUbU-$**^t ADVANCED PRACTICAL PHYSICS FOR STUDENTS ADVANCED PRACTICAL PHYSICS FOR STUDENTS BY B. L. WORSNOP, B.Sc. LECTURER IN PHYSICS, KING'S COLLEGE, LONDON EXAMINER IN PHYSICS, UNIVERSITY OF LONDON AND H. T. FLINT, Ph.D., D.Sc. READER IN PHYSICS, UNIVERSITY OF LONDON EXAMINER IN PHYSICS, UNIVERSITY OF LONDON WITH 404 DIAGRAMS AND ILLUSTRATIONS SECOND EDITION, REVISED METHUEN & CO. LTD. 36 ESSEX STREET W.G. LONDON First Published . . . April 1923 Second Edition, Revised J 9 3 7 PRINTED IN GREAT BRITAIN PREFACE THE course of Practical Physics described in this book is based upon that followed in King's College, London, by students who have completed their Intermediate Course, and who are proceeding to a Pass or Honours Degree. This has been extended, and it is hoped that the book will be useful to a wider circle of students of Physics than those imme- diately concerned with University Examinations. A number of well-known Physicists have contributed to the development of the King's College course, amongst whom we may mention Professors H. A. Wilson, C. G. Barkla, H. S. Allen, and W. Wilson, who formerly worked here in the Wheat stone Laboratory, and Professor O. W. Richardson, the present occupant of the chair. The general aim has been to provide with each experiment a short theoretical treatment which will enable the student to perform the experiment without immediate reference to theoreti- cal treatises. To aid this scheme an introductory chapter in the Calculus has been included. This chapter is an innovation in a book of this type, but it is hoped that the student will find here a bridge over that period during which his Physics demands more advanced mathematics than his systematic study of that subject has yet given him. We take this opportunity of expressing our gratitude to Professor O. W. Richardson, who has allowed us to make use of laboratory manuscripts and results of experiments. We are also greatly indebted to our colleagues and to Mr. G. Williamson, who have given us many suggestions, and to the Honours students of vi ADVANCED PRACTICAL PHYSICS FOR STUDENTS the past session who have supplied us with numerical and graphical results. We have been greatly helped by the ready assistance on the part of The Cambridge and Paul Scientific Instrument Co., Messrs. Elliot Bros., Gambrell, Ltd., Adam Hilger, Ltd., W. G. Pye & Co., and the Weston Electric Co., who supplied us with the blocks for many of the illustrations. B. L. W. Wheatstone Laboratory, H. T. F. University of London, King's College. March, ig2j PREFACE TO SECOND EDITION IN this revised edition we have removed the misprints and errors which occurred in the first. We have added to Chapter XXIV on Radioactivity, and have included a chapter which contains a miscellaneous collection of additional experiments. We wish to thank those who have made suggestions for additions and who have pointed out errors in the original text. The adoption of all the suggestions would have made the book unwieldy, but we hope that the additions made will increase its value to the student. To thank, individually, all students, colleagues, and others to whom we are indebted would add greatly to the length of this preface. But we feel that we owe our special thanks to Mr. Brinkworth, Dr. D. Owen, and Prof. Rankine for suggestions ; to Dr. K, G. Emeleus for suggestions and help with the proof ; and to Prof, E. V. Appleton, who has added several experiments with thermionic valves in the new chapter. B. L. W. H. T. F. February 21, 192J CONTENTS II. III. IV. V. VI. VII. VIII. IX. X. XI. XII. XIII. XIV. XV. XVI. XVII. XVIII. XIX. XX. XXI. XXII. XXIII. XXIV. XXV PACK INTRODUCTION TO THE DIFFERENTIAL AND INTE- GRAL CALCULUS - - - - I MEASUREMENT OF LENGTH, AREA, VOLUME, AND MASS - - - " " " 3 1 MOMENTS OF INERTIA AND DETERMINATION OF "g" 5 1 ELASTICITY - - * " - 80 SURFACE TENSION ----- 124 VISCOSITY ------ 149 THERMOMETRY AND THERMAL EXPANSION - - I78 CALORIMETRY - IQ^ VAPOUR DENSITY AND THERMAL CONDUCTIVITY - 217 MISCELLANEOUS EXPERIMENTS IN HEAT - - 243 REFLECTION - - - • -258 REFRACTION - - - " - 2J2 INTERFERENCE, DIFFRACTION, AND POLARISATION - 314 PHOTOMETRY _ _ - - - 380 SOUND - - . - - - - 39 2 MISCELLANEOUS MAGNETIC EXPERIMENTS - - 426 TERRESTRIAL MAGNETISM - 437 PERMEABILITY OF IRON AND STEEL - - 445 AMMETERS, VOLTMETERS, AND GALVANOMETERS - 456 RESISTANCE MEASUREMENTS - 4^9 RESISTANCE OF ELECTROLYTES - 526 MEASUREMENT OF POTENTIAL - 531 MEASUREMENT OF CAPACITY AND INDUCTANCE - 550 THE QUADRANT ELECTROMETER - 584 MISCELLANEOUS ELECTRICAL EXPERIMENTS - 605 ADDITIONAL EXPERIMENTS - &37 TABLES OF UNITS - 652 APPENDIX ------ 654 INDEX ------ 655 ADVANCED PRACTICAL PHYSICS FOR STUDENTS INTRODUCTION TO DIFFERENTIAL AND INTEGRAL CALCULUS The Differential Calculus 1. Any quantity % which may assume a series of values is called a variable quantity or simply a variable, and if its value docs not depend on that of any other quantity it is called an independent variable. On the other hand a quantity y, which bears a particular relation to x, assumes values which depend on the values of x, and for this reason is called a dependent variable. We may have for example : y = 2x - 3. Here y takes values which depend in a quite definite manner on those of x. We may also have a dependence defined by the relations : y = sin x, y = log x, and y m §•. Such expressions as 2% — 3, sin x, log m, etc., are called functions of x, and when we say that y is a function of % we mean that y depends on the values that % assumes. In case we do not specify definitely how y and % arc related wc write y = /(*)■ f(x) denotes any function of %. It is often convenient in Physics to show by means of a diagram the relation between two variables y and x. For example, a record may be required of the atmospheric pressure at various times. U Such a record is drawn automatically by a self-recording barometer so that it can be seen how the pressure and time are related. Here we have as independent variable the time and the dependent variable is the barometric pressure. 2 ADVANCED PRACTICAL PHYSICS FOR STUDENTS In fig. i, the curve represents the relation between x and y and the shape depends on the way y and x are connected. X Fig. i If y — 2X — 3, the curve becomes a straight line, and if y = sin x, we have the familiar sine curve, fig. 2. Fig. 2 A function is said to be a continuous function of a variable when the graph representing it is a curve in which there is no sudden change in value of ti\e ordinate at any point. In such a curve, if we approach a point where x = a, from left to right, we find a certain value for.y, and if we approach the point from right to left we find the same value. In fig. 3 we have an example of a function which is discontinuous at x = o. If we approach the origin from left to right the value of y is very great and DIFFERENTIAL AND INTEGRAL CALCULUS 3 negative in sign, while in approaching from the right y is very large and positive. In nature we are chiefly concerned with continuously varying quantities. If a train is at rest at a station at a particular instant, and is observed to be moving with a velocity of ten miles per hour ten minutes later, it must have possessed every possible velocity between zero and ten miles per hour during the interval. The speed is continuous, and if it depends on the lapse of time from the start it is said to be a continuous function of the time. We do not contemplate the possibility that the train could possess a speed of five miles per hour at one instant and at the next without any interval whatever a velocity of six miles per hour. If this were possible we should describe the speed as discontinuous, because it had no value between five and six. If this appeared to be the case we should consider that our powers of observation were at fault, and we should describe the motion as changing very rapidly between five and six miles per hour ; so rapidly that we had failed to detect the lapse of time in which the change took place. Discontinuous functions are of frequent occurrence in Mathe- matics. Consider as an example the case of y = -. When x is a very small positive number, let us say — -;, y is large and has the value 10 8 . 4 ADVANCED PRACTICAL PHYSICS FOR STUDENTS On the other hand if x = 6 , y is large in magnitude but negative, it equals — io 6 . As x passes through the value zero y suddenly leaps from an enormously large negative value to a very great positive value, and has no value between. This is represented in the diagram, fig. 3. The curve has two branches : they are the two parts of the rectangular hyperbola xy = 1. We shall not be concerned with such functions so we dismiss them briefly. It is to be borne in mind that our applications of the Calculus are to continuous functions only. The results we obtain must not be applied to discontinuous functions without closer examination. It is important to understand the meaning of the limit of a function. Suppose y depends on x, and that as x approaches the value a, y approaches the value b. b is called the limit of y as x approaches a, and we write : Lim. y ► b x > a If reference be made to fig. 1, as x approaches the value OM t y approaches the value M{9 X and M^ is actually the value of y when x = a. Cases occur in which the conception of a limit is not so simple. If we examine the curve 1 J x in the neighbourhood of the origin as x — > o, we obtain a different value of y according as we begin on the right or left of the origin. On account of the discontinuity the limit of y as x approaches zero is not definite. Another case occurs in which x may continue to increase to any extent while y continually approaches some particular value. We may turn once more to the curve y-l. X As x gets larger and larger, y gets smaller and smaller approaching the value zero. We may get as near zero as we please by making x larger. For example, we may make y as small as one-millionth by choosing x — 10 6 . DIFFERENTIAL AND INTEGRAL CALCULUS 5 This is a very important point in denning a limit. It must be possible to get as close to the limiting value as we please by choosing x properly, although it may not actually be possible to cause y to attain the limit. We have in our example a case in point, y is never zero however large x may be, but it is possible to make y nearer and nearer zero by increasing x. The former definition of the limit of a function is not very satisfactory. A limit is accurately defined as follows : The limit of a function of x is some number, b, such that as x approaches a particular value, a, the difference between b and the function may be made as small as we please by taking x sufficiently near a. 2. In describing natural phenomena by means of equations, simplifications are often brought about by neglecting certain terms in comparison with other more important terms. Suppose an equation is obtained which we may write : A x + B x + C 2 + D 2 = E x + F 8 . The suffix numbers denote the order of importance of the terms ; that is to say, 1 denotes that the term is to be regarded as of first importance, or it is of the first order of magnitude. The 2 and 3 denote that the terms are only of second and third degrees of importance, they are of the second and third orders. If we wish to include terms of the first and second orders we omit F 3 , while if only terms of the first order are to be considered the equation becomes : h x + B x = E x . Great care has to be exercised in thus drawing up a scale of magnitude, and this leads to a short consideration of infinitesimals. Suppose a quantity X is divided into 1000 equal parts, these again subdivided in the same way, and so on. We then have a series of values : X, — 3— 6 ,etc, 10 3 10 6 which provides a scale of magnitude. If circumstances do not permit of accurate observation of quantities less than those of the same order as X we regard — r» — / etc., as negligible. io 3 io 6 ° ° Generally, if / is a small fraction, i.e. small compared with unity : /X,/*X,/ 3 X, etc. are all small compared with X, and are said to be small quantities 6 ADVANCED PRACTICAL PHYSICS FOR STUDENTS of the first, second, third, etc., orders. If these small quantities have zero limits they are called infinitesimals. In equations between infinitesimals only the terms of the lowest order are to be retained, i.e. the terms of greatest magni- tude. This is made clearer by an example which has important Physical applications. In fig. 4, AB represents the radius of an arc, BP, of a circle which subtends an angle at A. PN is normal to AB. BT is also normal to AB cutting AP produced in T. It may be regarded as an axiom that : PN < arc PB < BT. We shall examine the order of the differences between these quantities if be regarded as of the first order of small quantities. By expansion of sin and cos in powers of we have : 03 05 sin 6 = -, -1 , — , . (-i) 3l 51 02 04 cos = I r + — . — ( 2 / 214! If only small quantities of the first order are retained : sin = 0. -» . cos = 1. / (3/ PN = a sin 0. PB = a 0. i 3 5 ^ .\ PB - PN ,= a(0 - sin 0) = a { - - - ( + .. . .1 This difference is of the third order. Thus up to considerations of magnitude of the third order PB = PN. DIFFERENTIAL AND INTEGRAL CALCULUS 7 Again BT =* a tan 0. A 3 2 = a(0 + -~+ — 5 + - • • )» as may be shown by division of the expressions for sin and cos0. Thus BT - arc PB = a quantity of the third order of magni- tude. Fig. 5 BN — a — AN = a(i — cos 0) = a quantity of the second order of magnitude. Thus, if we regard as of the first order and retain only this order in our equations we may write : BN = o, PN = arc PB = BT. and with the exception of BN = o this is true for the case when second order quantities are retained. Extensive use is made of these relations in Geometrical Optics in the first study of reflection and refraction in mirrors and lenses. In the case of a mirror, for example (see fig. 5), when the angle is small, i.e. when the rays from an object, P, strike the mirror at M not far from the pole, O, we establish certain formulas by assuming that and N may be regarded as being coincident. This is because we do not retain quantities of an order higher than 0. Thus NO = o by the foregoing considerations. Another important case is the calculation of the order of the difference between the sum of two sides of a triangle and the base when the base angles are of the first order of small quantities. 8 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Referring to fig. 6, we have as for fig. 4, PA — AN = a quantity of the second order in 0, and similarly PA 1 — A X N is of the second order in 1 .'. PA + PA 1 - (AN + A X N) = a quantity of the second order since and 1 are by the data small quantities of the first order. /. PA + PA 1 = AN + A*N = AA 1 to the first order. This result is made use of in the establishment of Fermat's Law of the extreme path which plays a fundamental part in Optics. (See for example Houstoun's " Treatise on Light," p. 17.) We consider as a final example the difference between a chord and an arc both subtending the same small angle at the centre of a circle. Thus, again referring to fig. 4, we require the difference between chord BP and arc BP. a arc BP — chord BP = ad — 2a sin — . 2 = a e - 2( --- ^- + ...) _ 1 0* 2 3! 8" = a quantity of the third order. We can thus regard the chord and arc as equal up to and including quantities of the second order. It should be noted that the successive orders are vanishingly small with regard to the terms earlier in the scale, e.g. in com- paring aO, bd*, c0 s , hf}2 fia if is of the first order, --- = — so that as > o 60 2 — > o ad a infinitely more rapidly than aO, and the same holds for any two consecutive terms in the scale. The ratio of two quantities of the same order will be a finite quantity— not a vanishing or negligible quantity, but the ratio of two quantities of differing order (higher order -^ lower order) is vanishingly small. We are concerned with small variations of this kind in the differential Calculus. 3. The Differential Coefficient Let y be a function of x, and suppose x varies by a small quantity which we denote by dx. In consequence of this varia- tion y will vary a small quantity, say dy. DIFFERENTIAL AND INTEGRAL CALCULUS 9 The ultimate ratio 4^- when dx becomes very small is called dx J the differential coefficient of y with respect to x. It is denoted by -j-y and written -j- and sometimes denoted simply by Dy. In accordance with our notation we may write : dy _ Lim. #y dx ~ Sx > o dx' In general the quantities dy and dx are of the same order of magnitude, and -^- is a finite quantity. In order to illustrate this, consider the relation : y = 2.x — 3. Let x become x + Sx, then the new value of y is 2{x + dx) - 3 i.e. y + <*y = 2 (* + **) — 3 = y + 2<S#. .*. <5y = 2<5z. • & - 2 Now no matter how small dx becomes, the ratio is always 2, for dy is of the same order as dx, and their ratio is finite and equal to 2. We have a simpler case still in the differential coefficient of a constant. A constant is a number that does not depend on the variable. Thus, if y = A it does not matter how x varies, y still remains = A. Thus there is no change dy corresponding to a change dx. Hence -j- = o if y is a constant. It should be noted that -~ does not mean dy -^ dx. g~ is a short . dy notation for the operation of finding the ultimate ratio — Nevertheless Physicists continually appear to use the coefficient as if it meant dy -^ dx, and it is not a rare occurrence to find an equation : dx written alternatively dy = x 2 dx. This is, in fact, a very convenient mode of expressing the result, io ADVANCED PRACTICAL PHYSICS FOR STUDENTS and it means that dy and dx now no longer retain the same significance. The second of these means : dy = x 2 dx. We have in the equation j- = x 2 an expression of the rate of variation of y with respect to a; at a particular point on the curve, which represents graphically the relation between y and x. The alternative equation means that in the neighbourhood of this point we can calculate a small change dy corresponding to a small change dx. This point rarely causes difficulty in practice, and it is obviously inconvenient to change to and fro from d to 8, but to be strictly accurate we must bear the distinction in mind. The definition of the differential coefficient gives the clue to its determination. We will not determine its value for more than one or two cases but be content with reference to a table of values of the important coefficients. The Differential Coefficient for x n where n is any Number. Write: y = x*. (Sx\ n i+ — f = x" dx _j_ n(n — i) m (toy I + "¥ + ^Tr^-i-»^+--- dy fi(n — i) = x n + nx*- 1 • dx + - L — — - - x n ~ 2 (dx} 2 + X*2 = nx»-*dx + n ^~^ • x"- 2 (dx) 2 + . . . 1*2 ty n(n — i) .'. v- = nx"* 1 -\ — • x*~ 2 dx + higher powers of dx. dx is a quantity which we have called infinitesimal. In the next step of finding the limit of ~- we shall suppose dx a quantity of the first order of magnitude. It is therefore infinitesimally small with regard to the finite quantity nx n ~K We thus neglect all quantities of higher order than nx n ~ x and have : dy Lim. dy dx dx >odx nx ' + DIFFERENTIAL AND INTEGRAL CALCULUS 11 Differential Coefficient of sin x. y -f. Sy = sin (x + dx) = sin x cos dx + cos x sin <5#. We need retain only quantities of the first order on the right. Thus : cos dx = I, sin dx — to, by equation (3). •*• y + *y = sin x + cos x y - dx. .*. dy — cos a; • dx. dy or -3^ = cos x. dx Similarly ■=- cos x = — sin #. Differential Coefficient of log x. y + <5y = log (X + <5*) = log a/ I + -^j = log* + log(i + ^. Retaining quantities of first order only : dy = dx* — "* x ; *1 = 1 dx x d , t or ~-\og X — _. dx x The same method of treatment can be applied to other cases. In the case of a function f(x) we write : dy^ _ Lim. f(x + dx) —f(x). dx ~~ to > o fa The Differential Coefficient of the Sum of two Functions. If y = sin x + cos # we have -r- = cos # — sin x. dx From the definition it follows that the differential coefficient is the sum of the differential coefficients of sin x and cos x. In the general case if : y=yi+y 2 12 ADVANCED PRACTICAL PHYSICS FOR STUDENTS where y x and v 2 are any two functions of x : dx dx dx And similarly if y =y 1 — v 2 dy _ dyj _ dy 2 . dx ~ dx dx Differential Coefficient of a Product. Let y = y^y 2 where y x and v 2 are any two functions : e.g. we might have : y — sinx x # n , sin # and #» are two functions of x. Suppose that x becomes x + Sx and in consequence y increases toy + dy,y 1 toy 1 + dy lt y 2 to v 2 + <*y 2 . Then y + «5y = (yx + «yi) (y« + <5y 2 ) Since jy — yiy2 •*• <?y =^i4y2 + ^1^2 + tyity* dy 1 dy 2 is a term of the second order, and the other terms are of the first order. Thus we need not retain it. Dividing throughout by dx. &««,& + Si. y,. dx yx dx 6x • ra Hence in the limit : In a product we differentiate one factor at a time, leaving the others unchanged, and add all the resulting expressions together. This is true for any number of factors, as may be shown in the same way. Thus, if y =ytyzy*y± dy dy, , dy 2 , dy 3 , dy* & -at ' y w< +y > £** + ™* %* +y ^ y * it- e S- y = sin x x % n dy -f- = sin x • nx n ~ x + cos x • *". dx DIFFERENTIAL AND INTEGRAL CALCULUS 13 The Differential Coefficient of a Quotient. We use the same notation as before and apply the same principles. y* + *y* y* v (i + ^) (retaining only terms of first order). *■" y* 2 fy y * dx yx dx 6x ^ y 2 2 jV — tan x — dy __ cos x cos * — sin x( — sin #) dx ~~ cos 2 # , sin# e S> y — tan # = J COS # = sec z #. Differential Coefficient of a Function of a Function. The expression : y — a sin 9 # -f- b sin x -f c in which a, 6, c are constant quantities, is a function of sin #, sin x is itself a function of x. Thus, y is a function of a function of x. We proceed to determine the differential coefficient -j~ in this complex case. Before attacking the general problem we will consider a special case. Let y = log sin x. and write z = sin x. 14 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Then y — log z and as above : dy = — dz, J z In this step we have regarded y as a function of z. But z = sin x. .*. dz = COS X'dX. <5y = - • cos x ' dx. dy z COS X " dx ~ sin x Note that from dy i ~ z dz we have : dy i dz dx ~ z dx Hence dy i dz dx ~~ z dx' Now . dz z dy dy ^ dz dx ~~ dz dx' The rule is : Differentiate the whole function first as if the inner function (in this case sin x) were the independent ' variable, and thus obtain ~, then multiply by the differential coefficient of the inner function. . This rule, which has been established in a special case, can readily be proved generally. Let y = F(z) where z =/(*). fy = Q .*: 'dx ~ dz ' dx Then In the limit : ~ = -j- • -j-. dx dz dx e.g. y = sin 2 * == (sin x) 2 . Put z = sin x. then y =z\ . & = dz dy _ dy dz ~ = 2Z, -5- = cos x. dz dx , — 2z cos x — 2 sm X cos X. DIFFERENTIAL AND INTEGRAL CALCULUS 15 It will be convenient at first to introduce z in this way, but with practice this intermediate step may be omitted. 5. The Second Differential Coefficient We have seen that when y = sin x t dy -£ = cos x. ax Thus-f- is itself a function of x and will have a differential ax coefficient. Let -f- be denoted by z. dx J z = cos x. . dz . . -j- = — sm x. dx Thus the differential coefficient of -f- is — sin x in this case. dx This is called the second differential coefficient of v, and is written Similarly we define higher coefficients -=-^, etc., but we shall not be concerned with higher orders than the second. As another example consider y =x n . ~ — nx n ~ x . dx d 2 v j+=n(n- i) x»-*, etc. 6. Applications in Dynamics When a particle describes a path under the action of a force or set of forces, its position will vary with the time. Thus if a stone is thrown vertically upward the position at any instant will depend on the time that has elapsed since the moment of projection. If this distance is measured by y, Suppose the time at a particular instant is measured by t and the corresponding value of the distance is y. At a small time 16 ADVANCED PRACTICAL PHYSICS FOR STUDENTS dt later y will become y -\- dy. We may say the average velocity . <5y is -jf- in this short interval. The smaller we make the interval ot the more accurately will this ratio represent the velocity near the position denoted by y. Thus proceeding to the limit in which dt ultimately vanishes we have : , ., Lim. dy dy velocity v — CJ ~ = -£ J dt >0 dt dt It is often convenient to measure the position of a point by its position measured from some point O along the arc it is describing (fig. 7). Fig. 7. ds In this case -j? is the velocity at P along the arc or the velocity in the direction of the tangent at P, where s measures the arc OP. dv In the same way -.— measures the acceleration of a particle moving with velocity v., dv d 2 s . ds But W = df> "^ V = dt Thus the acceleration along the arc is measured by the second differential coefficient. 7. A Geometrical Application Referring to fig. 1, let Pj denote the point (x- y) and P, the point {x •+■ ** • y + <5 3')- Then P 2 N = dy and PjN = M^ = 6x. The ratio ^ = |^ = tan V^P^ = tan v . ox r x JN As the limit is approached the line P 2 P t becomes closer and closer to the tangent at P x and finally actually coincides with it. T , Lim. dy _ dy Thus . ^ *■ — r-=tanv dx ► o dx dx DIFFERENTIAL AND INTEGRAL CALCULUS 17 where y now denotes the inclination of the tangent at Pj to the axis of x. If a curve in the course of its extent is of the character shown in fig. 8, the turning points BjBa are called maximum and B 2 B 4 minimum values of the ordinate. Bx is a point such that it possesses the greatest value of y for points in its immediate neighbourhood. It is not necessarily the greatest ordinate of the curve. Fig. 8 A similar remark applies to the minimum values. At such extreme points the tangent is necessarily parallel to the axis of x. Thus y = o and tan y also vanishes. Hence for maxima and minima : £-0. ax This relation is also of great use in Physical problems, as in the case of the minimum deviation of a prism or in the theory of the formation of the rainbow. 8. Integration The process of Differentiation is to derive from a function its differential coefficient or rate of variation with respect to the independent variable. We have the inverse problem in Integration, where from the differential coefficient we have to derive the function., This operation is more difficult and cannot be accomplished in every case. . There are many standard cases which can be readily recognized, *e.g. we have seen that for_y = #** Y-= nx"- 1 ax i8 ADVANCED PRACTICAL PHYSICS FOR STUDENTS so that when we are given that ax we remember that y has the value x n . But if y has the value x n + A, where A is some constant, and hence has a zero differential coefficient we still have : dx nx •»— 1 Thus, there is an uncertainty about the value of y to the extent of an unknown constant. Such a constant must always be taken Fig. 9 into account in integration and is called the integration constant. In Physical problems this constant usually has some special value determined by the conditions of the problem. We shall approach the Integral Calculus by a Geometrical consideration. In fig. 9, NM represents a curve, y =/{x). OA and OB are two fixed values of x and the area between the ordinates AN and BM, the curve NM and the axis of x is divided by drawing ordinates P^, P 2 Q 2 , P 3 Q 8 , etc., between A and B, so that AP X = dx x , P X P 2 = dx 2> P a P 8 = dx 3J etc. DIFFERENTIAL AND INTEGRAL CALCULUS 19 All these elements are to be reduced and ultimately vanish. Let x v x 2 , # 3 , etc., denote the values^ of the abscissae for points lying in the strips AP 1} P^, P 2 P 3 , etc., respectively. Of these the ordinate at X 3 , which denotes the point with abscissa x z , is a typical example : Let the ordinates corresponding to these abscissae be denoted by y x , y 2 , y 3 , etc We shall suppose that between all the points such as NQ X , QiQ2> Q2Q3. etc., * ne curve is continuously increasing or decreas- ing. At every point where there is a maximum or minimum, as at C, we shall draw an ordinate. The curve is drawn with only one such turning point, and this is sufficient for our purpose since the argument may be extended to deal with cases where several such points occur. From the points N, Q lf Q., etc., draw perpendiculars NR 1? Q X R 2 , Q 2 R 3 , etc., to the ordinates ¥ x Q lf P 2 Q 2 , P 3 Q 3 , etc., respec- tively, and from Q lf Q 2 , Q 3 , etc., draw perpendiculars Q^, Q 2 S 2 , Q 3 S 3 , etc., to the ordinates AN, P^, P 2 Q 2 , etc., respectively, as shown in fig. 9. We have thus two step-like figures which we will call the outer and inner stepped figures. Now consider the expression y z 8x 3 . This lies between Q 3 P 3 x P 2 P 3 and Q 2 P 2 x P 2 P 3 . It thus represents an area intermediate between that of the rectangle P 2 P 3 Q 3 S 3 and P 2 P 3 R 3 Q 2 . In the same way y x fa x lies between APiQ^j and AP^N. Thus, if we consider the sum : y 1 6x 1 -\-y i 6x 2 + y z 8x 3 + . . . +y n fi%n = tydx, where y n represents an ordinate in the last strip which terminates at B and has a width 8x n , we know that it lies between the outer and inner stepped figures. We have to inquire into the value of the sum when we make the number of divisions very large or, what is the same thing, the values d$ lt 6x 2 , 8x 3 , etc., very small. In performing this summation we meet with the difficulty that although each term becomes small, the total number in- creases, and the question arises as to whether the sum remains finite under these circumstances. It seems almost axiomatic that the sum will in the limit prove to be the area between the curve, the extreme ordinates and the #-axis. However small the values, 8x, become the sum concerned always lies between the outer and inner stepped figures. The difference between these areas is the sum of the rectangles : SjQ.RiNi, S.QtR.Qp etc. 20 ADVANCED PRACTICAL PHYSICS FOR STUDENTS There will always be one strip which has a width greater than that of any of the others, or at any rate its width will not be less than that of any other, though of course it is possible to have another strip of equal width. Let this be the strip lying between P r and P r+1 , and draw the two rectangles as before, viz. P r P r+1 Q r+1 S r+1 and P r P r+1 R r+1 Q r l In our figure the greatest ordinate is CD. Continue P r +1 Q r , , to T 1 and P r S,+i to T so that P r+1 T* = P r T = CD. Now all the small rectangles S^, S 2 R 2 , S 3 R 3 , etc., lying between AN and the maximum ordinate at C could be removed and placed within the rectangle TP r+1 . Thus the difference between the two stepped figures lying on the left of C cannot be greater than the rectangle TP r+1 . As the widths 6x lt 8x 2 , dx z , etc., are diminished the area TP r+1 diminishes and vanishes in the limit. Thus in the limit when the widths of the strips become infinitely small, the difference between the two stepped areas also diminishes indefinitely. In other words, the two areas become equal in the limit, and each is then equal to the area between the curve, the ordinate, AN, the maximum ordinate and the #-axis. The same process may be applied to the area between the maximum ordinate, BM, the curve and #-axis. Thus adding the two parts together we find that the total area is equal to the sum : Sv<5# in which 8x tends to zero. If there occur any finite number of maxima or minima in the curve, it may be divided at each and the same process applied to each partial area. When the several areas are summed the result applies to the whole curve. ydx and it is called the integral of y with respect to x. a and b denote the values OA and OB, and since these are definite abscissae the integral is called a definite integral. In some cases when we do not fix the limits we write simply J ydx and the integral is called indefinite. The lower limit is then tacitly assumed to be some convenient starting point, while the upper limit is any variable point which we denote by the variable x. DIFFERENTIAL AND INTEGRAL CALCULUS 21 9. Connection between Differentiation and Integration If we consider the area included between the limits A and X where X is any point with co-ordinate X, the magnitude of the area depends on the positions of the points A and X, and the form of the curve. The form of the curve is fixed by its equation, and when X is changed the area will vary by an amount depending on the change in X. The area thus depends on the value of X at which the summa- tion is stopped. If the student feels any difficulty about this point he should draw a semicircle and choose the diameter as x— axis. Choose one end of the diameter for the point, O, and determine the area up to an ordinate drawn at a point, X, taken on the diameter. If OX is denoted by x it will be found that the area can be expressed as a function of x. Thus we may write : fydx = a function of X, say, /(X) Write : = jydx. J a We are about to show what is the relation between A and v. Proceed a step farther with the integration, up to the point, Xi. Denote XX 1 by <5X. The area will alter by the amount, XX^Y, where YY 1 means the arc of the curve between Y and Y 1 . Write this change = dA. Then the new area is A + dA = jydx. = area up to XY + XX^Y. But A = area up to XY. /. 8A = XX^Y. = Y x <5X in the limit, where Y is an ordinate in the strip, XX 1 , and in the limit we shall suppose the strip to possess a vanishingly small width. _. T . dA dA We have used X and Y to denote the variables for convenience, but we have not specified any particular values for them ; they 22 ADVANCED PRACTICAL PHYSICS FOR STUDENTS denote any abscissa and corresponding ordinate so that we may just as well write : _dh y ~ dx Thus y is the differential coefficient of the integral which has been denoted by A. The problem thus resolves itself into determining A if we know its differential coefficient. The occurrence of the arbitrary constant is seen to be connected with the choice of the starting point. We may summarize this by stating that if : = lydx then y =^ J dx There is no general rule for passing from y to A, i.e. from a function to its integral. We can only perform the operation by recognizing a standard form, and many devices have to be studied for the reduction of forms not directly recognizable, to more familiar ones. For these the student is referred to textbooks on the Integral Calculus. Reference should be made to tables in Mathematical textbooks for the important standard forms. 10. Evaluation of a Definite Integral Suppose that A has been determined in this way, and that it is now expressed as a function of x. Then A = lydx = f(x) + constant. = /(*)+B(say) B is the constant which it is necessary to add for the reason explained above. B depends on the arbitrary starting point so that there is some connexion between B and a. Write x = b, i.e. perform the integration up to a fixed point b. Then fydx = /(&) + B. If we write x = awe have lydx. This must vanish, for it means the calculation of an integral DIFFERENTIAL AND INTEGRAL CALCULUS 23 of no extent, or referring to our illustration it indicates an area of no width. .-. fydx=f(a) +B = o. /. B = -/(a). This is the relation between B and a. Hence : fydx = f(b) - f{a). The rule is therefore : Find the function of which y is the differential coefficient and express it as a function of x. Substitute in it the values of the upper and lower limits and subtract the results so obtained. /•io xHx. # 2 is the differential coefficient of e.g. -3 3"' 10 IQ 3 j% x*dx= ^- = 333- A I. * sin xdx. sin x is the differential coefficient of — cos x. IT .-. P sin xdx = [" — cos x\ = - cos ^ - ( — cos o) = 1. This measures the area of the part of the sine curve from the origin up to its maximum ordinate at -, lying above the x axis. 11. An important application of the Integral Calculus is to the calculation of moments of inertia. (Chapter II.) A body is considered as made up of a large number of small masses 8m. Let r denote the shortest distance of dm from an axis. Then the product r 2 8m is called the moment of inertia of the particle 8m about the axis. If all the particles of the body are taken into consideration and we make the summation Er 2 <5m for the whole body, we obtain the moment of inertia of the body. Thus Sr 2 <5w or IrHm is the moment of inertia of the body about the axis. 24 ADVANCED PRACTICAL PHYSICS FOR STUDENTS If we have an area A divided into small elements, da, each of which has a corresponding distance r from some axis, we require sometimes the value fr 2 da, and this may be called the moment of inertia (shortly M. of I.) of the area about the axis. Such an expression occurs in the treatment of bending of beams, and will be employed in the calculation of Young's Modulus by bending. As an example of the calculation of a M. of I., consider the case of a thin rod of total mass M and length zl. Let its mass per unit length be m, so that M = ilm. Assume that the axis is through the e.g. and perpendicular to the rod (fig. 10). Z Ac - — x— \ G 1 * Fig. io The form of section is of no consequence, but the ends at A and B are perpendicular to the length. The element of length dx at distance x from GZ will have a mass mdx and its moment of inertia about GZ is mix • x z . Thus the total M. of I. = Jmx 2 dx the limits indicating between what limits x extends. Now CmxHx = [y^] =| ml* = *m(2t)l 2 = —.... (4) The application of integration to the calculation of moments of inertia is extended in Chapter II. 12. Oscillatory Motion .Many experiments make use of the fact that in a number of cases bodies slightly displaced from their position of equilibrium perform periodic isochronous vibrations about that position. Examples of this occur in the case of the simple pendulum, in cases of torsional oscillations, and movements of galvanometer needles. DIFFERENTIAL AND INTEGRAL CALCULUS 25 We usually require the complete period of the oscillation in making calculations. Such motion is best and most simply treated by means of the Differential Calculus. A simple harmonic vibration is by definition one in which the body moves so thajt it is under the action of a force tending to restore it to the position of equilibrium, the magnitude of the force being proportional to the measure of its displacement. Thus, if a point is moving to and fro in a straight line about a position O, and is performing S.H.M. it is always under a force directed towards O and proportional to thd distance OP ( = x). d 2 x But the acceleration is -j^' and if we write,: force = — kx where k is constant and the negative sign denotes the direction of the force, we have by Newton's Second Law of Motion d 2 x . m M = -**" y It is usual to write this in the form : dH dt* where * 2 = — ■• r m This is an example of a differential equation. We do not here consider any series of arguments leading logically to the solution of this equation, we merely state the solution and verify the truth of the statement. This equation and another slightly more complicated are so important in Physics at an early stage that the solutions should be remembered and the student prepared to apply them with ease. Consider x = A sin (j>t + a) (6) A and a are arbitrary constants not occurring in the equation. They occur for the same reason that B occurred in § 10 ; in fact, we are proceeding from a differential coefficient to the function from which it is derived and so are performing an integration when solving the above equation. It should be remembered also that in the complete solution of a differential equation there must occur the same number of arbitrary constants as the number of the order of the highest differential coefficient in the equation. In the present case the highest order is 2, and so we have A and a. ■[£•+#* -o. (5) 26 ADVANCED PRACTICAL PHYSICS FOR STUDENTS dx d 2 x 57 i - = -^ 2 Asin(^ + a) Now ~£ = pA cos (pt+«), ^ = —p 2 x. Hence the value chosen for x satisfies the equation. Since x = A sin {pt + a) we shall have the same value of x occumng T sees, later when a complete period has elapsed. /. x = A sin [p(t + T) + a] = A sin (pt + a). .\ sin (pt + a + pT) = sin (£* + a). This is true if pT = 2*, 47c, etc. Thus the first recurrence of the value of x is after an interval T = 2*/p, and this is the period of the S.H.M. described bv our equation. We note that T = 2tu -f- square root of the coefficient of x in the reduced equation / 7 \ O ^9 Fig. 11 Thus in considering any problem in which T is required we have only to write down the equation of motion and but it in this form ; we can then write down the value of T immediately We consider the case of the simple pendulum (fig. 11). The displacement is measured by the angle 6. The force along the tangent to the circle described by P is ?fx Sm - e ? T ™ 8 ° if 6 is smaU - Note that the m °tion is only b.H M. if 6 is small, since the force is only then proportional to the displacement. The acceleration in this direction is ~ dt 2 where s = arc OP. or since s = Id, it is / dH dt 2 DIFFERENTIAL AtiD INTEGRAL CALCULUS 27 The more complicated case occurs when the effect of friction has to be included. The force of friction depends on the sum of a number of terms proportional to the powers of the velocities. In slow motions, such as those occurring in the movements of a galvanometer needle, the term of paramount importance is that containing the first power of the velocity. As this discussion is for the purpose of considering oscillations such as occur in galvanometers we consider this as a typical example. The position of the needle or coil is defined by its angular displacement from a normal position. For example, let OH denote the direction of a magnetic field in which the needle normally sets. mH A 1 "^J 3 A B - — H Fig. 12 In the new position, A 1 !* 1 , let measure the angle BOB 1 . If is small, the restoring couple due to the field H is 2mUl sin = 2mMd, where / = half the length of the magnet. If the magnetic moment be M this couple may be written MHO. m denotes the pole strength. If I denotes the M. of L of the needle, neglecting friction we have : !»?--»=• <9) and the motion is simple harmonic. We assume in accordance with what has been stated above that there is a frictional force proportional to the angular velocity 4i and write the force = c -& where c is constant. at at The equation then becomes : T d 2 d dd __„ , v and may be simplified by dividing throughout by I and changing the notation of the constants^ 28 ADVANCED PRACTICAL PHYSICS FOR STUDENTS It will now be written : dH , „ dd , , a . . — + K Jt+ n*6 = o. .. (ii) The derivation of this result has been made by reference to a magnetic needle oscillating in a magnetic field, let us say, supported by a thread of negligible torsion. The same result is obtained for such oscillations as occur in the case of bars vibrating at the end of stretched strings under torsion or for oscillations in the electrical discharge of a condenser. We shall denote ~rr by 0" and ^ by 0*. Again, we do not attempt to deduce the solution of the equation, we merely verify a stated result. The solution of such equations can be obtained by writing 6 = Ae mt for on using this value of the equation is satisfied provided m* + Km + n 2 = o. This is evident if we substitute = m 2 Ae mt , = mAe mt . „ — K ± VK 2 — An* Hence m = 3— = m x or w 2 , say. Thus the complete solution is : = Ae m i* + ~Be m 2*. A and B are the two constants necessary in the complete solution. If we substitute the values of m x and m 2 we find : e = e iAe 4 +Be * 1 (12) K 2 When — > n* the indices of the exponentials in the bracket are real and is not periodic. merely changes in value with t according to the exponential law. Of course K and n depend on the particular problem considered. K measures the friction and n depends on the nature of the restoring force. If friction were not present the body would vibrate with period — »so n that this may be said to be the natural period of vibration if there is no friction. The case with which we are concerned is when the motion is oscillatory, and this requires that the indices of the exponentials within the bracket should be imaginary, i.e. £1 <n i m 4 Write * = V — 1, we then have for the terms in the bracket * DIFFERENTIAL AND INTEGRAL CALCULUS 29 By altering the constants we can put this in the i orm : C sin or more simply : >p K 2 .*+ Dcos yjn^- 5! 4 t, Esin(^J« a - — -* + <x) ■(13). Thus the solution is now to be written : 6 = Ee ^sin [yjn* - — • t+ a) where E and a are the constants. \ The student may, if he wishes, take 6 as given by this value, substitute in the equation and verify that this satisfies it. This form of the solution shows the similarity with the last case in which we had == A sin {pt -f a). _Kt_ The amplitude is now Ee a instead of A. It thus varies — K£ with the time, and since K is positive, the value of e a is less than unity, and the amplitude diminishes with the time. The motion is said to be damped, and the damping depends upon K. does not maintain the same amplitude during the motion, as in the undamped case. The curve illustrating the motion is drawn in fig. 13. B = Ee a where t has t — Fig. 13 When t = o, O = E sin a, and when t = - -r- \\ n 2 — -— - — a, 2 \ 4 this particular value. Other pairs of values (0, t) may be obtained in the same way, and the curve drawn as above. It will be noted that the curve crosses the 2-axis at the points AxA 2 . . . and that these points are equidistant. The interval k x k z is called the period of the motion. 30 ADVANCED PRACTICAL PHYSICS FOR STUDENTS It will be noticed that 6 is zero when sin Ujn »- — * + a) vanishes. For simplicity write q =» J n * _ J£ a The n sin (qt+ a) =o if' j* + a = o, n, 2tt, 3tt, etc. that is when t = - - n ~ a . 2 * ~ a » 3* - « ^ . ?' <1 ' q q ' * On the curve these times correspond to the points A, A v a 2 , A3, etc. In the case of point A x the oscillator is at the zero position, but is moving in a direction opposite to that at A or A 2 . The difference between the above values of t at A 2 and A 3 is 3 TC ~ « _ " — a _ 2n ? q q' Thus the period is L« _ j^2 . . . , / x >) Had there been no damping, the period would have been —. We shall call these T x and T respectively. At such points as B,, B 2 , B 8 , etc., the value of sin (qt + a ) is numerically equal to unity, and the amplitude is then measured by Ee 2 with the appropriate value of t. At B 2 the value of t is^- a and at B a half a period later the value is — - a + -T, = ^- - a. 2? ^2 x 2.q The corresponding values of sin (qt + a) are +1 and -1 • this means of course that the displacement is on the opposite side of the mean position in the second case. Thus SA =tf +-^ X>2 < -'2 In the same way B 2 C 2 JSIi B3C. B£~r e * = B^ = etc (15) Thus the ratios of successive maximum displacements are constant. KT The value-— 1 is denoted by x which is called the logarithmic decrement. CHAPTER I MEASUREMENT OF LENGTH, AREA, VOLUME, AND MASS. The measurement of the length of a body may be made by using one of the usual vernier or micrometer devices, such as the vernier calliper, the micrometer screw gauge, or the spherometer. The measurement of small objects may also be carried out by use of the travelling microscope or the micrometer microscope. In using either of these instruments care must be taken, when viewing the image of the ends of the object, that this image is in the same plane as the cross-hair or the small scale in the eyepiece, otherwise parallax errors may be introduced. The image seen should not show any relative movement with the cross-hairs when the eye is moved across the field of view. When focussing the cross-hair, as a preliminary adjustment, the eye should be unstrained. The microscope is turned to a bright distant object and the adjustment of the eyepiece should be made so that the distant object viewed by the one eye is in focus when the cross-hair as viewed by the other eye is also clearly focussed. The Comparator When a length exceeds a few centimetres, the travelling microscope or micrometer microscope are not used individually as measurers, but are replaced by an arrangement of two such instruments arranged at a variable distance apart on a fixed graduated bed. Each microscope may be moved in the usual manner in a direction parallel to the length of the bed. Each microscope is provided with a scale or fine cross-hair in the focal plane of the eyepiece. Each eyepiece is adjusted so that the scale or cross-hair is in sharp focus for normal vision, and is replaced in the carriage. An object to be measured is fastened rigidly along the bed, in the groove provided for it. The image of one end of the object as seen by the first microscope is brought into coincidence with the image of the intersection of the cross-hairs and the second microscope is moved until the image of the other end on the object coincides with cross-hair intersection in that microscope. Thus the images of the two ends of the object as seen by the 31 32 ADVANCED PRACTICAL PHYSICS FOR STUDENTS two microscopes normal to its length are formed at the inter- section of the two pairs of cross-hairs. The object is removed and a standard rule substituted. The readings of the standard scale seen opposite the two cross-hairs obviously enables the length of the object to be ascertained by subtraction. As an example of this Substitution method we will consider the experimental details of the following experiment. The Comparison of the Yard and the Metre Standard rules, engraved as finely as possible with inch and centimetre graduations, and of lengths one yard and one metre, are employed in this experiment. Place the yard rule on the bed of the comparator and focus the one microscope on a scale division near one end of the yard, moving the latter and the microscope until this is accomplished. Then move the second microscope until a scale division near the other end of the rod is sharply focussed : the number of inches included between the two being noted — 36 if the scale is sufficiently well graduated to allow of this. Then, taking care not to upset the arrangement of the microscopes in any way, remove the yard scale and substitute the metre, so that the graduations are in good focus. Move the scale so that the image of a division near one end is in coincidence with the cross- hair intersection in the one microscope. Under these circum- stances the second microscope will not be opposite a division. Coincidence of the cross-hair and the image of a scale division is brought about by a movement of the microscope, parallel to the length of the bed, which is measured on the vernier scale attached to it. The size of the gap between the two cross-hairs ± the move- ment of the microscope is then read off in cms. Care is taken in noting the movement of the microscope to see exactly the unit used in these graduations. In this way we obtain two measurements, one in each system, for the same distance and may calculate the number of inches to the metre, or cms. to the yard. The vernier scale movement on the microscope is often replaced by a micrometer screw capable of a much shorter range of move- ment. With such a screw traverse, the movement of the micro- scope may be readily measured to «ooi of a cm. Using sucli a comparator the values given below were obtained : (1) z yard = 91-5 cms. + -008 inch. /. 1 metre s= 39/331 inches. MEASUREMENT OF LENGTH 33 (2) 50 cms. = 19*6895 inches, i.e. 1 metre = 39*379 inches. (3) 1 metre = 39*375 inches + -002 inch = 39'377 inches mean value 1 metre = 39*362 inches. ! The Planimeter The estimation of the area of a plane figure may be carried out by one of the many geometrical methods or by the use of a planimeter, an instrument designed to measure such areas directly. / Of this class of instrument the Amsler planimeter is generally used. Fig. 14 It consists, essentially, of two arms AC and EB, hinged at A, fig. 14; AC is of fixed length and is provided with a needle point loaded above by a small weight as shown in the diagram. The second arm may be varied in length by sliding that portion of it which carries the tracer, B, into the slot provided in the other half, EA. By means of fine adjustment, S, the length BE may be set accurately at any division along the graduated face of BA. In addition to the tracer, B, this arm is provided with a small wheel to which is attached a graduated drum which moves past a fixed vernier scale, V. By means of the graduated drum and vernier, the rotation of the wheel may be measured to Tff ta of a complete turn. The axle of the wheel and drum is arranged parallel, to the length of EB and is provided with a worm gear, which moves a horizontal indicator, D, one division per revolution. When placed on a plane the instrument is supported at three points : the needle point C, the point of the tracer B, and the point of contact of the wheel with the plane. 34 ADVANCED PRACTICAL PHYSICS FOR STUDENTS To measure an area the needle C is placed at a point outside it, such that B may be moved round the boundary of the area. For the measurement of large areas this will be impossible, but the details for such a case will be seen later. Starting at any point on the boundary of the area, the tracer is moved carefully over its contour until it is finally in the starting position. During this operation the wheel will have rotated in general a definite number of revolutions plus a measurable fraction of a revolution. From this observation, the area of the figure may be calculated. The method of calculating the area will be best understood by first considering the theory of the instrument. — X Fig. 15 When the needle C is fixed in the plane of the area to be measured, any movement of B along the boundary of the figure will result in a movement of A along the arc of a circle with C as centre and radius CA = a cms. (see fig. 15). Further, the wheel, W, will roll a distance equal to the total displacement when the movement of EB is at right angles to its length : movement parallel to the length causes no rotation, the forces acting on the wheel due to contact with the plane, under these latter circumstances only produces a couple tending to move the axle parallel to its length about the pivots. So for any intermediate form of displacement, the rotation produced in the wheel will correspond to the component of the displacement at right angles to the length EB, and if n revolutions occur, 2ttwR will measure this normal displacement, R being the radius of the wheel. Now it will be shown below that the area £0 be measured is equal to 2imR x AB : since the distance of the wheel from A MEASUREMENT OF AREA 35 does not occur in this expression we must first show that this distance has no effect on the number of revolutions the wheel makes. In fig. 15, let B and B 1 be two positions of the tracer a very small distance apart on the boundary of the area to be measured, B B 1 K ; A and A 1 being the corresponding positions of the hinge. Let AB = b, and suppose that the centre of the wheel is at P (fig. 16). Draw AN 1 and PN normal to AfB' 1 from A and P and let AN 1 = ds : PN = ds' : AP = c. AL being parallel to A X B*, let the angle LAP = dtp. Fig. 16 Since BB 1 is a small distance, ds, ds', and d<p are also small. Now PN = NL + LP or _ ds' = ds -f c- dq>. ' The distance moved by P as B traces the boundary of the area BB X K is S ds' or S ds' = S ds + S cd<p. B finally returns to the starting point, and therefore 2 dtp = o i.e. S ds' = Sis. Thus a wheel placed at A would indicate the same movement as the one at P, or at any other point along EB as could be shown by the same process as above. The position oj the wheel on the arm AB does not, therefore, affect the reading of the instrument. To show that the area of a plane figure which does not include the needle point C is equal to (2imR)b let us refer the figures to rectangular axes with C as origin, as in fig. 15 • Let CAB be one position of the planimeter and CA X B X a second position such that BB 1 is a small displacement of the tracer. Let the area to be measured be BB 1 K. Referred to these axes let x, y be the co-ordinates of B. 36 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The area BB*K may be conveniently referred to polar co- ordinate r and 0, i.e. let CB be r, /_ BCX = then if /_ B X CB = dd, the area B*CB = \r • rdd = \rHB, i.e. S \rW = area of BB^ . (i) For as the radius vector moves round the figure on the boundary remote from O, the small area contains, in turn, each element of the area to be determined, + the external triangle from the boundary of the figure on the side near to O ; this latter area is deducted from the sum as the radius vector travels along this near boundary, for here 66 is negative. Now x = r cos y = r sin dx = — r sin • dd -f cos dr, dy = r cos • do + sin • dr whence xdy — ydx = r* dd. Or, the area of the figure, £ S rHB from (i) = \ S (xdy — ydx) (2) Let /__ ACX = a ; A*CA = dx ; AN X A J = 90 ; CA = CA 1 = a ; AB = A^ 1 = b. The co-ordinates x and y of the point B may be expressed as under : x = a cos a -f- b cos q>, y = a sin a + b sin <p. .*. dx = — a sin a dx — 6 sin 9? dg? d[y = a cos a da + & cos yip. .*. x dy — j/d# = (a cos a + 6 cos 93) (a cos a-da + 6 cos tp-dcp) -f (a sin a +b sin 9?) (« sin a da -f& sin 9>-ig> = a 2 da 4- & 2 dg? + ab cos (a — <p) d (a + 9) (3) But (a + <p) = 2 a — (a — 9?) or d (a + <p) — 2 dec — d (v. — <p). ,\ a& cos (a — ?>)*i (a + 9?) = a& cos (a — 9?) {2 da — d (a — ?>)} — 2 ab cos (a — • 9>) da — ab cos (a — 9?)-d (a — <p) and AN 1 = ds = AA 1 cos A X AN == a • d acos (a — 9?). so that aft cos (a — <p)-d (a + <p) = 2b' ds — ab cos (a — q>) • d (a — 9?), and equation (3) becomes * • dy — y • d# = a 2 dat.+b 2 d<p + 2b'ds — ab cos (a — ?>)• d (a — ?>). Now, when the trace moves round the curve, Sdy = o, Sda = o for the planimeter returns to the exact position of starting : also we have then 2 cos (a — 9) • d (a — 95) =0, for S^cos (a — <p) • d (a — 9?) « £sin ( a - g>) J MEASUREMENT OF AREA 37 But limits (i) and (2) are identical so that this = o (see page 22), i.e. S (x-dy — ydx) = E zbds = ibYds. Now Sis = 27twR, as already shown, and we saw in equation (2) above Z(xdy — ydx) = 2 x area enclosed in BB 1 K, i.e. 2 area of figure = 2& 2tcwR. /. Area of figure BB X K = b • 27mR. Thus to measure any area sufficiently small, C is fixed in the paper at a point outside the area and B is taken round the boundary. The rotation of the wheel is measured, as is the distance from the hinge to B, and the area is thus equal to the product of BA and 27mR. » To carry out the calculation, R and AB must be measured. The distance from the tracer point to the hinge may be estimated by holding the instrument parallel to a scale or squared paper, and estimating as nearly as possible the position of the axis of the hinge on the scale. The difficulty is of course in estimating the true position of this axis. The value of R may be obtained by measurement with a screw gauge, which must be used with great care, as the edge of the wheel is easily damaged by screwing up the gauge unduly. Another and safer way of finding R is to note drum-reading on the wheel, and move the wheel along a straight line ruled on paper, until the wheel has made several complete revolutions. The distance moved and the number of revolutions enable the value of 2wR to be measured directly. The two measurements described above cannot be made with very great precision, but are performed much more accurately in the construction of the instrument. The graduations on the arm BA, ' 100 d cm,' etc., are made in the construction, and signify that when this graduation is adjusted to the fixed mark, one revolution of the wheel corresponds to an area of 100 sq. cms. Thus, if the graduations are to be trusted, i.e. unless the instru- ment has been subjected to rough handling, the area of the figure is equal to«x (the number on the graduated arm opposite the fixed mark. The Case when the Needle is Inside the Figure. Now let us consider the case when large areas are to be measured, e.g. BEjE^ . . . En, fig. 17. The needle support is fixed at a central point C, and the tracer may then be made to trace the boundary line of the area. But in this case it will be seen that 2nriR • b does not give the true value of the area of the figure ; for in this case ~Ldq> = 2w and is not zero. 38 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Take a point B on the boundary, such that BA is at right angles to CW, the line joining C to the centre of the wheel, W ; and draw a circle with C as centre and CB as radius, shown in the broken line in fig. 17. Then if B is moved round this circle the wheel W will move round a second circle of radius CW ; the relative positions of the two arms remaining constant so that the axle of wheel is always at right angle to the radius CW. Thus, while B is moved round the broken-line circle, W is moving always parallel to its axis around the second circle. So that during the whole revolution the wheel will not rotate. That is to say, the circle BFjF^ etc., so drawn, is such that when traced by / E, =7 1 /^v *<> B 9 Fig. 17 the tracer, the wheel indicates zero movement. The area of the circle is u(CB) 2 and is quite definite in size, depending on the setting of the graduated arm. This is sometimes called the zero circle, or datum circle. If now we consider our area EjE^a . . . E 1]L and commence with the pointer at B : passing from B to E 2 via E x , B is moved outside the circle. W will move towards C, and a definite rotation in one direction is made by the wheel. We could bring the tracer back to B along the path E^B without altering the reading of W. This reading corresponds, in the way previously considered, to the area EaFBE!. However, having reached E 2 , continue along the boundary MEASUREMENT OF AREA 39 E2E3E4. To do this the tracer moves inside the zero circle and W will therefore move outside its circle, i.e. in the opposite direction to the previous movement. Similar movements occur round the figure. The planimeter therefore adds algebraically the area of the curve outside the zero circle. Having carefully noted the direction of rotation of W when B traverses such a part as E 2 , we can tell from the final reading of the dial D whether the figure is of less or greater area of the zero circle. Suppose n revolutions of the wheel are indicated and the arm is set at/the ' 100 □ cm. ' mark. If the indication of the n revolutions is in the same direction as the indication of the wheel when moving outwards, e.g. along BEiE 2 , the area^of the figure is 100 • n + nCB 2 sq. cms. if CB is measured in cms. W Fig. 18 The same difficulty as before is met with in finding CB. The instrument may be set along two lines drawn at right angles so that the point of contact of the wheel is at the intersection of the lines, and B and C are each on one line, fig. 18 ; CB is measured directly or WB and WC are measured and CB calculated. How- ever, the value of this zero circle is inscribed on a second face of the arm BA. Adjacent to the '100 □ cm.' mark is a number which gives a value of the area of the zero circle, not usually in sq. cms., but in revolutions. Thus, if in the case taken above, there are n revolutions in- dicated and the second scale gives m, as the equivalent area of the zero circle, the area of the figure is 100 (n + m) sq. cms. To become acquainted with the instrument and familiar with the method of using, draw several small regular figures, calculate the areas, and then find them, using the planimeter at different graduated scale-settings. Draw a circle of known radius, calculate the area. Move the tracer round the circumference when the movable arm is set at various graduations. Note the number of revolutions in each case. From the calculated area and the number of revolutions 40 ADVANCED PRACTICAL PHYSICS FOR STUDENTS observed find the area corresponding to one revolution, and so check the graduations. Repeat this process with several measured areas, and obtain a calibration of the instrument. Measure b and R and again calculate a value for the area corresponding to one revolution. This is not so accurate as the above method, but shows the value of the construction as described. Draw irregular figures on squared paper and find the areas by adding the squares. Compare these results with those obtained by the planimeter as calibrated. Measure the radius for the zero circle at each setting, and check the value of the graduated scale, by calculating the area and dividing by 2nRb. The quotient should agree with the graduations on the scale. The instrument may subsequently be used for area measure- ments when required, making use of the calibration if errors are found by these experiments. Note. When the boundary of the figure does not cut the zero circle the process is identical ; for, suppose ABCD be such a figure, and the zero circle is completely inside the figure as shown. The process is as before. Start at any point A, and, with the fixed point inside the figure, trace round the boundary in one direction, say along the path ADC. Join AC. Then, if the tracer be taken along CA the value of the area ADC may be calculated on the foregoing theory. If now the tracer be brought back along AC and thence via B to A, the wheel will indicate precisely the same as if the path ADCBA had been taken, for on reversing along AC the record on the wheel for the path CA will be neutralized. The foregoing theory shows that the sum of MEASUREMENT OF AREA 41 the readings for the two paths record the number of revolutions corresponding to the two areas, and is equal to that for the boundary of the figure. Hence, starting at any point and tracing the complete boundary gives a record of the number of revolutions n, which when added to the zero circle number enables the area to be evaluated. The Graduation and Calibration of a Tube Graduation. The method of graduating a glass capillary tube described below is one which could be employed generally in the etching of scales on glass. A length of glass capulary'tube is coated with a thin layer of paraffin wax, by warming the glass and applying a small block of wax to the heated surface. The waxed tube is again gently heated, in the Bunsen flame, and rotated so that when cooled it is evenly coated with a thin wax layer. The tube is then clamped to a board at such a height that the upper surface is approximately on the same plane as a metal scale which is fixed in line with the tube, and at the other end of the board. A beam compass is arranged so that the two needle points are from 50 to 100 cms. apart, depending on the length of the glass tube. One needle point is placed in a cm. graduation of the scale and the other needle point is drawn across the wax coating of the tube, removing a straight line of wax. The beam compass is moved 1 millimetre, and a short scratch again made on the wax coating. This process is repeated until the required length is marked in this way, the whole cm. marks and 5 mm. marks being made larger than the rest. By means of a steel point the cm. divisions, o, 1, 2, etc., are scratched on the wax coating. To etch this scale on the glass, a swab of cotton wool, fastened at the end of a stick, is dipped into hydrofluoric acid, and then applied to the wax-coated tube. Where the scratches of the graduations and numbers have removed the wax, the glass is etched by the acid. In this process care is taken, of course, to avoid any of the acid touching the skin or clothing. , One end of the tube is also given several scratches, and is covered with acid at the same time as the scale. After ten minutes or so examine one of these scratches near the end of the tube, by scraping away the wax : if the glass is sufficiently etched, the whole process is stopped ; if not, leave for a few minutes and then examine a second test scratch near the end, and so on until the etching is complete. Wash off the acid with 42 ADVANCED PRACTICAL PHYSICS FOR STUDENTS tap water and remove the wax, the last traces may be removed with turpentine or xylol. A little rouge, or lamp-black and shellac, rubbed over the etched scale brings out the markings. Calibration of the Tube. The inside of the tube is cleaned either by the usual method, using caustic soda, alcohol, ether and tap water, or by immersing it for about twelve hours in a solution of potassium bichromate and strong sulphuric acid (equal parts). The tube is finally washed well with tap water and dried. A thread of mercury, about one-third the length of the tube, is drawn into the bore, e.g. ip cms. for a tube 30 cms. long. By means of a small length of rubber tubing attached to one end of the graduated tube, the position of the thread may be varied at will by altering the pressure within the tube. The mercury is first adjusted so that one end comes near the zero graduation. By eye estimation the position of the other end may be determined to £ of a subdivision ; hence the length of the thread when in this position is obtained. The mercury is then moved, by gently blowing through the rubber tube, until it occupies the central third of the tube, and its length is again estimated in scale divisions. The length taken by the mercury is finally measured in a similar manner in the remaining third of the tube. The volume of mercury is constant, so therefore the length indicated will depend on the average cross-section of that part of the tube filled with mercury. Thus this pre- liminary test will show the general form of the bore. A short thread of mercury, about 1 cm. long, is next introduced into the tube, replacing the 10 cm. length. This is moved to occupy approximately the space between the o and 1 cm. gradua- tion. The tube is arranged horizontally on a sheet of mirror glass on the platform of a travelling microscope. The readings on the scale engraven on the glass tube, corresponding to the two ends of the thread, are taken by means of this microscope. For this, the cross-hairs in the eyepiece of the microscope are turned so that one is parallel to, and the other at right angles to, the length of the tube. The intersection of the cross-hairs is brought into coincidence with the meniscus, so that one cross- hair appears tangential to it. The vernier reading of the micro- scope is noted. It is then moved towards the middle of the mercury thread until a graduation on the tube is seen. The difference between the vernier reading under these circumstances and the last readings gives the distance between the end of the thread and the glass tube scale reading. The distance between adjacent scale readings on the tube is measured in like manner. MEASUREMENT OF VOLUME 43 From these readings the length of the mercury projecting beyond an engraved division on the tube may be calculated in terms of the graduation of the tube, and the length of the thread of mercury may be measured in these units. The mercury is moved to occupy the space between the i and 2 cm. graduations, and again measured. This is repeated along the length of the tube, and the result entered as in column 2 of the table. The mean value of this length is obtained, and in the third column the difference between the observed and the mean value is tabulated for each part of the scale (p. 44). From these observations we may calculate the correction to be applied at each part of the scale to convert the scale readings to the corresponding volume readings. Thus, if a tube is divided into 20 cms. the mean value of the thread length as calculated from column 2 gives the reading for all parts in a uniform tube which is 20 cms. long, i.e. the difference between the length of the thread between o and I as observed, and the mean value gives the correction to be applied to the scale reading to correct it to the true volume of thread equal to ^ of the total volume of the bore. For the bore between 1 and 2 cm. graduations, column 3 again gives the correction to be applied to this small part of the bore. To correct the total length from o to 2, to give -£$ of the volume, the sum of the first two terms in column 3 must be added alge- braically. So, to find the correction for any scale reading, the sum of the third column must be taken up to and including the difference term for that reading. This is done with the figures for a tube tested in this manner and the correction entered in the last column. With the corrections so obtained a correction curve should be drawn, plotting the correction as ordinate and the scale- reading as abscissae. This kind of calibration is often required in thermometry, where equal volume expansion of mercury is employed. In such a case the scale-readings will not be uniform. In calibrating the thermometer the instrument maker fixes a few points along the stem by comparison with a standard thermometer. This method automatically makes some allowance for the change in the cross-section of the tube. In the example taken the stem was divided into 1 cm. lengths for calibration : it is doubtful whether much is gained by such a small subdivision, but is used in the example above to bring out the principle involved. In practice, the use to which the tube is to be placed determines the number of points tested along the tube. With a thermometer 10 points along the length will be ample for most purposes, i.e. with a o° to ioo° thermometer, 44 ADVANCED PRACTICAL PHYSICS FOR STUDENTS the thread detached should be the length of io°C, or about 3 cms. This is described in greater detail later ("Thermometry "). POSITION OF THE THREAD IN THE TUBE LENGTH OF THREAD DIFFERENCE FROM MEAN CORRECTION — I I*040 —0*044 - -044 I — 2 I*O00 — *004 - -048 2— 3 I '000 — -004 - -052 3— 4 0-975 + *02I - -031 4— 5 0-950 + -046 + -015 5-6 1-000 — -004 + -on 6-7 0-980 + -016 + -027 7-8 I -000 — -004 + -023 8-9 I -000 — -004 + -019 9 — io 1-000 — -004 -f -016 10 — II 1*000 — -004 + -on II — 12 1-020 — -024 - -013 12—13 I-OOO — -004 — -017 13—14 0-980 + -016 — *00I 14—15 I-OIO — -014 - -015 15—16 0-975 + -021 + -006 16 — 17 I-OOO — -004 + -002 17—18 1-000 — -004 — -002 18—19 1-000 — -004 — *oo6 19 — 20 0-990 + -006 o-ooo mean 0-996. The Balance The balance, as seen in fig. 20, consists of two pans suspended by knife-edge supports, KK, from the ends of equal arms of the beam ST, which is pivoted on a central pair of knife- edges. The central knife-edge is made of agate and rests on small plates of the same material, and KK support such plates. The free and sensitive movement of the beam depends upon the sharpness of the knife-edges. It is therefore important that they should support weight only when in use. To release the knife-edges the central pillar Q supports an ' arrestment ' A. The fixed arm A carries at each end two points which fit into a pair of cups on the upper agate plates at K. When the beam is lowered the weight is taken from the knife-edges by this means. MEASUREMENT OF MASS 45 When releasing the beam, the latter should be in the horizontal position, i.e. the beam is only arrested when two outer knife- edges are opposite the supporting point of the arrestment. In such circumstances all three knife-edges are released with an absence of jolting. H.L.CRimTIU. Fig. 20 In the ordinary way the centre of gravity of the beam, etc., is just under the line of support. The lower the centre of gravity the less sensitive the instrument. The position of the centre of gravity may be varied by an adjustment of the 'gravity bob ' B. 46 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The two masses W may be adjusted to change the rest positions of the pointer which moves over the small scale at the base of the pillar Q. > A fuller account of the balance will be found in " The Theory of the Physical Balance," by J. Walker. In using the balance, several simple things should be remembered, viz., before using, dust the pans with a camel- hair or similar soft brush ; arrest the beam before changing masses in the pans ; use the rider to start oscillation. Never touch the pans or ' weights ' with the fingers, or place chemicals or wet vessels on the pans ! Final observations should be performed with the case closed. METHODS OF WEIGHING The Oscillation Method of finding the Rest Position of the Pointer The arrangement of knife-edge supports ensures that the friction is reduced to a minimum. Consequently, when the beam is oscillating it will have a long period of swing which is very slightly damped, i.e. the pointer moving just clear of the scale will take a long time in which to come to its final position of rest. This rest position may be estimated by the method of oscilla- tion as follows : Suppose the scale be graduated from left to right into, say, 20 graduations. The end of the pointer, being arranged as near to the scale as possible, is viewed by the eye or by the aid of a lens, taking care to avoid any slight parallax error. The position of the turning point should be estimated to at least J of a small division — if possible to T V of a division. Seven such turning points should be obtained — three on one side and four on the other side of the zero position. The mean value of the three on the one side and also the mean of the other four should be obtained. Then the mean value of the two means gives the position of rest. (See example in the table below.) For such damped oscillation the mean of an even number of observations on each side of the zero would give a value biased towards the side of the zero of the first observed turning point ; so an odd number is taken as stated. For if we assume that the damping is small and that the first swing through the rest position is to the left, we have, from a consideration of such a lightly damped oscillation, the angular deflection, 0£, the first T swmg to the left is after a time - and is (see page 28), 4 8l = e- A * where A is a constant, O the undamped oscillation. MEASUREMENT OF MASS This is, under the conditions of very small damping, 01 = O (i - A ~) to the left. The deflections are therefore : 47 TO THE LEFT OF ZERO TO THE RIGHT OF ZERO , (x_aT) ».(x-A3J) eo (x-A ? ) 4"^ mean O (i - A^) O (* - A^ i.e. the mean is o since the mean position left and right is an equal distance to each side of the zero. Zero Position of the Unloaded Balance As an example of the method of oscillation, the following determination of the zero position for the unloaded balance is taken. The beam is released from the arrestment and given a slight oscillation over, say, about 5 divisions of the scale on each side of the zero. After a few swings the oscillation should be steady, and seven readings, are taken, as under : READING TO THE LEFT READING TO THE RIGHT I. 8-8 3. 9-2 5- 9'5 7. 9-9 2. 12-0 4. II'l6 6. n-2 Mean of left hand reading = 9-35 Mean of right hand reading = n-6 ~ . . t , Q'^5 + H'6 Zero positions unloaded = ^-^ = 10-475 A mean value of three such determinations gives the rest point. In general, call this rest position for no load X. 48 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Sensitivity of the Balance The sensitivity of the balance is denned as " the angle through which the beam will turn for one milligramme difference in load in the two pans." It is usual in practice to measure the sensitivity in terms of the movement of the potater over the scale. This should be measured for no load, by adding one milligramme, by means of the rider, to one pan and noting the position of rest by the oscillation method as before. Load each pan with increasing equal masses up to the limits specified for the balance, and for each mass find the sensitivity i.e. change of rest position for one milligramme on one pan. If the beam were rigid and the knife-edges truly in the same line the sensitivity-load curve would be a straight line parallel to the load axis. For increasing loads there may be, however, a slight depression of the knife-edges at the pan supports, and a change in sensitivity as a result. Find the sensitivity-load curve for the balance and note the region, if any, of maximum sensitivity. The value of the load at which there is maximum sensitivity depends upon the use for which the balance is designed. Method of Gauss or Double Weighing As an example of this method we will consider a determination of the number of grammes equivalent to one ounce troy ; the pro- cess can naturally be repeated with any other unknown mass. Place the ounce troy in the left-hand pan and add ' weights ' (grammes) to the right-hand pan until the pointer remains on the scale when the beam is released ; the rest position of the oscillat- ing pointer is estimated in the manner previously used. Let this be y on the scale. y Now find the sensitivity of the balance at this load, by adding one milligramme and proceeding as before. Suppose the sensibility for this load is s. The mass of the ounce troy in grammes is the mass (to the nearest centigramme) in the scale pan + ( ^7*) m^S 1 "* 11111165 = M i» say, when x is the zero reading for no load. The ounce troy is then transferred to the right-hand pan and the process repeated by adding ' weights ' to the left until, to the nearest milligramme, M 2 , the mass as so compared is obtained. Suppose that the two arms of the balance are of slightly different length, the left-hand arm being a cms. and the right b. MEASUREMENT OF MASS 49 Then W being the true mass of the ounce troy, neglecting buoyancy, > aW = M x 6 «M 2 = Wft Hence W = VMA The difference between M x M 2 and W will be very small, so that we may take as an approximation W = i(M, + M,). Borda's Method, or the Substitution Method A second method, quite as accurate as the double weighing, is a simple method of substitution. It eliminates equally well the errors due to unequal length of the arms, etc. The ounce troy (the unknown mass) is placed on a scale pan and lead shot is used to counterpoise it. The position of rest when the counterpoise is complete is noted by the method of oscillation. The ounce troy is now removed and replaced by standard masses until balance is again obtained. From the sensitivity of the balance for this load we may estimate very readily to a milligramme the mass which has exactly substituted the ounce troy. Thus the mass is obtained, avoiding errors due to faulty construction of the balance, etc. Buoyancy Correction. When discussing methods of weighing'no account was taken of the buoyancy of the air on the ' weights ' and the mass to be compared with them. Suppose, as before, we find that W is the mass of the body, by one of the methods above ; the true mass allowing for buoyancy we will denote by M. Suppose that in the determination of W we used copies of standard masses made of a substance of density D. Let p be the density of the ' unknown ' mass and o the density of the air. We have really compared (1) (the true mass of the body M — the buoyancy on the mass) with (2) (the mass W of the ' weight ' — the buoyancy on the ' weights '). These two quantities are equal, (m-;..)-w-;W.. i.e. M =w(i -~\ a I -M' + -;-»--) 50 ADVANCED PRACTICAL PHYSICS FOR STUDENTS neglecting -= cf . i. = W + W ( I p-D> The observed value of W has therefore to be corrected by the factor W g -i).. This correction depends on the density of the ' weights ' and the substance and the density of the air. For most purposes the density of the air may be taken as •0012 grm./c.cm. and for general use a table may be calculated giving the value of the correcting factor I =- J o for the two common materials used in the manufacture of ' weights,' brass and aluminium, taking the density of brass = 8-4 grms./c.cm. Aluminium = 2-65 grms./c.cm. Thus : DENSITY OF SUBSTANCE WEIGHED (p) CORRECTION FOR BUOYANCY BRASS ' WEIGHTS 'D =8-4 ALUMINIUM ' WEIGHTS ' D =2-65 V p 2-65/ •5 •55 •60 etc. •00226 etc. etc. •00195 etc. etc. CHAPTER II MOMENTS O^ INERTIA AND DETERMINATION OF ' g ' Kinetie Energy of a Body Rotating about an Axis Let ABC (fig. 21) be a section of a body by a plane at right angles to the axis about which it is rotating, O being the point of inter- section of this plane and the axis. ,FlG. 21 If we imagine the body to be subdivided into a large number of very small particles of mass m x , m 2 , m 3 , etc., distant r v r 2 , r 3 cms. from the axis, it is evident that when the body rotates each particle will move with a velocity which depends on the distance r from the axis. Consider one such particle at P, of mass m and distant r cms. from O. If the body rotate with a uniform angular velocity w, in the direction of the arrow, and if v is the velocity of P in the path, we have v - = w. r The kinetic energy of this particle is %mv z = \mw % r i . For all such particles the total kinetic 'energy of the body is therefore ^MjW^r 2 + ^m z w 2 r 2 z + J^gze'Va 2 + . . . . i.e. Kinetic Energy = H\w 2 mr* = \w* Emr* The sum of such quantities as mr 2 , taking every particle throughout the body, is denned as the moment of inertia of the Si 52 ADVANCED PRACTICAL PHYSICS FOR STUDENTS body about the axis through O. If we denote this by I , then the kinetic energy of the body is ll wK.... (i) thus I replaces the mass, and tfre angular velocity replaces linear velocity in the corresponding case for linear motion where K.E. = %mv 2 . In the same way if the axis passed through the centre of gravity ; I being the moment of inertia about an axis through the centre of gravity, K.E. = $Iw 2 , 1 being of a different magni- tude from I . To express the moment of inertia of a body about an axis in terms of the moment of inertia about a parallel axis passing through tfee centre of gravity, we proceed in the following manner : In fig. 22 let ABC be a section of the body at right angles to either axis. Let G be a section of the axis passing through the centre of gravity and O be corresponding point of intersection for a parallel axis, the distance OG being fixed and equal to a cms. Fig. 22 Consider at any point P a small particle of mass m gm., OP being r cms. The contribution of this particle to I is mr 2 . Now produce OG to D, and from P drop a line perpendicular to OD meeting it at D. f 2 = OP* = OD 2 -f- PD 2 = PD 2 + DG 2 + GO 2 + 2OG • GD = PG 2 + a 2 + 2a • GD Thus I = Em (PG 2 + a 2 + 2a • GD) = SwPG 2 + Swa 2 + X2ma • GD. Now SwPG 2 = I, the moment of inertia about a parallel axis through the centre of gravity. Sma 2 = a 2 Zm = a 2 M, MOMENTS OF INERTIA AND DETERMINATION OF 'g ' 53 where M is the total mass of the body. Further, the expression 2aSGD«w = o, for by the definition of the centre of gravity the sum of the moments {GD-m) throughout the body, about an axis through G, is zero. Thus , . I = I+^ 2 m (2) Radius of Gyration. We have defined the moment of inertia of a body about an axis as 2#w 2 . Now if the whjole of the mass of the body were concentrated at one point distant K from the axis, we should have I - K 2 M. If the distance K were s o chosen that K 2 M = Emr 2 = 1 v - it is called the ' Radius of Gyration ' of the body about the axis of rotation taken. As with the moment of inertia K has different values depending upon the axis chosen. Moment of External Forces. Considering a rotating body as before, we may readily deduce an expression for the moment of the external forces applied to the body to impart a definite angular acceleration. Imagine a force applied to such a body as shown in fig. 21 Suppose the body to be subdivided into small particles as before, of which one at P has mass m and is r cms. from O. Then v, the velocity of the particles, is given by dd . v = r • -r. or rO at * the acceleration of the particle in its path is dv d*B ■■ dt= r aW ==rd this is occasioned by a force mr§, whose moment about O is mr 2 0. For the whole body to rotate with this angular acceleration the total external couple applied is thus Sw 2 = QXmr 2 = 10 (3) Thus the moment of external forces applied to the body is I'd. Calculation of the Moment of Inertia for a Solid about any Axis. The numerical value of the moment of inertia of a solid about any axis may be readily obtained by integration. Having calculated this value for an axis passing through the centre of gravity, the corresponding value for the case of the body suspended through a parallel axis may be obtained by adding the term Ma 2 as shown on page 52. 54 ADVANCED PRACTICAL PHYSICS FOR STUDENTS We will consider an example of such calculations which is often employed, especially in magnetism, and which illustrates the points already considered. Calculation of the Moment of Inertia of a Rectangular Rod about an Axis at Right Angles to its Length and passing through the Centre of Gravity. Let ABCD be the rectangular bar, with centre of gravity at G, and supported by an axis KK 1 passing through G, normally to the face AD, fig. 23. Let M be the mass of bar (assumed to be uniform) p the density of the material of the bar ol the length of the bar 2& the breadth of the bar 2d the depth of the bar. -y b Fig. 23 Let G be the origin of a system of co-ordinate, the z axis coincid- ing with the axis of rotation : the x and y axes being at right angles to this, the x axis being parallel to the length, the y parallel to the breadth. The most convenient method of finding I the moment of inertia about KK 1 is to consider firstly a very thin section of the bar cut at right angles to the x axis, and of thickness 3x. Through the centre of mass G 1 of this section, imagine an axis, LL 1 , parallel to KK 1 . TJie moment of inertia of this section about KK 1 is equal to the moment of inertia about LL 1 plus the product of the mass of the section and x 2 , where x is the distance between G and G 1 . Imagine a very small rectangular portion PQRS of EFHI, y cms. from the axis and of width dy (fig. 24). MOMENTS OF INERTIA AND DETERMINATION OF ' g ' 55 The mass of this parallelepiped is [dx dy 2cC\ p. The moment of inertia of the section EFHI about LL 1 is therefore fzddxdy gy* = 4^- p • dx. J -h 3 About KK 1 the moment of inertia is therefore ***'"** + 2d2b 9 xHx. 3 Hence the total moment of inertia of the whole body about the axis KK 1 is * 3 8bdlp pdx + 4 bd x (6 2 + I 2 ) '■pdx J M - j(b 2 + 1% EXPERIMENTAL MEASUREMENT OF MOMENT OF INERTIA When the moment of inertia of a body cannot be conveniently calculated it may be found experimentally by imparting to it a known amount of energy and observing the resulting rotation, or, if the body is small, the method of the moment of inertia tableonay be employed, whereby the change in moment of inertia in a given system, due to the body, may be directly calculated. The following typical" experiments will make these methods evident. Moment of Inertia of a Fly-wheel To find experimentally the moment of inertia of a fly-wheel about the fixed axis of rotation a mass is attached to the axle 56 ADVANCED PRACTICAL PHYSICS FOR STUDENTS of the fly-wheel by a cord which is wrapped several times round the axle. When the mass descends, it causes a rotation of the wheel. The mass in its descent loses a definite amount of potential energy. Neglecting friction for the moment, this loss is equated to the gain of kinetic energy of the mass and the fly- wheel, and an equation results in which all the terms are known or measurable except I, the moment of inertia of the wheel and axle about the fixed axis. Fig. 25 The wheel might be supported on a horizontal or vertical axis. The two usual types met with are seen in fig. 25. The process to find I is the same, so we will consider one of them — the vertical axis type. The mass m is attached to the axle at a point where there is either a hole or a pin. If there is a small hole in the axle as at P, then to the end of the cord a small ' pin ' is attached. This can be made from a short length of suitable-sized brass wire to fit easily in the hole, or if the axle has a pin projecting, a loop is made at one end of the string. The length of the string is so adjusted that when m is on the floor, or whatever solid object is to arrest it in its descent, the other end of the string may be just attached to the axle. So that when the mass descends, the moment it is arrested, the string leaves the axle. If w be the angular velocity imparted to the wheel, and r be the radius of the axle, the velocity of the mass m just before striking the floor is rw. So that, neglecting friction, we have mgh = fynrV -f £Ize> 2 (4) where h is the distance through which m has fallen. To measure the angular velocity, a chalk-mark is made on the circumference of the wheel, in a position which can be seen the moment the mass touches the floor. The number of revolu- tions, n, made by the wheel after the mass becomes detached is counted by observing the chalk-mark. The time taken for the wheel to come to rest whilst completing the n revolutions is also observed (t sees.). MOMENTS OF INERTIA AND DETERMINATION OF ' g ' 57 The wheel is finally brought to rest by the frictional forces acting against it. If this frictional force is constant, the wheel is uniformly retarded. It commences with a definite angular velocity and finishes with zero angular velocity, so that the initial velocity is double the average velocity. Now the average angular velocity = —7- radians per second, Ann i.e. w —- L -r* The linear velocity of the mass m is - — Another method, though inferior to the above, is to observe v directly by timing the descent of the mass. If the mass descends the distance h in t 1 sees, the average velocity is — ' and the final 2h velocity is -pr- As we have noticed above, the frictional forces are not always negligible, so that, for a more accurate determination of I, allowance must be made for the energy lost in overcoming friction. Let there be n x revolutions of the wheel during the descent of the mass, and let / ergs be the energy per revolution used in overcoming the frictional forces, then the total energy expended in this way is n-J. mgh = fwu 2 + \lw 2 + nj. Now we already know that the energy possessed by the rotating wheel, \Iw z , is used up in overcdming friction in n revolutions, i.e. fn = %lw 2 1 f = ^Iw\ n i.e. mgh = \mrhiQ*+ |I*Wi +— M (5) Experimental Details. Arrange the cord round the axle so that throughout the whole of the unwinding the cord from the axle to the pulley, T, is practically horizontal, or at right angles to the axle. Bring the mass m so that the bottom of it is level with a fixed point, and the string of such a length that it fulfils the conditions already stated. The distance h from the fixed point to the floor is directly * measured. The number of revolutions the wheel makes whilst the mass is descending may be determined by making a chalk- 58 ADVANCED PRACTICAL PHYSICS FOR STUDENTS mark on the axle and allowing the mass to descend slowly, counting the number of revolutions (%) during the descent. The mass is once more wound up and allowed to fall freely. When it is heard to strike the floor a stop-clock is started and the number of revolutions of the wheel before being brought finally to rest is counted, i.e. n and t are observed : m the mass is known and w = ^-7—-, hence the value of I is calculated by the aid of equation (5). The experiment is repeated two or three times with the same mass and the mean value of — taken. I is further checked by repeating with two other masses m 1 and m 2 . The cord used should be of small diameter compared with the diameter of the axle, otherwise the value of r in equation (5) is the sum of the radii of the axle and cord. Rolling Bodies The two following experimental methods of finding the moment of inertia of a body about a given axis depend upon observations of rolling bodies. The energy of a rolling body may be very simply obtained. Consider, for example, a cylinder rolling with a uniform linear velocity v cms. per second (fig. 26). AB, the line of contact of the cylinder and the plane on which it rolls may be regarded as the momentary axis of rotation of the cylinder. The kinetic energy of the body is therefore given by £ze> 2 (moment of inertia of the cylinder about the axis AB) where w is the angular velocity of rotation. Now, if a is the radius of the cylinder, m its mass, and I the moment of inertia about a parallel axis through the centre of gravity; the kinetic energy is : \ w 2 (I + ma % ) = |lrc; 2 + \mv* (6) MOMENTS OF INERTIA AND DETERMINATION OF ' g ' 59 That is, the kinetic, energy is equal to the sum of the kinetic energy of rotation and translation. Wheel and Axle on an Inclined Plane The moment of inertia of a wheel and axle about an axis passing through the centre of gravity and parallel to the axle may be obtained by observing its descent down an inclined plane, and applying equation (6). W 7& n>£acm3. /// / / $ m Fig. 27 The method of rotation will be apparent from a consideration of the section, fig. 27. R and R^are rails supported on a hollowed inclined plane. The axle of the* wheel rests upon the rails. The whole plane may be inclined at any angle to the horizontal. For each inclination the wheel and axle is allowed to roll down a measured length, I cms., of the plane, in a time which is measured by means of a stop-clock (t sees.). terns' Fig. 28 If the vertical distance between the starting position and the finishing position be h cms., fig. 28, and v be the final velocity acquired in the descent, we have, equating potential energy lost to kinetic energy gained, mgh = %mv 2 + £1 M (7) where m is the mass of the wheel and axle, and a the radius of the axle. The body starts from rest and moves with a constant accelera- tion ; the final velocity is therefore twice the average velocity. This latter is equal to -, i.e. v = —• The plane is adjusted by suitable means to one fixqd inclination. The wheel and axle is placed at a convenient marked starting 60 ADVANCED PRACTICAL PHYSICS FOR STUDENTS point on the rails. The position of the centre of the axle is noted by means of a vertical-reading simple cathetometer. The position of the centre of the axle is also noted when the wheel is against the stop at the other end of the plane. The length I along the plane between these two points is measured directly. The value of the mean time of descent for three experiments is obtained. The mass m is also obtained by means of a spring balance or an ordinary balance which is capable of weighing such a mass. a is obtained in the usual way by means of vernier callipers. 2.1 Hence, from (7), substituting the value - for v, 'ghP I^ff-x) The experiment is repeated for several values of h and the mean I is obtained. A D B 1 w H-H* , Fig. 29 Moment of Inertia 0! a Disc Supported on Strings A disc, usually made of wood, is suspended by means of a metal axle on two strings, as shown in fig. 29. The string is wound evenly on the axle AB on both sides until as much string as possible is wound up. If now the axle and disc, of mass m grammes,' is released, it will descend until the whole of the cord is unwound ; it will then rise again due to the string being wound on the axle in the other direction. Suppose that from the starting point to the lowest point reached the distance the centre of the axle moves is h cms., and that the linear velocity at the moment when all the string is just unwound is v cms. per sec, then, if r is the radius of the axle, we have as the energy equation (equating potential energy lost to kinetic energy gained) : mgh = %Iw* + \mv*, MOMENTS OF INERTIA AND DETERMINATION OF ' g ' 6l I being the moment of inertia about an axis passing through the centre of gravity and parallel to the axle. v w the angular velocity at the lowest point ^s = -, i.e. mgh —-S-j- + fmv 2 . I = (mgh - frnv 2 )^- If the time of descent of the disc is t seconds, the average velocity is - cms. per sec. 2.h v, the final velocity is therefore — . i — '*(£•- *) Experimental Details. Weigh the disc, then measure the distance between the position of the centre of the axle in the starting position and the final lower position. The value of r is equal to the sum of the radii of the axle and the cord which supports it, unless the cord has small radius compared with the radius of the axle. These radii are measured by means of a micrometer screw. The cord is wound evenly on the axle until the disc is at the starting point. Care is taken to ensure that the axle is horizontal, otherwise the disc fouls the cords in descent. The time of descent is measured several times by means of a stop-clock, and the mean value taken. The Bifilar Suspension In order to determine the moment of inertia of a body about an axis passing through its centre of gravity, we may make use of a bifilar suspension of known or measurable dimensions. The body is suspended with the axis of rotation vertical, and the time of vibration of the system, T, obtained by observing the time of 40 or 50 complete swings. If, for example, we wish to determine the moment of inertia of a cylinder about an axis through the centre of gravity, the cylinder is supported in two wire stirrups CE and DF (fig. 30), which hang at the ends of two very thin wires AC and BD, which are fixed at A and B, AB being 2d cms. The distance between C and D remains fixed and equal to 2d 1 cms. When the body is displaced slightly in the horizontal plane 62 ADVAN£ED PRACTICAL PHYSICS FOR STUDENTS it will perform oscillations whose periodic time T may be ascer- tained as already shown. If m is the mass of the cylinder and / is the length of the wire AC or BD, we may readily see that t = 2x /te: ^ mgdd 1 For, let the tension in the string be t dynes and O be the mid- point between A and B. j — i 3 Fig. 30 When viewed from above, fig. 31 (a) represents the relative positions of the four ends of the wires. When displaced, the state of affairs is seen in fig. 31 (b). Where A 1 and B 1 are projections of A,B, fig. 30, on the horizontal plane through CD, and C 1 , D 1 are the displaced positions of C, D. Consider the forces at D. Due to the tension on the string there is a force t, which has a horizontal value t cos a, where a is the angle between the string BD 1 and the horizontal. Now B^ 1 cos a = — - — » for, in reality, the point of suspension, B, is above B 1 , BB^ 1 being a right-angled triangle. So that along D^ 1 in the horizontal plane there is a force /-«^ (8) Of this force the component at right angles to the displaced body, i.e. C^ 1 , is effective in restoring the body to its original position, the component along the direction OC having no turning moment. From B 1 draw B*E at right angles to C 1 D 1 . The component MOMENTS OF INERTIA AND DETERMINATION OF *g ' 63 EB 1 normal to OD 1 is / sin EDW =%i^r = / 1 ' sa y» from (8) f 1 = EB 1 '- (9) A similar force acts at C 1 , constituting a restoring couple of moment/ 1 - OD 1 . The restoring couple is, substituting value of f 1 from (9), EB 1 * OD 1 . A O. 2d- 1T D (a) (b) Fig. 31 If the small angle of displacement, D 1 OB 1 = 0. EB 1 = OB sin = d sin 0. OD 1 = 2d 1 . 2dd x /.Moment of the couple = —j- t sin 0. is usually made very small, so we have for the value of the restoring couple: 2dd 1 mg ^ I = **%%. e (10) V wis, for when d is not very different from d 1 1 =£= — . We have already seen (p. 53) that the moment of external d 2 6 forces acting on a suspended system is I -^ ; this quantity is equal and opposite to the restoring couple, i.e. T d*d _ mgdd 1 dt* I This will be recognized as the equation of simple harmonic motion (p. 25), as we have the angular acceleration = It— X the angular displacement, ° . being a constant. 64 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The time of vibration is therefore 27C / II f TJ \ It is interesting to note that for bodies of the same dimensions and of uniform density the value of T is the same. For, let p be the uniform density, then I = Ewr 2 . Consider the body divided into small volumes v, : Smr a = Xvpr 2 = pE vr % . Also m = Spv = pSv —tt£ ♦ » i.e. independent of the density. The above experiment may be carried out using a metal and a wooden cylinder as the supported body. If the dimensions of the wood and metal cylinder are practically identical, the time T will be found to be the same within small limits. Suspend each in turn and find T by timing 50 swings, or by the method of page 118, and, from the formula above, calculate I. For such a regular solid an independent calculation gives a second value of I, which should agree very nearly with that already obtained. /ja *2\ For a cylinder, about the axis taken, I = M( 1 — J where 2L is the length of the cylinder, r the radius. The formula may be further tested by varying d, d 1 and I. It will be found that T 2 is proportional to t-^. Moment of Inertia Table Fig. 32 shows the essential features of the moment of inertia table. AB is the table suspended by a fairly stout wire, K, from the overhead frame. The circular table supports three or more masses which just fit into a groove, concentric with the circum- ference. When the wire support is vertically above the centre of gravity the masses, W, may be moved round the groove into any position without altering the moment of inertia of the whole. The masses are arranged so that the table lies horizontally. In that case the axis is through the centre of gravity and is normal to the surface of the table. If now the table is given a slight twist, a restoring couple is called into play in the wire, equal to, say, t per unit angular displacement, and the result is oscillations about the axis of support, of periodic time T = 2 ,yx ^ T where I is the moment of inertia of the system about the axis MOMENTS OF INERTIA AND DETERMINATION OF ' g ' 65 of rotation ; for) suppose the table be twisted through an angle 0, the restoring couple due to torsion is t0. We have already seen, page 53> that the moment of external forces on such a rotating body is 10" where 9 is— 57a • This couple is opposed by t0 : t0 and 10 are equal and opposite ; i.e. I 9 = - T0 T0. T fQ or = £ K This was shown in the introductory chapter (p. 25) to repre- sent simple harmonic motion whose periodic time, T is given by T= /-=V- yi If now a regular shaped body is placed symmetrically on the table at C, so that its centre of gravity is vertically above the centre of gravity of the table and therefore in the previous axis of rotation, the time of oscillation for the loaded table is \= 2rt y (i+k) where k is the moment of inertia of the regular body about the axis of oscillation : k may be calculated directly from the mass and dimensions of the body. 5 66 ADVANCED PRACTICAL PHYSICS FOR STUDENTS We have, therefore, from the two equations above : 8 _ 4" 2 (I + k) . Ts _ 4 * 2 (I) „ T, a I + k . k Hence -^ = —~- = i + j> or I = k T 2 _ Ta (12) If, then, a bcdy of unknown moment of inertia about a given axis is placed centrally on the table with the centre of gravity in the axis of oscillation, we may find the value of I lt its moment of inertia abbut this vertical axis passing through the centre of gravity, by timing the oscillations of the table when loaded by the body. If T 2 is the time of complete swing, Ts _ „ yr±n, and we have t =2«y-» i.e. from these two equations : T a T2 i. = il -r^' 1 (I3) Substituting, from equation (12) above, T 2 T* T2 T 2 T2 x i Y 2 x a — T 2 ~ T 2 — T 2 ^™' Experimental Details. The time T for a complete swing of the table is obtained by timing as many swings as possible, or by the method of p. 118. To find the value of I, the moment of inertia of the unloaded table about the wire as axis, a regular solid, such as a plain cylindrical ' weight ' is employed. The mass should be fairly heavy so as to cause as big an alteration in T as possible. A two-kilogramme ' weight ' is of the order to employ with the apparatus described. To ensure that this standardizing mass is arranged with its centre of gravity over that of the table, the lead weights W should be adjusted so that the table swings horizontally. Any alteration in the position of the masses, W, will not alter I, so long as the table is horizontal and the axis of oscillation is vertically through the centre of gravity. Having obtained T and T v using a cylindrical regular ' weight/ MOMENTS OF INERTIA AND DETERMINATION OF ' g ' 67 the value of k should be calculated. For such a flat cylindrical object, k = — - where a is the radius and M the mass of the ' weight.' By equation (12) I is calculated and the table standardized. Other regular solids may now be used and I, obtained by equations (13) or (14), and then by calculation a check is obtained on I x as previously obtained. The following results were obtained in the above manner: Table unloaded Table loaded with Table loaded with T 2000 grammes unknown body Time 9-375 sees. 97 sees. i5'5 secs - radius of 2000 gm. ' weight ' = 675 cms. k _ 2000 x 675' =45 . 5e2gm , cm *, 2 Ii = 45*502 X i5*5 2 - 9'375 s 97 2 _ 9 . 375 2 Ij = i'ii x io 6 gm. cm 2 . I, by approximate calculation, assuming a regular shape to the body = 9-86 x 10 5 The Compound Pendulum. Let ABC, fig. 33, be a section of a body, passing through the centre of gravity, and at right angles to an axis about which it may turn, the point O being the intersection of the axis with this plane section. The body is at rest when the centre of gravity, G, is vertically under O. 68 ADVANCED PRACTICAL PHYSICS FOR STUDENTS If the body is given a small displacement so that GO makes a small angle with the vertical, then, m being the mass of the solid, the restoring force has a moment mg OG sin = mm sin 6 (putting OG = a.) We saw (p. 53) that in such a case the moment of the forces is equal to Io^. *2 2 i.e. I -=- = — mga sin = — mgad for small angular displacements. This represents a simple harmonic motion (p. 25), whose periodic time, T = 2w \J — — • I = I + ma 2 , where I is the moment of inertia about a parallel axis through centre of gravity and is equal to mk 2 where k is the radius of gyration about this axis. Thus T = 2* Jft> + a2fH = 2nJ« a (l 5 ) \ mag > g v 3/ This result is similar to that obtained for a simple pendulum : k 2 in fact, a simple pendulum of length I = — + a would have the same periodic time, T. Such a simple pendulum is called the ' Equivalent Simple Pendulum.' In the case taken, if all the mass of the body were concentrated k 2 at a point, P, along OG produced such that OP = \- a, we should have a simple pendulum with the same periodic time. The point, P, is called the ' Centre of Oscillation,' O being called the ' Centre of Suspension.' Now since I = \- a a or a 2 — al + k 2 = o, the length a is not the only value for OG, which has / as the equivalent simple pendulum, for the above equation has two roots, a x and cc 2 , such that, <«1 + «2 = l \ (l6) ajOCa = k 2 ) Since a is one value, a x say, k 2 we have a + oc a = I or oc 2 = I — a = — • Thus if the body were supported on a parallel axis through the MOMENTS OF INERTIA AND DETERMINATION OF ' g ' 69 former centre of oscillation; P, it would oscillate with the same time T as when supported at O. From what has been seen above it is evident that there are an k 2 infinite number of points distant, a and — from G, for any k 2 point on a circle drawn from G as centre and radius a or — will satisfy the condition given ; so that any axis parallel to the normal at G on the curved surface of two cylinders, of which the dotted circles are sections, will be axes of suspension which give the same time. If the body were supported by an axis through G, the time of oscillation would be infinite. From any other axis in the body the time is +& g a 2 _1_ £2 . This has a minimum value when is minimum. a a 2 +k 2 a* — 2ak + k 2 + 2ak {a - k) 2 + 2ak Now = ■= a a a This is a minimum when a = k. The corresponding minimum Tj is T t = 2*:^-— (17) o and will occur for a series of axes parallel to that through G, and on the surface of a cylinder whose axis is the axis through the centre of gravity and radius, k. An experiment which brings out these facts may be performed by using as the body a rectangular rod of brass about 1 metre long. This may be suspended on a knife-edge at various points along its length. To facilitate such suspension it is convenient to have a series of holes drilled along the bar at about 2 cms. intervals (fig. 34), Level the knife-edge, and suspend the bat at, say, every other hole in turn, and time 50 swings at each hole, which is a measured distance from the centre of gravity of the bar (which may be obtained by simple balancing). Or the holes may be measured from one end of the bar. Having obtained a set of values for T, and the distance from the centre of gravity, plot a curve with the periodic times as ordinates and the distances as abscissae. A curve such as shown in fig. 35 will be obtained. The values of T near the minimum points, MM 1 , should be further investigated by taking the time for vibrations in every •See also page 118 for a method of timing. 70 ADVANCED PRACTICAL PHYSICS FOR STUDENTS hole, three each side of the approximate position, and the graph completed. Let the line CG, fig. 35, be drawn from C, which represents the centre of gravity oX the bar. Draw any line EABFD parallel to the axis. This cuts the curve in four points, which have the same periodic time, T = BC. It will be found that the lengths, FB, BA, and BE, BD, are equal ; i.e. FB and BA correspond to radii GQ, GP, and BE and BD to radii GO and GR, in fig. 33. Take either set in pairs, say BA, BD, these are corresponding lengths to a x and <x 2 in equation (16), i.e. AB + BD = /, the length = 2 *Vi Fig. 34 of the equivalent simple pendulum. Its periodic time T is numerically equal to BC. Hence from equation, T all factors except g are known, whence g may be calculated. If now a tangent is drawn to the curve, such as line LMM^L 1 , HM = HM 1 = radius of gyration about an axis through the centre of gravity : this may be measured directly. Further, by equation (16), k = V^ol^ = VAB • BD, so a second value of k may be found. The corresponding periodic time is, numerically, the length of HC = T lf say. Hence, once more, by equation (1 7), V g g may be evaluated or, as the direct exact measurement of k (MH or M^H) is difficult, as there is some doubt in the general case as to the exact location of M and M 1 , this formula be used to calculate on third value of k, Ti 2 8^ ft — 75 — 2 ■ MOMENTS OFlNERTIA AND DETERMINATION OF V 71 The mean value of k may be taken and the moment of inertia about a parallel axis through the centre of gravity calculated, for I = km, where M is obtained by direct weighing. L- - - C Ois^ortce. fromC.G D'wsfaoce Tfom OG Fig. 35 Hater's Pendulum From the preceding experiment it is obvious that if it were possible to obtain, for a rigid body; two parallel axes of suspension, along any line through the centre of gravity and on opposite sides of it, which have exactly the same time of swing, then the value of g could be very well determined by measuring the distance between such axes. This distance would be equal to the length of the equivalent simple pendulum, /. If the equal periodic time about these axes were T, Then = *■<$• I and T being measured directly. The Kater pendulum is one by means of which this may be realized in practice to a very close approximation. It consists of a long rod which is provided with two fixed knife-edge supports, K and K, and terminates at each end in a ' bob,' B and B. Usually, the one bob, B, is made of brass and the other of wood. M and m are two adjustable masses which may be fixed in any 72 ADVANCED PRACTICAL PHYSICS FOR STUDENTS position between the knife-edges. Their adjustment serves to move the centre of gravity to such a position that the time of swing is approximately the same from either knife-edge. The pendulum is supported on knife-edge, K and K, in turn, and the approximate periodic time, T and T lt is obtained by counting swings, timed by means of a stop-clock. The large mass M is moved until these times are approxi- mately the same. The small mass m serves as a fine adjustment to this purpose. Having adjusted the masses so that the time for a complete vibration is very nearly the same from both knife-edges, it will be realized that to obtain exact agreement for T and T x would be a most tedious experiment. Fig. 36 However, we can see in the following way that such exact agreement is not essential. Let a and a x be the distance from the centre of gravity of the pendulum to K and K. > ag y a + k 2 ag > a x g T*ag == 4*2 (a* + k*) T x 2 a lg = 4* 2 (a^ + k*) Subtracting (T z a-T 1 *a 1 )g=4n*(a*-a 1 *) £! = a T2 ~ <*i T i a _ I/H±Il* j. HjzIA 1 a\ g a*- ai * ~2\a + a 1 + a - a x ) '-"P*) a and a x may be made to differ by a fairly large amount by suitable adjustment of the masses, M and w. With a little care *p2 np 2 T and T x may be very nearly equated, and so the term — a ■— a x becomes small. T and T x may be measured very accurately by the method of coincidences, (a -f a-^ may be measured directly as the distance between the knife-edges, by a comparison with a metal metre scale by means of a comparator (p. 32). The important first term in equation (18), (R.H.S.) is thus carefully evaluated. MOMENTS OF INERTIA AND DETERMINATION OF ' g ' 73 The second term is small and no serious error is involved if a and a x are measured from the knife-edges to a point at which the pendulum may be balanced horizontally on a knife-edge. By this method a very reliable value of g may be obtained. Method of Coincidences. This method of timing a pendulum consists in hanging it by a knife-edge from a rigid support, in front of, say, a seconds pendulum of a standard clock, the height of the support so arranged that the tails of both pendulums are on the same level. At rest, viewed by a telescope from in front, the Kater coincides with the seconds pendulum. If both pendulums are of the same period and start oscillating together, when viewed through the telescope, they appear to move as one. If the periods are not the same, they will be seen to get ' out of step,' and at one point both will pass a fixed reference point together and going in the same direction. This will not again occur until one pendulum has gained or lost a whole swing. Suppose the seconds pendulum makes n complete vibrations, each of period T (2 sees.), and the experimental pendulum makes (n + 1) complete swings, of period T lt Then Tn=T 1 (n + i) (19) Tl _ n 1 _ 1 _i_ , v T — n + 1 ~" i — n n 2 * ' + n Suppose n = 500. T *-i-^ + T 500 250000 further terms are negligible. Hence T j is obtained in terms of T ; which in the case taken is 2 sees. Similarly the time about the other axis may be checked. In the coincidence method it may be observed that one is never quite sure within a few (say m) passages of the pendulum which is the correct coincidence. We can easily see that the error introduced by this cause is not appreciable when n is fairly large. Thus we know that in equation (19) instead of n we may put (n ± w), i.e. T (» ± m) = T t {n ± m + 1). T*_ 1 T X+ 1 n ± m 74 ADVANCED PRACTICAL PHYSICS FOR STUDENTS — : i _ _ _ approx. i i = i — - "(*±2) n n'' In practice the coincidence may usually be limited to one of about 6, i.e. m = 3. A small value for n = 500. • ii-.i_.i-i 3 . T 500 ^ 5oo 2> i.e. an error of about 1 in 100000 is introduced, due to the uncertainty. In practice a cross-hair in the focal plane of eyepiece of the telescope is a useful reference point against which to estimate coincidence. Sphere on a Coneave Mirror. An Approximate Method of Deter- mining *g', the Acceleration Due to Gravity A concave mirror is arranged horizontally, facing upwards, so that a small steel ball may be allowed to perform oscillations on its surface, in a line through the lowest point. The time of oscillation of the steel ball is obtained by timing as many oscillations as possible on the surface. The observation is repeated, and from these results a mean value of the periodic time T is calculated. Then if R is the radius of curvature of the upper face of the concave mirror, as measured by a spherometer, m the mass of the sphere, r its radius, g the acceleration due to gravity, it will be shown that T= 2, j5^- r ' (21) whence g = _? . !£. (R _ r ) (22) Consider the sphere in its position of equilibrium to be with its centre at B (fig. 37), and when displaced to the extreme position, with the centre at C. MOMENTS OF INERTIA AND DETERMINATION OF «g ' 75 We will consider the case of a mirror of large radius of curvature and the displacement BC to be small compared with R. The potential energy of the sphere at C is mg • AB. Now AB = OB — OA = (R — f) (i - cos 6), i.e. the potential energy a is (R — r) 2 sin 2 - • mg, or when is small as specified : P.E. = 2(R - r) (~fmg = \ (R - r ) 02 m S- The centre of gravity describes a circular path, BC in the BC diagram, so that ~ • = 0- R -r Hence the potential energy is i /T> . BC 2 i - (R - r) 75 i^wg = - mg (R-r) 2 2R — r -•BC 2 . At this point there is no kinetic energy. At B the whole of the energy is kinetic, and equal to ^mv m * + ^Iw m * where v m is the maximum linear velocity of the centre of gravity and w m the maximum angular velocity of rotation, and I is the moment of inertia of the ball about an axis through the centre of gravity, at right angles to the plane of the paper, i.e. K.E. at B is - mv m 2 + - 1 ?*£• 2 2 r 2 At any intermediate point P distant x cms. from B along the 76 ADVANCED PRACTICAL PHYSICS FOR STUDENTS arc, the total energy is equal to either of these quantities and is therefore a constant, 1 1 1 i x z i.e. - mx z H -x z + - mg-j=- . = constant, 2 2 r 2 2 & (R — r) x being the velocity at that P along the path. Differentiating we have : mxx + - X 'X + (R _) ) * , * = °- Dividing by x and rearranging mg * = f- =-. x, (R-,)(m+l) i.e., the acceleration is a constant times the displacement ; this was shown (p. 25) to be S.H.M., whose periodic time T is given as under : |(R _,)(„+!_) T = 2TC = 27^ I mg y mg !(R - r) (m + I) Now, I, the moment of inertia about the axis described, is 2 equal to — mr*. (R -r).Z Hence T = 2n W ^ (21) o T is obtained by observations as indicated above ; r is measured by means of a screw gauge, R, by means of a spherometer, not by an optical method, unless the front surface is silvered. Hence all the terms in (21) are known except g, which may be calculated. The above method does not yield accurate values forg. Atwood's Machine A modern form of Atwood's machine is illustrated in fig, 38. The two masses, M x and M 2 , are equal, and are connected over the pulley, N, by a strip of white paper in the form of tape, while an equally long strip, M^Ma, connects the other ends of the weights, so that for all positions of M x and M a there is an equal mass of paper at each side of the pulley. The vibrator, V, carries an inked brush, B, and as^ the paper passes below it a trace, somewhat in the form of a sine curve, is drawn on the tape. The complete period of the vibrator MOMENTS OF INERTIA AND DETERMINATION OF ' g ' 77 is usually about one-fifth of a second ; but this should be carefully tested by means of a good stop-watch before beginning the experiment. The curve then provides a time record. The apparatus is used to provide an exercise in the determination of the accelera- tion due to gravity ; but even in its best form the experiment has no claim to great accuracy ; it does, however, provide an instructive exercise in Mechanics. Mj stands on a platform, as shown in the diagram, and the mechanism of the apparatus provides for the release of the Fig. 38 vibrator and of M t simultaneously. Small weights are provided, which rest on the top of M x , and on release of the platform cause an acceleration of the masses. The rider can be removed by a second platform, P, after a velocity has been acquired. The trace on the tape records the acceleration and velocity beyond P, and from the former of these the value of g may be determined. There is always a frictional resistance to be accounted for, although this is reduced by making the pulley light and mounting it on ball-bearings. It is best to get rid of the retardation due to friction by making 78 ADVANCED PRACTICAL PHYSICS FOR STUDENTS loops of wire, which may be placed on the top of Mi and remain when M 2 passes through P. The necessary addition can be judged approximately by observing the fall of Mj after it has been given a small velocity. If Mj moves down uniformly without appreciable loss of speed, the frictional error is nearly corrected. A finer observation may then be made by allowing the brush to make a trace. If the line consists of uniformly spaced waves the velocity is uniform. When this has been adjusted the wire loops are left in position and are not taken into account in the calculations. Suppose a rider of mass, m, lies on M x (in addition to the wire loops). When Mj is allowed to fall, suppose it does so with an accelera- tion, /. Let Tj denote, the tension in the paper above M x . The lower paper strip is loose and is not supposed to exert any force on the two masses. Let I denote the moment of inertia of the pulley, and let T 2 denote the tension in the paper on the left of the pulley, i.e. the tension acting on M 2 . Denote the radius of the pulley by a dw and its angular velocity at any instant by w. Then -j~ is the angular acceleration, and we have : dw . From the forces on (M x + m) we have : (m + MJf = (m + UJg - T x and from considering M 2 Ma/ = T 2 — Mag, while the motion of the pulley is expressed by ; I^=(T 1 -T ! )a, from which we have : I/=(T 1 -T 2 )^. We may therefore eliminate T x and T 2 and find : j = g™ m + M x + M a + \ The quantity —is called the ' equivalent mass ' of the pulley, and its magnitude in grammes is engraved on the pulley. We may therefore find the value of g from this equation from observations which give the value of/. MOMENTS OF INERTIA AND DETERMINATION OF ' g ' 79 This may be determined from the trace. By removing P, the trace may be made long, M t being allowed an extended fall. The line drawn by the brush will consist of /^*B waves which open out uniformly. Mark these off in groups of five, as at A, B, C, etc., beginning at a point A, where the trace is opened out sufficiently to be distinct. Measure carefully the distances AC, BD, CE, etc., and divide by the time interval which elapses between these points. This will give the average velocity over the strips measured, and this velocity is the velocity at the points B, C, D, etc. The differences between these velocities should $all be the same, or very approximately so. Take the average of all the determinations and so obtain the average increase in velocity during five periods of the vibrator. Hence deduce the acceleration byj dividing by the time of five vibrations. This is the value of /. Repeat the experiment with the various riders ^q j provided. Fig. 39 CHAPTER III ELASTICITY All bodies, when acted upon by forces, are deformed a certain amount. The magnitude of the deformation produced by a definite applied force enables a value of the elastic constant of the material used to be calculated. We may, in a general manner, call the forces applied ' stresses,' and the deformations produced 'strains.' However, these two terms have, more often, a more precise meaning, depending on the mode of application of the forces. We shall recognize three ways of producing a deformation : (i) by uniform compression or extension, (2) by applying equal and opposite forces in one direction, i.e. stretching, (3) uniform shear. Deformation may be produced in any of these ways or by a combination of them. (1) Uniform Compression or Extension. If a body of volume V be subjected to a uniform pressure of p dynes per sq. cm., a contraction will ensue. This corresponds to a change in volume of <5V, say. The fractional increase in , . <5V volume is ^ • In this case the stress applied is p dynes per sq. cm!, and the strain is -= numerically. (2) Stretching. The most direct example of this type of deformation is seen in the case of a wire fixed at one end and supporting masses at the other end. In this case the force acting on the wire is the weight of the suspended masses and the reaction at the point of support. These are equal and opposite, acting in a direction which coincides with the length of the wire. Due to their action the wire will increase in length and at the same time will be reduced a very small amount in cross-section. The reduction in cross-section for a wire will not be of a sufficiently large amount to be readily measured directly. If 81 is the increase in length of a specimen whose original 8l length is /, the fractional increase in length, the strain, is -=■• I 80 ELASTICITY 81 The stress producing this strain is denned as the force acting on each unit of area normal to it, i.e. if a is the area of cross- section of the wire and the total mass applied is m grammes, the stress is — - dynes per sq. cm. (3) Shear. Consider a cube of material ABCDEFGH (fig. 40) fixed at the lower face and acted upon by a tangential force F at the upper face. As a result of this force the cube will take up a position shown in an exaggerated manner by the broken lines in the figure, the vertical sides being sheared through an angle of <p radians from AH and BG to AH 1 and BG 1 . For such a shear — — „„ „ is the stress, i.e. the tangential force per unit area. The strain is measured by <p, i.e. the ratio EE 1 AE if EE 1 is small compared with AE 1 . In all the above cases the stress is measured as a force per unit area ; in the c.g.s. system in dynes per square cm. The strain in each case is a ratio of like quantities, and has therefore no dimensions. Hooke's Law. If the stresses are below a certain limiting value which depends on the material of the body to which they are applied, the strain disappears when the stresses are removed. If the limiting value is exceeded, the material is strained beyond the elastic limit, 82 ADVANCED PRACTICAL PHYSICS FOR STUDENTS and such strain is permanent ; as the stresses are still further increased the result is a fracture of the material. For stresses below the elastic limit, it was established by Hooke that the strain produced is proportional to the stress applied, i.e. under such conditions we have stress -7 — — = constant. strain The constant has a definite value which depends on the material, and which, in the three cases taken, is called, (i) the ' bulk modulus of elasticity,' (2) ' Young's modulus,' and (3) the ' modulus of rigidity.' Young's modulus is the most readily obtained directly by experiment. The following notation will be used throughout in dealing with these elastic constants. (1) Bulk modulus = K = £ oV A (2) Young's modulus = Y = — = 61 T F A (3) modulus of Rigidity, n = — In addition to the above three elastic constants, we may add a fourth, which is concerned with stretching. We noticed that during stretching there is a lateral contraction of the specimen. The fractional lateral contraction produced is proportional to the longitudinal stress applied and the ratio of Fractional lateral contraction Fractional longitudinal extension is called ' Poisson's Ratio ' fp) . Thus, if the specimen is a cylinder of radius r and length I, and the changes produced in these dimensions are 6r and 61, we have r ""IT / The following relations between the elastic constants may be ELASTICITY 83 readily deduced (see for example, Poynting and Thomson's " Properties of Matter "). * -2( 3 K-f*) * (2) Thus, if any two of the constants are found experimentally, the remaining two may be calculated from the above equations. Determination of Young's Modulus for the Material of a Wire A direct method of finding Y is to support, vertically, a long length of the wire, load it with definite masses, and observe the extension produced. Fig. 41 It will usually be most convenient to obtain such a length that, when supported at the ceiling, the wire extends almost to the floor. A second wire, C, is supported in like manner from the same support, and carries a fixed load of sufficient magnitude to keep the wire taut. The wire D carries a platform P. At V a vernier scale is attached to the wires, one half fixed to one wire, the other half to the second wire. The wire D, whose Young's modulus is to be determined, should be free from ' kinks ' and should carry sufficient load to make it taut, so that any further load added merely causes a stretching of the wire, and does not simply straighten out bends and kinks. If a heavy platform is employed at P, this weight 84 ADVANCED PRACTICAL PHYSICS FOR STUDENTS may be sufficient. Read the vernier ; place one kilogramme on the platform and notice the extension. If this is due solely to the stretching of the taut wire, the vernier reading, on removing the kilogramme load, should be once more the same as at commence- ment ; if this is not so increase the load until on adding a further kilogramme the readings on the vernier scale have a definite value, which is reduced to another definite value on removing the load. Having obtained satisfactory repeats for this adjustment, the load on the scale pan P should be increased by equal increments and the vernier reading for each load noted. Having arrived at a safe maximum load, the latter is reduced by the same increments and the vernier readings again noted. The values obtained may be tabulated as under. LOAD VERNIER READINGS. EXTENSION LOAD INCREASING LOAD DECREASING MEAN FOR 6 KILOS 2 4 6 8 10 12 mean extension for 6 kilos = The mean vernier reading for each load being taken and tabulated as shown, we may obtain several values of the extension of the wire for a definite load. Thus, in the case taken, the loads were o, 2, 4, 6, 8, 10, 12 kilos. The difference between the vernier readings for and 6 kilo load gives the extension for 6 kilos. In the same way the difference between the readings for 2 and 8 kilo load, 4 and 10, 6 and 12, also gives the extension for 6 kilogramme increase in load. This is entered in the last column, the mean value, I say, of which is used in the calculation of Y. The radius of the wire is measured in at least six places, using a micrometer screw, and the mean value taken, r cms. say. The original length of the wire, about 7 metres, is measured directly (L cms.). 6000 x 981 Hence TVT' I in the case taken. ELASTICITY 85 An alternative way (due to G. F. C. Searle) in which to measure the extension of the wire is illustrated in fig. 42. The standard wire terminates in a frame A which supports a mass M, sufficiently large to maintain the wire in a stretched state. The wire to be investigated is also fastened to a similar framework B. The two are fastened by cross-pieces C and D, which prevent relative rotation of the frames, but allow the frame B to be depressed relative to A, when masses are added to the scale pan S. Fig. 42 A spirit-level L is supported at one end on a rigid cross-bar of the frame A, and at the other on the point of a micrometer screw V, which moves vertically through a rigid cross-bar. The micrometer screw has the usual circular division, which enables the movement of the head to be estimated to ts or rita of a complete turn, enabling the movement of the point of the screw (and hence the end of the spirit-level) to be measured to shs or ths of a millimetre. The level is first adjusted, when the wire is suitably stretched free from ' kinks/ so that it is truly horizontal. The load of, 86 ADVANCED PRACTICAL PHYSICS FOR STUDENTS say, one kilogramme is added to the scale pan S. The micrometer screw is moved a suitable distance over to scale G, so that the spirit-level is once more horizontal. The amount of movement required to bring this about is obviously equal to the elongation of the wire by the load added. The results may be tabulated as in the former method and the value of Y calculated from the mean of a set of observations. Bending of Beams The value of Young's modulus may be found by less direct measurements for substances not in the form of a wire. Consider a rod of any uniform cross-section, say rectangular, Fig. 43 bent into the form of a circular arc of fairly large radius. Take a section of the rod by a plane passing through the long axis of symmetry and parallel to it, and passing through the centre of curvature (i.e. the plane of bending). The layers of the material of the bar in the lower half will be compressed and the upper half extended. There will be one plane, therefore, at right angles to the plane of bending, which will remain of the same length as before the bending took place. This plane is called the neutral surface, and it will be shown to pass through the centre of gravity of the bar. It is represented in fig. 43 by NS. If we imagine the bar to be made up of a number of filaments along the length, then such filaments, as stated above, will be extended or compressed according to their position above or below the neutral surface. One such as shown at EF, fig. 43, or at P in the section diagram (fig. 44) a distance y above the neutral surface is extended. ELASTICITY 87 The strain in such a filament depends upon y, for, if R is the radius of curvature of the neutral surface, and the angle sub- tended at the centre O by NS, the unstretched length of the filament EF is the same as the present length of NS =R0. Also EF = (R + y)Q. Hence the elongation is yd and the strain is therefore ~L. Now if / is the force acting on the filament of cross-section a, we have from the definition of Young's modulus / or N Y a . / R 3 R -y» i.e. fccy •O « SO Cf s 3 ______ "-t= Fig- 44 Thus, the arrows in the lower part of fig. 44 show the type of forces acting on all such filaments into which we have subdivided the bar. This system of forces on the bar must have an algebraic sum of zero. Y le. S/ = oor-^- Say = o. Since ^ is not zero, R Say = o. Thus, the neutral surface passes through the centre of gravity of the bar. The forces have a definite moment about the neutral surface. Y The moment for the smgle force is fy = = .- ay 2 . For equilibrium the sum of such moments is equal and opposite to the external couple which set up the internal forces. If C is the external couple, we have Y „ Y . s r «r R*' 88 ADVANCED PRACTICAL PHYSICS FOR STUDENTS where * = Say 2 . From the similarity between this case and the corresponding sum in considering masses in connexion with moment of inertia, i is sometimes called ' the moment of inertia of cross-section.' It may be calculated in the same way as moment of inertia if area replaces mass. We have therefore for such a bent beam *Y = CR -....(3) Cantilever Consider a light beam fixed horizontally at one end and loaded with a mass m at the other. If the mass of the beam is small compared with the load m, the whole depression may be taken as due to the load. In fig. 45 let AC 1 be the unloaded position for the neutral surface, and AC the position taken when the load is applied at C. To obtain an expression for the depression of the end in terms of the dimensions of the bar, m and Y, it is convenient to refer to a system of axes with the end A as origin ; AC 1 being the x axis, and a line at right angles to AC 1 from A in the plane of the paper the y axis. We will assume that the curvature of the loaded rod is small, i.e. the total depression at C is small. Consider a section at B, fig. 45, x cms. from A. As already seen, across the face of such a section a system of forces exist. These forces on the segment BC are extensions above the neutral surface, and compressions below, constituting a counter-clockwise t"Y couple C = -p on BC. At the same time the force mg at C has a clockwise moment equal to mg(l — x) on BC. For equuibrium these two opposite couples are equal in magnitude, iY i.e: -^ = mg{l -x) (4) ELASTICITY 89 where R is the radius of curvature at B. {■+■©•}' Now R = jw- d*y dx* But in this case ^ is small — for the total depression is dx assumed to be small — so (~\ is negligible compared with unity, and therefore we have for such small curvature R=JL, d*y dx* d 2 v W / x 1 i.e. ^ = -W? (I — x) from equation (4) above. dx % *Y x The value of the total depression at the encf of the bar may be obtained by integration, and is the value of y when x = /, after such a process. Integrating we have BO -£( fc -3 (5) dy the constant of integration being zero, for when #=0,^=0. A second integration between the limits o and I gives L^Jo *Y L2 6 Jo where y is the end depression, i - e - y° iY 3 For a bar of rectangular cross-section, i about the neutral axis NS (fig. 44) is — , where b is the breadth and d the depth of the bar. 12W I* Hence ^» = 6^Y*3 v _4¥ 3 (6a) If the mass of the beam is not negligible, see treatment on page 94. 90 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Beam Supported at Two Knife-edges and Loaded in the Middle If the beam is now supported at the two ends and loaded with a mass m at the mid-point we have a reaction ^ at each 2 knife-edge. If / is the total length of the bar, the depression produced at the centre will be exactly the same as the depression at the end of a similar bar of length - and loaded at the end with m rrii .. a mass — , i.e. if we imagine the bar clamped at the mid-point and a force ^ applied at one end. The movement of the end is precisely the same as the depression in the middle in the actual case taken. Ar Fig. 46 I Such a depression may be obtained by substituting the length - and force -£ in equation (6a), giving for the depression y ° ~ *Y ■' 48 or Y==-^!_ ." >>(7) 4bd 3 y The value of Young's modulus for the material of a beam may be obtained in this manner. The beam, say of iron, and of about 1 cm. square cross-section, and about 1 metre long is supported on two knife-edges near its ends, and a load is applied at the mid-point by placing masses on a pan which is suspended from a knife-edge which rests on the bar at this place. The depression is measured on a vernier scale, one scale of which is fixed, the other moves with the beam. The load is increased, and the vernier reading (y ) for each load is tabulated as under. The value for — is obtained in each case, the results being tabulated as under. ELASTICITY 91 load m VERNIER READING DEPRESSION (y ). m 2000 gms. 3000 „ 4000 „ 5000 „ mean m Jo~ m A mean value of — from the series of observations is taken. y Q The distance I between the knife-edges is measured directly, being of the order of 1 metre ; this can be done with a good degree of precision. Now d occurs in the third power, and is only a small quantity, therefore many observations must be taken and the mean value used. For d approx. 1 cm., an error of *i mm. means 1 per cent error, and this is magnified in d 3 to 3 per cent. Substitute the values found, in equation (7) ^/m\ gl* W4W» Koenig's Method , Fig. 47 Another method, due to Koenig, of determining the value of Y for the material of a beam, is by means of the type of apparatus shown in fig. 47. 92 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The bar carries a knife-edge which supports the load on a pan P. At the ends of the bar are two mirrors, M x and M 2 , almost normal to the bar, but slightly displaced to enable a scale S to be seen in the telescope T, the light from S having suffered two reflections. The telescope carries a cross-hair in the eyepiece, and the apparatus is arranged so that a scale division, as seen in the telescope, coincides with the cross-hair. If now the bar is loaded with, say, i or 2 kilogrammes, the mirrors will be turned towards each other as a result of the depression produced, and the scale division viewed in the telescope will be altered* This difference is noted = x divisions, say. Then we will show that Y ~ WdH ••••• (8) where W = Mg, M being the load in grms., I = distance between knife-edges, D r= distance between scale and the more remote mirror, M 8 , m = the distance between the mirrors, b and d having the same values as before, b the breadth, d the depth of the bar. For a bent cantilever we saw, equation (5) or between limits o and I, b Now -2- is tangent of the angle through v/hich the beam has been bent. Let 9 be this angle. ti, W l % Then tan <p — -=- T »Y 2 Now for in the present case, I being the whole length of the bar supported by two knife-edges, we obtain the angle through / W which each end is turned by substituting V for I and — for W in the last equation, i.e. in the present case WZ* tan * = i67Y dxj i Y 2 ELASTICITY 93 For rectangular bar i = bd* 12 tan 9 — 3WZ 4bd*Y For small depressions the angle <p is very small and so tan <p = q>. <p = 3W 4bd*Y ® Now a value of <p may be obtained from a consideration of the movement of the mirrors. Let m x and m t be the original positions of the mirror (fig. 48). In the first case the image of the division at D is in coincidence with the cross-hair ; when the mirrors move, each through an angle <p, F is then seen in coincidence with the cross-hair. Fig. 48 Let us imagine the rays of light reversed, ABCD being the original path : when m x moves through q> and takes position m\, BC is moved through 2q> striking w 2 at E. Obviously m being the distance between the mirrors CE = m • 2q> very nearly. EG is a line drawn parallel to CD. The ray EF is swung round through an angle 49, for in addition to BE having moved through 2<p, m 2 itself has rotated through <p. i.e. since GEF = 49 FG = 49* D, very nearly. But DG = CE = 2mp. ;. x = DF = 2<p{m + 2D) <p = 2(m + 2D) .(10) 94 ADVANCED PRACTICAL PHYSICS FOR STUDENTS from (9) and (10) x 3W/ 2 2(w -f 2D) 4&i 3 Y 2&i 3 # y== 3 W/ a (m + 2D) ^ T P W In performing this experiment a mean value of — is obtained as in the last experiment, from observation of x corresponding to several loads and the other terms measured as before. Hence by substituting in (8) Y is obtained. Determination of Young's Modulus of the Material of a Bar by the Vibration Method We have already seen that the depression due to a load W at one end of a beam which is rigidly fixed at the other end is given by equation (6), page 89. This strain sets up an equilibrating internal stress equal to 3«Y * L fs—y which acts as a restoring force. If the beam be allowed to oscillate in a vertical plane the restoring force when depressed y cms. is therefore 3*Y Expressing the load as W dynes (m grms.) we may write, if y W is the acceleration of the mass m = — at the end of the bar, g W .. 3*Y 3»Yg ovy = -wi* > y This is a periodic motion, whose period T is -*■>£ w T -">fe£- <"> '3*Yg or, as the mass at the end of the rod is m grammes, \ nd r '3*'Y A more complete treatment of this case may be seen under. In fig. 49 let O be the point of support, OA a vertical section of the unloaded beam, and OB the section of the beam when depressed by a load m at the end, so that the end depression AB = z , and the depression of the centre of gravity shown ELASTICITY 95 is u , whereas any point P on the bar is depressed a distance # , P being a distance s from O ', the total length of the bar is I cms. , Now let us imagine the bar to be homogeneous, and of mass 6 per unit length ; so that its total mass is M = 13. We have, if i is the 'moment of inertia of cross-section,' considering the length (I — s), following the treatment of page 88, tY = R[W(Z - $) + (I - s)<5-g^^]|, taking into account the weight of the section of the bar concerned, Fig. 49 which acts through the C.G. of the section. Since — = — if the beam is not greatly bent, ,-Y g- = W(Z - s) +~(l - s)» 6-g, whence integrating for when s = o, x = o, i.e. the constant of integration is zero. Integrating between the limits s = o, s = /, x = o, to x = z =,(1^ ............. ...« I2 ) since M = d'l. This deals with a case of a steady depression. When the bar vibrates it is depressed below the position OB to some extreme position OC. To arrive at a value of the forces acting in the material of the bar in such a case let us assume that Wis increased to W\ causing the bar to be depressed into the position OC, where the extra depression of the end is z, of the centre of gravity of the projecting bar u, and of the point P is x. 96 ADVANCED PRACTICAL PHYSICS FOR STUDENTS XT -V 73 W . M ^ & Now tYz =l 3 \- o~ 3 8 and iY^ + ^/'W^M^ 3 o i.e. »Y* = — (W 1 - W) *=5^(W»-W). (13) In this case W — W is just balanced by the internal stresses in the beam which are thus equal to (W*-W)==-^s (14) This gives the value of the restoring force, hence the equation W of motion of the mass — 'neglecting the mass of the beam, is W dH__ 3*Y g dt* I 3 '* or * = -&*S.z or Z*W X - e - Ni**" ^5) This is identical with the result expressed in equation (n), which was deduced from very simple considerations. It does not, however, take into account the mass of the beam itself. This is most conveniently done by considering the energy of the system. As the beam oscillates about OB, the steady deflected position of equilibrium, we will express the potential energy of the parts, using OB as the reference position of zero potential energy. Thus, in the position OC the Potential Energy of the system is — VJz — Mgu + (Energy stored as strain in the beam). The last term is obviously equal to the work done in straining the beam, i.e. is / (straining force) dz (W 1 — W)dz = I ^tj — dz by equation (14) above 3*X* 8 * 2l* ELASTICITY 97 .-. P.E. is 3*Y a_ W 2_ M gw _ ^6) 2 I 3 If instead of obtaining equation (12) by integrating the previous equation between the limits o and I we had integrated between the limits o and - . we should have obtained the depression u 2 of the centre of gravity as under. *° *Y 48* +384 *Y . u may be obtained in a similar way to z (p. 96) to be u = W '- W ._L_Z3 and since by (13) »Y 48 16 So that in similar terms (16) becomes 1 75-** - Ws - f 6 Mgz (17) Now the kinetic energy of the system is the sum of the K.E. of the mass-— I = - — £*) and the total K.E. of the vibrating g V 2 g / beam. To obtain the K.E. of the beam consider an element of length ds at P. The K.E. of this element is £ d.ds(x) a x may be obtained by precisely the same method as z and u. * = ,4(t-iD(") so that x = -\---} z The total K.E. of the bar is 2 v ' I. 98 ADVANCED PRACTICAL PHYSICS FOR STUDENTS 2 / 6 L V20 36 ^ 252/J 280 280 Thus the tote/ Kinetic Energy in the system is iT 1 '-'-^ 11 ** (l8) The sum of the Potential and Kinetic Energy is constant. Thus, adding (17) and (18) we obtain I/W + 33_ M Ua + 3*Y 2 _ m _ 4 M^ = constant. 2\g I40 / 2l 3 16 & Differentiating I /W + _33 M \ 2i g + 3 *Y 2 ^ _ Wi _ 5 M q 2\g 140 / 2 I s 16 Dividing by £ we have (7 + ^ M )'" + 3 , J«-(w + 4lfe)-o....(i 9 ) This represents a harmonic medium whose period T is un- affected by the constant term W + -4 Mg as seen below.* The period is the same as that of \g 14° / ' or T -«W V « 3( y° ; •; (20) * We may write equation (19) : put * = *-^( w + i! M s) /. 'p = Z, "P=" z = 2*^1 W + -33_m > )/» 3^ ELASTICITY 99 Now, i, the ' moment of inertia of cross-section,' depends, of bd 3 course, on the form of the rod. For a rectangular rod, i — — (about the neutral surface), where b is the breadth and d the depth of the cross-section in cms. The Experiment may very well be carried out using an ordinary boxwood metre rule. A definite length, /, of the rod is projected from the top of a table, to which it is rigidly clamped, as seen in fig. 50. f /////////y///////y/yyy^/y/^/^y^ Fig. 50 w A mass — is rigidly attached to the end of the metre scale, so that it has no 'play/ i.e. there is only the one vibration, W namely, that of the scale itself. The value of — should be such o as to cause but small depression. The rod is made to vibrate, and the time T is obtained by timing 50 vibrations in the usual manner. The length of the vibrating rod is altered, and the periodic time again determined. This is carried out for several W lengths of rod, and also for several masses — at the end of the o scale. It must be remembered that M in the expression (20) for T is the mass of the vibrating part of the scale, and must be obtained for each length / employed. Of course, if m' is the total mass of the scale and L the total vn'l length, then M = -y- for the uniform rod. The results of the various experiments may be tabulated as under. joo ADVANCED PRACTICAL PHYSICS FOR STUDENTS / /». w 1 ' M W + ,33 M 8 140 T T* Mean 11 (YL T 2 U 33M\ 140/ _ i6it a (/ 8 /W + 33M\ g 140/ Note. W. g is the mass attached in grammes. The method works equally well for such substances as wood, where the correction for the mass of the beam is small, and for brass, etc., where— M is comparable with the values of W. RIGIDITY The modulus of rigidity is determined by observation of the twist produced in a wire by a definite couple, either statically, or less directly by torsional oscillation. ELASTICITY xot Measurement of the Rigidity of a Wire by the Static Method Let us consider a long wire fixed rigidly at the upper end and subjected to a couple, C, turning it in the direction shown in fig- 5i. . We may consider a section of the wire (fig. 52), and of this section an annular ring as seen. Consider a small rectangular segment EFGHIJKL in the undisturbed wire. When the couple acts, this rectangular parallelepiped will shear as we have pre- viously seen, the faces making an angle <p with the original vertical direction. This happens throughout the ring, and adjacent sections will behave in the same way throughout the length of the wire. Fig. 52 A vertical line, MN, along the wire (fig. 51) is moved by each ring through an angle <p, and so finally takes up a position MP when MP is inclined at an angle g> to MN. If I is the length of wire from the fixed end, and R is the radius of the wire, then since N is moved to P when the couple acts, the radius ON moving to OP, where the angle NOP =6 radians, we have arc NP = h = R0 • • --(21) Considering again the annular ring of fig. 52, of radius r and width &r, we have the shear produced by the couple C. Let /be the value of the shearing force over the area, 2-nr-Sr, of its upper and lower surfaces. From the definition of rigidity n f 2-nr ' dr n == «^— <p or f = 2isnqrrdr. Since this force has a moment fr about the axis of the wire, we have, replacing <p by the more easily measured term 0, from (21), ' 2w»0 ft — T 102 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The total couple throughout the solid section is therefore : Jo Jo I 2l a couple which is equal to the applied couple C for equilibrium, hence C-=£-' (») This may be re-written : C . ttR* -'I — • n. c -, the couple required to produce unit angular twist, is called the ' coefficient of torsion ' = t, say. We thus have : *R* 2 or «R* 2 tZ = «-"-^- .(23) xl = i • n. =i the moment of inertia of cross-section for the circular wire, and in equation (22) is measured in radians. Fig. 53 Experimental Details. The method of finding the coefficients is to fix a wire specimen rigidly at one end, apply a measured couple at the other, and measure the twist produced at a given distance I from the fixed end. The two types of apparatus usually employed are shown in figs. 53 and 54. ELASTICITY 103 Using the vertical wire type, pointers are fixed to the wires at different distances from the fixed end, and a couple is applied to the free end by adding weights to the scale pans S, S 1 . The amounts of twist, V a > 3 , at the distances, l lf Z a , Z 3 , from the clamped ends are observed on the circular scales shown. It will f) ft f) be found that -~ = ~ = -~ as we may expect from equation (22). L x / 2 / 3 Now consider a fixed length of wire I cms. from the support. If M is the sum of the masses applied in S and S', and the diameter D of the wheel at which the couple is applied is D, then Mg— = C in equation (22). 3 A series of values for 0, corresponding to various masses on the pans, is obtained. These results may be tabulated as under. MASS ON PANS ANGLE OF TWIST M M 6° 6 _ 100 15 6-6 200 30 6-6 300 46 6-52 400 61 6-55 500 77 6-50 104 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The last column of the table shows the ratio — to be a con- stant — or nearly so. For such a variation as the one shown in the table of results above, the mean value of — is taken and substituted in equation (24). Re-writing (22) we have „ D TmR* B . „. M-g • — = — j — 6 e in radians, M ,gV>l or n = —• • -s— 6 in radians. tcR 4 The length of the wire is measured directly. The arm of the couple, D, is measured by means of callipers. As R, the radius of the wire, occurs in the fourth power, especial care is taken to obtain its true value. Determinations of R are made with a micrometer screw gauge at several points along the length of the wire, and a mean value taken. ^ M Substituting the mean value of — from the table of results, all the unknowns of the equation are ascertained. However, as is usually measured on the circular scale in degrees, the M value of the ratio — must first be converted to radians. Assuming therefore now that is measured in degrees, the end result which converts to radians, etc., is M I g(i8o)D , x The same type of observations are necessary with the horizontal apparatus, which makes use of a shorter length of wire specimen. Maxwell's Needle If a bar, AB, is suspended horizontally by means of a wire whose modulus of rigidity n is to be determined, the value of n may be obtained in terms of I, the moment of inertia of the bar about the axis of suspension, and T, the time taken for the bar to make a complete horizontal oscillation. Consider a small displacement from the position of rest of the bar. We have already seen (p. 102), that in such a case the couple called into play in the wire to equilibrate the displacing couple is t = -j- per unit angular displacement, where tcR 4 i = , so that the couple exerted by the wire for an angular displacement 6 is t0 for that particular wire. ELASTICITY 105 It has been shown (p. 53) that the moment of such external forces on the oscillating bar is T dH Hence I '-!£- - rS dt* i.e. these couples are equal and opposite at any point. ^ = "I This will cause vibrations whose periodic time, T, is given by (p. 25) Hence, knowing T by timing a number of swings, if I can be , ascertained t may be evaluated and hence n determined since *R 4 n Fig. 55 / In Maxwell's needle (fig. 55) the bar is replaced by a hollow tube, of length D cms. Fitting in the tube are four equal-sized cylinders, each of length/ — j, two of solid brass, and two hollow. Let I„ be the moment of inertia of the hollow cylinder of length D, about the wire as axis, I ! be the moment of inertia of the solid brass cylinder about a parallel axis through, its centre of gravity, 1 2 be the similar moment of inertia for the hollow brass cylinder, tn x the mass of each solid brass cylinder, m 2 the mass of each short hollow brass cylinder, I the length of the wire in cms. In the first case place the cylinders in the order shown in 106 ADVANCED PRACTICAL PHYSICS FOR STUDENTS fi g- # 55 (a) and find the periodic time T x . Then arrange the cylinders as in (b), once more obtaining the periodic time by timing, say, 50 complete swings. Let this time be T 2 . Suppose that the moment of inertia of the complex bar in the first case is I', and I" in the second. We have T t = 2* JX T 2 = 2tc <J — Ti* - t,« =^-Vr -r) Now we have for the moment of inertia of each of the four units is given by the ' law of parallel axes,' I' = I + 2l 2 + 2 m 2 (^ + ali + zm t (^) 2 whence n D 2 D 2 / \ D 2 = 2w x -g - 2w a -g- = f m x — m 2 J — By such an arrangement I' — Y may be evaluated and hence t may be calculated. To enable an accurate measurement of T in both cases, the hollow frame carries a small mirror. A beam of light from a lamp is focussed on the mirror, and the reflected beam is directed on a scale. As the reflected beam, shown as a spot of light on the scale, passes a certain mark in one direction, a stop-clock is started, and, starting counting, 1, 2, 3, etc., as the spot again passes the scale reading in the same direction, the time for fifty complete oscillations is obtained, when the four short cylinders are arranged as in fig. 55 (a) and (b), whence T t and T 2 are obtained. The length I of the wire from the rigid support, and the length D are measured directly : m x and m 2 are obtained by weighing, and therefore all the factors for the determination of n are measured, for, 2 I' — F . .m, - m, D 2 and n 4w 'TV=Tr» = 4w TV _ 2tZ ~^R* ELASTICITY 107 Determination of the Modulus of Rigidity of the Material of a Flat Spiral Spring In this experiment the value of n for the material of the wire of a flat spiral spring is deduced from a knowledge of the periodic time of vertical oscillations of the spring when loaded with a known mass, and the dimensions of the spring. Let us consider a flat spiral spring, whose turns are closely wound with a wire whose radius r is small compared with the radius of the spring itself ; such a spring may be made by winding the wire on a wooden cylinder of suitable diameter. Let the ends of the spring be bent twice at right angles, the free ends, such as BC (fig. 56), being therefore along the axis of the cylinder. If the spring is clamped vertically at one end, and loaded with a mass, M, at the other end, as shown in the figure, the force, Mg, along the axis exerts a couple tending to twist the wire in the direction of the arrow. It was shown* on p. 102 that, under such circumstances, if I is the length of the wire from the fixed end, and R the radius of the wire, C = f^ 6 (25) 2 I In the case of the spring we may apply this formula when / represents the total length of wire, and is equal to 2ttRN, where N is the number of turns. If r = radius of the wire, R = radius of the cylinder on which the spring is wound -f «r, I = 2«RN, M = the load in grammes, to c -'r-iaar (25a) 108 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Take a section of the spring at A, shown in fig. 57 enlarged. When the couple is applied the arm AB is twisted through the angle 0, given by equation (25a) above, and takes up the position AB 1 . Fig. 57 The depression x, of the end B, is obviously equal to R0, approximately, when R is large, i.e.. * = R.***!C = 4NR!C If / be the restoring force on M due to the wire, the couple C =/R. ie - *5RF=' "• (26) The equation of motion of the moving mass is therefore M dt* ~ 4NR 8 ' *" This was shown (p. 25) to represent S.H.M., whose period is given under. T = ->/^ r w From observation of the periodic time of the spring oscillating in a vertical manner, n, the rigidity coefficient of the material of the spring can therefore be obtained. However, in the foregoing we have neglected the mass of the spring itself, and have also assumed that the total depression produced by M was due to twist alone. We must now consider these factors. Shear in the Spring We can very easily see that the shear effect in the type of spring taken above is of a negligible order when the radius of the wire is small compared with the radius of the spring. For this purpose again neglect the mass of the spring. If / is the force acting along the axis, at any point A in the spring there is an equal and opposite force,/, for equuibrium. Thus, for the small segment AD, shown enlarged in fig. 58, at A there is a downward force /, and at D an upward force /, constituting a shearing couple which would cause a depression ELASTICITY 109 of A to some lower position A 1 , through an angle 9', say, where from the definition of rigidity v An The area A is the area over which the forces / act, which, in this case, is the area of cross-section of the wire, nr 2 , „> - f i.e. icrht Fig. 58 The total depression due to this shear for a length / of the wire (28) Now, using the expression for the depression due to twist, given in equation (26), and substituting I = 2«RN in the ex- pression (28) above, we have /2tcRN Depression due to shear _ nrhi r^ Depression due to twist /4NR 8 ~~ 2R a r*n Take an average case, r = -05 cm., R = -8 cm. r* = 1 2R* 512 i.e. depression due to shear is about -2 per cent of the depression due to twist. no ADVANCED PRACTICAL PHYSICS FOR STUDENTS Thus, when r is small compared with R we may neglect the shearing effect. The Mass of the Spring The expression for the time of vibration is modified if account be taken of the mass of the spring itself. The total mass moving is greater than M. We may regard the problem as being similar to the case of an ideal massless spring, loaded with (M + m'), where m' is an additional load which just has the same effect as the mass of the actual spring used ; m x may be called the ' equivalent mass of the spring ' and T = 2tt l (M + m') 4 NR 3 ~ \ r*n Thus, using this formula, we may find n and m' by obtaining T for two loads, and solving for m' and n. However, we may treat the problem in somewhat more detail, and deduce an expression for the equivalent mass m', in terms of the actual mass m of the spring. It has been shown that the twist of the wire of the spring at the end is = f— , since C, the couple, is equal to MgR. The depression of the end due to this twist is R0. If this depression is x, * = 2 ™^! (29) In the vertical oscillations, the mass M at the end of the spring is depressed further than this. To arrive at a value of the forces called into play due to such further straining of the spring, suppose a bigger mass, M 1 , be applied, causing an increased depression z, i.e. total x -{• z. Then . x + z = ~^- * = ^-(M*--M)*» (30) To allow for m, we may consider the energy of the system with regard to the position of rest of the loaded spring. •The internal forces equilibrating the force (M 1 — M)g, axe therefore = —r^i- Therefore the equation of motion of the mass M is Mz= — = 2 *Y 2M/R» ■K**n as previously shown, equation (27), neglecting the mass m of the spring itself. ELASTICITY in When displaced a distance z from the position of rest (which is x cms. below the unloaded end) Potential Energy is —Mgz—mg (displacement of centre of gravity) + (energy stored as strains in the spring). The centre of gravity is lowered a distance which is equal to z half the depression of the end= - . The energy of the spring is equal to the work done in straining the spring, i.e. since the internal forces are equal to (M 1 — M)g = — rp-'^i from (30), the work done in depressing the end a distance dz is l—p~\zdz, i.e. energy in the spring is •ttrhi , _ nr*n z 2 2mF' z ' dz " iJR 2 ' 2 Total P.E. is therefore 7tr% 1 ___*2_Mg Z - -mgz. The Kinetic Energy is \ M(i) 2 + K.E. of the moving spring. If 6 is the mass of unit length of the spring wire, the second term above is / — (mass of a small element) x (velocity of the element) 2 . At a point P s cms. from the point of support measured along the length of the spring, consider a small element ds. The velocity of that element is j • z, its mass is dds, i.e. K.E. of moving spring is . 2J dds • j 2 = i*Ly-=I(™W since m-M. 2 l z 3 2 \ 3/ The total energy of the system in the position considered, and which is constant by principle of conservation of energy, is therefore ^ -(^ + ^, + 1(11+5),.- constant. H2 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Differentiating and dividing by i we have 5£.-(* + «.) + („ + !).-.; as shown on footnote, p. 98, such a system is a periodic motion whose periodic time T is the same as for or nr*n 2.1 t M + ™W i-e. T = 27r-JJi f (31) The effect of the mass of the spring is therefore the same as though M were at the end of a massless spring together with a load equal to | the mass of the spring. Experimental Details. A flat spiral spring is chosen, having a radius R which is fairly large compared with the radius of the wire. One end is clamped firmly in a heavy retort stand ; a mass is attached to the lower end of the spring, and the time of vibration of the vertical oscillations is obtained by timing 50 vibrations with a stop-watch. This is repeated for various loads. /, the length of the wire in the spring, may be obtained from a knowledge of N, the total number of turns in the spiral. If there are exactly N turns, the length is 2tcNR where R is the average value of several observations of the mean radius of the spiral, i.e. the value of the outside radius of the wire spring, minus the radius of the wire of which it is made. r occurs in the fourth power, so should be measured with extreme care. A number of values are obtained with a screw gauge at points along the length of the spring, and the mean value taken. / being equal to 2tcRN, we have from equation (31) i6k 2 R 8 N + 3 .. n = — jT-'—fr* - (32 ) The results of the several experiments may be conveniently tabulated as over. ELASTICITY "3 M. (M + tn/3.) T. T*. M + w/3. "£2 mean value of M H — 3 The mean value of — ^ for the series of loads (M) taken is obtained and substituted in equation (32), thus giving the mean value of n for the complete set of observations. Determination of Young's Modulus of the Material of a Spring The value of Young's modulus for the material of the wire of which a spring is made may be obtained by supporting the spring vertically and allowing a bar, which is firmly fastened to the lower end, to perform horizontal swings as in the case of Maxwell's needle (p. 102). It is shown below that in such a case the time of a complete horizontal oscillation is 8RNI where Y R r I is Young's modulus for the wire, is the radius of, the spiral, measured from the centre of the wire, the radius of the wire, the moment of inertia of the bar about the axis of suspension, N the total number of turns in the spring. The above assumes, as in the last experiment, that the spring is a flat spiral ; the layers are very close together, and each may therefore be regarded as horizontal. Let fig. 59 represent a horizontal section of a small length s of the spring, with NS the section of the neutral surface by the plane of the diagram. This neutral surface will be normal to the diagram. ABCD is the section of the element of the spring when the bar is in an undisturbed position. 8 H4 ADVANCED PRACTICAL PHYSICS FOR STUDENTS When the bar is turned in a horizontal plane through any angle y>, the section taken, in common with every other element of the spring, will become more curved, as shown in the broken lines. Consider a single ' filament ' of the material, as was done in the case of the bending of beams (p. 86). If a is the area of cross-section of the filament, and / is the force required to bend up the filament through a small angle Y = Strain of the filament Now, if R is the normal radius of curvature of the neutral surface, and PQ is a distance x cms. beyond NS, the original Fig. 59 length of PQ = (R + x)6 , where O is the angle between the radii from the original centre -of curvature O to the ends AC and BD of the element considered. Now, due to stresses similar to /, the element has a greater curvature, and consequently the neutral surface has a smaller radius of curvature R. If O 1 is the new centre of curvature, the new length of PQ is (R -f x) 0. However, the neutral surface is Of the same length as before, so that R O O = R0. The strain is therefore (R + x)S — (R + x)d -f- original length == — — f — — approximately. S = Rn0fl = R0- ELASTICITY H5 Strain is therefore * \R0 "" *W = * \S ~ Ro/ / 4Z • Y = The moment of this force on the filament about the neutral surface is fx. The total moment of the couple acting on the element considered is therefore **»-/*«■(£-£) If the angle subtended at the centre by i cm. of the wire be q> and <p Q , respectively, the above couple is "C = Yi{v - <Po)- If I is the length of the wire in the spring, the total change of angle, i.e. the angle through which the inertia bar moves is y, = l(<p — <p Q ) i.e. C-Yt*- This is the value of the- couple due to the internal stresses, called into play by the strains in the wire, i.e. Iv = — j- v The periodic time T is given by i = ^1, 1 = 2*RN 4 or I8INR T = aTc^J^Y - 32k5NR 1 ~" r*T* n6 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Experimental Details. The spring is set up as in the last experiment, and a rod, say, of rectangular cross-section is clamped to the end of it, so that there is no free play between the end of the spring and the rod, and the centre of gravity of the rod is under the suspension. The ' inertia rod ' is then given a displacement in the horizontal plane, and the subsequent horizontal oscillations are timed. T is obtained by timing 50 complete swings in the usual way. r and R are carefully measured as previously described, and the total number of turns of wire in the spring, N, is counted. The value of I may be calculated from a knowledge of the dimensions and the mass of the bar (see p. 54). Determination of the Modulus of Rigidity and Young's Modulus for the Material of a Wire by Searle's Method To find Y, Young's modulus, or n, the coefficient of rigidity, of the material of a wire specimen by this method, the wire is Fig. 60 fastened to two identical rods, &J& lt A 2 B 2 , at their mid-points, as shown in the diagram at C^ C 2 . These rods are usually square or circular in cross-section, and are supported by threads from points T 1 and T 2 , such that the axis of suspension and the axis of the wire intersect at the centre of gravity, and the sus- pended rods are a distance apart, which allows the wire to be stretched in a straight line, the whole assuming an H formation. If now the ends, B x and B 2 , of the rods are drawn together and fastened by a loop of thin cotton, the wire will be bent into the arc of a circle, subtending an angle 2<p at the centre ; each rod will make an angle q> with its original direction. The suspension of the two rods being such that the torsion is negligible, the only couple acting on the bars is that due to ELASTICITY 117 the bending of the specimen, CjCj. With this method of suspension we have a couple produced equal to ^ in the wire (p. 88), where * is the ' moment of inertia of cross section/ and R is the radius of curvature of the arc C^. Fig. 61 Now, if I is the length of the wire between the rods, R~i. 2q> If r is the radius of the wire (assumed circular), • = *^ 4 ' whence the couple acting is 4 * I ~~ 2/ Let I x be the moment of inertia of the rod, kj& x , about the axis of suspension, -57J the angular acceleration. The moment of the external force is, by the theorem given on p. 53, 1-j^- This is equal and opposite to the restoring couple exerted by the bent wire. The equation of motion for the rod is therefore This is simple harmonic motion (p. 25), and T =2K J5T \*Yr« ii8 ADVANCED PRACTICAL PHYSICS FOR STUDENTS If the rod AjBj has a length 2L and width 2a and mass M grammes, M L 2 + a* whence Y = 8tt/ L 2 + fl2 M. y 4 T 2 3 To evaluate Y for a specimen of wire, the arrangement described is fitted up. The thin cotton loop holding the rods together at B X B 2 is burnt, and the resulting oscillations are timed with a stop-watch, T being obtained by timing 50 complete oscillations. An alternative method of finding T, and which may be used in many experiments where T is required, is seen under. A reference point having been chosen — say, a chalk -mark under the oscillating bar — the time is noted at which the counting of the swing is commenced. After 5 swings the time is again noted, and the times at the end of every 5 is recorded in a table as seen below. (I) NO. OF OSCIL- LATIONS (2) TIME (3) NO. OF OSCIL- LATIONS (4) TIME (5) TIME FOR 30 OSCILLATIONS U 30 a a — u 5 V 35 b b -V 10 W 40 c c — w 15 X 45 d d — x 20 sm 25 y z 50 55 e f e -y mean time for 30 oscillations. = Tabulating in the second column the time for the starting point, 5, 10, 15, 20, 25 oscillations, and in the fourth column the time for the 30, 35, 40, 45, 50, 55 oscillations, the difference in any line between the fourth and second column value will give the time for 30 complete oscillations. The mean value of the last column being taken as the time for 30, T may be evaluated. The radius of the wire occurs in the fourth power. The value of the diameter is therefore measured in several places, say, six, and the mean radius, r, calculated. The other measurement for the evaluation of Y are straightforward. It will be noticed that in this method the value of Y is obtained by timing, and not by observing deflections as in some of the ELASTICITY 119 other methods. Another advantage of the method is that it requires only a small specimen of the material. To find the modulus of rigidity of the specimen, the rod A 2 B 2 is clamped rigidly in a horizontal plane and the wire acts as a support for the rod AjBj, which may therefore be made to perform horizontal torsional oscillations. The rod is displaced slightly from its position of rest causing a restoring couple to act on it, due to the twist of the wire. For a displacement 0, the restoring couple is as established on p. 102. We therefore have as the equation of motion of the rod T d 2 d _ _ nnr 4 * dt* ~~ 2.1 This is once more a case of simple harmonic motion, whose periodic time, Tj, is given by \ = ^ 2/1 If the rod is of square or circular cross-section, the moment of inertia about the axis taken in this experiment is the same as the last case. If of rectangular section, the moment of inertia will be obtained by calculation as before. Tj may be found as described for the first experiment; the length I and the radius r are already known. Hence n may be found by substitution in the formula below : _ 8tc/I Determination of the Bulk Modulus for Glass The bulk modulus, as shown on page 82, may be expressed as k--L V where p is an increase in pressure causing the small volume change 6V. In general, it will not be convenient to apply a uniform pressure to a body, but it is a simple matter in most cases to apply an extending force per unit area, or a pressure, in one direction only. If, as a result of such a pressure, the change in volume is <$V l , then 6V 1 — — • 6V where <5V is the change in volume when the same pressure is applied uniformly over the body. If a cylinder be supported at one end, and an extending force 120 ADVANCED PRACTICAL PHYSICS FOR STUDENTS be applied to the other, in a direction coincident with the axis of the cylinder, we have the extending force at one end and the equal reaction at the other. The change in volume in this case is of the type 6V 1 above. So that if the change in volume of a body be observed in such a case, the value of k may be calculated. The apparatus used to determine the bulk modulus of, say, glass is seen in fig. 62. A glass tube g is cemented to two caps of brass, A and B. The upper brass cap A is provided with two pegs, which act as a support for the whole apparatus. The lower cap B carries a hook to which a pan S is attached. I? a B S Fig. 62 Fitting in the upper end of the tube is a rubber cork, carrying a capillary tube, CD, which is graduated and calibrated in the manner described on page 41. The tube g is filled with water. Care is taken to avoid trapping air, and the cork, etc., is placed in position, resulting in a little water rising in the capillary tube, i.e. the whole of the glass tube and part of the capillary are filled with water to a definite level. If now the pan S is loaded with, say, 5 kilogrammes the volume of the glass will increase by a small amount. The difference in the levels on the calibrated tube enables the value of the volume change to be determined if the temperature remains constant throughout the observation. However, the apparatus is, by the construction, very much affected by small temperature changes ; it is a water thermometer with a very large bulb. To obtain a good approximation to the value of 5V 1 , the volume change, the reading on the tube is noted when S has no load ; 5 kilos are applied and the reading ELASTICITY 121 noted, the load is removed, and once more the scale reading is taken and the process repeated with 5 *& os ^ no load for 10 observations, each one being taken at a regular time interval as under. TIME FROM TIME FROM COMMENCE- LOAD READING COMMENCE- LOAD READING MENT MENT o min. o kilos. 5 min. o kilos. * 5 5* 10 I 6 I* 5 6* IO 2 7 • a* 5 7* 10 3 8 3* 5 8* 10 4 9 4* 5 9* etc. 10 A similar process is repeated with a load of io, 15, and 20 kilos. The results are plotted as a graph, using a fairly large scale for the ' scale readings ' as ordinates. O ^ Time in minute© . Fig. 63 The no-load curve will be a continuous line, and the short curves for 5, 10, etc., load will be at a vertical distance above it, depending on the load employed. The form of curve obtained in a particular experiment, using one type of glass, is seen in fig. 63. AA X , BB 1 , CC 1 are then drawn parallel to the axis of scale readings, and their values measured : the mean value of this 122 ADVANCED PRACTICAL PHYSICS FOR STUDENTS length gives the scale reading increase in volume of the glass tube g. The actual volume change is then obtained from the calibration curve of the capillary tube. This process is repeated for the curves giving the results for each load, the mean of the three ordinates being taken in each case. From each value of <5V the increase in volume SV per kilo load may be calculated. A mean value for all loads therefore gives the mean for all the observations taken, for the value of SV per kilo. The original volume of the glass tube may then be found by obtaining the product of the length between the brass caps and the area of cross-section. To obtain the cross-section, a little water is poured into the empty tube so that the surf ace becomes visible above the lower cap B. A measured volume of water is then run into the tube, so that the level is just below the upper cap. From a measure- ment of the increase in the level and the volume introduced from a graduated flask or measuring jar, the average cross-section may be calculated. Hence V is obtained. To find the stress applied to the tube we must find the internal and external radii of the tube. The load applied at the end causes a stress, a force per unit area, equal to /T -. m ^ — r-, where 7U(R 2 — f 2 ) m is the mass, R the external and r the internal radius of the tube. Now, Ttr 2 was determined above, when obtaining V. To find R, the external radius is measured with callipers at about 6 or 8 places, and the mean taken. Thus, if 6V is the mean value of the volume change per kilo- gramme load, i.e. for a force of 981 x 10 3 dynes in one direction, the change for a uniform stretching force would be $dv, 981 x io 3 and K = -r— — dynes/cm a . Poisson's Ratio for India-rubber k A length of solid india-rubber is supported at one end and loaded at the other. The rubber, of. circular cross-section, is marked at about six places along its length, and the diameter at these selected places is measured with a micrometer screw for each load placed in the scale pan at the free end of the india- rubber. ELASTICITY 123 The length of the specimen is also measured for each load. Then, since Poisson's Ratio has been denned as Lateral contraction / _ Original diameter ^ ~ Longitudinal extension ,. Original length (i may be calculated for each load, the mean value from the six measurements in the lateral direction being used. For a rubber cord of about f-inch diameter, suitable loads would be 500, 1000, and 2000 grammes. It will be found that, to some extent, the values obtained depend on the history of the specimen. The results when the load is increasing will be somewhat different from the values for the same load when decreasing. In all observations the readings should not be taken until about ten minutes after the adjustment of the load. CHAPTER IV SURFACE TENSION The surface of a liquid acts in many respects in a manner analogous to a stretched membrane. The well-known example of mercury resting on a clean wooden surface shows the effect to a marked degree. The mercury takes the form of a globule, as if it were surrounded by a membrane supporting it in this form. The examination of a water drop slowly formed at the end of a glass tube or tap from which it emerges provides another example of this phenomenon. The water in this case accumulates, as though it were collected in an invisible membrane, until of a definite size, when it is detached as a spherical drop. D i B A. tff Fig. 64 B These effects are due to forces existing in the surface of separa- tion of the liquid from the air and the other media in contact with it. The effect is generally known as surface tension. This term may be defined in two ways, depending on the point of view taken. If we imagine the surface of the liquid to be cut by a plane, there is a definite force per cm. acting on the line of intersection, at right angles to its length and parallel to the surface. This force, expressed as dynes per cm., is defined as the surface tension of the liquid. The value of this force depends on the liquid and the surrounding medium, but, unless otherwise stated, will be taken throughout this book to represent the force per cm. when air is the medium. For example, consider a frame ADEB, fig. 64, of width I cms., across which a film of liquid, whose surface tension is T dynes per cm., is stretched. If the film terminates at the lower end on a light rod AB, 124 SURFACE TENSION 125 since the liquid film has two surfaces, it will exert an upward force of 2T/ dynes on the rod. If the rod has a mass of m grammes it will be in equilibrium when 217 = mg. Now, suppose the rod be displaced a small distance 8x, to A*B 4 , against the surface tension forces, the work done is 2TI • dx, and the resulting increase in area of the surface is 2I • Sx. Thus, the work done per sq. cm. of surface is 2Tl6x „ ^^= Teigspersq.cn,., which may be taken as defining surface tension, i.e. the surface tension is the work done in enlarging the surface by one cm?" When a liquid is placed on a horizontal plane surface, the form it takes depends, for a given liquid, on the material of which the plane surface is made. Thus, if water is placed on a clean glass surface it spreads over it, whereas if the glass is greasy the water takes the form of globules. Bl u v-— < 1 • A Fig. 65 The angle contained between the plane surface and the liquid surface is different in each case. If we measure this angle in the liquid we have a measure of the angle of contact. Thus, in fig. 75, showing a section of a mercury drop on a glass surface, is the angle of contact, whereas for a liquid like water which ' wets ' the glass the angle of contact is zero — the water spreads over the surface. METHODS OF MEASURING THE SURFACE TENSION OF A LIQUID (1) Wilhelmy's Method To determine the approximate value of the surface tension of such a liquid as water, paraffin oil, or turpentine, the following method may be employed. A clean wire, preferably platinum, is bent into the form shown in fig. 65, making three sides of a rectangle of breadth / cms. 126 ADVANCED PRACTICAL PHYSICS FOR STUDENTS It is then suspended from a beam of a balance by means of a thin wire, and counterpoised when the upper horizontal arm, B, of the frame is almost immersed in the liquid. When a balance is obtained the frame is dipped under the surface of the liquid by lowering the beam of the balance, and then withdrawn again. A film of the liquid will be formed in the frame as seen in the shaded part of fig. 65. Due to the downward surface tension pull on each side of the film of length I there will be an apparent increase in weight = 2IT. This can be found experimentally by adding ' weights ' to the other scale pan> until on raising the beam it remains horizontal. Fig. 66 To perform this approximate experiment clean the frame by holding it in a Bunsen flame until red hot. Repeat the experiment as described above several times, and take the mean value, whence if m is the mean value of the added mass 2T/ = mg, T = — j- dynes per cm» Take care that the frame is the same height above the water surface when the balance is made, before and after immersing in water, to eliminate buoyancy errors. The surface tension effect on the two vertical limbs is eliminated, but the experi- ment may be regarded as one which gives an approximate value of T as described above. (2) By Weighing Drops The liquid whose surface tension is to be measured is allowed to form drops at the end of a narrow tube, C, fig. 67. If m is the mass of the drop, we have mg = KT, where K is a constant. SURFACE TENSION 127 From a simple approximate investigation of the case we see that mg = 7tfT as under. When the drop is about to break away from the tube we will assume it has the cylindrical form shown in the firm lines, D. The broken line indicates one of the subsequent forms. If r is the radius of the orifice, we have, at this level inside T the drop, an excess of pressure equal to — due to the cylindrical curvature of the liquid surface. This is equivalent to a downward T force — • T?r 2 =Tnr. J y The weight of the drop being mg, the total downward force is T7rr + mg. This is equal to the upward surface tension force over the circle of contact, i.e. T • 2nr, or T2rcr = jzTr + mg, or Tnr = mg. Fig. 67 This is deduced, assuming static conditions to hold in the actual case. To a closer approximation Lord Rayleigh showed that mg = 3- 8/T gives an expression of the relation between T and m. In practice the uncertainty may be avoided by taking the expression T = \f- and finding the value of K for the particular tube used. A tube is drawn out to a fine capillary of about *5 mm. at the end. It is connected to a burette by means of a short length of rubber tubing. The burette and tube are of course first made thoroughly clean, as in all the surface tension determinations, by the method given on page 42. A liquid of known surface tension, say water, is placed in the burette and 128 ADVANCED PRACTICAL PHYSICS FOR STUDENTS the tap is opened to an extent which allows the drops to form at the rate of about one every second. A large number, say ioo or more, are collected in a weighed vessel and m per drop is obtained, hence, knowing T, K may be calculated. The same tube is now used for the liquid of unknown surface tension, say paraffin oil. Some of the liquid is run through the apparatus to remove all traces of water, and the burette is then filled with an uncontaminated sample *of oil. The collecting and weighing is repeated, and knowing K, T is calculated. Find by the above means the surface tension of alcohol, paraffin oil, benzine, etc. Fig. 68 (3) Determination of the Value of the Surface Tension of a Solution in the Form of a Film The value of the surface tension can be obtained for, say, a soap solution film in the following way. Two pieces of copper wire, ABC and DEF, are bent as seen in figs. 68 and 69, so that AC = DF = about 4 or 5 cms. At the points A, C, D, and F a length of cotton thread is fastened, so that when the whole arrangement is suspended at B the thread takes the form of a rectangle about 5 cms. by 10 cms. (fig. 68). If now a film of soap solution be stretched across the thread, the latter will be pulled into the form shown in fig. 69, AD and CF taking the form of arcs of circles. Let AC = DF = 20 cms., GH = 26 cms., AD = CF = 2A cms., and the mass of the lower copper wire, DEF = m grammes. Let T be the value of the surface tension in dynes/cm. When stretched in this manner, there will be a tension in the thread equal to, say, / dynes. Suppose a is the angle which is included between the horizontal and the threads as shown in fig. 69. The film being two-sided, the vertical force upwards due to surface tension on the length 2a at the bottom of the film = 4«T. SURFACE TENSION 129 The resolved part of the tension / at D and F is / sin a upwards, i.e. total upward force of 2/ sin a. The downward forces are mg, the weight of the lower copper wire. We thus have for equilibrium : mg = 2/ sin a + 4«T (1) Considering the equilibrium of half of the film as obtained by a vertical dividing line through the mid-point of the film, we have the following equal and opposite forces acting : Fig. 70 (a) A force '/ cos a ' at *^ e to P an ^ an e(ma * one at * ne bottom = 2/ cos a. (b) due to the two-sided film a surface tension effect = 4™, i.e. 2/ cos a = 4TA (2) From (1) and (2) we have tan a = _ mg — 4aT 4Th (3) 130 ADVANCED PRACTICAL PHYSICS FOR STUDENTS From the geometry of the case (see fig. 70), if O is the centre of the circle of which CHF is arc, a = z. OCIC So that tan a = ^§» and if HK — d = a — b, r -d tan a = h Eliminating r since h 2 = (2r — d)d, h 2 + d 2 r = 2d h 2 -d 2 We have tan a = ,, 2dh Equating this to the value given in (3) we get mg — 4aT ^ h 2 - d 2 4T& ~~ 2dh 2 d(mg) = 4T(A a - d 2 ) + 4*T • 2d, whence T = /T . ,f -, 2(h 2 — d 2 + 2ad) Putting d — (a — b) Using the very simple apparatus described, a fair value of T maybe obtained simply from a knowledge of a, b, h, and m. A film is stretched across the string very easily by placing the frame horizontally in a flat dish containing the soap solution. Care must be taken to avoid excess of soap solution spreading over the part of the apparatus other than the thread. None must be allowed to remain on the lower copper wire, for obvious reasons. The measurement of the dimensions of the film can be made by means of ordinary dividers. The method does not justify the use of a travelling microscope. The measurements should be taken fairly rapidly so that they are all obtained before the condition of the film changes appreciably. In an example 2a = 3-5 cms., 26 = 1-2 cms., 2h = 11-7 cms., m = 1-51 grms., which gives T = 23 dynes/cm. (4) By Measurement of the Rise 0! a Liquid in a Capillary Tube If a clean, fine-bored capillary tube is depressed into a liquid Which ' wets ' it, and is then clamped vertically, the lower end SURFACE TENSION 131 of the tube being just below the surface of the liquid, it will be found that a column of the liquid remains in the tube, so that the surface in the latter is a height h cms. above the free surface of the liquid in the vessel which contains it. Suppose r is the radius of the tube and p the density of the liquid. The forces acting on the liquid in the tu£>e are : (1) The weight of the liquid. This is equal to (the volume of the liquid) X gp. Now the volume of the liquid is equal to V = nr 2 h + (volume of the meniscus) for a uniform tube. If r is small the meniscus is practically hemispherical, hence i.e. the downward force is V = *r*h-+ \{nr 2 ) r --t^ 3 ' 3 j 7tr '(* + 0*p (5) (2) The Upward Surface Tension Force The line of contact is the intersection of the glass wall and the liquid surface, i.e. a circle of radius r. If a is the angle of contact the upward force, from the definition of surface tension, is equal to 2-nr • T cos a (6) For equilibrium, the forces (5) and (6) are equal and opposite , 2tztT cos a = itr 2 f h + - Jgp, i.e. or T = gpr =H> 2 cos when the liquid wets the glass as in the case of water and clean glass a = o, ^ T -££(»+£) (7) To find T, using a capillary tube, a glass tube is cleaned thoroughly. This may be done by using nitric acid (in which the tube is boiled) and caustic soda ; the tube is washed in tap water and dried in alcohol and ether, or the glass tube is allowed to stand for several hours, overnight if possible, in a concentrated solution of sulphuric acid (one part) and potassium bichromate (one part). It is then washed in tap water and dried. It 4s not advisable to use distilled water. 132 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The tube is then heated and drawn out to a capillary. A length of uniform bore is chosen and clamped vertically in a vessel which is brimful of the liquid whose surface is just above the top of the containing vessel, as shown in fig. 71. Care is taken to avoid touching the tube or the liquid in this adjustment, for even small traces of grease cause a large variation in the value of the surface tension. The tube is viewed by means of a travelling microscope, provided with a vertical traverse, the lower end of the meniscus is focussed and the vernier reading of the microscope noted. The free surface of the liquid in the containing vessel is next focussed. If the liquid surface is just above the top of the vessel, this level may be viewed very readily, and from the vernier reading on the microscope scale in this position the value of h may be obtained. 1 1 —4 Fig. 71 To find r, the tube is broken at the point at which the meniscus rested, and viewed horizontally by the microscope. By arranging the cross-hairs »in the eyepiece to be tangential in turn to the two ends of a diameter, the internal radius may be measured on the vernier attached to the traverse. Alternatively, a weighed amount of mercury (Density, D =13-6) may be introduced into the tube and its length observed by means of the microscope, when the tube is horizontally on the bed of the microscope. Hence r, the mean radius of the tube, may be found if I is the length occupied by the mercury of mass m, for {nr 2 )lD = m. The experiment is repeated, using tubes of various diameter, and a mean value of T obtained. An alternative method of measuring h, which also overcomes the difficulty of viewing the liquid through the glass beaker, is one which makes use of a pin bent twice at right angles so SURFACE TENSION 133 that the point is displaced about one centimetre from its original position. The pin is attached to the capillary tube by means of a rubber band (see fig. 72). The point is in this way well removed from the curved surface of the liquid round the tube itself. It is adjusted to coincide with the free surface of the liquid. , . . . The image of the upper end of the pin is brought mto coinci- dence with the cross-hairs of a microscope, and the vernier reading on the vertical traverse is noted. The meniscus is next viewed, and the distance between it and the pinhead determined by the subtraction of the vernier readings. A subsequent measurement, by means of the microscope, of the vertical distance between the pin point and head enables the value of h to be obtained. Fig. 72 (5) Surface Tension Determination from Measurements of Bubbles This method is suitable for measurements of surface tension of soap solution. Inside any curved film in equilibrium there is an excess of pressure over the outside, by an amount which depends on T, the surface tension, and R, the radius of curvature.* Consider a bubble with excess pressure p inside : take a section through the centre ; then there is an equivalent force over this section = p X (area) = ^>tcR 2 . The opposite surface tension forces, since there are two surfaces to the bubble, is 2T x (circumference of section) = 2T-2tuR. * It can be readily shown that inside a cylindrical./*/** (which has two surfaces) 2T this pressure excess == -j^- If now the film has an equal curvature in the other direction at right angles— 2T making spherical bubbles, there is a further pressure excess of ^, making a total ±£ = p the excess of pressure. 134 ADVANCED PRACTICAL PHYSICS FOR STUDENTS For equilibrium these are equal and opposite, i.e. 4tcTR = ttR 2 £ P = 4T R A suitable form of apparatus with which to obtain a measure- ment of p and R is seen in fig. 73. This consists of a fairly wide U-tube, containing water, sealed to a T-piece, T. One arm of the T-piece is bent at right angles and terminates at B, where AB is parallel to the U-tube limbs. To the other arm at C is attached a piece of rubber tube, R, having a glass rod, P, which just fits it, and which can act as a piston. P c^* / /^ T " A V-> D F Fig. 73 B -D- The piston P is withdrawn and the end B immersed in the soap solution ; then P is advanced slightly, causing a bubble to form at B. The excess of soap solution is drained off this bubble by touching any excess with a pencil, or the side of the^ dish which contains the solution. The rod is then further advanced, so that a bubble of, say, 1 to- 2 cms. diameter is blown. It will be noticed that the water in the U-tube takes up a position as in the diagram indicating a pressure inside the bubble in excess of the atmospheric pressure. This difference in level may be measured, using a microscope with horizontal and vertical traverse. Having read this, the microscope is moved until the bubble is focussed, first the image of one side, and then the other being brought in coincidence with the cross-hair. The difference in reading of the vernier of the horizontal traverse giving the dia- meter D, of the bubble, p = hg dynes per sq. cm., for a water manometer. R-5. Hence T = teD SURFACE TENSION 135 (6) Quincke's Method This method is most readily followed for the determination of the surface tension of mercury. A large flat drop is formed on a horizontal platform, and from its dimension, as measured by means of a travelling microscope, the value of T, the surface tension, and 0, the angle of contact of mercury on the platform may be calculated. £ $ --B--AB Fig. 74 We will assume that the radius of the flat drop, R, fig. 74, is large, so that the drop is truly flat, i.e. the pressure at a point just above and just below the upper surface is the same. Consider a section as in fig. 75, obtained by cutting it by a vertical plane passing through the centre, LMRN, and two parallel vertical planes, AINL and ESRM normal to LMRN. Let ACDE be the horizontal plane of maximum area in the drop, so that a tangent plane at AE is vertical. M T ;« ^*^s L sfff/// /Ve /^T "-- C VV 'ff " " I N Fig. 75 By considering the forces on the upper portion, AECDLM, which is in equilibrium, a value of T may be obtained in terms of the dimensions. The horizontal forces acting on this portion of the drop due to the remainder of the drop are (a) T-LM, left to right, (p) The hydrostatic pressure over the surface LMDC, from right to left varies from o at the upper surface to gph 1 at CD, and is of a mean value " — , i.e. total force is ^— 2 2 X area LMDC, where p is the density of mercury, i.e. equating (a) and (p) T-LM = £- p - l • JMh\ 2 T== ^i 2 2 or .(8) 136 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Considering now the equilibrium of the whole slab, we have in addition t© the corresponding terms, T • LM and ^--, where 2 A is now the total height (fig. 74), a term T- IS • cos a= -T« LM cos (fig. 75), from left to right. or T-LM(i -cos«) =££^LM 2 or, substituting the value of T from (8), . 6 h sin — = — -==— - (q) The experimental details necessary to carry out such a deter- mination are as follows : A clean glass slab, provided with three levelling screws, is first of all arranged horizontally, by means of a spirit-level. Mercury is placed on its surface in the form of a circular drop (fig. 76). -*- ^ Fig. 76 N.B. — The mercury should be as free as possible from impuri- ties. To ensure this it should, if possible, be submitted to one of the methods of purification described on page 532. To obtain the values of h 1 and h, use is made of a travelling microscope, which is provided with a cross-hair. The cross-hairs are arranged so that one is vertical in the eyepiece ; this can be done by viewing a thin wire plumb line. The edge of the drop is then focussed. When the vertical cross-wire is tangential to the image of the side of the drop and the intersection of the cross-hairs is at the point of contact, such as A, figs. 74 and 76, the position on the vernier of the microscope is noted. Then by making use of the vertical movement the upper and lower surfaces of the drop can be focussed and the corresponding vernier reading will enable the values of h x and h to be ascertained. Note For liquids which wet the surface of the glass a simple modifi- cation enables the method to be utilised. SURFACE TENSION 137 A concave lens of over one metre radius of curvature is supported concave side downwards, on three legs inside a glass box, having one plane glass side. The box is filled with the liquid, and an air bubble is blown under the concave surface, using apparatus similar to that shown on page 141. A narrow bent tube is connected to R in place of the U-tube, etc. shown in fig. 79. The slight concave surface enables this air bubble to be blown without much trouble. The section of such a submerged bubble would be similar to that shown in fig. 74, when inverted. Then, if h 1 is the distance between the lower surface of the large air bubble and the plane AB, and if h is the total thickness of the bubble as measured from the lower surface to the plane of contact with the lens, all measured through the plane glass window by means of a microscope, equations (8) and (9) will give T and for the liquid in the box. (7) Rayleigh's Method This method depends on a measurement of the wave length of ripples formed on the surface of the liquid whose surface tension is to be determined. The velocity v of a harmonic disturbance on the surface of any liquid is given by* „= !A g+ ^I, (10) N2« 5 ^ Xp when x is the length of the wave, T the surface tension, p the density of the liquid, that is » = ^+£- T ) (I0 «> The surface tension is therefore seen to increase the effective 4.7C 2 value of g. For large waves the term ^--T can be neglected, x p and the velocity of propagation is <J-s. For waves of less than 1*5 cm. wave length, the value of the term involving T becomes more important, and when X is suffi- ciently small the velocity becomes more nearly equal to -^ Xp i.e. the first term in equation (10) becomes negligible cf. the second when X becomes very small. Equation (10) shows that when x = o, v = 00 , and when * See Poynting & Thomson's " Properties of Matter." 138 ADVANCED PRACTICAL PHYSICS FOR STUDENTS X = oo , v = oo , between these values of x there is a minimum value of v, corresponding to a value of x, which is obtained when _X _ 2ttT 2n S ~~ Xp ' (T~ X = 2n-J— ; (n) this mmimum velocity is therefore '-GJ V2 For water x OT = 17 cms. from equation (11), for T =75, p=^i, g = 981, and hence v m = 23 cms. per sec. Waves having a smaller wave length than this critical value which corresponds to the minimum velocity are called ripples, and the more important term is the one involving T. Such ripples cannot be viewed and X measured directly, but the following method, first used by Lord Rayleigh, may be employed with success. The liquid is placed in a large flat dish, say, a large porcelain developing dish. As in all surface tension experiments, the dish is cleaned thoroughly before introducing the liquid. The ripples are made by having a tuning fork electrically maintained, arranged at one end of the dish as in fig. 77. Attached to one prong of the fork and dipping in the liquid is a thin sheet of aluminium foil, D. If the frequency of the fork is n, and X the wave length of the ripples we have v = n\. But the ripples cannot be directly observed and measured, as the phase change is rapid, and a general illumination results. If these ripples are viewed by intermittent light, the frequency of the flashes of illumination being the same as the source of the ripples, then they appear stationary and may be measured. (Cf. stroboscope, p. 404.) Alternatively, if the observer has intermittent views of the surface of the liquid, such views being of same number per second as the vibration number of the tuning fork which causes the ripples, between each view the ripples will have moved forward a distance equal to a wave- length, and if illuminated by a constant source of light they will appear stationary. To obtain such intermittent glimpses of the surface, the latter is observed through the prongs of a second tuning fork of the same frequency, and maintained in vibration by the same circuit. Two thin pieces of aluminium foil, A and B, are attached to the SURFACE TENSION 139 prongs of the second fork, so that direct vision is impossible when the fork is at rest, but is obtained when the prongs of the fork are at the position of extreme separation. Thus, for each complete vibration of the fork a view is obtained of the surface, and an apparently stationary train of waves is seen when viewed in this way. The wave length can be obtained by direct measure- ment of the longest possible number, m say, by dividers adjusted over the surface. When performing this experiment with water, the measurement will be found to be somewhat complicated due to shadows cast by the water, and the exact setting of the dividers over the surface for the stretch of m waves will not be as easy as in the case of a more opaque liquid. Fig. 77 A method which has been found to give satisfactory results is to suspend an incandescent bulb about two metres above the surface of the water. This casts a series of shadows on the bottom of the white porcelain dish. The dividers are arranged also to cast a shadow and adjusted so that the two ends of the dividers' shadow coincides with the corresponding parts of the first and last of the m waves. In such a case the magnification of the shadow of the waves due to the obliquities of the incident beam is compensated by a corresponding distortion of the shadow of the ripples, and the kind of thing shown in fig. 78 results : the length of, say, ten or more ripples may be obtained, whence x may be calculated. Now, if n is the frequency of the fork, or T=^(2M a X*-£). 140 ADVANCED PRACTICAL PHYSICS FOR STUDENTS A suitable experiment by such a method is to find the variation of surface tension of a salt solution with concentration. Use first of all pure water. Make sure that the aluminium plate is clean, and that no soft wax, by which it is fastened to the tuning fork, adheres to it, and so contaminates the water. Measure the value of X and calculate T, n being known. A suitable frequency for the two tuning forks is about 60. Then repeat the experiment with a sodium chloride solution having \, 1, ij, to 5 grm. molecules per litre, and plot a graph showing the increase in the value of T with concentration of the solution. Fig. 78 (8) Jaeger's Method In this method air bubbles are formed in the liquid under investigation. The bubbles in this case have one surface only, and therefore the excess of the internal pressure over the external assuming a hemispherical form is * 2T The value of R is fixed, and equal to the radius of the orifice from which the bubbles emerge (approximately). The following apparatus, as seen in fig. 79, is used. A thin glass tube, C, drawn out to a fine capillary of less than *5 mm. ('2 to *5 mm. is satisfactory), and is fixed vertically in the liquid. It is connected either rigidly, as in the diagram, or by means of a piece of india-rubber tubing, to glass tube, AB, which terminates in another rubber tube, R, which is clamped in a screw pinch cock, T. Sealed into AB is a glass T-joint which leads to a manometer U. Air is sent in at D at a steady rate, and consequently, bubbles are formed in the liquid. To ensure that they are always formed at the same depth, h cms., below the surface of the liquid, a scratch is made on C, and this is adjusted to coincidence with the liquid surface. When the bubble is of the same diametei as the orifice at G, SURFACE TENSION 141 it becomes unstable and breaks away. During the formation of the bubble the manometer rises and the maximum difference in heights between the two columns, H cms. , is observed for many bubbles, the pressure of the incoming air being constant ■ * When a good agreement for H for a series of bubbles is obtained, the tube C is taken out and the value of R determined by viewing the end with a microscope. If the capillary is very small, a microscope carrying a $mall scale in the eyepiece is used, and the value of the scale divisions in the eyepiece scale determined by comparing with the image of a small mm. scale viewed by the microscope. B R Dtf /*%& d^ T CL-jl U > \ B i Fig. 79 The tube is rotated and other values of the ' diameter * taken. These measurements should be taken before commenc- ing the experiment to ensure a regular tube. A good method of producing the stream of air is to allow water to enter, one drop at several seconds interval, into a Winchester, closed at the top by a cork, which is waxed into the neck of the bottle and provided with an entrance and exit tube. The types shown in figs. 79 and 80 serve very well : the air is trapped inside the bottle by water in the bottom of the bottle in the form shown in fig. 79, and in the U-tube entrance in the type illustrated in fig. 80. In this way the pressure gradually rises until of sufficient magnitude to form a bubble of maximum diameter. This detaches and an interval then occurs, during which the pressure again rises, as is seen in the manometer U, and the whole process is repeated. By slowly increasing the pressure, its maximum value corres- ponding to H cms. difference in the manometer may be easily *It -might be. found advisable to draw out the open end of the manometer U, to a capillary tube, to damp the oscillation of the liquid. 142 ADVANCED PRACTICAL PHYSICS FOR STUDENTS measured either directly on a scale or by means of a travelling microscope. If only one large bottle is available, air is compressed into it and the outflow tube of rubber connected to B. By varying the compression on this tube by means of a screw clip the condi- tions described above may be obtained. Let H be the maximum pressure difference established in the manometer, and h the depth at which the orifice is placed below the free surface of the liquid, then the pressure of the air inside the ^bubble, when the latter is about to leave the orifice, is n + gaH, where n is the atmospheric pressure and a the density of the liquid in the manometer. "3 \ / k VW/ Fig. 80 The pressure outside the surface of the bubble is n+ gph, where p is the density of the liquid under investigation. Then, P being the excess of pressure inside the bubble over that on the outside 2T p = (n -f-goH) - (n + g?h) = -^ , or* T = ?£(aH-p/*) (12) The values of R, H, and h, are obtained as already described, and T is calculated. The method is a good one for obtaining relative values of surface tension. As an example, find the variation of surface tension with concentration for a salt solution containing from o to 5 gramme molecules per litre. * The result expressed in equation (12) above is developed assuming the bubble formed is hemispherical and of the radius of the tube. Ferguson ("Phil. Mag.," No. 28, 1914, page 128 et seq.) has deduced an expression for T without making this assumption and arrives at the result : where I »Vb/ SURFACE TENSION 143 By altering the temperature of, say, water in the beaker, the change of surface tension with temperature could be obtained. (9) Capillary Tube Method (Sentis). A capillary tube is drawn out to about *5 mm. bore as in Jaeger's method. It is immersed in the liquid under investiga- tion, and then withdrawn and clamped vertically. Some of the liquid will emerge at the lower end and form a drop as shown in fig. 81 (1), so that the distance from A, the meniscus in the tube, to B, the lowest point of the drop, is h x cms., and MN is zr. If now the lower end of the tube is surrounded by a vessel, C, containing the liquid, the column will fall in general, but the meniscus may be brought to the original level by raising C until the free surface of the liquid in the beaker is h 2 cms. below the meniscus, fig. 81 (2). Fig. 81 From a knowledge of h x , h 2 , r, and p, the density of the liquid, T, may be Calculated from the formula T=f{r(/> 1 -^)-^.. To establish this formula we may assume that the portion of the drop, shown in section as MONB, is hemispherical. This approximation is a safe one when the radius of the capillary tube is small, and r small compared with h v Consider the forces acting below the horizontal plane of maximum area shown in section as MN. The length OB = r, and hence the column from MN to the meniscus is (h t — r) ; of this a length, h 2 , is supported by the upper surface tension forces as shown in fig. 81 (2), and hence the pressure at MN due to the liquid column is g?{h 1 — h 2 — r), which contributes a downward force g9{h 1 — h 2 — r)nr 2 . The 2 weight of the hemisphere - T:r 3 gp also acts downwards. 144 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The surface tension acts vertically in the circle of section of the drop and plane MN, and has a value T • 2izr ; hence T27tr = gpfa — h t — r)*r* + -*r 3 gp, To make a determination of T for a liquid, the cleaned capillary about 25 cms. long is drawn out to a radius of about «5 mm. at the one end, and is almost entirely submerged in the liquid, so that the latter fills the greater part of the bore. It is slowly withdrawn and is clamped vertically. By means of a travelling microscope, the length MN = 2t is measured and the microscope is then focussed on the meniscus. A small glass beaker is raised under the tube, until the liquid in the beaker just touches the drop. The reading of the micrometer screw, which raises the platform carrying the beaker, is noted. The beaker is then further raised until the meniscus, as viewed by the fixed micro- scope, again acquires its original level. The micrometer screw reading is again noted. The difference between these two readings is equal to (h x — h t ). P is determined in the usual manner, whence T may be calcu- lated from the formula deduced above. If the form of adjustable table with micrometer or vernier attachment is not available, some simple convenient method may be devised for the measurement of (h x — h t ). For example, two microscopes may be used. With the first the value of MN is observed, and then the point, B, is viewed and its image brought into coincidence with the cross-hairs; the second microscope is adjusted until, viewing the tube conveniently at right angles to the first, the image of the meniscus is in coincidence with the cross-hairs in the eyepiece. The beaker, C, is then introduced and adjusted until the meniscus is again as before, producing an image in coincidence with the cross-hairs of the second stationary microscope. A pin is adjusted to coincidence with the free surface of the liquid in the beaker which is then removed. The first microscope is then moved a distance which is measured on the vernier scale attach- ment, until an image of the point of the pin is in coincidence with the cross-hairs, the vertical distance moved by the microscope is {h x — h 2 ). Care is, of course, taken that the capillary tube does not move during the experiment. SURFACE TENSION *45 (10) Anderson and Bowen's Method i A method by which to determine the value of the surface tension of a liquid, and the angle of Contact with glass was described in the " Philosophical Magazine," April, 1916. Another method not so readily adaptable to general laboratory use is seen in the same magazine, February, 1916. A small rectangular sheet of thin cover glass is cleaned (by standing it in concentrated sulphuric acid and potassium bichro- mate, etc. as described previously) and dipped into the liquid whose surface tension is to be measured. It is withdrawn and clamped vertically. The liquid takes up the form shown in the diagram, fig. 82.* Ar N O i O l h Fig. 82 The drop has two curvatures, making the equivalent of a cylindrical lens, concave at the upper half and convex below, the centres being at O and O 1 . The upper limit of the drop may be at O or any point, N, above. The drop is tangential to the glass plate at O. If A is the focal point of the concave lens, OA the axis, f x the focal length, B the focal point of the convex lens, and OB the axis of the lens of focal length, / 2 , OO 1 = r, = P Pi P* n T surfaces h cms. the radius of curvature of the concave (assumed symmetrical and equal), — the radius of curvature of the convex surfaces, = the refractive index of the liquid, = the density of the liquid, = the pressure in the liquid at O, = the pressure in the liquid at O 1 , = atmospheric pressure, = surface tension of the liquid. * It was established by the original experiment that for water, glycerine, olive oil, and turpentine, the angle of contact is zero, and hence the form of fig. 82 represents the section of the drop. 146 ADVANCED PRACTICAL PHYSICS FOR STUDENTS We have, as the lens is a thin one, using the lens formula /■>(!*- *) (7 ~ 7). (page 278.) 7i = { *- x) k (I3) rr { *- J) k --m Since the pressure inside a cylindrical surface is greater than the pressure outside by p = T Q-), (see Poynting and Thomson's " Properties of Matter"), and here we have a cylinder of radius r, i.e. R x = r ^ = U -J- 1 (i5) ^ = n+l .. (l6 ) Now p2-pi= gph, Pt ~ Pi = T ( - 4- - J by (15) and (16) above. Hence t(J- + ±-J = g9 h. But by (13) and (14) above 1 _ 1 1 _ 1 *1 2/lOx-l) ' y2~2/ 2 ((X -I) 1 also T i.e. 2(|* ^)(7x + 7 2 ) = ^ (17) or T - 2gp ^^ ~ I ^ 1 ^ 2 (/i+/ 2 ) If one side only of the glass sheet is wet, using the same notation we have T _ gP%-i)A/ 2 (/!+/■) (l8) The apparatus used to obtain T, is a collimator illuminated by a sodium flame, and adjusted to give a parallel beam. The light passes through the cover glass and the liquid lens, and is viewed by a low-power microscope provided with a vertical traverse, and a traverse parallel to the axis of the microscope. The usual type of travelling microsope will need a littie modification to make this latter condition possible. If a parallel beam of light be sent from the collimator from SURFACE TENSION 147 left to right, and normal to the plate, a virtual image of the horizontal slit will be formed at A by the upper half of the liquid lens. The distance OA may be measured by using the low-power microscope, arranged with its axis parallel to the direction of the incident beam. The microscope is first focussed on the image at A, and then moved backwards a measured distance until the glass plate is in focus. The distance moved being = OA = f v In the same way OB and/ 2 may be measured by noting the difference in reading of the microscope when the glass sheet is focussed and then when the image of the slit is coincident with the cross-hairs. It will be found most satisfactory to use one side of the cover glass only, i.e., dry the other side before making the observation, and allow the incident beam to fall on the dry side. As the incident beam is parallel, it will be found, of course, that the observing microscope must be moved in a vertical x direction to enable a focus of first A and then B to be made. This distance, h, is noted. The refractive index and the density may be obtained from tables, or by one of the many methods available. Thus, having measured / x , / 2 , and h, knowing jx and p, T may be calculated for the liquid used. Measure in this manner the surface tension of water and turpentine. The Variation of the Surface Tension of a Liquid with Temperature The variation of surface tension with temperature may be obtained by Jaeger's method, which enables a good comparison of the relative values of the surface tension at different tempera- tures to be made. The details of the experiment are as described on page 140. The bubbles in this case are formed in the liquid at different steady temperatures, and H is determined for each. A large beaker is rilled with the liquid, say water, and heated to about 90°C, and then allowed to cool. The value of T being obtained every io°C. The liquid is well stirred before each observation, and if a large volume is taken will remain sensibly at the same temperature throughout the observation. A curve is plotted, showing the decrease of T with temperature. Capillary Tube Method The decrease of the value of the surface tension may also be investigated by a capillary tube method. Either form of appara- tus shown in fig. 83 or fig. 84 may be employed. One form of apparatus is filled with the liquid and immersed in a water bath whose temperature can be regulated either by a 148 ADVANCED PRACTICAL PHYSICS FOR STUDENTS thermostat or by manipulation of the Bunsen burner and a stirrer. The bath is raised to boiling point and allowed to cool, so that at about 90-95 °C. the whole of the apparatus and contents are at the temperature of the bath. The difference in level TJ Fig. 83 B between the two surfaces, A and B, is measured as quickly as possible with a travelling microscope. This process is repeated at different temperatures, say, every io°C, and from a knowledge of this difference in level, h and the radius of the tube as measured B Fig. 84 by the method given on page 132, the value of T at each tempera- ture may be calculated and a graph representing this relation plotted. The value of the density (p) is obtained, for each temperature, experimentally, or from tables, and is used in the evaluation of T by the above methods. The essential to success in this and every other surface tension experiment is that the glass, etc., is clean. CHAPTER V VISCOSITY WpEN adjacent layers of a fluid move with a relative velocity, forces, known as viscous forces, are brought into play tending to reduce this relative movement. If we consider a fluid whose upper layer is moving with a velocity v in a fixed direction, the state of affairs shown in fig. 85 will be reached, where intermediate layers, between the upper layer AB, which has a velocity v, and the lower layer CD, which is at rest, have a velocity shown by the arrows. L*ve» .in N/v©-no«N» Fig. 85 The force, F, acting on any area in a plane at right angles to the diagram, and parallel to EF, is proportional to the area A ; v and to the velocity gradient, in the case taken -y 1 i.e. at constant temperature. F oc A x (vel. gradient). Taking the normal to EF, in the plane of the diagram, as a y axis we have F-'.A*. W where t\ is a constant for the liquid and is called the coefficient of viscosity. In the case of a liquid flowing down a tube, the axial stream is moving with a definite velocity and the layers in contact with the walls of the tube are at rest and, provided that the pressure difference which is causing the flow is not too great, the result is the regular type of motion already considered. If the pressure exceeds a certain limit, the liquid no longer proceeds in this regular manner, i.e. no definite stream-line flow takes place. The result in this case is called turbulent motion. 149 150 ADVANCED PRACTICAL PHYSICS FOR STUDENTS We will assume, in the experiments that follow, that the pressure applied is below this critical pressure, and that the motion is therefore regular. The Determination of the Coefficient of Viseosity for a Liquid by Obser- vation of the Flow of the Liquid through a Tube The value of the coefficient of viscosity of a liquid, such as water, may be obtained by measuring the quantity passing per second x through a tube of uniform radius, when a definite pressure difference exists between the ends of the tube. Consider, as in fig. 86, a section of such a tube, whose radius is R cms., and imagine a thin tube of liquid in it of radius, r, and thickness dr ; the area of cross-section of such a tube is 2nr -dr. Fig. 86 If P be the pressure difference between the ends, then the force actmg on the tube ABCD, due to this pressure, is P • 2nr • dr. Over the curved area of such a hollow cylinder there are viscous forces which are of a magnitude dependent on the value of the distance r from the axis of the tube. Fig. 87 Fig. 87 shows the form of distribution of velocity of the layers for various values of r ; as shown later, equation (3), the curve of fig. 87 is a parabola, and hence j- is proportional to r, and therefore the viscous force, F, is also proportional to r by (1). dr ex P ress the rate of cn ange of this force with r, then VISCOSITY 151 the value of the difference between the magnitudes at the two dF surfaces of the thin tube ABCD, since the thickness is dr, is j- • dr. In steady flow we have, therefore, the equal and opposite forces, which may be expressed as : -($-> 2izrP - dr, -^=27^ (2) dr Now equation (1) gives The area A is the area over which F acts, i.e. is 2itr • I, I being the length of the tube. Equation (2) becomes r [ -117 2tzH J = 27crP. it V dr J dr Integrating ,dv r % „ , n — 2l\Ttrl -j = 27C — • P + C, dr 2 C being the constant of integration. But when r = o, -j — .*. C = o, i.e. 7 dv _, — 2vjZ 3- = rP. dr Integrating once more — j 2t\1dv = j r¥dr, — 2lf\V = — -f B, PR 2 When r = R, v = o /. B = Hence » = -— (R 2 - r 2 ) (3) In one second, a column of liquid, v cms. long, and 2-nr • 87 cross- section flows, down such a hoUow tube, i.e. the volume passing per second = j-y- (R 2 — r 2 ) I 2itr • dr. Thus, for the whole tube of radius R, we have the sum of such expressions. 152 ADVANCED PRACTICAL PHYSICS FOR STUDENTS If Q is the total amount of liquid passing per second /.r p 27c — (RV - r*) 8r o 4h TQ p I |~RV 4 J< W 8 I r> (4) A suitable form of apparatus to use in an experimental deter- mination of T) by this method is seen in fig. 88. The liquid, say water, is contained in a large bottle, B, standing a suitable distance above the level of the table. The water flows from Fig. 88 this reservoir to the union, X, thence through a capillary tube of known length to the union, Y, and so on, via a length of india-rubber tubing to a graduated jar, J, where the thermometer, C, measures the temperature of the emerging water. From the unions, X and Y, two lengths of india-rubber tube make connection to the manometer M. The difference in the levels, E and F gives in cms. of water the value of the pressure difference between the ends of the experimental tube, K. A pinch-cock L enables the flow of the liquid to be regulated. In order to maintain a constant difference of pressure between the two ends, X and Y, whilst the water is flowing, the bottle B is closed by means of a tight-fitting india-rubber cork through which a glass tube passes, to a point well below the surface of the water. 'This end being open to the atmosphere allows the entrance of air bubbles as water flows through the tube. The lower end of the tube remains at atmospheric pressure, and so, VISCOSITY | 153 until the whole of the water above this point has passed through the tube, the manometer will remain with a uniform average difference of level. The flow should be so arranged that the emergent water issues as a slow trickle or succession of drops, avoiding a rapid stream of water, which might cause the flow in the tube to become turbulent, in which case the formula which is developed, assum- ing a regular flow, will break down. In adjusting the apparatus, the water passing is not collected. When everything is steady the tube is inserted into J, as seen in diagram, and the time is taken in seconds for a definite amount of water, say 500 c. cms., to pass. From this, Q is obtained. If h is the difference in level in the water manometer, P = g?h, approximately, or as shown later, equation (n), more exactly. Before determining the remaining unknowns, I and R, several values of Q, corresponding to different values of P, should first be obtained, for we have _kR^ P ^"8 l"Q' p By varying P and Q, the mean value of -~- can be obtained, and this, not one value of P and corresponding Q, used for the computation of 73, provided that the temperature of the room remains constant during the experiment. The value of / may be obtained by direct measurement of K. Now R, wjiich occurs in the formula in the fourth power, must be obtained as accurately as possible. A suitable method is the measurement of the length and mass of a column of mercury in the dried tube as described on page 132. Unless the liquid is passing through the tube with a very small velocity, equation (4) must be modified to allow for the kinetic energy imparted to the liquid. This reduces the effective pressure P. To arrive at such a correction, consider the case of a liquid flowing through an irregular tube whose two ends are at different levels. Take two points, B and C, in the tube such that the cross- sections are A t and A 2 ; the velocity of the centre of the liquid, v x and v 9 ; the pressure in the liquid, P x and P 2 ; and the dis- tances of the centre at these places are L x and L 2 cms. below a fixed horizontal plane. 154 ADVANCED PRACTICAL PHYSICS FOR STUDENTS If the liquid moves a distance dx x at B, it will advance a dis- tance dx 2 at C, such that A 1 dx 1 = A 2 dx 2 (5) The work done on the liquid between B and C is therefore PiAji*! — "P 2 A 2 dx 2 (6) This work must be the equivalent of the energy gained in the liquid. Gain of Kinetic Energy The mass (A^xjp of liquid enters at B (p being the density of the liquid, assumed constant). This has a K.E., I A x dx x ^v x 2 . Similarly for the mass of liquid leaving, and since condition (5) holds, the g ain of kinetic energy is Aiteiftpw," - IpvS) (7) Fig. 89 Potential Energy The P.E. of the mass of liquid at B is gL 1 (A 1 ^ 1 )p, and at C is gLgAjjtaaP. For a gain of P.E., C must be higher than B, L 2 <Lj,i.e. the gain of P.E. is A 1 dx 1 (pgL 1 - P gL 2 ) (8) Equating (6) to the sum of (7) and (8) we have, using (5) A x dx 1 (P 1 - P 2 ) = A l dx 1 {$ 9 v 2 * - i pVl *) + A 1 ^( P ^L 1 -p^L 2 ), re-arranging terms, Pi + *P» x 2 ~ gtU = P 2 + |pV - £pL 2 , i-e. P + f p*> 2 — g P L = constant (9) Applying to the case shown in fig. 90, where the liquid is contained in a wide tank, BC, and flows .through a uniform tube, CD, with a constant maximum (central) velocity, v, we may find the effective pressure difference between the two ends of the tube using equation (9) above. VISCOSITY 155 If II is the atmospheric pressure we have, applying (9) to the point B and C, where C is just inside the tube and is at pressure p', H + ipv v i-gpL B = p' + l ? v* - £ P L C For such a wide cross-section as the tank we may assume the surface to be fixed. This may be further brought about by the device shown in fig. 88. i.e. v B = o. We will further use this surface as reference plane for measure- ments of L, i.e. L B = 0, L c = L cms. We therefore have II =P' +$ p v*-g P L, or p' =11 +g p L-|p*;2, the pressure at the open end, D, being n, we have the pressure difference between the ends of the tube, CD, p, = p* ~n i- e - P =£pL-| P t/ 2 ( I0 ) Thus, the effective ' head ' is reduced due to the gain of kinetic energy by the liquid. TT C D Fig. 90 It was seen (p. 151) that the velocity of the stream at a distance r from the axis of the tube, was 2ril\ 2 J' i.e. when r = o, we obtain the value of v used in equation (10) above, i.e. the maximum velocity, PR 2 v Now we saw (p. 152), 473/ P 7C R* Hence _2Q rR2 ' and the effective pressure, p, is given below, *=* p ( L -i^r.) («) 156 ADVANCED PRACTICAL PHYSICS FOR STUDENTS An alternative apparatus for the determination of tj is seen in fig. 91. The horizontal capillary, DE, is fixed in a cork which closes the lower end of the tube, CF, which has con- strictions at A and B, on which are scratches on the glass. The whole vessel is filled with the liquid and the ground-glass stopper, S, placed in the neck above A. The volume between the scratches is calibrated, so that its capacity, V, is known. The stopper, S, is removed, and the time, t cms., is taken for this volume to flow Fig. 91 through the tube, i.e. when the level reaches A a stop-clock is started, and when the level reaches B, the clock is stopped. p, the pressure, is taken as due to the average height, h, of the liquid above the capillary tube level, i.e. p = gph, or as shown on more precisely in equation /n), The other terms are measured as for the first form of apparatus described. Determination of the Viscosity of a Liquid by the Coaxial Cylinder Method The value of the coefficient of viscosity for a liquid such as glycerine may be obtained, using the apparatus shown diagram- matically in fig. 92. The liquid, say glycerine, is placed in the cylinder, AB, which may be rotated by hand or by a small motor. A belt driven by either means passes over the pulley, P, and the VISCOSITY W rotation is imparted to the cylinder by the crown bevel wheels, W. The revolutions may be counted by the revolution counter, R. Hence, if a number of revolutions, n, be timed, the angular velocity, w, may be calculated. Immersed in the glycerine is a second solid cylinder, CD, which is suspended on a phosphor-bronze suspension, which carries a mirror, M, and which is maintained central by the pivot S. Due to the viscous forces in the liquid, the inner cylinder will experience a couple, C, which turns it through an angle, 0, such that the torsional restoring couple just balances the turning moment due to the liquid. If t is the restoring couple per unit angular twist in the sup- porting wire, and 6 be the constant deflection in radians, we have t0 = the restoring couple due to the torsion of the wire. To obtain a value for the couple due to the viscous forces, we will first consider the co-axial cylindrical surfaces. Let fig. 93 represent a section normal to the axis of these cylinders, of radius R 2 and R, as shown. Take any thin cylindrical ring in the liquid having points, A and B, on the same radius. If the liquid rotated as a whole, there would be no velocity gradient — no relative motion. Thus, for no relative motion, when A moves to A 1 , B moves to C. If w is the constant angular velocity, AO = r, and BO = r + Br ;, AA 1 = rw ; BC = (r + 8r)w. Now, in the actual case, the outer cylinder moves with an angular velocity, Cl say„ and the layer in contact with the inner cylinder is at 158 ADVANCED PRACTICAL PHYSICS FOR STUDENTS rest, i.e. the liquid does not have this constant angular velocity. Actually, the particle at B moves with some larger angular velocity {w + dw), and B moves to B 1 where BB 1 = (w + dw) (r + dr), thus having an excess over the no-relative motion velocity equal to (w + dw) (r + dr) — w(r + dr), i.e. = 6w(r+dr) i.e. the velocity gradient is — ^~ — '-, or since dr is negligible cf . r, the velocity gradient ~ = r • — in the limit. Thus, considering the forces on the cylindrical shell, if / is the length of the inner cylinder from the surface of the liquid to Fig. 93 the lower extremity, the area of the cylindrical shell is (2^)/, and therefore from (1), F = yi(2*rl)£> where F is force due to the viscosity over the curved surface of such an imaginary shell. The moment of this about the common axis is C = Fr — 2itrHt\r ^—» dr substituting the value for -j-. Integrating we have J2nlridw =J~-JT> i.e. 2ir/i)tt> =* - — % + B, where B is a constant. Now, when r = R lt w = o, B r = R,, w = H, 2R X * VISCOSITY 159 i.e. 2 ^n=§[^-^ 5 ], Also we have L== R^-R** (I2) C = T0, Wh6nCe " - ^OR.'R. 1 ' " - (I3) In the actual case taken (12) must be modified to include the couple due to the viscous effect on the under surface of the inner cylinder. If this effect were truly due to the parallel circular plates, we might readily deduce an expression for it. But the stream-lines in the liquid will not be composed of two such regular patterns. The effect at the end of the cylinder, CD, will be a gradual transition from the one to the other. Some end correction is therefore needed. Let us assume that the couple exerted on the cylinder, CD, due to the under surf ace and the lower portion of the curved surface, and also the "effect due to the support, S, which rests on AB, for an angular velocity, CI, is K CI, then the total couple will be _ WRi 2 R a 2 Ka C ""(R,»-R 1 *)- Q + K0 ' where V is the length of the curved surface of the inner cylinder, measured from the liquid surface to the lower limit of the regular stream-line between the cylinders ; the second term deals with the rest of the curved surface and the lower surface, etc., as stated. If now we have the space between the Cylinders filled to a level, such as L 1} we have, if l x ' is the corresponding length and B x the deflection ^=°mw+«i- <-> For a second case, where the length of the cylinder immersed is Z 2 ', corresponding to a level, L 2 , in the figure, if 2 is the deflection *- pg£g! + K] ( I5 ) K may therefore be eliminated from (14) and (15) and, if the angular velocity, Cl, is the same for both determinations, we have by subtraction T (0 i- e *>- fi R 2 2 -Rx 2 L i 2 J ( } 160 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The length to be measured is therefore the difference between the two levels, Lj and L 2 . Thus, by maintaining the speed of rotation constant, and observing the steady deflection corresponding to two levels of the liquid surfaces, knowing L^ = // - /,', and the dimensions of the cylinders, tj may be calculated in terms of t, which itself may be evaluated by observation of the twist of the wire, when the liquid is not in the cylinders, and loads are applied at G, as indicated in the diagram. For liquids whose viscosity is fairly large, the torsion control will not be sufficient. In this case the restoring couple is in- creased by adding masses, m, m, thereby increasing the restoring couple by m D, where D is the diameter of the wheel G. In many cases it is advantageous to eliminate the torsional restoring couple entirely. This may obviously be done by adding masses, m, m, of such magnitude that the inner cylinder is brought to its rest position, as observed by the reflected beam from the mirror, M, on the scale, which was previously used to measure B x and 2 . In that case, if masses of total value, M x , are supported on the wheel, G, C t = Mj • — , and for the second case with the liquid at level, L 2 , C 2 = M 2 -. Then equation (16) simplifies to (M, - M a )| - "gp%^ (// - V). This latter method of working will be found most satisfactory for glycerine and similar liquids, whilst the former method, using the torsion of the fibre, will be satisfactory in the case of less viscous substances, such as water. Viscosity of a Liquid (Oscillating Disc Method) The viscosity of a liquid may be determined by timing the period of oscillation of a flat circular disc in air, and finding the logarithmic decrement in the liquid and in air. O. E. Meyer has shown that with such a disc, the coefficient of viscosity for a liquid, tq, is 7u P T(y 4 + 2r 3 ^) 2 \\ 7c °j + V 7c °j j " * ' * (I7) where I is the moment of inertia of the disc, and attachments about the axis of suspension, p the density of the liquid, T the time of a complete swing in air, ■n = VISCOSITY x6i r the radius of the disc, d the thickness of the disc, x*the logarithmic decrement in the liquid, x the logarithmic decrement in air. The development of the above formula is beyond the scope of this book, and may be found in " Poggendorf Annalen," No. 113, page 55. Without any discussion of the development of the result, we will use it as an empirical formula which agrees with determina- tions by other methods. It is an excellent method whereby to study the determination of the logarithmic decrement of an oscillating system. Logarithmic Decrement Consider a suspended body to oscillate about the suspension as axis in a simple harmonic manner. If I is the moment of inertia of the body about "the axis of suspension, F the restoring force per unit angular displacement, we have the equation of motion, IB -f F0 = o. If now a frictional resistance acts on the body so that K is the resulting opposing couple per unit angular velocity, the above equation is modified into the form I'd + Kb + F0 = o. By the method given, p. 28, trying as solution = djt, we arrive at the result, -J£« / /F K 2 \ e=6 .e *i cos^yjy-— t + aj, when O and a are arbitrary constants, being the angular displacement at any time t. The period time, T, of such a motion is T 2" I F K* \ I 4I 2 K The amplitude is o e al * , and decreases exponentially with the time. O is the amplitude when the friction is eliminated, i.e. when K = o. In the case where the friction resistance, K, is very small, K 2 the value of K 2 is correspondingly much smaller and -p- becomes F negligible compared with j . 162 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The periodic time becomes The oscillating body therefore performs vibrations in equal times, but the amplitude gradually dies out (see p. 29). Starting from the undisturbed position of the body, let a x be the angular displacement measured to the first turning point (measured to the right, say). Let a 2 be the angular displacement following, on the other side of the zero (to the left, say), also measured from the zero position. a 3 the next swing in the original direction measured from the zero. • a/ of+fit* +ir«/i *"* _ • The first deflection 0^ is after a time t = — ; <x 2 after time -^- ; 4 A 5T 0.3,-^— ' and so on, 4 Thus K T To the right : a x = d e " 2l ' 4 _K 3T To the left : a 2 = e »*■ ' 4 _K 5T To the right : a 3 = d e »i ' 4 K 7T To the left : a 4 = 6 e ~»i ' 4 a 2 a 3 a 4 a 5 a n+i So that Log^=log^...=log^-=g.± = x,say, (18) K T x = -y , — is called the logarithmic decrement. Now we have logfe • «t . ?• . . . X -*^) = log (^) n \ a 2 a 3 a 4 a »»+iy V a 2y Since — 2 = - 3 . . .-^- =-°^ a s a 4 a n+I a. *. n • log ^i = log ( -^— ) Again, since VISCOSITY 163 OCi Oa . a l _ a l + « 2 a 2 a 3 ' a 2 a 2 + a 3 2L* _ <^3_ <*2 _ «2 + a 3 a 3 a 4 a 3 a 3 + a 4 Hence . a x 1 , (x.i + a 2 log — = r lo g * « „ «2 + a 3 x a » + a .n+l_ n+2 w a 2 + a 3 a 3 + a 4 ' ' a n+1 + a x== Li og *i + " 2 (20) n a n+1 + a B+2 (i.) The value of x, the logarithmic decrement, may be obtained, as in (18), by observing the value of the first swing, say to the right, and the successive swing to the left, whence x = log, -i, oc 2 or if logarithms to base 10 are used x = 2-303 log 10 j 1, a 2 X here depends on the observation of swings. It will be seen that a x and a 2 cannot be very accurately observed, hence the method of equation (19) may be used. (ii.) By observing the 1st and the (n + i)th J swings, the error is reduced to - of that in the first method corresponding to n equation (19). (iii.) The third method of finding x, set out in equation (20), depends on the observation of swing from left to right (| periods). Here no knowledge of the zero reading is required. This last method is obviously the one to be recommended, for most purposes, of the three discussed above. It will be seen that, if we replace <x. 1 + <x 2 by p 1? <x 2 + a 3 = p 8 . . . a n + a n+1 = p„ that equation (20) becomes >=>£; (2I) Since T, the time for a complete swing, is constant when the damping is small, we may find X from equation (21) by observing Pi directly, and then allowing the system to oscillate a measured time until the amplitude is appreciably reduced, say, from half to quarter of the original, then measure p B+1 . The number of swings between the observations may be found from the time, for the number of complete swings may be obtained by dividing the time by T; hence (n + 1) is known, p refers to half* a 164 ADVANCED PRACTICAL PHYSICS FOR STUDENTS complete swing, so that twice the above quotient will give the number of swings between observations. An application of the above may be seen in the following method of finding the logarithmic decrement, which is especially advantageous if there is uncertainty in measuring the ends of swings, and hence p v Suppose we read an even number of successive left and right deflections, cc v <x 2 , a 3 , a 4 . . . a m . Then Pi = «i + a 2 > P3 = «3 + a 4 , Ps = <*s + a 6 , etc., omitting p 2 , p 4 , etc., for convenience in tabulating. After a timed interval, when the deflection is reduced to about half the original deflection, obtain the same number of left and right readings. From a knowledge of T, suppose the first of the second set of readings is a n , and a«, a B+1 , a B+2 . . . are observed, we may obtain Pn = *»+*n+l» Pn+2 = a n+2 ~t~ a n+3> e ^C. We have since h = li = hzj = ^ P 2 P 3 P» Pi _ Pi P Pn \U and in same way for the other pairs giving Pi _ P 3 P 5 Pn P«+2 Pn etc. +4 01 h = Pi + V* + Pi • ■ • = ,*(«-!) P« P»+ P« +2 . . . Pn+4 Hence (»_i)x = log, Pi + P 3 + P 5 • • • 2-3026 x Pi + P 3 + P 5 ■ „_. , } (n-i) gl ° P„ + P B+2 +p a+4 . . . ~l 29) The practical details of this are given below. Experimental Arrangements A suitable form of apparatus with which to make a deter- mination of t) for a liquid is seen in fig. 94 The flat disc is suspended horizontally by a phosphor-bronze suspension which is attached to a rod rigidly fastened to the centre of the disc. This rod carries a cross-bar whose ends VISCOSITY 165 have a serew thread, along which two masses may be screwed to balance the disc horizontally. A small concave mirror is fastened to the rigid support. The time of oscillation of this system is first obtained in air. This observation is carried out by aid of the usual lamp and scale arrangement. A beam of light from a lamp is directed on to the mirror, and is brought to a focus by the latter on a scale about a metre away. As the spot of light passes its rest position on the scale, during its oscillation, a stop-watch is started and stopped after 50 complete swings, hence T, the time for one swing (i.e. the interval between the spot of light passing the zero at consecutive times in the same direction). Determination of I. The value of I may be obtained from the time of swing. If t is the restoring couple per unit angular u ■jm Fig. 94 displacement due to the suspension, we have, in air, where the logarithmic decrement is small 2n £ (23) The value of T has been obtained as above. A small ring of thick copper wire is now placed symmetrically on the disc and T 1 , the time of a complete swing, is obtained from a determination of the time of 50 swings, for the loaded disc. We have T' 2tc P + r (24) where V is the moment of inertia of the copper wire ring about the axis of suspension. If a is the radius of the ring of wire, and if the centre were in the axis of suspension, I x f = ma 2 where m is the mass of the wire ring. 166 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Squaring (24) and (23) and dividing, we have -Z 2 = t + v = ,r_ , ^i 2 Hence I = T2 ' ma \ TV2 <-p2 r, d, and p may be readily found by the usual methods. The logarithmic decrement is now found for the disc oscillating in air. Use is made of the method embodied in equation (22) above. The disc is shielded from draught in an empty glass dish, as in the figure. A straight vertical wire is placed in front of the lamp and by the mirror, and a lens an image of the wire is focussed on the scale. The disc is given a displacement, and the oscillations are observed by means of the light spot. As the spot passes the position of rest a stop-clock is started. The first turning point of the spot of light is read off, on the right, say, then one turning point on the left, and so on, tabulating as under 8 or 10 readings on each side. The disc is then allowed to perform the oscillation until the amplitude has decreased to half or quarter (depending on the time) of the original amplitude. At this stage the reading at the turning of the light spot at the right is observed, and the stop-clock stopped. The next reading to the left is taken, and so on for 8 or 10 on each side. The time being t seconds between the starting of the clock and the first observation of the second set of swings, T being the periodic time of the system in air, there have been — it complete swings or ^ half periods, hence n as shown under. The above process is repeated in the liquid whose coefficient of viscosity is to be determined, say paraffin oil. Thus, in the table opposite, which gives the observations for the disc in air and paraffin oil, we have, if T = 3*07 sec, n, the number of the £ swing at the commencement of the second set of observations for air = 2 ( — — j = 400, \ 3'07/ * i.e. for air _ 2-3026 An-A x °- ( 4 oo-i) logio Wiy 399 ss '0027525. VISCOSITY i67 AIR PARAFFIN OIL SCALE READING LEFT RIGHT 2-4 45 2-5 45 2.8 44.8 3-o 44-5 3*i 44'3 3*3 44-2 3-3 44-o 3'6 43-8 37 437 4-o 43-6 AMPLITUDE 42-6 42-5 42.0 4i-5 41-2 40-9 407 40-2 40*0 39-6 Total 411-2 Time Interval = 615 sees. 16.5 30-4 16-5 3o-4 16-5 30*4 16.5 30-4 16-55 30-3 16-6 30-2 167 30-2 167 30-2 16-6 30-2 16-6 30-2 13-9 13-9 13-9 13-9 137 13-6 13-5 13-5 13-6 13-6 Total 137-1 SCALE READING LEFT 5'i 6-o 6-5 7-2 7-8 8-5 9-1 97 10-2 II-O RIGHT 47-2 46-6 46-0 45-2 44-5 43*8 43-2 42-6 42-0 4i-5 AMPLITUDE P 42-1 40-6 39*5 38-0 367 35-3 34*i 32-9 3i-8 30-5 Total 361-5 Time Interval = 123 sees. 20-7 20-9 21-0 21-2 21-4 21-5 21-6 21-8 22-0 22-5 31-3 31-2 31-0 30-9 307 30-6 30-5 30-2 30-1 30-0 io-6 10-3 io-o 97 9*3 9-1 8-9 8-4 8-i 7*5 Total 91-9 In the same way the value of the logarithmic decrement in paraffin may be calculated, 2 - 3026 log 10 3 6l -5 X = (n -1) 91-9 n being obtained as above from the time interval measured in that case from the commencement of the first set of observations to the commencement of the second set. 168 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Thus, by the methods given, we have a value for I, p, T, r, d, X and x ; hence substituting in equation (17) tj may be calculated. Repeat the experiment, using water and paraffin oil, and calculate the value of y\ for each. FLOW THROUGH A VERTICAL TUBE The variations of the coefficient of viscosity of water with temperature may be investigated by means of the apparatus shown in fig. 95. A T ■ i , 1 1 — - ■ C B iH i____lL____i Fig. 95 AB is a copper cylindrical vessel into which the capillary tube, CD, is fitted by means of a rubber cork. Water is heated in a boiler to a temperature 5 or 6 degrees above the one at which a determination is to be made, and is transferred to AB, the end, D, being closed by the rubber stopper shown. The water is stirred and the temperature noted by the thermometer, T. It is not very difficult to arrange that the water cools slowly and regularly about the desired temperature. The water is then allowed to flow into a graduated vessel and the time taken for a measured amount to pass. This process may be repeated, using water at various temperatures. Care must be taken that the temperature, throughout the flow, is practically constant, otherwise unreliable results will be obtained. The radius and the length of the tube may be measured as before. VISCOSITY 169 Then, as shown under, we may obtain the value of tj for each temperature from the following formula : — using the same notation as before. In the case of a vertical tube, such as the one described, account must be taken of the gravity attraction on the .liquid in the tube ; the direct application of the results of equation (4) dealing with a horizontal tube being thus inadmissible. Consider a vertical tube of length, / cms., and in the liquid imagine a thin cylindrical tube of the liquid of radius r, and thickness dr. Let p be the pressure difference between the two ends of the tube, then we have in the steady state of flow pzitr • 8r-\- 2-kt • drlpg the downward forces, opposed to -=— • 8r due to viscosity where F = yjA • -j-, dr i.e. 2-*r(p + gpQdr = — rf( tjA^V or integrating r(p + gpV)dr = — 2-qldv, since A = 2twI. Hence, once more integrating, r 2 R 2 where C, the constant of integration = — r(lgp + p), for when r = R, v — o, i.e- V = (R'-r*)^±^ (25) Through such a tube the quantity of liquid passing per second s= v (2izr • dr) , i.e. the total quantity per second, Q, is 27W • v dr. Substituting from (25) above and integrating, 7U R* p+gtl In the case taken we may apply the result of equation (9), page 154, p + %pv 2 — gph — constant. If AB is the upper surface of the cylinder which contains the liquid (fig. 95), the cross-section being large, the velocity of «/ 170 ADVANCED PRACTICAL PHYSICS FOR STUDENTS AB will be small compared with that at C. Talcing AB as the reference plane for measurement of L, h being the distance between C and AB, we have, i.e. the pressure, p' , at C is P' = n +g?h -|pt> 2 , where v is the velocity at the centre of C. The pressure at D is n, therefore p = p 1 — n ; hence p = gph ~ \pv 2 , and thus we have the value below O - - JL 4 Z ph -jpfl a +g?l * ~ S' 7] I Q.-^fl^leL- (26) The correction %pv 2 may be obtained from (25) above, i.e. To a first approximation, putting p = gph, R 2 — r 2 or at the centred where r — o, R 2 gpH 41? To the same approximation n _ « R* g P H 2Q Hence we amend equation (26) ,R* ^ 2Q 2 \ This equation is used to calculate -t\ in the experiment already described. Viscosity of Air Consider two parallel circular plates, one suspended by a fibre, and the other rotating at a constant speed. If the space between the plates be filled with any gas a velocity gradient will be set up in the layers of the gas parallel to the rotating plate. The layer in contact with the rotating disc will move with the latter. VISCOSITY 171 Due to the viscosity of the gas, the adjacent layer will also acquire a velocity comparable with the former. Thus throughout the space, the air strata will acquire a motion, just as in the case of a liquid flowing through a tube. The layer of the gas in contact with the suspended plate will therefore experience a force tending to rotate it in the same direction as the constantly rotating parallel plate. Due to the force, a couple will act on the plate, which will therefore turn through a definite angle of such a magnitude that the restoring couple due to the torsion in the suspension just balances the displacing couple of the viscous drag. A & o! Qd b R B * 1 Fig. 96 It can be shown that each stratum of air moves as though it were a solid, i.e. it moves as a whole. To obtain an expression for the deflection of the suspended plate in terms of ij, the coefficient of viscosity of the gas, etc., let us assume that the edge effect is negligible, and the gas between the plates behaves as though the plates were of infinite dimensions, an assumption which is justified by using a guard ring round the suspended plate. Let d be the distance between the discs, w the angular velocity of rotation of the moving plate, R the radius of the suspended plate. Consider a stratum shown in fig. 96 by the horizontal broken line, between EF, rotating, and CD, which is suspended. (The parts, AB in the diagram represent a guard ring to eliminate the end effect, CD being of radius R.) In the stratum considered take a point, P, on a circle of radius r cms. about the axis of rotation. Points Q and R are the projections of P on CD and EF. The velocity of R is r - w ; the velocity of Q is zero. The velocity slope is therefore -3-. If uniform, the value of the velocity of a P is ~ • wr, y being the distance of the stratum considered, below the plate CD. We have seen that by the definition of viscosity F ~ At) Tr ~ " T A ' 172 ADVANCED PRACTICAL PHYSICS FOR STUDENTS In the stratum considered, let a second circle of radius, r+dr, be drawn ; between the two circles there is an annular ring of width, dr. The area of the ring is 2nr • dr. The viscous force acting on such a narrow ring is, from the equation above, equal to F, where f = 2^.<*.^t. a The turning moment about the axis is a Such a moment acts on the suspended disc on the projection of this area. Fig. 97 The total couple is the sum of such couples taken over the entire area. Let this couple be C, then C = A*r» &dr*= ^ *-* = ^R^. Jo & » 4 2d Let the suspended plate be turned through an angle, 0, due to this couple. The equilibrating couple due to the torsion of the suspension is t0, where t is the restoring couple per unit angular displacement. This gives 7crjR%; t9 = ^3- < 2 7) A suitable form of apparatus is essentially of the form shown diagrammatically in fig. 96. A brass plate is rigidly connected to a shaft which may be rotated by means of a belt drive on a pulley, which is on the shaft. A small counting indicator serves to record the revolutions of the disc, which is steadily turned by hand or by a small motor. At a distance, which may be adjusted, above the plate is a mica disc, suspended by a phosphor-bronze suspension, and arranged inside a guard ring of brass. The suspension carries a small mirror which serves to measure the deflection by the VISCOSITY 173 usual method of lamp and scale ; the whole is enclosed in a brass case which serves as a shield. A preliminary experiment will give an indication of the most suitable speed with which to rotate the plate, for any given distance, d, between the plates. The zero of the spot of light having been determined, the lower plate is rotated until a full scale deflection is obtained. The speed of rotation is maintained constant. When this is steady, the counting gear, having been read, is thrown into action, and a stop-watch started. Maintaining the spot constant by steady rotation for as long as possible (at least several minutes) the value of the deflection is noted. The counter is then thrown out of action and the stop-watch stopped. In that way the number of revolutions n, in a known time t, is obtained, hence As R occurs in the fourth power, several values of the diameter are obtained in all directions, as accurately as possible, and the mean value calculated. d is measured by means of a cathetometer. From the value of the steady deflection and a knowledge of the distance between the mirror and the scale, may be calculated in radians. is half the value of the angle subtended at the mirror by the length of scale, moved over by the spot of light, i.e. tan (20) may be obtained, knowing the linear deflection and the distance between the mirror and the scale. Thus, from this experiment all the terms in equation (27) are known except t and 13. To obtain t the suspended plate is twisted from its equilibrium position, and the simple harmonic oscillations set up are timed. If T is the mean value of the periodic time for, say, 50 complete vibrations, we have (p. 105) = 27t \7 (28) where I is the moment of inertia of the mica plate about the axis of suspension. If now the mica plate is loaded by placing on it, symmetrically with regard to the suspension, a circle of brass wire of radius a cms. and mass m grammes, the moment of inertia has been increased to T, where I' = I + ma 2 . The time of oscillation, T', of the loaded plate is next deter- mined by timing 50 oscillations, then - 2 *VT ■<«9) 174 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Squaring (28) and (29) and subtracting, we have T' 2 - T* = 4 TC2 ( r - 1 ) , Anhna 2 T = — . ^'2 ^2 Thus, by equation (27) 2d An 2 ma z t i.e. 7rR 4 (T /2 — T 2 ) 2nn _ qmaH d 73 -~" -R4(T' 2 - T 2 ) n ' Having obtained the deflection in degrees, <p° say, we have = i8o"' so that tc ma 2 t<p d •n 45R 4 (T' -T> The Determination of the Viscosity of a Gas (by the Flow though a Capillary Tube) A simple method for determining the viscosity of a gas which may be described as ' The Constant Volume Method,' has been described by Prof. A. Anderson (" Phil. Mag." Dec, 1921, pp. 1022—3). The apparatus is illustrated in fig. 98. It consists of a bulb, V, from which a tube, DO, projects downwards and is connected by rubber tubing to a glass tube, AB. These tubes contain mercury up to certain levels marked at A and O. Just below the bulb a capillary tube leads from the tube, DO, as shown at D. This capillary tube is provided with a piece of rubber tubing which may be closed by the pinch-cock, P. The whole apparatus is mounted on a stand and is of about the same size as the constant volume air thermometer. The arm, AB, is mounted on a carriage which is readily adjusted by means of a rack and pinion which is regulated by the turn- screw, C. The volume of the gas within V, down to some mark, O, and extending to the end of the capillary tube is determined. The pinch-cock is then opened to the air and the mercury brought well beldw O by properly adjusting AB. The pinch-cock is then closed, and the air compressed by raising AB until the mercury stands at O. A few minutes' interval is allowed to elapse so that the tem- perature of the gas, which may have been disturbed in this VISCOSITY 175 compression, once more attains the temperature of the room. The difference in level of the mercury in the two tubes is observed. The pinch-cock is opened for a definite accurately measured time and then closed. During this interval the level of AB is continually adjusted, so that the mercury always stands at O, and the gas consequently maintains a constant volume. At the end of "the time the difference in level of the mercury is again noted. From the observations made, viz, from a knowledge of the volume, V, of the gas, the two pressure differences, p x and p 2 respectively, the time interval, t seconds, and the baro- metric pressure, P, the viscosity is determined from the formula : 71 =Vt ttR 4 -7- !Og ( (*» + P) ifil ~ P) 8/V ^ ~ 5e (p t + P) {P2 - P) ' where R = radius of capillary tube and I = its length. r KJ Fig. 98 The above formula may be obtained from an extension of the result obtained for the flow of a liquid through a capillary tube. Suppose, in the first case, that the gas enters the capillary at a fixed pressure P x , and leaves at a pressure P 2 . Let V x and V 2 be the volume entering and leaving per second; then p x V x = P 2 V 2 . The volume of gas passing any point in the tube per second depends on the pressure at the point. The velocity is consequently variable along the axis of the tube, and therefore the method of the liquid flow cannot be applied to the full length of the tube. Consider a small element, dx of the tube. Let p be the mean *-"rf*-8 176 ADVANCED PRACTICAL PHYSICS FOR STUDENTS pressure in the element and dp the difference in pressure at the ends. Equation (4), page 152, for the liquid flow, is yi ~ l 8 „' If q x is the volume of gas passing through the element, the above formula may be applied (dx is very small), and since in this case — = — ^- (since the pressure decreases with increase in x), dp it R* Now, PjVj = P 2 V 2 = qjp, since the mass of gas passing any point is constant, i.e. q x = * * ; therefore PiVi^ _dp nJR* p ~~ dx'S 7) ' F 1 V 1 dx = -~^ pdp. Integrating over the limit of the tube P 1 V 1 /'W= -Z R * T'p-dp, Jo ° 1 J Pi i.e. W = ¥ -^r^-T6-T (30) If V is the volume available to enter the tube at a pressure P x , we may write dV We may apply Boyle's Law to the instantaneous values of P and V at entry, i.e. PV = constant, „dV , „dV . ^dV „dP *di +Y dt =°> he - v Tt=~ Y W whence from (30) (P x » - P 2 2 ) * R* _ _ dP / 16 71 ~ dt ^ dP where -=- is the rate of change of pressure at the end of entry, dt and V the fixed volume. s P a in the experiment is the constant atmospheric pressure VISCOSITY 177 at the end of the capillary, and Pi has values, say, p x and p 2 at the beginning and ending of the period of observation of t seconds duration. Re-writing (31), we have ifiWVo* " + 2P J >x V^+P, p,-p J ' * R4 * - * rion Pl + p - 2 T a i6^V 2P a L g Pi - P 2_L ~ 2P 2 t 10g £ 2 -P 2 10 \^-Pj» which becomes, on writing P 2 = P = atmospheric pressure, _/tcR*\ PJ 71 " \«V J. (p* + P ),( fr - P) lg, "¥rrp)(^ 2 -p) The term ^%. may be calculated once for all for the apparatus and is the ' constant of the apparatus.' The Constant Pressure Method The determination of yj may be made with constant pressure difference between the ends by maintaining the levels of the mercury columns in the two tubes at a fixed difference. The point, O, would be chosen at the commencement near the bottom of the tube, and the time taken for a volume corresponding to a length of tube between the original and final positions of O measured. If p is the total constant pressure inside V, we have v » = tf l6 V (30) ' in which all terms except tj are known. The difficulty in this modification lies in keeping the mercury levels a fixed distance apart as the gas is driven out. CHAPTER VI THERMOMETRY AND THERMAL EXPANSION The Comparison of a Thermometer by means of a Standard It is recommended that each student before beginning his experiments on heat should choose a thermometer, test its accuracy, and use it when required throughout his experiments. The method of comparison is to immerse it together with a standard thermometer in a bath and observe temperatures over a suitable range simultaneously by both instruments. It is convenient for the following experiments to have two thermometers, one reading from o° C. to 35 ° C. and the other from o° C. to ioo° C. Both should be calibrated in this way. A large water or oil bath should be carefully heated over a Bunsen flame and constantly stirred. The thermometers should be placed in the liquid so that the mercury thread shows just above its surface, and with their bulbs close together. A record of temperatures at intervals of 5 should be taken over the range, and a curve drawn with temperature corrections as ordinates and with readings from the thermometer to be calibrated as abscissae. By means of this curve the readings of the thermometer in later experiments can be reduced to that of the standard. A better method of heating the liquid is to place it in a bath standing in a box lined with cotton wool and to supply the heat by passing an electric current thrbugh a resistance coil immersed in the liquid. If the current is drawn from storage cells, and a variable resistance included in the circuit, it is possible to adjust the current so that the bath is maintained for a long time at a constant temperature. The most convenient form of stirrer is a small propeller driven by a small electric motor. The Calibration of a Mercury Thermometer It is impossible in practice to obtain a perfectly uniform bore in the stem of a mercury thermometer, so that it is not sufficient in an accurate instrument to divide the interval between the fixed points into a number of parts of equal lengths. The makers of thermometers usually attempt to make some correction for this lack of uniformity by adjusting the distance 178 THERMOMETRY AND THERMAL EXPANSION 179 between consecutive divisions to suit the bore of the stem at the various points. But, in spite of this, unless the thermometer is exceptionally carefully constructed, errors remain and a cali- bration has to be made if accurate observations are required. In a given thermometer, as a rule, the divisions will be unequally spaced at different parts of the tube, and the bore will vary from point to point. The first step is to divide the tube into segments consisting of five or ten degrees each, and over each to find the average distance between each division. We assume for the sake of definiteness that we are considering intervals of ten divisions. Measure each of these, beginning at o° by means of a micrometer microscope, and deduce the average length per degree for each of the ten intervals up to ioo°. We are assuming that the thermometer is divided into degrees Centigrade from o° to ioo°. When this has been done a thread of mercury is broken off from the main column of length equal to that of about io° on the thermometer scale. This thread may be obtained by connecting a small jet, made by drawing a glass tube to a narrow bore, to a gas-pipe, and lighting the gas at the narrow end, adjusting the supply to produce a flame about half a centimetre long. If this flame be applied cautiously at the point of the thread where it is desired to sever it, the thread will divide. During the application of heat the thermometer must be rotated to avoid fracture due to unequal heating. The thread is moved by gently jerking the thermometer until one end is at o° and the other near io°, and its length measured. The thread is then moved so that one end lies at io°, and the other near 20 , and so on up to ioo°. If it is difficult to get the detached thread down to zero, owing to the projection of mercury from the bulb past the zero mark, the bulb should be cooled by wrapping it in wool and moistening with ether. This will clear the tube and allow the thread to be moved down without its re-joining the main mass of mercury. Denote the ten lengths of the thread by l lt Z 2 , l z , etc. Let these be reduced to their equivalents in degrees. This is readily done since the average width of one degree is known in the different parts of the scale. Denote these equivalent lengths by t x , t 2 , t z , etc., and the mean of these by t. If the tube had a uniform bore of the same length as that of the actual instrument between o° and ioo°, the readings would be t, 2t, 3*, etc., instead of t lt t x + 1 2 , t x + 1 2 + t z , etc. Let the values added respectively to t x , t 2 , t 3 , etc., to make them equal to / be d v 6 S , <S 3 , etc. i8o ADVANCED PRACTICAL PHYSICS FOR STUDENTS Then t = t x + 8 X t = t z + <5 a , etc. The corrections to be applied near io°, 20 , 30 , etc., are therefore : or t-t x , 2t — t x — t 2 , 3* — h — h — *a> etc., <5i + d 2> ^1+^2+ 5 3- The correct temperature corresponding to t x is (t x + d x ), and corresponding to the point [t x + 1 2 ) it is {t x + t 2 + 8 X + <5 2 ), and so on. We have assumed up to the present that the fixed points at o° and ioo° are correct. A table is drawn up as shown below. The correction to be applied near io 9 is — -023, for the thermometer column is too long at this point, near 20° the correction to be applied is (_ .023 + -032) or + '009. The correction would have been + -032 had the point near io° been correct, but since that was not the case both errors come in. We correct similarly for other points by adding errors algebraically, and recording in the last column the amount to be added to the thermometer reading to obtain the correct temperature. Note that in the example given the thread was not very near to the mean length of io°. It is convenient to arrange this as closely as possible to io°. Strictly, the error —023 ought to be applied to the recorded temperature of 10-231, but the error will probably not vary very rapidly in the neighbourhood of any given point. Hence we take the corrections in the last column as applied at io°, 20 , etc. Mean length of thread, as deduced from column 4, 10-208. REGION OF TUBE . MEAN LENGTH PER SCALE DIVISION LENGTH OF MERCURY THREAD EQUIV- ALENT OF THREAD IN DEGREES DIFFERENCE FROM MEAN CORREC- TION TO APPLY TO UPPER READING O - IO° IO°- 20° 20°- 30 3O - 40 4O - 50° 50 - 6o° 6o°- 70 70 - 8o° 8o°- 90 90°-ioo Q •2385 •2389 •2391 •2389 •2398 •2350 •2355 •2353 •2347 •2367 2-442 2-431 2-449 2-435 2-438 2-426 2-404 2-393 2-392 2-408 IO-23I 10-176 IO-243 10-193 10-167 10-323 10-208 10-170 10-192 10-173 8 X — 023 <5 a +'°32 <5 3 —035 <5 4 +-oi5 <5 6 +-041 6* —"5 0- -ooo <5 8 +-038 6 9 +-016 <5io+'°35 023 + .009 — •026 — •Oil + •030 085 085 —047 03I + •004 THERMOMETRY AND THERMAL EXPANSION 181 Draw a curve with thermometer readings as abscissae and the corrections to be applied to obtain the corrected readings as ordinates. The curve should pass through the #-axis at o° and ioo°, since these points have been assumed to be correct. If, however, the fixed points are incorrectly placed, the errors must be found in the usual manner with ice and steam. Suppose that the zero correction is d , while that at ioo° is ^100- The upper fixed point must be corrected for pressure, latitude, and height above sea-level. The barometer must also be corrected, owing to the fact that it is probably not read at the temperature at which the instrument was standardized. This last correction may be made by the following formula : If h denotes the height at o°, and h t that read at t°, h = h t {i — -0001622). For latitude x and at a height d feet above sea-level, the length of the column which produces the standard pressure at o° and at sea-level in the standard latitude of 45 ° is : L = (760 + 1-94562 cos 2X -f- -000045466^) mm. Under this pressure, L, the boiling point is ioo° at the height and latitude of the place of observation. h h Thus, h is equivalent to -j- atmospheres or ^- x 760 mm. under normal conditions. From this and the following table the correction to the boiling point may be made. CORRESPONDING MAXIMUM VAPOUR PRESSURE TEMPERATURES 720 mm. 98-493° c. 725 98-686 730 98-877 735 99*067 740 99255 745 99443 750 99.630 755 , 99815 760 1 ioo-ooo 765 100-184 770 100-366 775 100-548 780 100-728 The zero on the scale actually records the reading the ioo° records (100 — 5 100 ). Thus a correction is required for this. d n , and 182 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Plot on the curve the two points (0,0), (100, <5 100 — <5 ), and join them by a straight line. The ordinate of this line at x x° is (3 100 — <5 ) • — , since each degree, even if correct as regards bore, would register only = — 10 ° ° , we must add J 100 to it an amount, — — , on account of the errors at the 100 fixed points To x° we must add 100 Thus, if we draw on the same graph on which the first set of results was plotted this second curve the difference between the two ordinates taken algebraically will give the true reading corrected for errors due to bore and fixed points. Re-draw a new curve, showing as abscissae temperatures as recorded by the thermometer and as ordinates the difference of the orcjinates of the two curves, and the resulting curve will give the amount to be added to any recorded temperature to give the true temperature. Throughout the taking of measurements the temperature of the detached thread should remain constant ; and in order to be sure that this condition holds, place another thermometer close by and observe whether it varies or not. Do not handle the thermometer under examination more than is necessary, and only do so by holding it at the tip away from the bulb. Newton's Law of Cooling The object of this experiment is to verify Newton's Law of Cooling, which states that the rate at which a body cools is proportional to the difference of temperature between itself and the enclosure in which it is placed. The constant of proportion- ality depends on the surface exposed and the thermal capacity of the exposed body, and the law is true for small differences of temperature only. The apparatus required is a small metal thimble, into which water at about 8o° C. can be placed, provided with a cork through which a thermometer passes for noting the temperature of the liquid. The enclosure consists of two calorimeters, one inside the other, and containing water in the space between, to provide an enclosure at nearly constant temperature. A thermometer THERMOMETRY AND THERMAL EXPANSION 183 placed in this water, which should be stirred occasionally, gives the temperature during the experiment. Observe the temperature recorded by T x and T a (fig. 99), at Fig. 99 intervals of half a minute during the initial stage of the fall, and, as the rate decreases, the interval between observations may be increased. The record of T 2 should not vary very much. Tabulate the results thus : TIME (MINS.) RECORD OF T , RECORD OF T 2 72-5° C 12-2° C. •5 7i-3° C 12-2° C. 1 70-1° C. 12-2° C. i-5 69-0° C. 12-2° C. 2 67-8° C. 12-2° C. 2'5 667 C. 12-2° C. 3 65-6° C. 12-2° C. 3'5 6 4 -5° c. I2'2° C. 4 63-5° c. 12-3° C. 4-5 62-5° C 12-3° C. — — — Draw a curve showing the relation between the temperature Tj and the corresponding time. Make the temperatures the ordinates and times the abscissae. The rate of fall of temperature may be obtained from this curve by measuring the tangent of the angle of inclination of the tangent to the curve to the axis of t. As explained in the introductory chapter this measures the value of -^ at the various points of the curve* it 184 ADVANCED PRACTICAL PHYSICS FOR STUDENTS According to Newton's law these values should be proportional to the differences between T x and T 2 . If T 2 does not vary very much its mean value may be regarded as the mean temperature of the enclosure. x If T 2 varies too much to permit this approximation, draw a curve showing the relation between T 2 and the time on the diagram which shows the relation between T x and the time. Then at each time we can determine the value of (T t — T^ and the corresponding value of -— from the same graph. Make another table containing two columns, one for values of (T x — T 2 ) and the other for the corresponding values of — j^- Draw a curve with the values of (T 2 — T 2 ) as ordinates, and those of -~ as abscissae, when, if these two quantities are proportional, the result should be a straight line. The Use of the Weight Thermometer The weight thermometer consists of a glass bulb, B, drawn out at the upper end into a capillary stem, A. / 1 \ B KJ Fig. ioo A convenient size is obtained by making B about 6ne and a half to two inches long, and between a quarter and half an inch wide. It forms a good exercise to make the apparatus from a piece of glass tubing. If this is done, care must be taken to get rid of any blob of glass likely to accumulate at the bottom of B, otherwise in the subsequent heating B will be very likely to crack. The apparatus measures the expansion of any liquid placed in B relative to glass. In order to deduce the real expansion it is necessary to know that of B. The coefficient of expansion of mercury has been found with great accuracy, and may be taken as -0001818. THERMOMETRY AND THERMAL EXPANSION 185 We may therefore use mercury to find the expansion of B, and then use B to determine the expansion of other liquids. We shall consider the determination of the expansion of water. (1) Determination of the Expansion of the Weight Thermometer Carefully weigh the apparatus. Surround B by wire gauze and warm carefully in order to drive out air. Place the end A under clean mercury contained in a dish and allow B to cool so that a little mercury enters. Then, when sufficient has entered, boil the mercury so that the space above it may become full of mercury vapour. Once more place A below the mercury, when B will fill as the vapour condenses on cooling. It is best to carry out the process gradually, heating and cooling B several times, and allowing a little mercury to enter at a time. Warm up the mercury first to prevent the bulb from cracking when it enters. Allow B to cool gradually, and finally surround it with ice, keeping A immersed all the time. Bring up a small weighed, empty dish, and remove the mercury at A, replacing it by the empty dish. Remove the ice and allow B to acquire the temperature of the room. Mercury will, of course, flow over into the dish. As soon as the flow ceases, weigh the bulb, B, and the dish. Let the total weight of mercury within B at o° be denoted by W . Immerse B and as much as possible of the stem in boiling water, and catch the mercury that flows out in the little dish. After the bulb has remained in the water for a quarter of an hour, to allow it to assume the temperature of the boiling water, remove it, dry and carefully weigh it. Suppose that w is the amount that has flowed out. As a check re-weigh the dish and again determine w. Let p denote the density of mercury at o° C. and p t that at temperature T. If V denote the volume of the apparatus at o° and V x that at T, while p denotes the coefficient of expansion of glass, £ = * + P T. Wl But V =-^, Po - 1 + Pi - Wfl pi W() l^^' 186 ADVANCED PRACTICAL PHYSICS FOR STUDENTS • s =, W ° ~ w * +ocT x " W ' T T W — ze> te> — a — W * W T « =H7^ P + * i.e. W — w W Q —w T W W I p ■ W -a>; p T W -ze> r The ratio ^ _ ^ is small, and since p is also small we may often neglect the first term on the right and use simply a - p = w ,1 . P W -ze> T (2) The Coefficient of Expansion of Water The value of this coefficient varies considerably throughout the range o° to ioo°. In this experiment the average value between two temperatures, say, 20 C. and 6o° C. will be deter- mined. If both temperatures are above that of the room the experimental difficulties are not so great. We assume this to be the case though the method is quite general, and exactly the same precautions have to be taken as in (1) if water flows out on moving B from the lower temperature enclosure to the balance. Carefully fill the weight thermometer with water that has been recently boiled to get rid of contained air, and place it completely immersed in water at temperature t x . Remove it, dry, and weigh. Place the thermometer in water at t 2 and repeat. Use the formula given above to deduce a for water, making use of the value of p previously determined. Determination of the Density of Water at Various Temperatures by means of a Glass Sinker In this experiment a solid is weighed while totally immersed in water at different temperatures, so that by the principle of Archimedes the weights of the fluid displaced by the solid corresponding to the various temperatures are known. Let V denote the volume of the solid at o° C. and a the co- efficient of expansion of the solid, so that at temperature t° the volume is V (i + at). If Pt is the density of the water at temperature, t, the loss of weight due to immersion is V (i + a*) p t . This value is observed by the balance ; let it be W«. Then p ,=- — ^i THERMOMETRY AND THERMAL EXPANSION 187 In order to carry out the experiment, a long Wire is attached to the scale-pan of a balance and passed through a hole in the base of the balance to support the solid, which hangs in the water. Since during some part of the experiment the water will be at a temperature considerably above that of the immediate surroundings of the balance, it is necessary to use a wire about 40 cms. long, so that convection currents may not disturb the equilibrium of the balance. The wire should have a diameter not greater than T V mm., so that surface tension may not cause any appreciable effect where it enters the water. In practice thin copper wire is often employed, though it is preferable to use a short length of platinum wire, specially treated to diminish surface tension effect for immersion in the water. ' # A stirrer is necessary to keep the temperature of the liquid uniform, and as soon as the whirls, due to stirring, have died away, the balance is made and the temperature taken by a thermometer immersed in the water with its bulb as near as possible to the solid. It is preferable to heat up the water to the highest desirable temperature and allow it to cool down. In this case the weights in the scale-pan will require to be continually diminished. The weights should be adjusted before an observation so that the solid appears a little too heavy. After a short interval the scale pointer will cross the zero position, and at this instant the tem- perature of the water should be observed. In this way a series can be obtained very conveniently for temperatures above atmospheric. The solid is first weighed in air so that the values, W*, corres- ponding to different temperatures may be obtained by subtraction. As a sinker it is usual to employ a glass bulb containing lead shot, and a may then be taken as -000025. When it is not possible to reduce the temperature right down to zero it may be taken down to some convenient low temperature, t . By finding the weight, W tQ , of the submerged vessel at this temperature, the volume, V tQy may be deduced by the help of the density table appearing on p. 188, taken from Kohlrausch's " Physical Measurements," v - W ">. since V '»~7r The densities at the other temperatures may then be deduced by the formula W« p < V t0 {i + «(*-g}' 188 ADVANCED PRACTICAL PHYSICS FOR STUDENTS DENSITY OF WATER BETWEEN o° AND 20 C. PER CC. TEMP. DENSITY TEMP. DENSITY 0° 0-99988 11° 0-99965 1° 0-99993 12° 0-99955 2° 0-99997 13° 0-99943 3° 0-99999 14° 0-99930 4° 1-00000 15° 0-99915 5° 0-99999 16 6 0-99900 6° 0-99997 17° 0-99884 7° 0-99993 18 0-99866 8° 0-99988 19° 0-99847 9° 0-99982 20° 0-99827 10° 0-99974 In the laboratory it is convenient to begin at a temperature of about 8o°, and make observations about every io° C. A curve should be drawn exhibiting the relation between temperature and density. Alternatively the experiment may be made in order to deter- mine the coefficient of expansion of water for varying intervals of temperature from the formula : p = p*(i + p*). The Constant Pressure Air Thermometer In this form of thermometer temperature is denned by means of the equation V< = V (i + at) ; t denotes the temperature, V t and V the volumes occupied of a certain mass of gas at two temperatures, the former at t°, and the latter at a convenient fixed point : the zero of the scale. By choosing the melting point of ice as t = o°, and the boiling point of water as t = 100 °, under standard conditions we can find the value of a. We may therefore say that the equation is assumed, and that t is defined by it, or -w-y It is assumed that the pressure remains constant throughout. The diagram shows a simple form of constant pressure thermo- meter. THERMOMETRY AND THERMAL EXPANSION 189 The mercury reservoir is adjusted so that the mercury stands at the same level in the two tubes, CD and EF. When this is the case the pressure is equal to that of the atmosphere in both tubes. Suppose it is desired to determine a certain temperature with this instrument, say the melting point of wax. First surround the bulb completely with powdered ice and turn the three-way tap, T, so that B and CD are open to the room, and adjust G so that the mercury stands at the zero division on the graduated scale of CD. Fig. ioi Wait for about ten minutes to allow the bulb to cool exactly to o° C, and turn the tap so that B and CD are connected to each other but cut off from the atmosphere. In this way the zero on the scale is made to correspond to o° C. Now immerse the bulb in boiling water, and lower G until the mercury stands at the same level in the two tubes, and observe the scale reading. Read the barometric height, and deduce the boiling point of the water. Surround the bulb with warm water and adjust its temperature to the melting point of the wax. To do this, put a small piece of the wax in a small test tube and immerse it in the water. Heat the water until the wax melts, and then let it cool a few degrees, and then warm up very slowly, keeping the water igo ADVANCED PRACTICAL PHYSICS FOR STUDENTS stirred until the wax begins to melt again and then take the reading of the thermometer. Of course G must be adjusted so that the level of the mercury is the same in both tubes. From the observations made we can deduce the melting point of the wax. Note the temperature also by means of an ordinary mercurial thermometer. Theoretical Considerations The fundamental equation of gas thermometry, whatever may be the form of thermometer, is simply : Total mass of gas in the instrument = constant. In practice it is not possible to maintain all the gas at the same temperature ; some of it is necessarily remote from the point of application of the body examined. These remote regions are described by the term ' dead space.' In our apparatus the dead space extends from fl to the level of the mercury in C. We shall suppose that the scale readings are in ccs., beginning at the zero and extending downwards. Let the reading corresponding to the case when boiling water surrounds B be denoted by x b , and let x w be the reading when the wax is melting. Let the volume from the top of the bulb at the point where it is immersed to the zero of the scale be denoted by v, with a suffix to indicate the temperature at which it is measured. The temperature of the dead space, which has a total volume (v -f x), will vary from one end to the other ; but we shall make our calculations by assuming that this temperature is uniform throughout and equal to that measured by placing a thermometer in a position approximately midway between the two ends of® this space. We shall denote this by the letter t, and when the bulb temperature is t we shall write t t for the corresponding tempera- ture of the dead space. Let the volume of the bulb together with that part of the tube which is immersed be V at the temperature zero, let p denote the coefficient of cubical expansion for glass, and p the density of air at zero. The mass of gas in terms of the quantities measured when the bulb is at o° C. is : V Po + «V Pr • The temperature of the dead space is, of course, not necessarily at o°, it has some value t . Or we may write for this mass : THERMOMETRY AND THERMAL EXPANSION 191 ••( T -*SHH3) '■' When the bulb is at a temperature, t, let x t denote the reading on the scale. This denotes the volume between the zero of the scale and the mark x t at the temperature at which the apparatus was graduated. This is often done at about 15 C, and the laboratory temperature is usually in this neighbourhood. We shall not introduce any great error into our calculations if we regard this as measuring the true volume of this part of the apparatus under the conditions Of the experiment. The temperature of the dead space is now t t , and the total volume is therefore : v (i + p«r t ) + *t The mass of gas in the dead space is : v (z + PT f )p x tPo + I + OCTf 1+ OCT Hence the total mass is measured by : V (i + Pflpp , v (i + Pt,) Po x tPo i+att "*" 1 + <xt, "t" 1 + <xt, {2) If the expressions (1) and (2) be equated, since they denote the same quantity, it will be found that : *. = v o(I + .Tof^i* + f«{i±^ -f±^n . . o) l_I+otf V (i+aT 1 + aTjJ w; The second term in the square bracket may be written : ?o(« - P) ( T f - T o) V. (i + oct ) (1 + ax t )' where we have neglected the product a p in the numerator. In practice the difference between t t and t is not very large, and the construction of Jhe apparatus provides that ^is small. We may thus neglect this term in comparison with the first term in the bracket without introducing any great error. The student is recommended to find an approximate value of the two terms, so that he may better appreciate the effect of this neglect of the second term. We then have : - *• = v «<« - «« • TT#- (4) If we apply this to the case when the bulb is surrounded by boiling water, of which the temperature is b, corrected of course 192 ADVANCED PRACTICAL PHYSICS FOR STUDENTS for any variation of the barometric height from normal, we have from (4) *» - V (a - p)6 Z + aT& Hence from (4) and (5) 1 + ub I + a6 £ *6 (5) .(6) I + a^o I + orf -» + T - lr + » , a a The value of — may be taken as 273*1. j t and t h may be observed on a mercury thermometer, although this is introducing into the experiment the mercury scale. Equation (6) is a linear equation in t, which is thus determined from the reading on the scale of the air thermometer. This enables us to deduce the temperature of the melting wax. Fig. 102 The Constant Volume Air Thermometer The diagram illustrates a common form of the apparatus. The bulb, B, is connected by a capillary tube to rubber tubing, DE, and to the glass tube, EF. THERMOMETRY AND THERMAL EXPANSION 193 EF slides against a scale, SS, by means of which the height between the mercury levels in the tubes, CD and EF, can be read. EF can be clamped in any desired position, so that the level of the mercury in the tube CD stands always at a definite mark, C. In this case the temperature is denned by assuming the relation : p t =p (i+<rt), or t=l.t±Zl±*. p t denotes the pressure within B at a temperature t, and po that at a standard zero position — the temperature of melting ice, while the volume of gas remains constant. The pressures are measured by adding the atmospheric pressure to that due to the mercury column of length, h. In practice the part of the apparatus containing the air expands with rise of temperature, and there is the dead space, HC, to be alloweoVfor as in the last experiment. Let V denote the volume of the bulb and immersed portion of the apparatus, v that of the dead space at o°. Suppose that the dead space remains at temperature t during the experiment when B is surrounded with ice, and t& when it is surrounded by steam. We shall show how a may be determined experimentally. If p denote the density of air under normal conditions, and p that at pressure, p, and temperature, t, _ Po P_ 9 - I + at ' 76 The mass of gas contained in the thermometer is : -^-•V + 76 ^o (l+aTo) (7) expressed in terms of the conditions prevailing when B is at the temperature of melting ice. Similarly the mass of gas expressed in terms of the conditions prevailing when B is at temperature, t, is : Vq(i + PQ-/>iPo ■ Vo(t + t*t)P#o ,™ 76(l+0C/) ^ 76(1+ OCT,) W Hence on equating (7) and (8) : Again on account of the smallness of the ratio ~ we have as * an approximation : *-*'T+* (I °) J 3 194 ADVANCED PRACTICAL PHYSICS FOR STUDENTS When the bulb is at the temperature, 6 i of boiling water, i + $b Po =Pb I + <x& (II) We may assume the value -0000232 for p and thus calculate a from (11) by observing the pressures when the bulb is surrounded by melting ice and by steam respectively. Equation (10) then enables us to deduce the temperature corresponding to any pressure, fi t . Take a mercury thermometer and immerse it close to the bulb, observing its readings and the corresponding pressures. From the latter deduce the temperatures from (10) and draw up a table recording these in one column opposite to the records of the mercury thermometer in a second column. Draw a graph with air temperatures as ordinates and mercury temperatures as abscissae, exhibiting the deviations between the two temperature scales. In order to calculate the value of —^ if this is necessary, first adjust the mercury to the mark C, immersing B in watei that has come to the room temperature. Fig. 103. Read off the pressure, P, to which the air is now subjected. Carefully raise EF so that the mercury approaches the bend at H, and so fills nearly all the dead space. Let the pressure within the bulb be now P.. Then, since the conditions are THERMOMETRY AND THERMAL EXPANSION 195 isothermal and the whole volume, v , has been very nearly filled with mercury in the second case, P 2 V = PxCVo + v ), *V P x We may read the temperatures t and t t by means of a mercury thermometer placed close to the dead space and obtain a closer approximation to the value of, a. In the second form of apparatus (fig. 103), into which the first may be readily converted, the volume of the dead space is made negligible. _ In order to measure the difference of level between C and F, two tubes, WW, connected by a rubber tube containing water are adjusted so that the level on the left is the same as that at C, and consequently this is the same on the right at C 1 . The distance, C*F is readily observed on SS. CHAPTER VII CALORIMETRY The Specific Heat of a Solid by the Method of Mixture The student will be familiar with the principle of the method of mixture. The main object in this description is to give an account of the method of making a correction for the error arising from radiation. If W is the water equivalent of the calorimeter and contents, m the mass of the solid, and s its specific heat, and if t x is the initial temperature of the calorimeter, t % the final temperature and T that of the solid initially, then if there has been no loss of heat we have : ms (T - jQ = W(* 2 - t x ). In practice there is a loss or gain of heat from or to the calo- rimeter, which should be added on the right-hand side of this equation, since all the heat from the solid has not been retained in the calorimeter. We may make the correction by the consideration that the final temperature, t 2 , would have been t 2 + At, where At is an interval of temperature which must be small if the experiment is to be successful. Hence the corrected equation is : ms{T - t 2 ) = W(t 2 + At - t x ). At will be small if during half the experiment the calorimeter gains heat, and in the other half loses heat. This can be arranged by adjusting the initial temperature so that the room temperature is approximately a mean between it and the final temperature. A preliminary experiment is made to find out roughly the temperatures that will be attained during the experiment. The same amounts of the materials are used in a second case, but the calorimeter is cooled down by adding small pieces of ice or warmed up, as may be necessary, so that t x and t 2 may lie at nearly equal temperature intervals below and above the temperature of the room. The temperature of the calorimeter is noted immediately before immersing the hot body, and then at quarter or half minute intervals until the maximum temperature is attained, the 196 CALORIMETRY 197 observations being continued beyond this point at definite intervals. These results should be plotted on a graph (fig. 104). The curve obtained will be similar to ABC. Had there been no losses or gains on account of radiation, the curve would have been similar to ADE, the final -temperature remaining constant at the level DE. Ms Fig. 104 We can, by applying Newton's Law of Cooling, derive the curve ADE from ABC. Suppose the axis of t to be divided into small intervals, OM x , MjMji, M 2 M 3 , etc., of magnitudes 8t x , 6t z , dt z , etc., and let the average temperature during these be : X , 2 , 3 , etc. Then the loss by radiation in 8t x is kd x 8t x , where k is a constant. .-. p^i = QxMi + ke x dt x . Similarly the loss during MiM 2 is kd 2 8t 2 , and consequently P 2 M 2 = Q 2 M 2 + kd x dt x + &0 2 # 2 . This process may be continued to any extent, and if we make the intervals sufficiently shorty M x , Q 2 M 2 , etc., are not sensibly different from 6 X , 2 , etc. Thus, denoting by © the ordinate of the upper curve at a time, t, and by that of the lower at the same time : ©! = ©! + ke x 8t x , © 2 = 2 + ke x 8t x + kd 2 8t 2 ; and generally : © rt = 0„ + k{6 x dt x + 2 # 2 + . . . + B n 8t n ) = n + &(area of lower curve from OA to ordinate B ), i.e. © = + k I ddt 198 ADVANCED PRACTICAL PHYSICS FOR STUDENTS We can thus make the correction by carefully drawing the lower curve, calculating the area up to the point 0, which it is desired to correct, and add its product by k to 0. We require in this experiment the ordinate corresponding to any point P 1 along DE, beyond the point B. We therefore determine graphically OAQ x M and apply the correction to Q X M, thus obtaining P*M. In order to calculate k, find th6 rate of cooling along BC, corresponding to a mean temperature 0. k = — • j- (Newton's Law.) It will usually be convenient to note the fall for four or five minutes, and make the deduction from it. Reynault's method of making the correction is actually to make the correction of the ordinates and draw the curve ADE. For this purpose we require to know the rate of cooling at any particular temperature. Find the rate at one period, as above, and plot jr against on a curve. In doing this it is assumed that the relation is linear in accordance with Newton's Law. At the temperature of the room ~ vanishes ; for if the calo- at rimeter and surroundings were at the same temperature then there would be no loss due to radiation. Fig. 105 We therefore have two points on the graph, and by joining dd these by a straight line we can determine -j- at any value of 6 at (fig- 105.) Divide ABC into sections very nearly straight, as AR, RQ, etc. dd Note the mean temperature over AR and from the graph for-r dd note the rate -5- for this temj^erature. Multiply this by the CALORIMETRY 199 time, ON, and add the result to the ordinate, NR, thus obtaining NR 1 . Let this correction be <50 v In the same way find the amount <50 2 , lost during the interval, NM. Add the sum (<50 x -f <50 2 ) to MQ and so obtain MQ 1 . Continue this process until the maximum ordinates along DE are attained. The curve obtained in the experiment should attain this horizontal branch very nearly, and the ordinate is the quantity Speciflc Heat of a Liquid by the Method of Cooling The rate of loss of heat of a body depends only on the tempera- ture of the body and that of its surroundings, on the area, and on the nature of the surface exposed. If the difference of temperature between the body and its surroundings is not large, the rate of emission of heat is propor- tional to the temperature difference. This is Newton's Law of Cooling. Suppose a mass of liquid, M lf is enclosed within a calorimeter of mass m x , and let S x and s x denote the specific heats respectively. The thermal capacity of the system is (M^ + m^^. If the temperature fall from t t to t 2 in n x seconds, the average rate of loss of heat is (MjSj + m x s x ) . — - In the case of the second liquid under the same conditions, let n 2 denote the number of seconds required for a fall of tem- perature from t x to t 2 , and the loss of heat per second is 200 ADVANCED PRACTICAL PHYSICS FOR STUDENTS We have by Newton's Law (M X S 1} + m x s x ) h-U = (M 2 S 2 + m x s x ) h-h :. s x = n x (M 2 S 2 + m x s x ) (i) _ m x s x n 2 M x "M7 The apparatus for carrying out the determination consists of a small calorimeter fitted with a rubber stopper through which a thermometer may pass (fig. 99.) A calorimeter of aluminium of about an inch diameter and three inches high serves the purpose very well. The calorimeter should be supported by threads, or should stand on a non-conductor within a double-walled enclosure, the thermometer passing through a cork in the lid of the enclosure. In order to secure a uniform temperature, the space between the walls of the enclosure may be partly filled with water. In this case care must be taken that the inner box does not float, or it may happen that it will touch the calorimeter and there will be loss of heat by conduction. The enclosure may consist of two calorimeters — an outer large one fitted with a lid, and an inner smaller one standing on blocks. First fill the aluminium calorimeter about two-thirds full of water, and warm it to a temperature about 70 C. by immersing it in hot water. Place the apparatus in the position shown in the diagram, and take readings of the thermometer at intervals of half or whole minutes down to a temperature below 30 C. Note from time to time the temperature of the enclosure, which should hardly vary during the experiment. Fig. 107 Some time must of necessity elapse between the observations on one liquid and those, on another, and although it is not difficult to maintain a constant enclosure temperature throughout each set of observations, it often happens that the mean temperatures recorded by the thermometer, T 2 , are appreciably different in the two cases. CALORIMETRY 201 This difficulty may be avoided by using a second aluminium container similar to G, and suspending it by the side of G inside C. The records of the temperatures of the two liquids are then made almost together, and the enclosure temperature is the same for each. Make up a table containing the liquid temperatures opposite the times of observation, and in a third column record the enclosure temperatures. Draw on the same graph as illustrated in fig. 108 the curves, one for each liquid, with the differences of temperature between liquid and enclosure as ordinates, and with the times for abscissae. Let AB denote the curve for paraffin (say) and A B that for water. Draw the horizontal lines, T X AA and T a BB to cut the curves at A, A and B, B as shown. Let t x denote the temperature of the liquids above that of the enclosure in the first case, and t 2 the corresponding temperature in the second. Then in the time that elapses between the instant measured by TjA until that measured by T 2 B the paraffin cools down the interval (f r — t 2 ), and the water cools down the same amount during the interval between T^o and T 2 B . Denote the two periods of cooling by n x and n z . In this case the value of S a is unity, and for aluminium the value of s x is -219. By weighing the liquids and calorimeters we obtain sufficient data to give the value of s 2 for paraffin by means of the formula (1). Determination of the Specific Heat of a Solid by means ol Joly's Steam Calorimeter A metal jacket, J, enclosed in a casing of felt surrounds a platinum pan, P, suspended by means of a fine wire attached to one arm of a balance, whose base is shown at BB, 202 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The upper end of J is closed by a light metal disc, D, through a hole in which the wire passes. This disc is free to move, and when oscillations occur in P it finally settles down so that the wire passes through the hole without contact with the disc. Just above D, a small coil of wire carrying a heating current round the suspension prevents condensation on it and also on D. Fig. 109 In the first place, J is allowed to attain the temperature of the room, and the inlet and outlet pipes are then closed. The pan is balanced in the usual way, and in the meantime water is boiled in a container ready to supply J with steam by means of I. When a good supply is obtained O is opened and steam passed through I. When the steady state is attained it will be found that additional weights are required to counterpoise P on account of the condensation of steam on it. Suppose w grammes are condensed and let the initial temperature of P, which has been observed by a thermometer placed in T, be t lf and that of steam t z . Once more allow the apparatus to dry and weigh the solid S. When steam is again passed into J with S in the pan, a greater amount of steam will be condensed on account of S. Let this now be W and suppose the initial temperature now is T v Previously the scale-pan condensed w grammes of steam and rose in temperature through the interval (t 2 — *,). In the second case it rises from Tj to t%. The amount of condensation w ' per degree rise of temperature is -r- — r~, hence the weight of (t 2 — *i) w steam condensed in the second case is : -j- r-r x {t 2 — Tj). CALORIMETRY 203 Denote the mass of the solid by m and the specific heat by s. The mass of steam condensed by the solid is : w(t t - TJ W- ('• ~ h) Hence we have the equation : ms(t t - TO =fw - H' ! Ti) ] L, (*, - tj by means of which the value of s may be calculated from a knowledge of the value of L or, conversely, L may be determined if the specific heat of the solid is known. L denotes, as usual, the latent heat of steam. A correction to account for the differences in apparent weights of the solid in air and steam has been neglected. The temperature of the solid is supposed equal to that of the apparatus and surrounding air. An interval of certainly not less than twenty minutes is required to allow the solid to acquire this temperature. Bunsen's lee Calorimeter Description and Preparation for Use A diagram of the apparatus is shown in fig. no. The calori- meter is represented by ABCFED. It consists of a test tube, B, Fig. 1 10 fitted into the glass jacket, A, which is drawn out at its base into the tube, CFE. This tube ends in a cup, E, closed by a cork, through which passes the narrow glass tube, D. A is filled partly with clean mercury and partly with distilled water containing no air. 204 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The surrounding jacket, J, is a calorimeter closed with the cork or wooden stopper, S, which supports the apparatus. In order to keep J and its contents at the freezing point, it is placed in a larger vessel, standing on non-conducting blocks and packed round with a mixture of ice and snow or with flaked ice. By cooling the inner surface of B sufficiently a layer of ice may be formed round the outside, as indicated at I. On melting one gramme of ice the volume diminishes by •0907 c.c, so that if heat be added at B the amount may be determined by noting the change of volume as a result of the partial melting of I. The change of volume is observed by noting the movement of the end of the mercury column at D along the capillary tube. If this has been previously calibrated the change can be observed directly. In order to fill the apparatus, remove the capillary tube, D, and the stopper in E, and introduce into A sufficient distilled water to fill it to about half. Invert the apparatus with the open end of the test tube downwards and carefully boil the water, continuing until A is about one-third full. While this is proceeding, boil some distilled water in a large beaker, and towards the end of the evaporation of the water in A, place the end, E, well under the surface in the beaker. Cease boiling the water in the calorimeter, and allow more to flow over from the beaker. In this way the inside of the calo- rimeter and the tube, CFE, become filled. Clean mercury must now be passed in to lie below the water in A. Introduce it gradually from a pipette held under the surface of the water in E, allowing displaced water to overflow. Take care that no air bubbles are introduced with the mercury, particularly when it becomes necessary to tilt the apparatus to allow water to pass over the mercury in A towards the tube. Fill up with mercury to E, place the stopper in position, and by carefully adjusting it make the end of the thread coincide with any desired position along D. The apparatus should then be placed in a calorimeter containing water and ice to reduce the temperature as nearly as possible to zero. This will probably take an hbur at least, and the progress of the fall may be tested by placing a thermometer in B. When the temperature is about 2° C. introduce cooled ether into B. The ether may be cooled by placing it in a cooled test tube, and standing it in the calorimeter with the apparatus. Draw air through the ether and cause it to evaporate, con- tinuing until a cap of ice surrounds B. CALORIMETRY 205 The solidification will cause D to move farther along the capillary, and enough ice should be formed to cause more expansion than is likely to be required in succeeding experiments with the apparatus. The evaporation of the ether may be brought about by some such device as that indicated in fig. 111. T n J=& Fig. 111 When sufficient ice has been formed the remaining ether is evaporated, and a current of air further drawn through to remove all traces of it. Now place the apparatus within the jacket, J, and stand it in the vessel, K, packing it round as described above with ice and snow. Leave this standing for an hour or two until a steady state has been reached, and the movement in the capillary tube is only slight and steady. It is not possible to maintain the end of the mercury thread quite steady with this arrangement, so that the slight motion must be accounted for in determining results from observations. The capillary tube may be calibrated by the method described on p. 41. Calibration of the Apparatus By placing warm water within B of known mass and tempera- ture, we may note the movement of D at various parts of the capillary tube for a known absorption of heat. In performing the experiment it will be sufficient to calibrate the tube for one particular region, and in using the instrument again a slight pressure on the stopper or a slight easing of it will drive the end of the thread into the calibrated strip. Heat up pure water to about 25 C, and transfer carefully to B. Allow the apparatus sufficient time to become steady, and note the displacement of the end of the mercury thread. Let this be /, and let the time be noted between the insertion of the water and the return to steady conditions. This will be denoted by t. 206 ADVANCED PRACTICAL PHYSICS FOR STUDENTS In order to correct for the small creep of the thread, observe the rate of motion just before adding the warm water, and also just after the absorption of heat. All these measurements are to be made with a travelling microscope mounted conveniently opposite the capillary tube. If the rates of creep are respectively p and p 1 , then the average rate may be taken as \{f> + p 1 ) during the time t. Thus the displacement of the thread due to absorption of heat from the water is / — %(p + p x )t. We shall call this quantity L. If m is the mass of water added and its initial temperature, then a motion of L units corresponds to the absorption of md calories. Determination of the Specific Heat of a Substance Let M grammes of the substance be heated to a temperature 1 , and let s denote the specific heat. The substance when placed in B will transfer to the ice M0 x s calories. If the mercury thread moves a distance, L 1 , in the calibrated region, the heat absorption is -j— L 1 calories. Tu ,,., m0L l Thus M0 x s = — j — , md L 1 S= Wi'L- The correction for Creep must again be applied by observing the motion just before and just after the insertion of the mass. L 1 is the corrected length. When a solid is put into B, a pad of cotton wool should be placed at the bottom of the tube to prevent breakage when it falls. In order to facilitate removal it is a good plan to tie a light thread round it. This will introduce only a slight error. During the absorption of heat, and generally while the apparatus is in use, the end of the tube, B, should be stopped with a plug of cotton wool. In the case of determining the specific heat of a liquid the experiment is almost exactly a repetition of the calibration. In order to dry the tube after liquid has been put in, a roll of clean blotting paper may be used. The Determination of the Density of Ice In this experiment we require to know the volume per unit length of the capillary tube. We have assumed that this has been previously calibrated. It thus remains to determine the CALORIMETRY 207 shrinkage due to absorption of a definite quantity of heat. The experiment may be performed in conjunction with the calibration just described. Let L denote the latent heat of fusion of ice, and suppose warm water added to B imparts k calories. k The amount of ice melted is -=- grammes. Let dv denote the shrinkage as measured by the movement of the mercury thread. Then -=- grammes of ice have become ■=- grammes of water L L, at o° C. Let d denote the density of water at this temperature. k The volume of water is j-^ c.c, and the volume of the ice is thus : Gi + »)«* Hence the density of ice is : A L . grammes per c.c. &+* The Latent Heat of Fusion of Ice It is assumed that the student is familiar with the principles of the determination of the latent heat of fusion and has carried out the experiment without making corrections for radiation. We are concerned in this description chiefly with an account of how this correction may be made. Care is taken, as in the determination of specific heat, to adjust the initial and final temperatures so that the room temperature is the mean of the two. *In addition, care must be taken that the calorimeter is not cooled down so low that the dew point is reached, otherwise there will be a deposit of dew on the apparatus, and a liberation of latent heat in consequence. We may make the correction for radiation as in the experiment on specific heat, but an alternative method will be described. Note the temperature when the ice is placed in the calorimeter, and at intervals of half a minute until it is melted, and finally at intervals of one or two minutes during which the calorimeter is absorbing heat by radiation. 208 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Let the temperatures observed in this second period be 1 1> 1 2> I 3 • • • ■*• n+V Then the change of temperature due to radiation during these intervals will be : »e t = AftCTi + T 2 ) - t ] <50 2 = AftCT, + T 3 ) - t ] <50 o = A[£(T n +T B+1 ) -*„], by Newton's Law of Cooling, where A is a constant depending on the calorimeter but not on its temperature, and t is the temperature of the surroundings. Thus the total change of temperature A0 is given by : A0 = A{ Tl + 2 Tw+1 +(T 2 +T 3 + ..+T n )-^ J But A0 = T B+1 - T 1( so that A can be calculated. In the first part of the experiment during the melting of the ice, let the observed temperatures be : t x , t 2 , . . . t n+1 . Then we have : At AJ * 1 + J n+1 + (*, + t 3 + . . . + t n+1 )-nt Q }. Thus, since the observed minimum temperature is t n+1 , the corrected minimum is (t n+1 -j- At). In the above the <50's and dt's are to be treated algebraically, for some will be negative and some positive if the initial tempera- ture is adjusted in the manner described. Thus, if W is the water equivalent of the calorimeter and contents, and m the mass of ice melted, we have : wL + mt n+1 = W{*i - {t n+1 + At)\. The Latent Heat of Vaporization. (Berthelot's Apparatus) One form of this apparatus is depicted in fig. 112. Its essential feature is the condenser spiral, C, with the receptacle below. This is immersed in water in a calorimeter and the calorimeter shielded by packing it round with a non-conductor and placing it in a convenient vessel, or better still by standing it on non- conducting blocks in an empty larger calorimeter and enclosirig within the larger vessel (see fig. 112). The condenser is dried and weighed. It is placed in the water and allowed to stand until the temperature becomes steady as recorded by the thermometer, the uniformity of temperature CALORIMETRY 209 throughout the calorimeter being procured by means of a stirrer. A quantity of vapour is introduced into the condenser, and the spiral provides a large area of contact with a cold surface, so that liquefaction takes place and the liquid collects in the receptacle. The water is constantly stirred, and its temperature observed by means of the thermometer. Fig. 112 When a suitable rise is obtained the supply of vapour is cut off and the thermometer watched until the maximum temperature is reached. In this interval, immediately after cutting off the supply, the end of the condenser outlet tube must be closed by a cork to avoid convection effects. The condenser is removed, dried, and weighed, and the mass condensed thus determined. Denote this by m, so that if L denotes the latent heat the supply of heat on condensation is mL calories. Let T 2 denote the temperature of vaporization, and T 2 the final temperature of the calorimeter and contents, while T denotes the initial temperature of the calorimeter. Let s denote the specific heat of the liquid, and W the water equivalent of the calorimeter, condenser, and remaining calori- meter contents. Then : W(T 2 — T ) =wL + m(Tj — T^s ; so that L may be determined. One of the weak points of this form of apparatus is the mode of introduction of the vapour. 210 ADVANCED PRACTICAL PHYSICS FOR STUDENTS As the diagram shows, the liquid is vaporized over a small gas ring in the reservoir. It thus easily happens that the vapour gets superheated, and, in addition, it is difficult to shield the calorimeter effectively from the heat of the flame. The screen, S, is introduced to reduce this effect. It is preferable to use an electrical method of heating, and this is done in the more recent forms of the apparatus. Kahlenberg's heater is illustrated in fig. 113. The liquid is contained in the vacuum flask, H, and is heated by passing an electric current through the platinum wire, PP. For ordinary laboratory practice a test tube may take the place of H without introducing much difficulty in shielding the calorimeter. Fig. 113 Correction for loss due to radiation in the calorimeter, C, must be made by one of the usual methods (p. 197). The Heat of Solution of a Salt When a salt is dissolved by a liquid the solution is accompanied by an absorption or liberation of heat. The amount varies with the proportion of the salt to the solvent, i.e. with the resulting concentration, and with the amount of salt dissolved. The number of calories absorbed or liberated when one gramme of a salt is dissolved in a certain amount of solvent is said to be the heat of solution for the particular concentration. The experiment described below is designed to measure this quantity. CALORIMETRY 211 The apparatus necessary is illustrated in fig. 114. It consists of an outer protecting calorimeter of metal, C, which carries a cork through which the inner vessel, A, is supported. This vessel also carries a cork through which pass a thermo- meter, stirrer, and thin test tube, B, into which the salt may be placed. Fig. 114 The vessel, A, and stirrer, S, are usually of glass since many solutions attack copper. There is uncertainty about the specific heat of the glass, so that the water equivalent of the vessel, A, and its contents should be determined in a separate experiment. For many purposes, however, we may assume the specific heat of the glass to be «i6. The pure solvent is placed inside A, and it should surround the lower part of the tube, B, which contains the salt. The apparatus is allowed to stand so that the salt may acquire the temperature of the solvent. Half an hour should be allowed for this, and during this interval the salt may be occasionally stirred by a clean dry glass rod. The weight of solvent is obtained in the usual way before the insertion of B. Let this be denoted by W. Suppose the water equivalent of A and its contents is w, and the specific heat of the solution s. This quantity must be determined later by the method of cooling (p. 199), or by any other convenient experiment. It will be necessary, also, to make a correction for radiation by one of the methods previously described (p. 197). Let the initial temperature of the salt and solvent be denoted by t and the final temperature, corrected for radiation, by t v 212 ADVANCED PRACTICAL PHYSICS FOR STUDENTS and suppose that the weight of the salt dissolved is q. The heat of solution being Q, we have : qQ= {(W+q)s+w}(t-t ), for (W -f q) is the weight of the solution. The value of Q will be positive for the liberation of heat, and negative for absorption. In order to mix the salt and solvent, the glass rod used above for stirring should remain in the salt until just before mixing. At this instant observe the temperature, t , recorded on the thermometei^and push the rod through the bottom of the thin tub<T?T This tube must, of course, be clean and dry, and the salt will then fall into the solvent and be dissolved. The rod should then be removed, and care taken not to carry with it any of the solvent. The solution is assisted by the stirrer, S, and observa- tions of T taken every quarter or half minute in the initial stages, and later at longer intervals. A graph is drawn showing the variation of temperature with time from the instant of mixing, and this curve corrected for radiation as explained above, or we may use the non-graphical method. To find the Ratio of Specific Heats at Constant Pressure and Constant Volume for Air. (Clement and Desorme's Experiment) Apparatus and Experimental Details A glass reservoir, provided with a tap, T, giving a wide opening to the air is connected to an oil manometer, G, and to a pump; Fig. 115 an ordinary bicycle pump is convenient. A small excess pressure is applied to A, the difference between it and the atmospheric pressure being measured by the manometer (fig. 115). CALORIMETRY 213 In the first stage of the experiment the temperature under these conditions is allowed to become steady. In the second the tap, T, is opened and closed suddenly by giving one half-turn. By this means the pressure falls to that of the atmosphere in so short an interval that we may suppose there is no passage of heat to A during this expansion. The condition of the expansion is therefore adiabatic. Finally the temperature is allowed to return to that at the beginning of the experiment, during which process the pressure in A increases, though it does not recover its original value. Theory. Suppose that gas occupying the volume below the dotted line remains in the flask all the time. Denote its volume by V while that of the flask is.V. Let the initial pressure be P , and let that immediately after the adiabatic expansion be B. Then BV' = P V ', where r — ratio of specific heats, the value of which we require. The flask is open to the air so that B is the atmospheric pressure. It is in order that we may ensure a fall of pressure from P to B during a short interval that the tap is wide. In the final stage let the pressure become P when the tempera- ture has attained a steady value. We have now passed from volume, V , and pressure, P , by an isothermal process to volume, V, and pressure, P. /. PV = P V . /. P'V «= P 'V '. Hence by dividing this by BV f and P V f , we have : B or (0- p r-x B . p * log r — log P_ Po Let the difference in heights of the manometer be h initially, and h finally. Then if pressures be measured in terms of heights of the liquid columns : p = B + K, P = B + h, 214 ADVANCED PRACTICAL PHYSICS FOR STUDENTS r = j—X (approx.), Po by expansion in logarithmic series and neglecting higher powers of ~ and -4g — than the first. This is permissible since the values of h used are only a few centimetres. 5 cms. is a convenient value for h . Hence r — -, — 2-r- «o — A The result may also be obtained by another method. In fig. 116 let AC denote any curve relating the pressure and volume of a gas. The elasticity is denned to be the ratio : stress strain The stress will be measured by a slight change in pressure, and the strain by the corresponding slight change in volume per unit volume. Let us consider the volume, v, represented by BC. Let a change of pressure, dp, denoted by FA, bring about the change in volume denoted by CF. We shall record this by &o, but 6v = — CF on account of the diminution of volume on the addition of pressure. >/OU.C»KA«: Fig. 116 CF CF Thus, the elasticity, E, is measured by AF -7- — » for — J v v denotes the change of volume per unit volume. .-. E-— flt dV When we proceed to the limit and make the changes very small we have : E = — v-r-- dv CALORIMETRY 215 For a gas the value of E depends on how the change is made, and we shall consider two cases, first the case of an isothermal, and then that of an adiabatic change. Let AC denote the adiabatic curve and AB the isothermal ; the former is steeper than the latter. Let E* denote the adiabatic and E«, the isothermal elasticity. From the formula for E we have : E = — v x slope of curve. Thus E^ == — v x slope of adiabatic, and E fl = — v x slope of isothermal. If we start at A with a particular volume, v, measured for each curve by P A, we have : E$ slope of adiabatic E^ — slope of isothermal The changes of pressure and volume in the experiment are small, so that the curves, AB and AC, are approximately straight, and the slopes can be measured by : p£ and gg respectively. E*__AF •* E„~~AE Now, AF is the change in pressure during the adiabatic part of the expansion, viz., h , and AE is the change during the isother- mal part. In our case the atmospheric pressure is that at the end of the adiabatic expansion, i.e. that at C, and since the point, B, on the curve represents the final state, CB denotes the pressure, h. Thus AE = h Q - h. ~,, Ea h n i.e. xxiua E, ~ ' h — h For the adiabatic Differentiating we expansion log^> + have : idp pdv we have : pv r — constant, r log v = 0. +■^-0. V dp dv ~ P - — r. V In our case p and v denote the values of the co-ordinates at A, id -£- i: dv axis of x. dj> and j- is the tangent of inclination of the curve AC to the 216 ADVANCED PRACTICAL PHYSICS FOR STUDENTS . Similarly, for the isothermal case we have : pv = constant, and £»_£ dv v p and v are the co-ordinates of A, but the slope is now for AB. Thus the ratio of the slopes is r. ■ • - -* = r = h ° E«j h — h In carrying out the experiment make about six independent determinations, and increase the pressure cautiously so as not to expel oil from the manometer. CHAPTER VIII VAPOUR DENSITY AND THERMAL CONDUCTIVITY Vapour Density. (Victor Meyer's Method) The vapour density of a substance can be found by measuring the volume of the vapour produced from a small quantity of the solid or liquid whose weight is known. In Victor Meyer's method this volume is found from the volume of air displaced by the vapour. The apparatus (fig. 117) consists of a vertical glass tube pro- vided with a bulb at the lower end, A, and a side tube, ST. The side tube dips under water in a beaker and a rubber cork closes the upper end of the tube. Fig. 117 As it is necessary, on introducing the vapour, to allow a small bottle, D, to fall the length of the tube, it is advisable to place a little asbestos at the bottom of A to prevent breakage. The tube, A, is surrounded by a larger tube containing a liquid which boils at a higher temperature than the substance to be experimented on. 217 218 ADVANCED PRACTICAL PHYSICS FOR STUDENTS In the case of the determination of the vapour density of ether, water may be used in the bath. The various parts of the inner tube should be kept at a constant temperature during the experiment, and in order to maintain this condition the outer tube is screened from draughts by surrounding it with a cylinder of asbestos or cardboard which fits it above the bulb. Before beginning the experiment the inner tube must be quite dry. If necessary it should be warmed over a Bunsen flame while a current of air is blown through it. The apparatus is set up as shown in the diagram, and heating is kept up until no more bubbles come out from the side tube. The substance is weighed and enclosed in the small bottle, D, and suspended close to the upper end of the inner tube. When everything is quite steady the bottle is allowed to fall ; vaporiza- tion takes place, and the cork of the bottle is blown out. Air T B Fig. 118 passes over into the side tube and may be collected in the burette, B. It is better to collect it by the method illustrated in fig. 118, in which case the gas collected can be brought to atmospheric pressure by raising or lowering the burette. In the other case it is necessary to correct for the height of the water column, L. In order to cause no disturbance on introducing the substance to be vaporized a piece of thread or thin wire should be passed through the cork and be held by a stop-cock, E, which pinches a piece of tubing, F. The bottle is allowed to fall by opening E, and then closing it immediately. A better method is to use the apparatus shown at G. By turning the wire, H, through 180 the bottle will be caused to slip off the hook. VAPOUR DENSITY AND THERMAL CONDUCTIVITY 219 The air collected at B is over water at a temperature, T, and will be saturated with water vapour at this temperature. Let B be the saturation pressure at this temperature. If v is the measured volume of air, and H the total pressure, then v the volume under normal conditions is given by H - B 273 v = v x — 7 — x L ^f 76 273 + T If w — weight of substance enclosed in D, the density of the vapour is : w __ w 76 273 4- T v ~ ~v H -B 273 1 c.c. of hydrogen at o° and 76 cm. pressure, weighs -0000900 grammes, and its molecular weight is 2. Hence the molecular weight of the substance examined is : 2 w 76 ^ 273 + T •00009 v H — B 273 At the conclusion of the experiment remove the stopper from the end of the inner tube to prevent any sucking back of water from B into the bulb, A, as the apparatus cools. It is important to cause the bulb, A, to be heated by the steam from the bath, and it should be adjusted to prevent actual contact with the water in the bath, and to be out of reach of splashes when boiling takes place. Vapour Density. (Dumas's Method) A large flask is cleaned and dried, and fitted with a cork provided with a bent piece of glass tubing drawn to a fine point so that it can be easily sealed by the application of a Bunsen flame (fig. 119). The flask is suspended from the arm of a balance and weighed. Let the observed weight be W lt and let w x denote the weight of air displaced by the wall of the flask, while w 2 denotes the weight of air displaced by the closed flask, so that w 2 — w x denotes the weight of air within it. This will be denoted by w a . Hence if W denote the real weight of the flask : W 1 = W-ze' 1 (i) Now introduce a small quantity (5-10 c.c.) of the liquid to be vaporized into the flask. The actual quantity will depend on the size of the flask and will be discovered by trial. The amount given is of the right order when a flask of capacity about a half litre is employed to find the vapour density of chloroform. This liquid is very suitable in the case of a laboratory exercise, since it has a high vapour density. The flask is placed within 220 ADVANCED PRACTICAL PHYSICS FOR STUDENTS an enclosure, J, over a sand bath, with the tube, T, projecting through the lid. Heating is continued until the liquid is vaporized and no more issues from T. This may be tested by placing a polished surface near the end of T. It will become dimmed if vapour is still coming out. When the steady state is reached, T is sealed off. Fig. 119 Let this happen at a temperature, t°, measured by means of a thermometer hanging close to A, within the enclosure, J, and let the volume of the vapour be V t , and density ? t . Allow the flask to cool, and weigh. Break off the end of T under water. The flask will fill with water, the space occupied by the condensed vapour becoming negligible under the new conditions. Preserve the broken pieces from T, and after drying the outside of the tube re-weigh the flask and water, noting the temperature of the water, t °. Let W 2 denote the weights in the scale-pan when the flask and vapour are weighed, and let w v denote the weight of the vapour. Then W 2 = W + w v - w 2 ; .-. W 8 — Wj = w v — {w t — Wj) by equation (1) VAPOUR DENSITY AND THERMAL CONDUCTIVITY 221 Let V t0 denote the volume of the flask obtained from the weight of water it contains at t °. Then if p denote the coefficient of cubical expansion of the glass, which may be taken as '0000232, V t = V i0 {1 + ?(<-«„)}, If the barometric pressure be P, we can reduce p t to normal conditions by the formula : (273 + *) , 76 273 P (273+J) . 76 w % H =~m T' Pt 2 73 P V #0 {1 + fi(t-t Q )}' w a denotes the weight of air filling the flask at the temperature of the air within the balance. Let this be t 1Q C. Then if d a denote the density of air under these circumstances, w. = d.V* = da V f() {1 + p (V- - t Q )\ -^•^.^f^r-v^i + p^-^)}, d — -001293 gm. / c.c. From these two formulae, since w v = w a + W 2 — W 1( we may now calculate p . One of the difficulties of the experiment is to drive out all the air from the flask and replace it by vapour. It frequently happens that on attempting to fill the flask with water some air is left behind. More of the liquid is required for driving off the air in this case. If the volume of air left over is small, we may apply a correc- tion. At the temperature t , of the water, let the volume of air be v. This may be determined by filling up the flask with water from a measuring flask. The total pressure to which the mixture of vapour and air is subjected is the sum of the partial pressures, P 1 and p\ of the vapour and air respectively. The weight of the vapour is now w v \ obtained by subtracting the weight of air of volume, v, from w v , determined as above : wj> = p% V <0 denotes as before the total volume of the flask, i.e. the volume of the water after the bubble of air has been replaced. We can find P 1 by remembering that the air of volume, v, and at atmospheric pressure occupied a volume, V t , under the partial pressure, p 1 , the temperatures being respectively t ° and t° C. 222 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Thus aP = V t p l 273 +* 273 +r p 1 = P -P 1 ; .-. P x V t = PV t - 273 1 1 • Pv, * * 273 + 1 273 + * 76 I 273 + 1 J or p = w J 273. P L 273 + 1 273 + *oJ 76 L 273 + 1 The term, — » will be small if the experiment is to be 273 + 1 y successful at all, so that it will only be necessary to change the value, w v , to the actual weight of vapour, to, 1 . It should also be remarked that the volume, v, is a mixture of air and water vapour, whereas it has been assumed to consist of air only. We shall not, however, further consider this correction, which is small and affects a term which must be already small. The discussion shows how the error affects our formula, and it would be sufficient to measure the volume of the bubble, multiply by the density, -001293 gm./c.c, and subtract from w v . After a few attempts the bubble will usually be sufficiently small to be neglected altogether. Conductivity of a Copper Bar The apparatus consists of a bar of copper, CC (fig. 120), with two holes bored well into it to carry thermometers, T x and T 2 , mercury being placed in the holes to ensure good thermal con- tact. At the ends of the bar are two metal boxes, through one of which, A, is passed steam, and through the other, B, a steady stream of water from a constant pressure head. The shelves, LL, within B, serve to prevent any flow of water straight from inlet to outlet. The temperature of the water is taken just before it enters and leaves B by the thermometers, T 8 and T 4 . The apparatus is allowed to attain a steady state, when all the thermometers will record steady temperatures. It is usual to pack loosely round CC and the boxes some cotton wool, the whole being enclosed in a felt-lined wooden box, through which T t and T a project. T 3 and T 4 are kept as close VAPOUR DENSITY AND THERMAL CONDUCTIVITY 223 as possible to the box, and it is a good plan to wrap the T-pieces loosely with wool. In this way the heat from A is transmitted by conduction to B, and the heat passing across in any time, t, is noted by collecting water as it leaves B and weighing it. If the mass collected is m the rate of transmission of heat is *» (T 4 - T 3 ) t '• T. T, T, f 1 j-L— 3 B Tc -L— } C A Fig. 120 T 3 and T 4 denote the initial and final temperatures of the water. Since this quantity of heat is transmitted from T x to T a —a distance, d, say, the amount is k • (T 1 — T a ) • j, where A is the area of section of the bar, and k the conductivity A is measured by finding the diameter of the bar if it is cylindrical, or by measuring its breadth and height if rectangular. We have, therefore : k (Tj — T a ) • j = m (T 4 - T 8 ) In another form of apparatus the cold water is passed through a metal spiral wound round the end instead of through the metal box. Good thermal contact is made between the spiral and bar,^ and the temperature of the water measured at entrance and exit as before. Thermal Conductivity of Rubber Tubing The apparatus required consists of a length of rubber tubing, B, a copper heater, A, for producing steam, a calorimeter, C, thermometer, T, and a measuring glass, as illustrated in fig 121. The method of procedure is as follows : A quantity of water is introduced into the calorimeter, C and weighed. 224 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Steam is passed through B until a rise of temperature of the calorimeter and water of from io° to 20 ° C. has occurred. The initial and final temperatures are noted and also the time of passage of the steam. The tube is removed and the length that has been immersed is noted — I (say). Two pieces of cotton should be tied round the rubber at the points where it enters and leaves the water. Fig. 121 Let the initial temperature be T and the final T 1 . If there had been no loss of heat by radiation the final temperature would have been some other value, T 1 + A T, and it is necessary to make the correction AT. In order to do this, observe temperatures, at intervals of half a minute or some convenient period, beginning at T and ending at T 1 , during the passage of the steam. These will be denoted by t v t a , ... t n . x . If the temperature of the room be t we have for the change of temperature due to heat radiated or absorbed, as the case may be, according to Newton's Law of Cooling : T +t x ^c^-4 dd 2 = C J — t [, VAPOUR DENSITY AND THERMAL CONDUCTIVITY 225 dd n = C . . \ ... c *oj, where 3d denotes the change in any interval, the suffix denoting which interval. Thus AT = S.50 = C | T ^ T1 - w* + (*i + *i + • • • + *«-i)|. C is a constant depending on the calorimeter and contents, and must be determined if M is to be calculated. When the final temperature, T 1 , has been attained, cut off the supply of steam and allow the calorimeter to cool, observing temperatures, T^ T 2 , . . . T B , at equal intervals. Then as before if 8t denotes the loss of temperature in each interval we have : S# = C |3i±I-« - (»-i) t + (T a +T 3 + . . . + T.-0 But S# = T x — T n , so that C can be calculated and may be employed in the above case. The ranges of temperature in the two cases should be as nearly as possible the same. If M denote the water equivalent of the calorimeter and con- tents the heat transmitted through the tubing is MfP+AT-T). We can connect this with the conductivity, k, in the following way : Let the outer and inner radii of the tubing be r x and r 2 , and con- sider a portion of unit length of the tube between radii, r and r + 8r, at which the temperature is t. The rate of change of temperature at the distance, r, is — -j-. ar The negative sign expresses the fact that the temperature diminishes as r increases. Thus if Q denote the quantity of heat transmitted per sec. per unit length, i.e. across an area, 2nr, r» t, dt J 2-ak J r is 226 ADVANCED PRACTICAL PHYSICS FOR STUDENTS where the integration is to be taken between the limits, r x and r 2 for r, and between the inner and outer temperatures of the tubing for t. Let the steam temperature be t, the outer temperature is taken as the mean of T and T 1 , /. ,- i<T+T') = 2 -filog/J. But Q = y (T 1 + AT - T), . . M T> + AT - T , r, ■■ * - 55 x 2 ' 3 ° 3 x utt ' lo e» f, 2 (Change being made to logarithms to base io.) The value of r x may be determined by means of a screw gauge, and in order to find r 2 , place a length of the tube of 5 to 10 cms. in water in a measuring glass and note the volume, v, displaced. Then if L denote the length of tubing : v = tcL (rS - r 2 *). All the quantities except r 2 are known, so that this value can be determined. Another way of determining the radii is to cut the tube clean, normal to its length and use it as a rubber stamp, pressing it lightly on a clean sheet of paper. The impress of the outer and inner circumferences will be distinct and the diameters may be measured by means of a travelling microscope. The Conductivity of Glass The conductivity of glass in the form of a tube may be found by the method described in the last experiment. A different arrangement of apparatus is required, but the theory is identical in both cases. Steam is passed through a jacket, J, round the tube, B. Within B a stream of water is caused to flow from a supply which provides a constant head of pressure. Within B is a spira> made of cord or rubber so that as it pro- gresses up the tube the water is caused to traverse it spirally. This is important as the temperature at any cross-section of B VAPOUR DENSITY AND THERMAL CONDUCTIVITY 227 must be the same throughout the section. The rate of flow is adjusted to cause a difference of temperature of about 20 be- tween T and T 1 . The thermometers are enclosed in T-pieces as near as possible to the ends of B, the T-pieces being covered with felt or cotton wool to prevent loss of heat by radiation before the temperature is taken. Fig. 122 In order to measure Q, water is collected on exit for a measured time. Conductivity ol Cardboard by the Method of Lees and Chorlton The apparatus consists of a retort stand provided with a clamp (fig. 123) and metal ring, AB, from which hangs a cylindrical slab of copper or brass, DE, of diameter about 12-5 cms. On this rests a hollow cylinder, C, of the same diameter, provided with inlet and outlet tubes, G and H, through which steam may be Dassed. Towards the base of C and into DE holes are bored so that the thermometers, T x and T t , may be inserted and the temperatures read The two cylinders are nickel-plated in order to produce a surface of uniform emissive power. Suppose a thin slab of material of the same diameter as the cylinders is placed between them, and let the loss of heat that is radiated from the edge of the slab be small enough to be neglected in the calculation. Then all the heat transmitted across the slab is radiated from DE during the steady state. Let A denote the area of cross-section of the slab and d its thickness. Let the thermal capacity of DE be denoted by W, and let the thermometers, T x and T 8 , record temperatures, T x 228 ADVANCED PRACTICAL PHYSICS FOR STUDENTS and T 2 in the steady state, while the temperature of the surround- ing air is T . The heat transmitted through the slab per sec. is : k. Tx d •A, where k is its thermal conductivity. A B \ H G E 'T 2 F 12 I Fig. 123 The heat radiated per second from DE is C • (T 2 - T ), where C is a constant. Thus .T, A = C (T 2 - T ). We may readily use the apparatus to give comparative results. If two sheets be cut of the same diameter as DE, one of glass and the other of cardboard, about 1 mm. thick, and if they be inserted between the cylinders we may observe two sets of temperatures, one for each. Let the letters without dashes be used to describe the experiment with glass while those with dashes correspond to cardboard. In the latter case we have : h'~ T/ - T 5 d' •A = C(T 2 '-T / ). VAPOUR DENSITY AND THERMAL CONDUCTIVITY 229 On dividing we have : k' _ T 2 ' - T ' d' k T 2 — T n d T t -T s T 2 - T„ d T/ - T 2 ' ' d and d 1 are measured by means of a screw gauge. In order to find T place the thermometer below a sheet of cardboard, FF, to protect it from direct radiation from DE, but it should be placed directly below DE, so as to give the tempera- ture of the air which rises upwards to DE. An absolute value may be found for k by determining the rate of fall of temperature of DE at the temperature, T 2 . This may be found by removing C and allowing a Bunsen flame to play on DE until T 2 registers a temperature about io° above that recorded during the steady state. Observe the temperature recorded by T 2 as the slab cools to about io° below that recorded during the steady state. The observations should be made every half-minute, or more fre- quently if the change is rapid, and a graph drawn relating the time and temperature. From the graph determine the value of the rate of change at various temperatures. Plot a second graph with the rates, -^-as ordinates and the corresponding temperatures, T, as abscissae (fig. 124). From this determine the particular rate for the temperature, T a . Denote this "&\ Weigh the slab and determine its water equivalent, using for the specific heat of copper the value -094, or of brass the value -09. Denote this by W. Hence the loss of radiation is also expressed by W( — J and we have the equation : by means of which k is determined. / 230 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Make a table of the observations during cooling as follows : TIME (MIN.) TEMP. (T.) AVERAGE TEMP. RATE— . dt * I I* 2 2* 3. Forbes's Method of Determining the Conductivity of a Metal Bar This experiment consists of two parts, in the first of which a metal bar is heated at one end until the steady state is reached. In the second part a bar of the same material and cross-section, but shorter, is allowed to cool under similar external conditions. In order to maintain constant external conditions the experi- ment should be performed in a part of the laboratory sheltered from draughts. 3 Fig. 125 The bar (fig. 125) usually has one end curved and dipping into a convenient molten metal contained in a vessel on the other side of a screen, which protects the bar from direct radiation from the source of heat. The metal may conveniently be molten lead or solder. The bar is provided with a series of holes which lie regularly along its length into which thermometers fit. VAPOUR DENSITY AND THERMAL CONDUCTIVITY 231 If these are small and contain mercury or the molten liquid, it is found that the process of conduction is not disturbed by their presence. The holes near the hot end should contain the molten liquid and the remainder mercury. By observation of the temperatures at various distances along the bar measured from the scifeen, the temperature slope may be found. This is best done by plotting a curve (fig. 126) and calculating the slope, -=-, from the inclination of the tangents at various points. e 3£ Fig. 126 A thermo-junction may be used alternatively to find the temperatures by dipping one junction successively into the holes. For the calibration of the junction the reader is referred to p. 545. The importance of maintaining steady external conditions will be appreciated in this part of the experiment, and great care will be required to maintain the whole length of the bar simultaneously steady. The bar is assumed to be sufficiently long that its end is at the external temperature, O . A convenient length is one metre and its section may be about 2 cms. square. In the steady state all the heat passing a section, B, escapes from the surface between B and the end. The rate of flow at B is : dx> where k is the conductivity and A the area of section. This rate of flow is calculated from the second part of the experiment by determining the rate of cooling of the portion of the bar beyond B. The equation thus obtained serves to find k. The second bar may be conveniently 10 cms. long. It is heated to the temperature of the molten metal, but so that its surface is not damaged and remains similar to that of the first bar. To do this and at the same time to prevent sudden cooling of the molten mass, the rod is wrapped in several layers of paper and completely immersed. 232 ADVANCED PRACTICAL PHYSICS FOR STUDENTS It is provided also with a hole to carry a thermometer and is suspended under the same conditions as prevailed round AB, and its temperature observed at successive times so as to include in the range those values which prevailed along the bar. A curve is drawn showing the relation between the temperature, 6, and the time, t, for the short bar. This bar is similar to the long bar and cools under the same conditions, so that the rate of cooling for both is the same. d6 From the curve we can deduce the rate of cooling, — , in the usual manner. We require for the purpose of the calculation the rate of cooling for points along the bar in the first part of the experiment. We dd must correlate the values of x and —=- , x denoting distances measured from the screen. This may be done from the curves. Take a series of values of x from the curve illustrated in fig. 126 and observe the corresponding value of 0. From the curve relating and t take the values -=- for these particular values of 0. dd We then have the corresponding values x and -=- at Plot these on a curve as illustrated in fig. 127. Fig. 127 de This curve will cut the axis at a point, C, where -^ vanishes, at or where the temperature of the bar is equal to that of the sur- rounding air. If 5 is the specific heat and p the density, the rate of loss of heat between two points separated by distance dx is AdXpS -57- dt • The total loss per second between B and the end of the bar is thus : VAPOUR DENSITY AND THERMAL CONDUCTIVITY 233 i the Aps 5— ax b « * Ap Vb -s- = Aps x shaded area of fig. 127, graph measures the distance from the hot dx where OP on end up to B. This area may be found by the planimeter or calculated if carefully drawn on squared paper. •■• * A GD B =Apsxs - S = shaded area and ( •-=- ) denotes the temperature slope at the point B. ■••*•(&- * We can thus determine k, the values of p and s being given as constants of the apparatus or determined in the usual way. These values may be taken to be 8-93 gm. per c.c. and -094 respectively when the bar is of copper. The Determination of the Conductivity of a Bar of Metal by Angstrom's Method In this method heat is supplied to a long bar by alternately heating and cooling it at one region in regular periods. In this way the temperatures at points along the bar fluctuate periodically, and on account of surface radiation the temperature Fig. 128 amplitudes diminish as the distance from the region of supply increases, while the maximum and minimum values occur at later timefe with the increasing distance. The bar must be chosen so that it is sufficiently long to allow 234 ADVANCED PRACTICAL PHYSICS FOR STUDENTS us to neglect the effect of the terminal faces. The fluctuations should die away at a short distance from the cooler end, which thus has the same temperature as the air surrounding it. When the heating is continued long enough the periods develop themselves completely, in which case the mean temperature at any point of the bar preserves some constant value. We first consider the theory of the experiment. Let us consider a bar, AB (fig. 128), with one end, A, exposed for a definite time, T x , to a current of steam and for a succeeding time, T 2 , exposed to a current of cold water. Suppose that this process is continued regularly until the steady fluctuations throughout the bar are developed. These fluctuations will have a complete period of (T x + T 2 ) which we may denote by T, and for the sake of convenience we shall write Let the conductivity of the bar be, k, p its density, s its specific heat, A its cross-section, P its perimeter, and its temperature at a point, P, distant x from a convenient origin. We shall choose this to be at A. Consider a point, P 1 , distant dx from P, as in the diagram, and let its temperature be (0 + 66). The heat flowing into the element, PP 1 , of the bar across a plane at P drawn normally to its length is equal to : , P — P i dx A P ersec -> where the subscripts denote the points at which is measured, i.e. the heat flow amounts to : - k '~Sx~' A -- k 'Tx' A = F ^ s ^' in the limit when dx is made infinitesimal. Thus this expression denotes the flow from left to right at the point P, of the bar. Since depends on x, in the general case this flow will also depend on x, i.e. F depends on x. Again consider the element PP 1 . We have calculated the flow, F, into it at P, and the flow out at P 1 will be F + — • dx. dx Hence the total flow of heat into PP 1 ^ _d¥ dx dx J, d * 6 A * = ft ' -j— z • A • 6x per sec. VAPOUR DENSITY AND THERMAL CONDUCTIVITY 235 This heat is used up, partly in warming up the part of the bar concerned, and partly in radiation from the surface. If the temperature is changing at the rate, -j-, t denoting the time, the first part amounts to : and the second to : h • P • 6x (0 - O ). h denotes a constant for the surface and O is the external temperature. We shall, however, suppose that is measured in degrees above the surrounding temperature, so that we may write the last expression simply : h-V'Sx-B. We, therefore, arrive at the equation : kAdx ^ = h • P -dx • 4- A -gsdx -=? ax* at or 2 -*&-=•■ <*> where K =~ , H = -^- (3) ps Aps Any function which satisfies certain conditions can be expanded as a Fourier Series. Students of Physics should make themselves acquainted with the Fourier Analysis and they may be referred to Carslaw's work on this subject. For the present purpose we need nothing more than the statement that the expansion is possible. Fourier's Series is the following : f(t) = A + A x cos pt + A 2 cos 2pt + A 3 cos 3pt + . . . ) , * + B x sin pt + B 2 sin 2pt + B 3 sin ipt + . . . J w Here f(t) is any function existing over some interval from t = a to t = b and satisfying certain conditions with regard to continuity, etc., which we need not enter into here. We merely mention that the series is applicable to the function we shall use in our experiment. With regard to the values of the A's and B's the rule is that: 236 ADVANCED PRACTICAL PHYSICS FOR STUDENTS A B =^ / /(«) cos -7 — — u du; b — aj a b — a B„ = T -/ t(u) sin T w. du; b — a J t JK ' b — a n has the integral values, i, 2, 3, etc. 2tc The value of p in the above series is in this general case r When f(t) is periodic and has the complete period, T, we take the limits, a and b, at the ends of this period and write a = o, b = T, so that the series represents f{t) from t = o to t = T, and on account of its periodic character represents it also from T to 2T, etc. In this case : . 2tc P = x ' I I /* T Aj L-J f{u)du; 2 /* T A B = =r / /(w). cos npu du ; 2 f T B„ = — / /(w) sin w^)w • ^w. Ic/ The periodic function with which we are concerned is that which expresses the fact that from t = o to t = Tj the temperature has some value, $ lt and from t = Tj to t = T 2 it has the value 2 . Such a function can be represented by a Fourier Series and we shall assume that the particular expression is (4), with coefficients calculated according to the rule. These, however, are not required for the experiment. It is shorter and convenient for our purpose to write (4) in the form : a Q + a x sin {pt + r x) + « 2 sin (2pt + r 2 ) + (5) And let us suppose that this is the expansion which expresses the temperature at A. Now the temperature, B v at the point whose distance is x t from A is a fluctuating function and so is that for any other point, %. We may therefore write for 6 lt some such value as (5), viz. B x = C ' + C/ sin (pt + a/) + C 2 ' sin { 2 pt + <V) + • • • • (6) and for the point, x % 6 2 = C ' + C ± ' sin {j>t + d{) + C/ sin ( 2 pt + «■*) + . . (7) The quantities C ' C/ . . . , <V 6 2 ' . . . , will differ from C " Cx" . . . , 5/ 5/ . . . , since they correspond to different values of x. VAPOUR DENSITY AND THERMAL CONDUCTIVITY 237 ■■**'. We have to solve the equation (2), and we have a clue from the experimental observations of diminishing amplitudes and lagging maxima and minima. Moreover, the value of is known at the point A where x is zero. This value is given by (5). The solution is in fact : 6 = a,?-*** + a 1 e- a ^ sin (pt + p,* + r x ) + a#-** x sin {2pt + M + r 2 ) + etc (8) Where : a » = ? K (a n * - p n *) = H, \ {g) 2Ka n p tt = -np J When x = o this expression reduces to that of equation (5), so that the solution satisfies the end condition. The reader maj' verify that the solution satisfies equation (2) by substitution. He will observe that the coefficients of the separate sine and cosine terms and the terms independent of trigonometrical functions all vanish if (9) holds. Now if we consider the two points, x x and x 2 , at a distance, I, apart, by equations (6), (7) and (9) we have : C/ = a x e~ a **\ C x " = a x e-"***, d x " - <V = p^. For S x = $ x x x -f r x and d x = $ x x 2 + r x Hence «. x l = log ~i and a^ = ' ^ x log ^V But by the last of the conditions (9) TW-Olog^ K = ■?— = TT-s «. (10) and in the same way : K = pr-, (11) T(d n ' -d n ")\og^ fl In this formula we note that the constant of radiation, h, does not appear and we are not troubled with the difficulties always associated with it. The conductivity, k, is given by Kps so that if our experiment is performed carefully and as a consequence the quantities, C n and 6 n , accurately known, we have to rely on the accuracy of the knowledge of the density and specific heat. Both these are accurately known. This is the reason of the importance of Angstrom's Method. The student is recommended to refer to a translation of the original paper in the " Philosophical Magazine," series 4, vol. 25, p. 130, 1863. From (11), we observe that each coefficient, C n , gives rise to a value of K. As the carrying out of the experiment will show, the coefficient, C x ; gives the most reliable result but the value corre- 238 ADVANCED PRACTICAL PHYSICS FOR STUDENTS sponding to C 2 should also be worked out. It remains to describe the experiment and to show how to find the C's and <5's. The bar, which may be of copper, iron or brass, has one end, A, inserted in a chamber so arranged that steam and cold water may be passed in alternately as described above. The total period is to be measured and care exercised to reproduce the conditions exactly in each successive period. Steam from a conveniently large tin flask and water from the tap will set up the right conditions. T\me or the f>o»r>r« =c , 8- oc^ CX.\ &, So 3G Fig. 129, The steady temperature fluctuations will be the more easily attained if the bar is sheltered to avoid draughts in its neighbour- hood and consequent troublesome convection effects. The temperature at points on the bar should be observed by means of a thermo-junction placed in a small cavity in the bar. Two such cavities will be required and should contain mercury, or it will be sufficient to hold the junction in contact with the bar at the points concerned. The thermo-junction must be calibrated in the usual way (see p. 545). When this is done read the temperature at convenient intervals by means of the galvanometer in the circuit and obtain as many readings as possible throughout two complete periods, or more. Plot a curve for 6 against the time very carefully on squared paper as illustrated in fig. 129. Suppose that this refers to the point, x v Repeat the process for x v VAPOUR DENSITY AND THERMAL CONDUCTIVITY 239 We shall determine the necessary quantities from the graphs. In order to see how to use the graphs multiply the equation (6) by sin pt and integrate both sides over a complete period from any arbitrary time, t=t x to t=t x +T t where, of course T= — P We thus have : 0! sin pt dt = Co' / sin pt dt + C/f sin (pt + <5/) sin pt dt. + C 2 ' f sin (2pt + <V) sin pt dt + etc. (12) In doing this it is well to remark that we are integrating the series on the right-hand side term by term, adding all the in- tegrated terms and equating to the integrated function on the left. This process is not always legitimate, but on account of the properties of the particular series we may apply it in the present case. On performing the integration the only term on the right-hand side which does not vanish is the second, and this has the value : ^TC/cos <V- Denote the integral on the left by S,. Then S, = £ TC/ cos d x '. In the same way if S« denote the value of ^,+ T B x cos pt dt, J t x we find S^iTC/sin V- We can determine C x and 6 X from these two equations, for tan V=|^ .(13) and C t '» = -^-(S,* + V) (14) In the same way if we multiply the series by cos 2pt and sin 2,pt and integrate we find : X cos 2pt dt = £ TC a ' sin <5 2 ', and as before we can express C a ' and «5 2 ' in terms of the quantities, S ac and Sg,, the meaning of the latter being of course J] 0j sin 2pt dt sm $ TC a ' cos <5 3 '. Jto. j* . 240 ADVANCED PRACTICAL PHYSICS FOR STUDENTS This suggests a graphical method of determining the necessary quantities. Measure as many ordinates of the curve, fig. 129, as is con- veniently possible and multiply each by sin pt or sin -=r . T is the complete period of the periodic heating and t is the value of the time appropriate to each ordinate. Plot a new curve with these new quantities as ordinates and times as abscissae and thus obtain fig. 130, Cutue sWxjoinQ \V»e uatia\tor> of 0sinSTTP ,jj\tVt \ke rime -fa* the \x>\r\X X.| T Fig. 130 Draw any ordinate A X B X for some arbitrary time, t v and construct C X D X the ordinate at one complete period later. Measure the area between these ordinates, the curve and time axis carefully with a planimeter. This gives S,. VAPOUR DENSITY AND THERMAL CONDUCTIVITY 241 Go through the similar process to find S c . Multiply Bx by sin 2pt and cos 2pt to find S 8 , and S 2C . All this has to be repeated for the second point, x 2 , and we then have all the data necessary to deduce k from formulae (10) and (11). C?otoe sWou/mq tW uwriatlons of ■& coe -=^r UJ»m the Hme at Fhe ^pomf 0C t Fig. 131 Another way of making the calculation is to make two ex- periments with different periods, T' and T". If <x n ' and <x„" are the values of a n in these cases respectively we find from (9) the result : K |t/2 „ /i '\ „ /2 V 2 «» 2 . 2a « a n * a n " — a, a n ' and a n " are derived as before, and 16 242 ADVANCED PRACTICAL PHYSICS FOR STUDENTS It is to be noted that we do not in this case require the values d. The density and specific heat may be taken from tables or be measured by any of the usual methods. In drawing the curves it is a good plan to have two complete periods shown to give a means of testing the accuracy of the areas, S, and S c . Always test the end of the bar remote from the heat supply to determine if it remains at the air temperature. It will suffice to place a thermometer close to this end. A longer bar must be used if the fluctuations continue right to the end. Angstrom used a square bar of side 2-375 cms., and the length, I, between the two points, x x and x 2 , was 5 cms. It has been found, however, that a cylindrical bar of diameter from 1 to 2 cms., and of length about 60 cms., will suffice, with such a range of temperature as described above. In the original experiment Angstrom heated his bar at the central region and, of course, had similar conditions on both sides CHAPTER IX MISCELLANEOUS EXPERIMENTS IN HEAT Determination of the Radiation Constant. ("Phil. Hag.'*, ser. 6, 1905, p. 270) By Stefan's Law the total radiation from a black body is propor- tional to the fourth power of the absolute temperature, or R = oT*. It is the object of this experiment to determine a. Apparatus The diagram (fig. 132) illustrates the arrangement of apparatus which consists of a blackened hollow metal hemisphere, B, about ten inches in diameter, fitted into a wooden box, W, lined with tin. Fig. 132 This fits on to a table, of which the top, DE, is shown, containing a small hole at S, which lies at the centre of the hemisphere. B is heated to a uniform temperature measured by the thermo- meters, T x and T t , by passing steam through the box above the hemisphere. The black surface of B is the radiator, and the heat is received by a small disc of silver placed at S, and blacked on the upper surface to prevent reflection. It is better to fit the disc in a vulcanite frame rather than to allow it to touch the table directly. From the disc are lead away two wires, one of constantan, and the other of silver, to a galvanometer and second junction. 343 244 ADVANCED PRACTICAL PHYSICS FOR STUDENTS S is thus one of the junctions of a thermo-electric couple, the other is placed in a tube containing oil standing in a calori- meter, C, containing water or ice. The junctions to the galvanometer are kept in a canister, A, packed with cotton wool to prevent any electrical effect due to difference of temperature at these junctions. It may be further necessary to include a resistance in the circuit to keep the deflection on the scale if the galvanometer is too sensitive. A rise of temperature due to absorption by the disc is thus recorded on the galvanometer. Theory Let Rj = radiation absorbed by the silver disc per unit area per second, and R that emitted. Let the temperature of the radiator be T 1} and of the disc T. If the whole enclosure, the disc included, had temperature T lf there would be equilibrium, and the disc would both emit and absorb R x in unit time. The energy absorbed would arise, of course, from B. This same energy falls on the disc when at the lower temperature and is absorbed, but the energy emitted is now R. Thus the gain of energy per sec. = (R t - R)A, where A denotes the area of the disc. /TV Let m denote the mass of the disc, s its specific heat, and -57 at its rate of change of temperature. Then we have : ws ¥ = ^f-- A = X (Tl4 " T4) ' or Jms _ dT AfV-T*) dt* where J = Joule's equivalent (4*2 X io 7 ergs per calorie). All the quantities on the right are measured, and hence a is calculated. Experimental Details It is first necessary to ascertain the" relation between the readings of the galvanometer scale and the difference in tempera- ture between the two junctions. In order to make this comparison the disc is surrounded with cotton wool, and the cold radiator placed above it. The calorimeter, C, is then heated and the difference in temperature between it and the disc recorded on a graph against the readings of the deflection. MISCELLANEOUS EXPERIMENTS IN HEAT 245 We thus obtain : difference in temperature per scale division = AB BC' J By measuring the temperature of C, we can then deduce from the galvanometer deflection the temperature of S from this graph. V f u Oivmie*-*? on Scale Fig. 133 Secondly, we require the rate of rise of the temperature of the disc when the radiator is put on. As soon as the temperature in the enclosure, B, has become steady, the box is placed over S, keeping the latter at the centre, and the galvanometer is read at equal intervals of time ; these may conveniently be every 5 or 10 sees. The table shows a record of observations made every 10 sees. TIME SCALE DIV. TIME SCALE DIV. O 227 40 190 10 218 50 l8l 20 208 60 173 30 199 70 165 These results are plotted on a graph. Scale Divisions Fig. 134 dT Draw a tangent to the curve and measure the value of -=- as at close to A as possible, since errors soon arise by conduction from the silver disc. 246 ADVANCED PRACTICAL PHYSICS FOR STUDENTS m and s are found in the usual way. It will be convenient to regard these as constants of the apparatus, and to record them on the occasion of making the apparatus. It is inconvenient to make determinations at each experiment. Or s may be measured by Regnault's Method from a piece of silver identical, with that in the apparatus. The Rise of Boiling Point of Solutions The object of this experiment is to determine how the tem- perature at which a solution boils depends on the concentration, to determine the rise of the boiling point above that of the solvent, and to compare experimental results with those given by the thermo-dynamic formula. If P denote the osmotic pressure of a solution, i.e. the pressure due to dissolving one gramme molecule of the substance in one litre of the solvent, S, the molecular weight of a salt, and s, the number of grammes in ice. of the liquid, we have : P _ 2R0 s S where R is the value of the gas constant per gramme molecule, and is the temperature of the solution, i.e. for any particular salt and solvent and for a temperature 0, P ocs. All this is true for dilute solutions only. A calculation based on the second law of thermo-dynamics shows that the rise of temperature of the boiling point, T, is given by the formula : T- £?' where P is the osmotic pressure, L = latent heat of vaporization of the solvent, p = its density, and = temperature of boiling of the solvent. For purposes of the calculation it may be assumed for the case of water, p = — at boiling point. P = 22-3 x 10 6 dynes per gramme molecule per litre at o°C. If = B.P., the approximate value for P at temperature is : — - x 22*3 xio 1 dynes. 273 ° y MISCELLANEOUS EXPERIMENTS IN HEAT 247 e is in absolute units, and the value for any particular concentration may be deduced by simple proportion. L, for water at ordinary pressures, may be taken as 537 X 4*2 X 10 7 ergs, in which units it must be expressed for the equation. Suppose w grammes of solute are dissolved in W grammes of solvent. Fig. 135 . W W The volume of the solvent is — c.c. or P 1000 p litres. If M denote the molecular weight of the solute, we have ... . 1000 pw . litres, i.e. w — gramme mole- cules per litre. w , , • W — gramme molecules m M 6 1000 p 248 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Thus the depression according to the theory should be : 1900 pw JP0 ioooze>P0 MW ' L P ~ MWL ' Verify this expression by taking several values of w. A modern form of apparatus for carrying out this experiment will be described. It is represented diagrammatically in the figure (135). A is a glass tube which holds the solvent from which proceed two tubes, B and C, provided with corks. The thermometer passes down B, and the bulb is immersed m the solution or pure solvent in A. Weighed quantities of solute may be added through C. Through the cork, H, passes the metal tube of the condenser, F, which is cooled by passing in water by one of the tubes, T, and allowing it to flow out by the other. The small tube, G, carries garnets, which are placed at the bottom of A, and which together with a short wire through the base of A, assist the commencement of boiling and tend to prevent superheating. A stands on a cylinder of asbestos, D, which rests on a plate, E, of pipeclay, and the liquid is heated by two burners placed below E. The thermometer is set and its reading taken when the pure solvent is boiling, and the difference noted when the solute is added and the solution boils. i We have first to weigh the vessel, A, when detached from F, and not carrying the thermometer. Add a weighed amount of solute, w, to a convenient quantity of solvent. After solution, and after taking the boil- ing point of the solution, F is again removed and also the thermometer, and A is again weighed. Since w is known, we can obtain W, the weight of solvent. Care must be taken not to remove any of the liquid with removal of the thermometer. The thermometer bulb must be placed in the solution ; if it is above it attains the temperature of the steam coming off from the solution, and this is not at the temperature of the solution, but at the temperature of steam appropriate to the prevailing pressure. The Beckmann Thermometer # The magnitude of the change of temperature is not very great, and great care is necessary to measure B it accurately. The most convenient form of thermo- meter for this purpose is one of the Beckmann Fig, 136 type. MISCELLANEOUS EXPERIMENTS IN HEAT 249 A common form of this thermometer is made of Jena glass, having an apparent coefficient of expansion for mercury of magnitude—^—. It is illustrated in fig. 136. A special feature is the large bulb, B, containing a compara- tively large quantity of mercury from which a fine capillary tube extends, bent as shown at A, where the capillary is widened. The mercury in the bulb being of considerable volume gives a large additional volume on expansion, and this causes a large movement in the fine capillary tube when the change of temperature round the bulb is only slight. If, however, the quantity of mercury in the bulb were fixed the range of temperature for which the instrument could be used would be very small unless the capillary were inconveniently long. The thermometer is used to measure a smalt change in temperature above or below some particular temperature. Suppose that we require to measure a small change of temperature above T . It would be best to arrange so that the mercury extended from the bulb up to the zero of the scale when at the temperature, T . A slight increase in temperature would then drive the mercury along the capillary, which is marked in hundredths of degrees at intervals of sufficient width to enable an experimenter to estimate to thousandths. Suppose now that it is required to read temperatures slightly above another temperature, T lt which is greater than T . If we could withdraw sufficient mercury from the bulb it would be possible to arrange that the mercury extended up to the zero mark at the temperature, T v and slight increases would drive the thread along the fine tube as before. Of course, the expansion per degree rise of temperature in the second case is not strictly the same as in the first, since the initial volumes are not equal in the two cases. In order to see how this effects the observations, let us suppose that the graduations on the stem are correct at t ° C, i.e. with mercury filling the bulb and extending to the zero mark, when the surrounding temperature is t °, a rise of temperature of i° C. would cause an expansion to the first degree mark above the zero. Let the volume of mercury at t ° under these conditions be denoted by V , and let the coefficient of apparent expansion of the mercury in glass be a. Then the volume of the capillary per i° is V oc. Now suppose that mercury is drawn off until at temperature, /, the mercury fills the bulb and extends to zero. The volume drawn off is V a(rf — t ). 250 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The remaining mercury had a volume at t ° of V {i-a(*-* )f, and on further heating of one degree would expand by an amount V o^i -ce(* -t )}. Since the volume of the capillary per i° is V a this expansion will be registered as {i — <x.{t — t )\ of a degree. The degrees on the seale are too large for the temperature t. A correction could be made by dividing the scale readings by {i — cc(t — t )\, and it would Abe sufficient to add to the observations an amount <x{t — 1 ) per degree registered on the scale to obtain the corrected rise. As has been stated a has the value - — , so that if we suppose the thermometer correct at 90 C. and we record a rise just above ioo° C. we should add to each degree an amount ? = -0016 . 6370 This is an appreciable amount since we can record up to thousandths of a degree. The correction can be neglected when we are recording to hundredths of a degree. The widening of the capillary at A enables the change in the quantity of mercury in the bulb to be made. Suppose it is necessary to read a temperature between ioo° and 101 C. The thermometer is placed into a bath at a tempera- ture of about 102 C. to 103 C, and so that the mercury extends from the bulb up to the widened part when at this temperature. If a quick jerk is made the thread will break at the point, where the capillary enters the wider part. When cooled to about 100 C, i.e. when held in steam, the temperature of which may be determined by observing the atmospheric pressure, and referring to a book of tables, the top of the thread should sink to the lower end of the capillary. It will not lie exactly at zero, and the position occupied by the end at this known temperature is recorded. It may be necessary to vary the upper temperature when the thread is continuous before the end will sink conveniently towards the zero at the temperature of steam. When this condition has been attained the small rise of temperature is readily recorded. It is not necessary to bring the mercury thread exactly to zero before reading the small temperature change, and if the correction is to be applied it may be made by giving the value to t which corresponds to the temperature for which the apparatus is set, just as if it had been set at the zero. MISCELLANEOUS EXPERIMENTS IN HEAT 251 If it is more convenient to observe a diminution of temperature below a certain point, it is only necessary to bring the top of the mercury thread at this temperature to a point high up on the scale. A slight depression will cause the top of the column to sink, and the interval may be measured as before. In such accurate temperature measurements it is necessary to make a correction for the emergent column. This is done by reference to Grutzmacher's tables taken from the " Zeitschrift fur Instrumentenkunde " (1896), p. 220. A reference to this table will show how important such a correction is. GRUTZMACHER'S TABLE VALUE OF ONE SCALE DIVISION WITH THE THREAD CORRESPONDING TEMPERATURE IN DEGREES ALL OUTSIDE AND VALUE IN DEG. INTERVAL. CENT. WITH AT MEAN TEM- ENT. PER SCALE THREAD TOTALLY PERATURES BELOW DIVISION. IMMERSED. °c. °c. °c. °c. —35 to —30 •982 •977 to 5 •997 15 •995 45 to 50 I-OII 26 1-015 95 to IOO I-02I 32 1-032 145 to 150 I'027 138 1-045 195 to 200 1*028 44 1-053 245 to 250 1-024 50 1-055 The second column gives the correction due to the variation of mercury within the bulb, and it shows that the thermometer reads correctly at a temperature between 5 and 45 , for in this region lies the point where the value of one scale division is i° C. The fourth column gives, in addition, the correction for the fact that part of the mercury is at a different temperature from that in the bulb. v In order to make a correction, choose the part of the table which describes most nearly the condition of the experiment, e.g., at a temperature in the neighbourhood of ioo° with a mean thread temperature about 30 , one degree recorded by the apparatus is approximately 1-032°. Two thermometers will be required, one adapted for use in the neighbourhood of o°C. and the other adapted for use near ioo° C. Molecular Weight by Depression of Freezing Point The thermo-dynamic formula for the depression of freezing point is T-f 252 ADVANCED PRACTICAL PHYSICS FOR STUDENTS where L is the latent heat of fusion of the solvent, p its density, and is the temperature of fusion. P is the osmotic pressure. Let W denote the weight of solvent used, and let w grammes of solute be added of molecular weight, M. Let P denote the pressure for one gramme molecule per litre, i.e. 22-3 xio 6 dynes/sq. cm. By using this value and p = i, L = 80 x 4*2 X 10' ergs, 6 = 273 Absolute, calculate the corresponding depression T . W In the present case the volume of solvent is litres, and the r 1000 p w number of gramme molecules dissolved is^rv m The corresponding depression is T 1 , where looop w Pe __w_ . I000 .Pf . 1 " W M L P ~ WM lou L It will be the object of the experiment to calculate M. The vessel, D, with the stirrer, S, is weighed and a quantity of the solvent introduced, and the vessel again weighed. This gives W. The thermometer is placed with its bulb in melting ice, and the top of the mercury thread adjusted so that it stands near the upper end of the scale. This temperature is o° C. and small differences from this point may be read from the thermometer graduations. These may be corrected by Griitzmacher's Table. A weighed quantity of the solute is introduced into the solvent through the side tube, C, and solution brought about by help of the stirrer. The vessel is placed in an enclosure, E, and surrounded by a freezing mixture, F. The thermometer, B, must be kept in ice until D has cooled down to o°, and then the transference must be quickly made to D. Otherwise the thread will rise beyond the scale, and mercury will flow into A. During solidification the mercury will stand at a definite mark if the solvent is pure, and the difference, T 1 , can be measured. In order to set the thermometer ready for use in the experiment it must be placed in water cooled to within a degree or two of the freezing point. The thread of mercury which must be continuous from B to A is broken by a quick jerk at the top of A, when the temperature of the water is attained. In this way the thread will extend nearly to the top of the scale, when the bulb is at the temperature of melting ice. MISCELLANEOUS EXPERIMENTS IN HEAT 253 For the temperature at which the solution freezes the fall below the freezing point of water is obtained by subtracting the second reading with the bulb in the solution from that with the bulb in ice. The degrees thus read will require correction on account of the emergent column. / Fig. 137 In order to make the correction, use Grutzmacher's Table, e.g., if a temperature difference recorded is 1-5° C. with an average room temperature 15 C, the degree as read on the scale is really only -995°. Thus the difference is : i-5 x -995 = 1-492°. I ° T ° The thermometer reads to , so that we can estimate— — 100 1000 and it is therefore necessary to take account of such an error as that arising above, viz. of -008° C. Determination ot the Mechanical Equivalent of Heat by Friction Cones The apparatus consists of two vessels of gun-metal of the shape of truncated cones, one fitting within another. The inner cone contains water, and is helcj in position by a measured 254 ADVANCED PRACTICAL PHYSICS FOR STUDENTS couple applied by means of a weight hanging over a pulley and connected to a disc attached to the cone. The outer cone is rotated by means of a cord to a large wheel rotated by hand. Fig. 138 It is possible by steady turning of the handle to cause the weight to hang almost steady, so that a steady couple is applied to the inner cone by friction, and its amount is measured by multiplying the force and the diameter of the wheel to which the cord is attached. A counting device is fixed to the outer cone so that the number of rotations- is recorded. If n is the number of rotations, w the weight in dynes applied to the disc, and a its arm, the work done is 2tfnwa ergs. The rise in temperature of the water is measured by means of a thermometer, so that if M represents the total water equiva- lent of the cone and contained water, and if the rise in temperature is T, the heat developed is MT calories. If no heat is lost by radiation and conduction, the mechanical equivalent, J, or the number of ergs necessary to produce one calorie is given by : 2ntW>a J = MT A common form of apparatus in use in laboratories is shown in the figure, it was designed by Dr. G. F. C. Searle. An improved design is described in the " Phil. Mag.," Sept. 1920, by H. P. Waran. In order to prevent radiation losses it is a good plan to cause a rise in temperature of not more than io° C, and to cool the water about 5 below the room temperature initially. The two MISCELLANEOUS EXPERIMENTS IN HEAT 255 cones are mounted in a metal case lined with cork to diminish conduction losses. The heat is conducted through the walls of the inner cone to, the water, and it is necessary to stir the water during the experi- ment to assist the flow of heat throughout its extent. This may be done by means of the thermometer. One of the features of Waran's improvement is the automatic stirring of the liquid which consists of an oil with good conductivity and known specific heat. The Determination of Joule's Equivalent of Heat by Callendar and Barnes's Electrical Method The principle of the experiment is to supply electrical energy to a wire surrounded by water, and to measure the heat developed in the water by noting the rise of temperature. Fig. 139 If we express the electrical energy in ergs or joules We can thus deduce that required to generate one calorie. Inside the glass tube, H (fig. 139), is fixed a helix of manganin wire of resistance about 9 ohms. The ends of the wire are joined to the terminals, C and C 1 . Water from the tank, B, enters H by a side inlet tube at one end and after flowing round the wire comes out from a similar outlet tube at the other. Thermometers, which enter the ends of H as shown, penetrate the inflowing and outflowing water, and serve to record the temperatures of the water before and after it has received heat from the wire. The vertical tubes, AA, allow air bubbles which may flow in with the water, to escape. The level of B is variable for the tank is movable up and down its stand. The makers of the apparatus recommend that the rate of flow 256 ADVANCED PRACTICAL PHYSICS FOR STUDENTS should lie between 55 c.c. and 65 c.c. per min., or about 1 c.c. per sec. B should be adjusted to produce this rate. The tank consists of an inner and outer chamber. Through the tube, E, water enters the outer chamber from the supply, and comes out through M to enter H. The water flows over from the outer chamber to the inner and escapes to the drain by means of F. Thus water is supplied at a constant pressure to H, and the flow is made steady. The outflowing water may be collected in a measuring glass, or, of course, it may be collected in a weighed beaker and the mass per second flowing out deduced. The apparatus gives best results for a current of approximately 2 amperes, so that a voltage of from 20 to 25 should be applied. To obtain a steady source ten or twelve accumulators should be used and connected through an adjustable resistance to C andC 1 . In this case, with the rate of flow indicated, a difference of temperature of the ingoing and outgoing water of about 8° C. is maintained. If possible the mean temperature of the water should be that of the room, to avoid errors due to radiation. When it is not possible to arrange this exactly, it is necessary to apply a correction. Let the temperature of the room round the apparatus be t , and the mean temperature of the water at entrance and exit be t. Then the number of calories lost per second in radiation is m{t — Q x -05, where m is the outflow per second. Or we may make the correction by adding -05(t — t ) to the difference of temperature recorded by T and T 1 . This correction has been obtained experimentally by the designer. Measure the resistance of the spiral by means of a Post Office Box. Connect the resistance, R (fig. 140), in series with the supply of current and the ammeter, A, which has a range up to 3 amperes. If a voltmeter is available there is no need to measure the resistance of the helix, for the difference of potential between C and C 1 may be measured by connecting it through the key, K, in parallel as in the diagram. Adjust the current to about 2 amperes, and if necessary make slight variations from this to adjust the temperatures so that radiation losses are made as small as possible. Let m denote the rate of outflow per second, C the current in amperes, E the voltage supplied between C and C 1 , and T the difference in temperature of the two thermometers, corrected if necessary according to the rule given above. MISCELLANEOUS EXPERIMENTS IN HEAT 257 The supply of electrical energy is EC joules per second, and the heat developed is ml calories per second. Thus, J, the heat equivalent, is given by: JwT = EC, EC or J =^ joules" EC -f_fl.g-ftfla.il ».tuuuuuuub>- ++■-+- GLh Fig. 140 If the resistance of the wire is measured, instead of EC we must write C 2 R. The accuracy to be expected is between one-half and one per cent. The following is an example of an experiment carried out in the laboratory, and indicates the order of the quantities : Temperature of room, 17-3° C. Temperature of water at inlet, i6*8° C. Temperature of water at exit, 25-35° C. Mean temperature of water, 2i'07° C. Temperature difference of water hi the two cases, 8-55° C. Radiation correction = -05 x (21*07 ~~ I 7*3) = ' x 9° C. Corrected temperature difference, 8-74° C. Volume of water flowing out in 2 mins., 115 c.c. m = — - gms. per sec. 120 ° r Current, 2 amperes. Resistance of wire, 8-92 ohms. ,\ J = -J 2— = 4-260 joules. — x 874 120 '^ *7 CHAPTER X REFLECTION The Sextant The instrument consists of a graduated arc, SS (fig. 141), with two radial arms, A and C. A third arm, B, moves about an axis through one of its ends at right angles to the plane of SS. It is fitted with a clamp and tangent screw, so that it can be accurately adjusted, and carries a vernier at its end which moves over the scale of SS. Fig. 141 A plane mirror, M lf is attached to B, lying with its surface in the direction of B, and in a plane normal to that of the scale. The axis about which N B turns lies in the surface of this mirror, which is called the index glass. The second mirror, M 2 , fixed to the arm, C, is called the horizon glass, and its plane must also be perpendicular to the scale. It consists of a plate fif glass only one-half of which is silvered. At T, on the arm, A, is fixed a telescope with its axis parallel to the plane of SS and passing through the centre of M 2 . Suppose the movable arm is turned so that the mirrors are parallel, and T directed towards a distant object, on which it is focussed. 258 REFLECTION 259 * Only one image will be seen of the object, for the light from M x and M 2 is brought into the telescope in the same direction, M 2 T. The rays by the two reflections may not he along the same line as those seen directly through M 2 , but since they are parallel there is only one image formed by the telescope (see fig. 142). ^vfi Fig. 142 If the index glass be rotated through an angle, A, by turning the arm, B, then the rays reflected by it into the telescope no longer come from the* same object as that which supplies the ray, QM 2 . Two superposed images are seen in the telescope, and the angle between PM^ and QM 2 is 2A, since on turning a reflector through any angle a beam of light is rotated through double this angle. Thus every degree on the scale, SS, corres- ponds to a difference of direction of two degrees. The scale is marked to give directly the angle between the rays, PMj and QM t , i.e. to measure the angle subtended at the instrument by two distant objects. In order to measure this angle one of the objects is observed directly through M 2 , while the other is made to produce a super- posed image on that of the first by rotating M x into the proper position. The angle through which the index glass is turned from the parallel position is then one-half of the angle subtended. Before making any measurement the instrument must be tested to see that it satisfies the following conditions : (1) The plane of the index glass must be normal to the plane of the scale, (2) The axis of the telescope must be parallel to the plane of the scale, and (3) The index and horizon glasses should be parallel, and at the same time the vernier should read zero. It will not be necessary, as a rule, to adjust for the first of these, but in order to see that the instrument is satisfactory in this respect look at the image of the scale in M^ Since M x passes through the centre of this scale the latter and its image will appear to intersect at the edge of the mirror and, if the- adjustment is satisfactory, to lie in the same plane. Both M x and M a are attached to frames which can be turned 26o ADVANCED PRACTICAL PHYSICS FOR STUDENTS through small angles by means of screws. If necessary Mj may be adjusted until the test is satisfied. The second condition is tested by observing two objects and causing them to coincide at the centre of the field of view. The axis of the telescope is a line joining the centre of the object glass to the centre of the eyepiece, or to the centre of the field of view. Perpendicular to this axis lies one of the cross-wires. Tilt the instrument until the images lie near the edge of the field, and note if they still coincide. Then tilt it so that they lie near the opposite edge. If coincidence persists the axis is correctly adjusted. If this is not the case the telescope can be adjusted by means of the screws. Observe an object through the telescope and make its image appear in the field of view by reflection in M x . If it is possible to cause them to coincide, the two mirrors are parallel, and on account of the first adjustment this will mean that the third condition is partly satisfied. By means of the screw attached to M 2 the two images can be made to coincide if they do not do so at first. When these conditions are satisfactorily arranged it will probably happen that the pointer does not read zero when a distant object is viewed. To correct for this it is only necessary to note the zero error and apply it in all the observations. Coloured glasses are provided for diminishing the brightness of any object such as the sun. These can be made to intercept the light immediately before falling on the mirrors. Experiment i Place two candles at as great a distance as is convenient, and measure the angle they subtend at the instrument. Also find the angle by measuring the distance to each candle and their distance apart. Let the candles be C a and C 2 , and let S denote the sextant. The distances to be measured are a, b, and c, and if s = \{a + b + c) REFLECTION 261 tan J-J(* -»>('-*)■ 2 M S ( S _ a ) Check the values of obtained by the two means. Experiment 2 \ Let a trough of mercury or a carefully levelled mirror be placed so that the image of a lamp can be seen directly, and also by reflection, and measure the angle subtended at the sextant by the object and its image by holding the plane of the instrument vertically, at as large a distance from the lamp as possible. The elevation of the lamp is half this angle. Li„-"" Fig. 144 The diagram (fig. 144), illustrates that the angle, CBD, is measured since the instrument is of necessity above the surface, AE. Actually, we require the angle, LAL 1 , but since we use a distant lamp, the two angles do not appreciably differ. Thus, the elevation may be measured by half the angle CBD. Measure the horizontal distance between L and A, and deduce the height of L above the floor. Check the result by actual measurement. Measurement of the Angles of Crystals by Wollaston's Goniometer The goniometer is a convenient instrument for measuring accurately the angles between the faces of small crystals which are too small to be examined by means of a spectrometer. It consists of a circular circle, S, which may be rotated by the large milled head, B, and its position read off by means of a fixed vernier, V. The crystal is fixed by soft wax to a plate, P, carried by an adjustable support, D, which may be rotated by the smaller head, A. 262 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The edge of the crystal formed by the two faces between which it is desired to measure the angle is adjusted so that it lies parallel to the axis of the circle. This adjustment is first made approximately by eye. In order to make the adjustment accurately, view the upper corner of a distant window in both faces. On rotation the images will move in a vertical plane if the edge is parallel to the axis. This may be tested by noting if each image moves in a direction parallel to the edge of the window as seen directly by the eye. Place the instrument so that the axis is parallel to a tall, distant window, and turn the screw-head, B, until the graduated circle comes against the stop. Fig. 145 The eye is placed close to the crystal so that an image can be seen by reflection in one of the two faces. The axis is then rotated by the smaller screw-head, A, until the top of the window, as seen by reflection, appears to coincide with the bottom, as seen directly. When the adjustment has been made the angular position of the circle is noted. By means of B, the circle is now turned until the top of the window, as seen by reflection in the other face, coincides with the bottom, as seen directly. The second face now occupies a position parallel to the first, and if is the angle between them, the circle has been rotated through its supplement. This angle (180 — 0) is read off from the circle and deduced. It should be noted that the crystal must lie as close to the axis of the goniometer as possible, for the motion of the crystal from the first position with reflection in one face to the second position, with reflection in the adjacent face, consists both of translation and rotation, unless the crystal is on the axis. The amount of translation may be sufficient to cause an error in the angular measurement. If the window is a long way away the error is only small. REFLECTION 263 The Determination of the Radii of Curvature of Spherical Mirrors (A) Concave Mirrors The most convenient method of determining the radius of curvature of a concave mirror is to place a pin point in front of it and to locate the position in which the image of the pin appears to coincide with the pin itself. The method of parallax is employed to ascertain when coincidence is attained. " The rays from the point of the pin falling on the mirror are reflected back from the surface along their original paths and must therefore strike it normally ; consequently, the pin point lies at the centre of curvature of the surface. Another method consists in locating a series of pairs of con- jugate points for the surface and using the formula : z -+i = * (I) v u r v ' A pin is set up as object and another pin adjusted until the image of the first coincides with it. We can then measure a pair of values, u and v. Several pairs of values are obtained, and the above formula then gives r. Take the average of four or six observations. (B) Convex Mirrors Method 1 In the case of a convex mirror the image is virtual, and it is not convenient to locate it by a pin placed in a particular position since the image lies behind the mirror. Q C Fig 146 A pin, P, is set up in front of the mirror, CC, and in between them is placed a plane mirror, M, so that the image of P in both can be observed. The mirror is adjusted until the two images coincide (fig. 146.) By the simple law of reflection in the plane mirror, M, we know that the image of P in M, say, Q, lies at the same distance from M as P does, but on the other side of it. We can thus calculate the distance, AQ, for AQ = MQ - MA = MP - MA. 264 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Then v = — AQ, adopting the usual sign convention, viz. directions measured from A towards the object are positive, and in the opposite direction they are negative. Thus, using the formula (i) we can again deduce r by measuring AP and AM. Several pairs of values should be obtained, and they should give the same value of r. Method 2 Another method is to set up a pin and form a real image of it by a convex lens. The image is located by placing a second pin, Q, so that there is no parallax between it and the image. Then place the convex surface between the lens and second pin, and move the surface until an image is formed coincident with the first pin, P. The rays after passing through the lens are directed to the point, Q, but strike the surface normally, and are therefore reflected back along their path. The radius of curvature of the convex surface is MQ (see fig. 147). Fig 147 Method 3. By means of a telescope, metre rule, and small millimetre scale The diagrams (figs. 148, 149, 150) show the arrangement of apparatus. S is a small scale placed horizontally in contact with the surface of a convex mirror, along a line dividing it into two equal parts. E e Jhr- 3- Fig. 148 The scale, RR 1 , is mounted at some convenient distance away, usually about 60 or 70 cms., and below its middle point is fixed a telescope, T, focussed on the image of the scale, RR 1 , in the mirror. The apparent length of the image is read off by means of SS 1 , which will be sufficiently well focussed to make this possible. The distance from the centre of the mirror, P, to the middle REFLECTION 265 of RR 1 , is measured, say, d, and from these two measurements, together with the length of RR 1 , 2I, it is possible to calculate the radius of curvature of the surface of the mirror. Let rays from R and R 1 strike the mirror at L and L 1 , and be s ± T J Fig. 149 reflected down the telescope at O. Then LL 1 will denote the extent of the image, and the point, B, at which these two lines meet OP will be conjugate to O for reflection in the mirror. We may say that a point source at O will have a point image at B, so that if PB = x, we have : — J -uJ — _ 2 % a r r denoting the numerical value of the radius of the mirror. R Fig,. 150 But if we take LL 1 as approximately straight, since the image is of small dimensions, and denoting LL 1 by 2c I _ d + x C X Hence so that x d\c J' or 2 _ r r = 1 I — 2C The result may be verified by means of a spherometer. 266 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The Focal Lines formed by a Coneave Mirror When light diverges from a point and falls on a mirror, it is supposed in the elementary theory that after reflection the rays pass through a single point or appear to proceed from a point. This is approximately true if the dimensions of the mirror are small compared with the distance from the source. A closer approximation to the truth is that the rays after reflection pass through two lines or appear to come from two lines, situated in parallel planes and perpendicular to one another. These are the focal lines and it will be shown how to calculate their positions theoretically, while it will be the object of experi- ment to verify the result obtained. We shall take the case of light falling obliquely on a concave mirror. If the point source lies on the axis of the mirror we have symmetry about this axis, and the two lines degenerate into a point or circle through which the rays pass. Fig. 151 In the diagram, MM 1 denotes the concave mirror, and C its centre of curvature. The complete circle of which the section, MM 1 , forms a part is drawn for convenience. P denotes the position of the point source of light, and the diameter is drawn through P. The extreme rays, PM and PM 1 , are drawn and the reflected rays, MB, M X A, are drawn intersecting at F 1? and cutting off from the diameter the strip, AB. REFLECTION 267 The mirror is a part of a sphere, so that rays falling on the mirror from P, whether in the plane of the figure or not will pass through AB. AB is thus a focal line; and is denoted by F 2 . If we imagine the figure to be rotated about the diameter, the rays, M X A and MB, will still intersect at a point but now out of the plane of the diagram. For a small rotation the point of intersection would be on a line through F x normal to the figure. Thus, a second focal line is through F x perpendicular to the plane of the figure. Let the angle of incidence at the point M 1 of the mirror be i, and let PD = p. Denote the distances of the focal lines from D by p ± and p 2 , i.e. DF X = Pl , and DF 2 = p 2 . Let the radius of curvature, CD, be R. Let the mirror subtend an angle, r, at C, a at P, and p at F v We shall regard its dimensions as small in comparison with R, P and pi, so that these three angles will be small and may be measured by drawing perpendiculars from M on to the corres- ponding lines, and dividing this perpendicular by the distance from the point concerned, MM 1 cos i e.g. a = , p since the normal from M, on the line, PM 1 , makes an angle very nearly equal to i with MM 1 , and MM 1 is small and is regarded as straight. In the same way _ MM 1 cos i P ~~ "* Pi (for cos i = cos (i + di) when 6i is small), ,., MM 1 while r = — — • R From the triangles OMFi and POM 1 ^OPM 1 + ^OM x P = ziOFiM + z.OMF x .'. a + 2* = 2(» + di) + p, or. 2<5*' = a — 0; and in the same way from triangles, CQM and PQM 1 , M ' = a — r, ;. 2f = a + 0, 2MM 1 or — — - = cos 1. MM ix. f+i), \P Pi/ (- 1 + ~) \P Pi/ i.e. I - -\ 1 cos * = ^ 268 ADVANCED PRACTICAL PHYSICS FOR STUDENTS This is the formula concerning the position of the first focal line. In order to determine p 2 , we note that : A PM X A = A PM J C + A CM^A, i.e. | • pp 2 sin 2,i = \ pR sin i + | Rp 2 sin *', which may be rewritten as : 2 cos i i i ^ R p p 2 So long as i remains constant, p x and p 2 vary with p just as u and w vary together in the formula : - + -=-• v u f When * is made zero we obtain the usual formula, - +— = * , p Pi r' for both Pi and p 2 , so that the two lines coincide. To find p x and p 2 experimentally, use a small hole in a screen, with a lamp behind as a source of light. Find R first by adjusting the mirror so that an image is formed on the screen at the side of the hole as in the experiment on concave mirrors, page 263. The distance from screen to mirror will then be R. Allow the light to fall on the mirror at angles of about 20 , 30 , and 40 , and find the positions of the focal lines by means of a sheet or card or white paper held in a clamp. Measure the distances from the mirror to these lines, thus obtaining p x and p 2 . In order to find i, it is convenient to mount M on a stand carrying a pointer moving over a scale of degrees. Read off the mark against the pointer when the image of the hole is thrown back near the object, and turn from this position to any required incidence. If no scale of this kind is provided, measure the distances, PF X and PF 2 , say, a and b. We then know three sides of each of the triangles, PMFi and PMB, and can determine i from the usual trigonometrical formula for the tangent of half the angle of a triangle. Compare the values calculated in this way with those deduced from the formula derived from the above theoretical considera- tions. Searle's Methods of Determining Optical Constants Accurate methods for the measurement of radii of curvature of polished surfaces, of focal lengths, and for the localization of the cardinal points, have been described by Dr. G. F. C. Searle. REFLECTION 269 For the original accounts the reader is referred to the " Philoso- phical Magazine," Feb., 1911, pp. 218-224, or to the "Proceedings of the Optical Convention, 1912," pp. 161-172. The Determination of the Curvature of Spherical Surfaces For this purpose a table is mounted on a tripod stand (fig. 152), two of the feet of which carry screws for levelling. The table is horizontal, and can rotate truly about a vertical axis. It carries a millimetre scale on the top which can be clamped in any position, and a carriage bearing the surface slides along it. The figure shows a lens system in the place where the carriage slides. Fig. 152 It is essential that, as the carriage slides along the scale, the centre of curvature of the surface should move along a line which intersects the axis of rotation of the table. In the figure the scale is shown, and also the wooden slider which acts as the carriage. On the slider is a metal mount which may be screwed to the carriage. This mount carries a horizontal spindle, each end of which is turned to a conical point. One of these ends is provided with a screw thread, so that it will fit a brass plate carrying three screws. The screws fit into a second brass plate to which the lens or mirror to be examined is fixed by a little wax. The arrangement thus provides a convenient means of adjusting the surface. The edge of the carrier is smooth and straight, so that it can slide along the scale, and an index mark serves to record the position. In preparing the apparatus for use the spindle is first set accurately parallel to the edge of the carrier. The tip of a pin is held just in contact with one of the conical 270 ADVANCED PRACTICAL PHYSICS FOR STUDENTS points of the spindle, from which the brass plate is removed. The carrier is taken from the table top and replaced so that the other conical point lies near to the pin. If it is possible to bring this point and the pin into contact, the spindle is parallel to the edge of the board and scale. The spindle is rotated until this adjustment is possible, and the mount then firmly clamped to the carrier by means of the screw. It is also necessary that the axis of the spindle should intersect the axis about which the table turns. The scale is adjusted and the spindle moved along it until one point lies as nearly on the axis as can be judged by eye. A microscope is then brought up and focussed on the point, with the scale lying normally to the axis of the microscope, The table top is turned through 180 , so that if the axis of the table passes through A (fig. 153), and P denotes the first position of the spindle point, the second position will lie at P', where PM = M'P'. The slider has to be moved through the distance, P'P," in order to bring the point once more on to the cross-wires. If this distance is recorded and the slides moved back half this amount, the point will be at M', the foot of the perpendicular from the axis to the line P'P". X- M 1 — 1 — X p u iM .M u Fig. 153 The microscope is then traversed so that it is focussed on the point. Its axis is thus directed along MA. Turn the table through 90 , so that M' moves to M" ; and move the scale along a direction perpendicular to itself until the point once more comes on the microscope cross-wire. The point then lies on the axis of rotation of the table, and the axis of the spindle will intersect that of the table as it slides along the scale. Mount the surface to be examined on the brass plate, and attach it to the spindle. If the surface is one of the faces of a lens, the side not under examination should be blacked by covering it with vaseline and lamp black to absorb rays striking it. Set up an object in some convenient position and view its image in the surface. If on rotating the spindle about its hori- zontal axis the image does not move, the centre of curvature lies on this axis, and the preliminary adjustments are complete. Since the image remains stationary the rotation of the surface serves to bring up fresh parts of the sphere exactly into the place occupied by the part moved away, i.e. the centre lies on the axis of rotation. REFLECTION 271 The object may conveniently be a fine brightly illuminated line drawn on a piece of ground glass. Set up this object again so that rays fall nearly normally on the surface, and examine the image formed by reflection in it by means of a microscope. By moving the carriage a position can be found in which a rotation of the table to and fro produces no displacement of the image. This occurs when the centre of curvature of the surface lies on the axis of revolution of the table, for rotation then has the effect of replacing One part of the surface by another, and reflection occurs as before. The position of the index mark is noted. If the lens is moved until the vertical axis of the table is a tangent to the surface, a rotation to and fro will not displace the centre of the lens. Place in the centre a few grains of lyco- podium powder, and focus the microscope on one of these. The lens is moved and the focussing repeated until the movement ceases. The index reading is again noted. The difference of the two gives the radius of curvature. Preliminary observations may be made by eye until the motion appears to cease in the two cases, and only the final exact adjust- ments need be made by the aid of the microscope. CHAPTER XI REFRACTION Introductory Remarks Rays of light in passing from one medium to another usually undergo a deviation from their course in the first medium. On the wave theory this is accounted for by the fact that the light travels with different velocities in the two media. The refractive index with respect to two media is defined to be the velocity of light in the first divided by the velocity in the second, and we denote this by x yL 2 . When light travels from air to another medium, say glass or water, we shall write a \L g or n«, unless there is no doubt that we are dealing with air and some other stated medium when we may write simply \x. We have by definition, I and i^s = ~' 3^1 In particular ■"" = ^T a result which will shortly be found useful. In many experiments in this and following chapters it will be necessary to furnish a bright source of monochromatic light. The most convenient way to provide such a source is to use a Mecca burner, which consists of a Bunsen burner rather larger than the ordinary type of burner provided with a wide end over which is stretched a gauze with a wide mesh. If a small bead of soda glass is placed on this gauze and the Bunsen made to roar as much as possible a quite satisfactory yellow flame will be produced. It is a great advantage that there is no crackling in the flame as in the case of the use of common salt, when small pieces of hot salt are thrown about falling on the bench and on the slits of spectrometers. In the case of the latter serious damage to their shape may result. The slit may be illuminated directly, or better still, an image of the brightest part of the flame may be thrown on to it by a short focus convex lens. 272 REFRACTION 273 The student will do well to pay attention to the small point of illumination of the slit. It is important always, but assumes greater importance in the case of experiments on interference and diffraction which will be described in the next chapter. The difficulty of discovering Newton's rings, interference fringes, and diffraction bands is almost always due to a lack of care to obtain the best possible illumination of the slit or whatever may be used as a source of light. Determination of the Refractive Index of a Plate by its Apparent Thickness The apparatus necessary is a good travelling microscope and a plane cover glass. Set the microscope with its axis vertical and focus it on the metal platform. Note the reading on the vertical scale. Insert the cover glass over the point on which the instrument was focussed, again focus on the metal and observe the scale reading. Usually the metal surface, though dark, is easy to observe, but if desired a thin sheet of white paper may be placed over it and the surface of the paper used instead of the metal surface. Raise the microscope until the upper surface of the cover glass is sharply focussed. There will usually be specks of dust on the surface to assist this setting of the microscope, but if any difficulty arises place a small drop of ink on the surface and focus the extreme edge of the drop. The drop need be no larger than that made by a sharply pointed pen. Note the scale reading when this third adjustment is made. If we describe the scale readings by (1), (2), and (3) respectively ; the difference between (1) and (3) gives the actual thickness of the glass and that between (2) and (3) the apparent tfrickness. Consider a point, P (fig. 154), situated at the lower surface of *8 274 ADVANCED PRACTICAL PHYSICS FOR STUDENTS the glass and from which rays of light originate. Only those making small angles with the normal to the surface will enter an eye of microscope placed above P. If one of these rays, PO^ 1 , makes an angle, *, with the normal in the air and an angle, r, in the glass, sin*" OO 1 0*P O x P H = so that \l = 7^7 = , ., . f sm r O 1 ? 1 OO 1 ~~ Q 1 ? 1 ' For small values of * and r we may write : O 1 ? 1 = OP 1 , O x P = OP, OP _ thickness of glass OP 1 — apparent thickness' for all rays inclined at such small angles appear to come from P 1 , so that OP 1 is the apparent thickness. Determination of the Refractive Index of Liquids by Total Reflection When a ray of light passes from a medium of refractive index, Pi, to another of refractive index, \t 2 , with an angle of incidence, *', and of refraction, r, we have : \x 2 sin * ll * 2 — nj ~~ sin r We may not always find a corresponding value of r for a given i unless — is less than unity. In the case when — > i, the value — sin % must not exceed {*2 t*2 unity. In the limiting case when sm * = — , the corresponding value of r is 90 , and i then measures the critical angle. For values of i greater than this critical value the surface acts as a perfect reflector. If i is slowly increased a value is finally attained when the refracted ray suddenly disappears. In this case if the second medium is air so that \l 2 = 1 we have : 1 — = sm *. Hi This formula may be used to determine y. f by mounting a small rectangular trough, CD (fig. 155), with sides of plane glass on the table of a spectrometer, so that a parallel beam of light may be passed through it from a collimator, GH, and received in the REFRACTION 275 telescope, AB. The light is suddenly cut off when the air cell, EF, consisting of two plates of glass mounted parallel to one another and cemented together with a thin air space between, is turned so that the light falls on the air at the critical angle. 8 * It will be noted that the critical angle is that for air and glass, but the apparatus is used to determine the refractive index of the liquid in the trough. Let the ray, ABCDE, be incident from the water on the glass and be totally reflected at the glass-air surface at the critical angle. Then if FBG and FTOG 1 be the normals to the glass at B and D, we have : — = sin z BCH, where CH is the normal at C and aV . g denotes the refractive index from air to glass. Also V)V-g ^aV-a _ sin ABF sin ABF av. w ~ sin CBG ~~ sin BCH' *'• olXw ~"sinABF '"( J ) We have thus to measure the angle ABF, and from it we deduce the value a [i Wf the refractive index from air to water. As a source of light use a Bunsen flame containing sodium* and illuminate the slit of the collimator, which must be adjusted for parallel light (p. 279). Focus the telescope on the slit and turn EF (fig. 155), until the light just appears. EF is attached 276 ADVANCED PRACTICAL PHYSICS FOR STUDENTS to a pointer which moves over a scale of degrees. Note the position of the pointer. Turn EF from this position into another where the image again disappears. Let AA 1 denote the axis of the telescope and EF the first position. The second will be E X F X if L AOE = L AOE 1 . The Fig. 157 second position is reached either by turning through the angle EOE* or EOF*, and ^EOF* = * - iLEOE*. When the apparatus is used there is no uncertainty concerning which angle is measured. The angle we require is that between the ray and normal to EF or ET 1 , i.e. the angle between OA and the normal to EF or E*F*. This angle is half the angle between the two normals, and this is the same as half the angle E x OF. Thus we have to note the angle through which EF turns and take one half of it to find the angle ABF of formula (1). To Find the Refractive Index of a Liquid, using a Lens and a Plane Mirror In the experiment a plano-concave lens is formed of the liquid under examination (fig. 158), and its focal length found experi- mentally. The refractive index and radii of curvature of the two surfaces enter into the formula for the focal length, so that it is possible to deduce the index, ja, from a determination of the focal length, /, and the radius of the curved surface of the lens. The liquid lens is made by placing a drop of the liquid on a plane mirror and laying a convex lens of from 10 to 20 cms. focal length on the drop. The liquid is squeezed into the space between the mirror and lens and we have a combination of two lenses one of glass and the other of liquid, giving a combined focal length of, say, F. REFRACTION 277 If the convex lens has a focal length, /', then : 1=1+1:. We may therefore deduce / from a knowledge of F and/'. To determine /', place the lens horizontally on the mirror and adjust a pin, held in a stand above the lens, until the inverted 1 Fig. 158 image and object appear together and there is no parallax between them. The distance from lens to object gives /' (com- pare p. 263). Now place the liquid and lens on the mirror and again find the position of coincidence of object and image. This gives F, so that we now have /. For the purpose of substituting in the above formula, F and/' must be given their appropriate negative signs, and / will turn out to be positive. If r is the radius of curvative of the curved liquid surface, and n the refractive index of the medium, ~f-~T~ < a > since the second surface has zero curvature. r can be found by measuring the radius of the surface of the lens in contact with the liquid by means of a spherometer, or by any of the methods described in the last chapter, so that n can be calculated from the equation (2). If it is preferred, the determination of r may be avoided. If water be used and its index regarded as known and having the value, 1-33, we may make a water lens as above, and calculate its focal length, /*. But 7=T-> or ix = 1 4- •33/" / ' where p is the refractive index of a liquid other than water. 278 ADVANCED PRACTICAL PHYSICS FOR STUDENTS A convenient liquid to use for the experiment is aniline. Care must be taken to prevent it from getting at the back of the mirror since it dissolves the varnish protecting the silvering. Determination of the Refractive Index of a Lens by Boys' Method This method of finding the refractive index of a lens consists in measuring its focal length and the radii of curvature of both surfaces. The value of {* may then be determined from the equation : T- <*-<}-!> where / denotes the focal length of the lens and r and s the radii of curvature of its surfaces, with the usual convention regarding the signs of the quantities in the formula. In a convex lens let v x and r 2 denote the radii of curvature numerically and F the numerical value of the focal length. Then : **-o-*(Z + k) Determine F by any of the methods described below, p. 293. In order to find r lt set up a pin with its point on a level with the centre of the lens. Two images will be seen by reflection in the faces of the lens, one erect from the front surface which acts as a convex mirror and one inverted by the concave back surface. It is the latter which is required for the experiment. Move the pin until its inverted image is coincident with it, as judged by the method of parallax. When this is the case the rays must strike the back surface normally and be returned along their incident course. If an eye be placed on the side of the lens remote from P (fig. 159), it will receive the transmitted part of the ray, PB, and will see the image of P in the direction, CBQ. C __ A Fig. 159 Q will thus be the image of P in the lens, and OP and OQ are conjugate distances. Thus, writing u = OP and v = OQ, we have : 1 i_ _ 1 , . OQ OP F * [3) REFRACTION 279 This enables us to determine OQ from OP and F. Q is the centre of curvature of the left-hand surface of the lens since QAB is normal to this surface. Thus OQ gives the value of the radius, say, r x . Turn the lens round and repeat the process to obtain r 2 , the radius of curvature of the other face of the lens. Let OP = d x in the first case, and let d 2 denote the corresponding distance in the second. Then from (3) : I *x 1 ~1 x 1 ~~F 1 1 d 2 1 ~F I F = (*- -1) U7 + 1 d z ■1) Hence so that we may calculate ^ from the experimental determination of F, d v and d z . It is sometimes difficult to see the image by reflection at the back surface of the lens, but by holding it over the surface of mercury or floating it in the mercury, the image may be made to stand out and be easily located. The Spectrometer The spectrometer consists essentially of a telescope and colli- mator. The latter is a system of lenses mounted in a telescopic tube with an adjustable slit at one end, and it serves the purpose of rendering rays from the illuminated slit parallel on emergence. Both are mounted on a rigid stand, the collimator being fixed, and the telescope rigidly attached to an arm which rotates about the centre of the stand. Both are mounted horizontally with their axes in the same plane. P I . I _ a .' t A 1 B P Fig. 160 The centre of the instrument is occupied by a table provided with screws for levelling. Underneath the table is a metal scale of degrees on which can be read off the positions of the telescope and of the table. In the diagram, T denotes the telescope, AB the table, and C is the collimator. PP denotes the axis of the instrument and the table and telescope rotate round it. The telescope is fitted with a Ramsden eyepiece carrying cross-wires. 28o ADVANCED PRACTICAL PHYSICS FOR STUDENTS Before using the spectrometer for any experiment certain adjustments have to be made in addition to those made in the construction of the apparatus by the maker. In the first place, the eyepiece of the telescope is adjusted so that the cross-wires are distinctly focussed. The telescope should then be taken to an open window and focussed on a distant object such as a distant telegraph post, care being taken that there is no' parallax between the image and the cross-wires. When this is done the telescope must not be readjusted again during an experiment or it will be necessary to repeat this process. It may happen that a second observer whose sight differs from that of the first is unable to focus the cross-wires easily. He may readjust the eyepiece provided that he does not alter any other part of the telescope, for then he is not causing it to be out of focus for parallel rays. He merely gives himself convenience in focussing easily and leaves the cross-wires and image without parallax. The telescope is now turned towards the collimator, the slit being made vertical and illuminated with monochromatic light, and the collimator is adjusted until a distinct image of the slit falls on the cross-wires. The apparatus is now adjusted so that parallel rays pass from collimator to telescope. 2*^ 6 "ill - F \ " -T- 1 i -*—^ Fig. i6i When a prism is used in the spectrometer it is necessary to adjust it so that its refracting edge is vertical. The screws on the table enable this to be done. They are shown at D, E, and F, placed at the corners of an equilateral triangle, and the table is usually ruled with lines as shown (fig. 161), to assist in setting the prism with one face normal to one of the sides of the triangle, for example EF. The table should be made as nearly horizontal as possible by the use of a spirit-level, and the prism then placed with one face perpendicular to EF.' REFRACTION 281 Now suppose light from the collimator falls on this face and is reflected into the telescope. If this face is vertical the slit will now appear to lie in the same part of the field of view of the telescope as when it is seen directly. The three screws should be adjusted to restore the image to its direct position if necessary. Fig. 162 * Let the light be reflected into the telescope by the other face bounding the refracting edge. If further adjustment is necessary it must be done by the screw, D, for this will not disturb the previous adjustment, since it does not turn the face perpendicular to EF out of its vertical plane. The two faces are now vertical and the instrument is adjusted. It is sometimes necessary to arrange one face of a prism so that it lies normal to the collimator or telescope. This may be done by turning the telescope from the position in which the slit is seen directly without the prism, through a right angle, so that the axes of the telescope and collimator are perpendicular to each other. The prism is now placed on the table of the instrument and the table rotated until, by reflection in the face concerned, an image of the slit is thrown on the cross-wires. The face now lies at 45 to the axes of the collimator and telescope and a further rotation of 45° will bring it either perpendicular to the collimator or telescope. In making measurements with the spectrometer the slit should, as a rule, be narrow, and the cross-wire should lie accurately down the centre of the image of the slit before the position of the telescope or table is noted on the metal scale. 282 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Tangent screws help in the accurate final setting of the telescope and table, and verniers on this scale serve to measure the angles to an accuracy of one minute of arc. A modern form of the apparatus is illustrated in fig. 162. The Refractive Index of a Prism by the Method of Minimum Deviation When a prism with a refracting angle, A, causes a minimum deviation, D, in light passing through it the refractive index is measured by the formula : sin £ (A + D) sin I A Thus the determination of jx consists in measuring A and D. There are two methods in common use for measuring an angle of a prism. One of these consists in allowing the light from a narrow slit to fall partly on one of the faces of the prism bounding the angle and partly on the other. An image of the slit can be seen in the telescope when the latter lies on either side of the prism. Thus in the case illustrated (fig. 163), the telescope will receive rays from the direction AO on one side and from AK on the other. The telescope has to be turned through the angle KAO in turning from one direction to the other. F D L Fig. 163 Since DA and AO make equal angles with AC, we have : ^- EAC = z. CAO, and similarly, /_ EAB = ^ BAK. Thus £_ OAK = 2 ^ BAC. To measure A we need only focus the image of the slit on the cross-wire in the two positions and halve the angle through which it is rotated. REFRACTION 283 Suppose AB is the face which reflects the light into the telescope along AK. If the prism is rotated until the face, AC, now lies parallel to AB the rays will once more be reflected in the direction, AK, but now by AC. Some of these will enter the telescope if there is a sufficiently broad pencil of them. But the prism has been rotated so that AC moves round through the angle CAB 1 . Thus we measure, by means of the table, the angle (180 - A), and A can be deduced. Measure the angle of the prism by both methods. It now remains to find D. Set the prism so that with A as refracting angle, light is refracted through it and received in T. It will be found that as the prism is rotated the telescope has to be rotated to keep the image of the slit in the field of view. Rotate the prism until T is as close to the position directly opposite the collimator as possible, with the slit in the field of view. When this is the case the angle between the telescope and direct position is as small as possible. Note the position of T and then remove the prism and observe the slit directly. Again note T's position, and hence find D. It is, of course, necessary to adjust the spectrometer and edge of prism m the way described in the section on the adjustments of the spectrometer. The final movements of the telescope or table must be made carefully with the tangent screws. The Dispersive Power of a Prism The dispersive power of a medium is measured by : t*B — v.* w =- v- — 1 t* B and n R are the refractive indices for blue and red rays and u has the value, £ (^ + ^). J * The refractive indices may be found by the method of minimum deviation. As a source of blue and red rays a discharge tube containing hydrogen may be used. The^ tube should be held vertically, and the slit illuminated by it directly, or an image of a bright part of the tube thrown on to it by means of a short focus lens. 284 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Three well-marked lines can readily be seen, one in the red, a second of blue-green colour, and a third in the violet. Use the first and third of these — they are known as the C and H r lines respectively, while the second is the F line. The Refractive Index of a Liquid by Total Internal Reflection within a Glass Prism A glass prism with one face unpolished is mounted on the table of a spectrometer. The table is levelled and the edge between polished surfaces set normal to its plane (see p. 281). The angle between these faces is measured in the usual way by allow- ing light from a wide collimator slit to fall on both faces and by measuring the angle between the two reflected beams. Light from a sodium flame is allowed to fall on the unpolished surface, or an image of the flame is thrown on it by means of a lens to cover the whole matt surface. It is first necessary to find the refractive index of the glass prism and then to coat one of the bright faces with a thin layer of the liquid. Glycerine is a very convenient substance with which to carry out the experiment. The theory of both parts of the experiment is the same. Theory ABC denotes the prism, of which AB is the unpolished side. This side acts as a collection of point sources of light of which S denotes one. Rays from it strike AC and are reflected and re- fracted there. Those like SD falling at an angle of incidence less than the critical angle are partially reflected and partially refracted so that the ray, EF, issuing from BC is less intense than the incident ray, SD. A Fig. 164 Rays falling at angles greater than the critical angle suffer no refraction at AC, so that the emergent ray from BC is scarcely less bright than the incident ray. In any case there will be a REFRACTION 2 85 marked difference between the former and latter group. The ray, SG, is drawn for critical incidence so that the direction of the emergent ray, HK, stands between those for the faint and bright rays. * We get a similar state of things for any other point in AB, and for each point the rays striking AC at critical incidence give rise to rays parallel to HK on emergence. Such a group of rays is brought to a focus in the focal plane of a telescope placed to receive them. Similarly, a group of rays, parallel to EF, will correspond to all rays falling on AC parallel to SD, and these will fall on the telescope in a direction different from that of HK, and will form a line in the focal plane not coincident with the former. This is true for all the directions of rays from BC. We shall thus have a multitude of parallel lines in the field of view divided into two groups of different intensity by the critical direction, HK. The effect will produce a field sharply divided into bright and dark halves by the direction, HK, making a with the normal to BC. If the telescope is turned to face AC the field will be similarly illuminated on account of the reflections that have taken place on BC. The issuing critical rays will make an angle, a, with the normal to AC. If the cross-wires of the telescope are set on the dividing line of the field when it is directed towards BC, and again when towards CA, we can deduce the value of a by observing the angle through which the telescope has been turned, since the telescope is rotated through 180 — C + 2a = (say). Thus 6 + C ~ l8o ° a = 2 where it is assumed that the telescope is moved from side, BC, towards CA in a counter-clockwise direction as seen in the diagram. From C and a we can calculate \l from the formula given below. Suppose that AC is coated with a substance of refractive index, Pi, and that the index of the prism is p. We have on referring to the diagram : v-i = v- sin c, p + c = C, where = z. MHG since the points C, G, M, H, are concyclic, sin a = {x sin 0, .*. l*i = V- sin (C — p) = {i sin C (i — sin 2 p)* — cos C sin a = sin C (n 2 — sin 2 a)* — cos C sin a. 286 ADVANCED PRACTICAL PHYSICS FOR STUDENTS When (*! = i, i.e. when there is no layer on face, AC, we have : „ /r + sin a cos C V ■ • 2 ^2 = 1 -T- ) + sm 2 a> V sinC J This is the formula from which n may be calculated. Fig. 165 shows the state of affairs when the critical angle is large. This will be the case when the media on either side of AC are nearly of the same optical density. A Fig. 165 The formula above will still apply if the value of a has its sign reversed. The above method of measuring a can be used in finding {*, but when one side is coated it would be necessary to clean that side and coat the other when the telescope is turned. This would make it difficult to keep the prism fixed and would be inconvenient. It is therefore best to use an eyepiece, such as the Gauss eye- piece with cross-wires that can be illuminated. Fig. 166 The light from these may then be reflected in the face opposite the telescope and the image made to coincide with the object. When this is the case the telescope stands perpendicularly to the face and the angle between this direction and that in which the division of the field is viewed measures a. REFRACTION 287 After finding jx for the glass, place a few drops of the liquid on one of the polished faces and press over it a thin plate. This ensures that the face is covered with liquid, It is better for the sake of definition of the two halves of the field to allow light to fall at grazing incidence on the prism surface, say, AC. Then the rays entering the prism make angles less than c with the normal so that the field is now only half illuminated and the edge corresponds to the direction, HK. AB should be kept dark by covering with a sheet of dark paper. If the light is incident externally on the liquid film, it must enter by the edge, AD (fig. 166), any ray such as P would not reach AC at grazing incidence. Calibration of a Spectroscope The spectroscope already described had only one prism, but it is an advantage sometimes to use two mounted together on the spectrometer table. The experiment may be carried out with one only, but a wider spectrum is obtained with two. S .L 0. i %\ ^rj>k * Fig. 167 If the prisms, P 2 and P 2 , are used to produce a series of spectral lines due to some source, S lt the telescope, T lf will have a definite position on its scale corresponding to each particular wave length. To calibrate a telescope is to find the values of the wave lengths corresponding to the different parts of the scale. If a curve is drawn whose co-ordinates are the wave lengths and corresponding scale readings, such a curve is a 'calibration curve,' and may be used to determine any wave length from the division of the telescope scale at which it is seen. 288 ADVANCED PRACTICAL PHYSICS FOR STUDENTS First make the usual adjustments for parallel light. Place one prism, P x , in the position corresponding to minimum deviation and then put in P 2 also in the position of minimum deviation. Examine the spectrum of a Bunsen flame containing sodium light. Fix the telescope so that the sodium line lies on the cross- wire and note the position on the metal scale of the spectrometer. A graduated scale is provided consisting of close rulings cut on an opaque screen so that the lines are transparent. It is fitted at the end of a second collimator, C 2 , which is adjusted to direct rays down the telescope after reflection on one face of P 2 . The telescope is already set for parallel rays so that by adjusting C 2 so that an image of the scale, ss, appears at the cross-wires, it is ensured that parallel rays emerge from C 2 . The position of the sodium line is taken as a point of reference and scale divisions are read so many to the right or left. If the scale, ss, is moved accidentally no error is then caused and the calibration curve will still be of use. In a good spectroscope the sodium lines will be separated and will appear as two very close together. Use one of these as the reference line. Measure the scale positions of a series of lines of known wave length extending over the visible spectrum. Use only sharply denned lines. The table gives the wave lengths of lines which may be pro- duced conveniently in the laboratory. In order to produce the lines from the metals use a spark between poles made of these metals connecting them to opposite terminals of a Leyden jar charged by an induction coil. The salts are volatilized in a Bunsen flame as with sodium. Examine the inner cone of a Bunsen flame ; it is due to carbon monoxide, and contains several series of lines. The spectra of light from discharge tubes containing various gases should be examined as well. Lines from a Neon lamp should be examined also. The field will be observed to contain many lines, the yellow line of wave length 5853 units is a bright line and it is a good exercise to determine from the curve the wave lengths of the other lines, afterwards comparing with standard tables. Plot the curve on a large scale on squared paper. The Auto-Collimating Spectrometer In this instrument the telescope acts also as the collimator. The apparatus is almost identical with the ordinary spectrometer, REFRACTION TABLE OF WAVE LENGTHS 289 SALT OR METAL Lithium Chloride or any Salt of Lithium Any Salt of Sodium Salt of Potas- sium Strontium Chloride DESCRIPTION OF LINE Thallium Chloride Helium Hydrogen Red Double Yellow Red Extreme Violet Blue (Not to be confused with the bands) Green WAVE LENGTH TENTH METRES) Red (C) Blue-Green (F) Violet (H r ) 6708 /5890 I5896 7668 4044 4047 4607 5351 6678 5876 4471 3889 6533 4861 4340 except that there is no separate collimator and the telescope is modified. On looking into the eyepiece of the instrument, the field is seen to be divided into two parts, the lower half is dark and the upper bright, but crossed by a pin which extends from the upper edge downwards across about half of the bright part of the field. Near the eyepiece end the tube of the telescope is provided with an opening by means of which a slit just within may be illuminated. Just below the dividing line of the two halves of the field of view a small right-angled prism is placed which deviates the light from the slit down the centre of the telescope tube. Its position is denoted by the dotted rectangle, P, in fig. 168, 19 290 ADVANCED PRACTICAL PHYSICS FOR STUDENTS If a polished surface such as the face of a prism is placed in front of the object glass and set at right angles to the axis of the telescope we may, by focussing, make the rays leaving the object glass parallel, so that they will be returned by reflection at the face of the prism and come to a focus in the focal plane of the telescope. *S ET? Focus the eyepiece carefully or^ the pin. Illuminate the slit by means of a small electric lamp — a four-volt lamp, supplied by two accumulators, is convenient for the purpose. The slit will be observed to be of the shape of an inverted T. The image must be brought so that the horizontal bar lies along the edge bounding the two halves of the field, when the end of the vertical bar will coincide with the point of the pin. In order to obtain this result the face of the exterior prism must be accurately normal to the emergent rays. The table is provided with three screws in order to level to the prism, and the process described on p. 280 must be followed. The positions of the slit and pin are such that when the image of the slit lies at the same distance from the object glass as the pin, the latter is at the principal focus. The prism is slightly tilted to throw the image of the slit on to the upper half of the field of view. To Find the Refractive Index of a Prism by means of the Auto-colli- mating Spectrometer Place the prism on the table and adjust the faces bounding the refracting angle as described above so that the light reflected normally by both throws an image of the slit into the field of view just below the pin. When this is so the faces are vertical and consequently so also is the edge of the prism. REFRACTION 291 Use monochromatic light by placing a sheet of yellow glass between the bulb of the electric lamp and the slit. Observe the position of the table when the face, AB, reflects the light normally. Rotate the table until the light falling on AB and refracted there strikes AC normally and is returned along Fig. 169 its path. The table has been turned so that the face, AB, has turned from a direction normal to the rays from the telescope into the position at which refraction takes place, i.e. it has turned through the angle of incidence, i. The angle of refraction in this case is the same as the angle, A, and is so marked in the diagram. This angle may be measured by setting the face, AB, normal to the rays and reading off the position of the table. Then by rotating the table until the rays strike AC normally we turn the table through an angle (180 — A) . We may therefore calculate ja, since we know the angle of incidence and refraction in a particular case and _ sin i _ sin i sin r sin A" The Pulfrich Refractometer This apparatus is shown in figure 170. It is designed to measure refractive indices of solids and liquids to an accuracy of about r~ per cent. The principal part of the apparatus consists of a prism having two plane polished faces at right angles to each other. One of these is horizontal and the other vertical. On the horizontal face is placed the substance whose refractive index is required. If this substance is a solid it must have two faces cut perpendicu- larly to one another, both of which are cut accurately plane, so that one may rest on the horizontal surface of the prism and the other stand vertically. Optical contact is brought about by placing a few drops of a liquid on the horizontal surface of the 2Q2 ADVANCED PRACTICAL PHYSICS FOR STUDENTS prism, which has a refractiye index higher than that of the solid to be experimented upon, and standing the solid on it. The makers recommend monobromonaphthalene for this purpose. In the case of a liquid, it is contained in a glass cell cemented to the prism. Fig. 170 Light is directed into the liquid in a direction almost parallel to the horizontal surface so that light entering the prism makes the critical angle, c, with the normal. Let it emerge from the prism at an angle, i. (Cf. fig. 171.) Suppose that the refractive index of the substance to be examined is n and of the material of the prism, ii - Then sin c = — » sin * sin (| - c) REFRACTION 293 i.e. cose = sin* sin 8 c -f cos 2 c = 1 = ^- + sin 2 i (V_JV t* — V(Jt 2 — sin 2 *' The substance must have a refractive index less than that of the material of the prism if it is to be examined by this method. The apparatus measures the angle, *. For this purpose a telescope is attached to a circular scale and the rays are received by it. Since no rays within the prism make an angle greater Fig. 171 than c with' the normal on entrance to the prism, the angle, \i ~~ c ) ' measures the minimum angle at which rays strike the vertical face. Corresponding to this, i measures the minimum inclination of the emergent rays to the normal. Thus in the telescope the rays emerging in this direction bound the field of view. The apparatus is arranged so that the rays are deflected down the telescope, and when its cross-wires lie on the dark edge of the field the scale reads off the angle, *', to an accuracy of one minute of arc. no, of course, is an instrument constant and has the value 174. In addition a microscope screw is divided so that the value of the refractive index may be measured for different wave-lengths to a still higher degree of accuracy. The prism and specimen are surrounded by a metal water jacket and thermometers are provided for reading the temperature. A table is supplied with the instrument giving the value of t* corresponding to different values of *. Determination of the Focal Lengths of Thin Lenses by means of Pins Convex Lens. Method, 1 Support the lens vertically and place a vertical pin behind it so that its point lies on a level with the centre of the lens and on the axis of the lens. 294 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The adjustments are rendered easier by placing a sheet of white paper behind the pin. Set up another pin so that it coincides with the image. When this is the case the image and second pin will not appear to move relatively to each other when the eye is moved horizontally from side to side. Arrange the lens and first pin so that the image is not at the same distance from the lens as the object. Then without moving either pin displace the lens until a second position is found at which the pins occupy conjugate points. Fig. 172 Let the lens have been moved a distance, d cms., while the distance between the pins is D cms. In the figure we have evidently : F^i = u = — — , and C^Pa = v = — ^— (numerically). Thus from the equation : 1 _ 1 1 v u ~~ f we have : / — — — ^n — ' since in the formula we have to write : D +d v = — — . 2 Determine several different values of D and d, and calculate the value of /from each ; the result should give a constant. Method 2 Put up a plane mirror immediately behind the lens and parallel to it. Then place a pin in front of the lens as in the first case and adjust it to make it coincide with the image formed by refraction in the lens and reflection in the mirror. The distance from the pin to the lens is equal to the focal length of the lens, REFRACTION 295 for the rays are reflected back along their path and must therefore strike the mirror normally. They leave the lens las parallel rays, and must therefore originate from the principal focus. Fig. 173 Method 3 Set up a plane mirror, P^ 1 , at some distance from the lens, and a pin on the other side of the lens. The pin should be mounted so that its centre is on a level with the centre of the lens and should be adjusted until the position of the image of the j>in is made to coincide with the pin. The way this is brought about may be seen from the diagram. The image formed by the lens evidently lies on the surface of the mirror. B Fig. 174 If the distances be measured we have the positions of a pair of conjugate points. Make several observations for different distances and calculate/. In making experiments with a convex lens it is useful to know the focal length approximately before making an accurate determination of it. Sometimes time is wasted in trying to locate a real image when the object is so placed that a virtual one is formed. It should be noted that if the object is at a distance from a lens, which is less than the focal length, the image is virtual. Focus a distant object — a lamp or window, if not too close, is suitable — and measure the distance from the lens to a well- defined image thrown on to a sheet of paper. Unless the lens has a very long focal length this distance will be approximately/. Concave Lens Method 1 Put up the lens and place a convex lens of known focal length in contact with it so that their axes coincide. 296 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Note if the combination will form a real image of a distant object, and measure approximately the distance from the lens to image so formed. This will indicate whether the combination acts as a convex lens. If it does not the convex lens is too weak, and a stronger must be chosen. Determine the focal length, F, of the combination by any of the above methods, and let / denote that of the convex lens, f 1 that of the concave. Calculate f 1 from the formula : Fig. 175 Method 2 Set up a concave mirror behind the lens and a pin in front. An inverted image of the pin will be seen on looking through the lens, and it may be made to coincide with the pin by suitably adjusting the mirror. The diagram illustrates that this is brought about by the normal incidence of the rays emerging from the lens on the mirror. The virtual image of P is thus at P 1 , the centre of curvature of the mirror. Remove the lens after noting its position and the distance of P from it, and again adjust the pin until it coincides with its image formed by the mirror. We thus locate P 1 and can measure v, the distance from the position of the lens. On substitution in the formula : 1 __ 1 _ 1 v u ~J we calculate /. Repeat this for several different cases. The Focal Length of a Lens Combination In the case of a thin lens, in which no account of the separation of the two surfaces is taken, there are two points on either side of it called principal focal points at equal distances from the lens if it is situated in a medium which is uniform all round it. REFRACTION 297 These two points are distinguished by calling bne the first and the other the second focal points. The former is the position of an object corresponding to an image at infinity while the second is the position of an image corresponding to an object at infinity. A plane perpendicular to the axis at the position of the lens is called the principal plane and if / denote the focal distance, we have the relation : v u f* connecting the distances of the conjugate points from the principal plane. If the principal plane and focal points are known it is a matter of simple geometrical construction to determine the image of any object. When the lens is not thin, or when it is necessary to deal with a system of lenses, Gauss has shown that the formula is the same but that in this case there are two principal planes, the first and second, separated by a finite distance. The distance of the first principal plane to the first principal focus is called the first focal length, and the distance to the second principal focus from the second principal plane is the second focal length. If the lens system is situated in a medium the same on both sides the two focal lengths are the same and we shall denote either by /. When the object is at a distance,- u, from the first principal plane and the image is at a distance, v, from the second, the formula is still : 111 v u ~~~ f* A very important property of the principal planes is that if a ray is incident on the system towards a point on the first principal plane, the emergent ray is directed from a point in the second principal plane which lies on the same side of the axis and at the same distance from it as the first. It is now possible by a geometrical construction to determine the position of an image corresponding to any object. This is illustrated in the diagram (fig. 176). The ray, BI^, parallel to the axis must pass on emergence through the second focus, F a , and it appears to emerge from L 2 , where PiL x = P 2 L 2 . Pj and P 2 denote the principal points and PiL^ and P 2 L 2 are the principal planes. 2 g8 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Again the ray through F 2 emerges parallel to the axis from M 2 > where P 1 M 1 = P 2 M 2 . We thus find B 1 and can draw in the image, ATO 1 . If the media on either side are different, F 1 F 1 is not equal to P 2 F 2 . Two other important points for the lens system are the nodal points. They are defined to be points on the axis such that a ray directed to the first nodal point emerges as a parallel ray directed from the second. ] B h ^ V ** \F<, A 1 A F^ Mi ^X! ^ B 1 Fig. 176 When the medium is the same on both sides of the system the first and second nodal points coincide with the first and second principal points. We have, therefore, the line, BP^ parallel to PaB 1 , and the magnification, m, is A^ 1 = A X P 2 = v AP X AB u just as in the case of a thin lens. A direct determination of / requires the location of a pair of conjugate points, but we do not know the position of the principal points so that it is not possible to find v and u for substitution in the formula : v u f We may, however, determine / by measuring the magnification of an object by the lens for two different positions. From this formula we have : — = 1 Thus if u x and m x denote the distance of an object from the first principal plane and the corresponding magnification, we have : m x f REFRACTION 299 and for a second position : 1 Wo /. / = u. I I m x m z Thus the actual lengths, u, are not required for the calculation, but only the distance between the two positions of the object, which is given by the difference between u x and u 2 . If several values of — are observed and positions corresponding to u noted on the optical bench, a graph may be plotted of — against u. Fig. 177 The relation is linear, and if two points, P t and P 2 , be chosen on the line as far apart as is convenient : P 2 N 2 - P X N X NoN, < It is convenient to use a transparent glass scale as object and to illuminate it by monochromatic light. A similar glass scale is used as a screen so that the apparent length of a certain number of divisions of the first scale seen in the image can be read off at once on the second scale. The screen is adjusted as usual by noting when there is no parallax between the image and scale. For accurate work it is necessary to use a low-power microscope to measure the image. Another method which is instructive is to adjust the positions to obtain a magnification of magnitudes 1 and 2. 300 ADVANCED PRACTICAL PHYSICS FOR STUDENTS When the object and image are equal in size their distances from the corresponding nodal points are both 2/. This may be verified from a diagram or from the formula. Now keep the screen fixed and move the object and lens until the magnification is 2. It may be verified that in this case the object is — / from the first nodal point, and the image is 3/ from the second. The lens system has thus been moved through a distance, /, and this may be measured directly. Suppose the diagram (fig. 176), denotes the case for a magnifi- cation unity. In this case AP X and A X P 2 are both equal to 2/. We know the value of /, so that since A is a definite known point we can locate Pj and similarly P 2 is known from the point, A 1 . The lens system is sometimes encased in a tube, and in that case the distances from P x and P 2 to the ends of the tube should be recorded. If the lens is merely a thick lens of glass the distances of the principal points from the faces of the lens should be recorded. Determination of the Principal Planes of a Thick Lens Let DAEB denote a thick lens and let C and C 1 be the centres of curvature of the faces, DBE and DAE respectively. There exists a point, O, such that all the rays which pass through it emerge from the lens parallel to their original direction. This point lies on the axis, CC 1 , and is called the centre of the lens. The rays passing through O after their first refraction, were directed originally towards a point, P. Thus P and O are conjugate points with respect to the first surface. REFRACTION 301 The rays after the second refraction appear to come from P 1 , so that P 1 and O are conjugate points for the second surface. These points, O, P and P 1 , are fixed points for the lens and P and P 1 are the principal points. If the distance of an object is measured from P and denoted by u, while the image is measured from P 1 and denoted by v, we have the relation : 1 _£ _ 1 v u ~ f' as in the last experiment. The quantities, v and u, are to be given positive signs if measure- ment is made towards the source of light, and negative if it is made in the opposite direction. Parallel light incident on the lens on the right is brought to a focus at F 1 where PT 1 =/. Light originating at F is parallel on emergence where PF = — /. Let a source of light be located at a point, S, where FS = x, x is to be regarded as a number of cms., and is not given a sign. The focal distance is likewise measured by the number F. The lens, when convex, has a negative focal length, i.e. in the formula / = - F. The image will be formed at I, say at distance, y, from F 1 . Then u = (F + x), and v = - (F + y). Thus ^ » F +y F + x F or X y = F 2 . The disposition of the conjugate points shown in the diagram is such as occurs in the case of a convex lens. The object and image both lie within the focal distance or without it. In this case x is taken positive when measured from F towards the right, and y is positive when measured from F 1 towards the left. This relation enables us to find F, for although it is not easy to locate P and P 1 , which are required if the first formula is used, it is easy to locate F and F 1 , and hence to measure the distance of object and image from them. To locate F, set up a plane mirror on the left of the lens and put up a pin in such a position that its image coincides with it. The rays emerging from the lens must in this case be parallel and strike the mirror normally so that object and image lie at F. Repeat this for the point, F 1 , leaving the former pin in position. Leave pins at F and F 1 and do not move the lens. Place a third pin at a point such as S, farther from the lens than F, and locate its image, I. 302 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The distances, x and y, are now determined and we can calcu- late F. Hence since P and P 1 lie at distance, F, from F and F 1 respec- tively, the principal points are located. Repeat this for several values of x and y. "Piano-Convex Lens In this case any ray, R, striking the lens at P in the figure, where P is the pohit of intersection of the axis and curved surface, is not deviated but gives rise to the parallel emergent ray, R 1 , for it is just as if refraction took place in a slab of glass bounded by the face, A, and the tangent at P. "R 1 appears to Come from P l , so that P and P 1 are the principal points. These may be located as above. Fig. 179 We may, also, calculate the distance AP 1 . For AP 1 is the apparent thickness of the widest portion of the lens, AP _ t •'• '""AP 1 d This gives d if \l and t are known. It forms a useful exercise to determine P and P 1 by the method described above, and then to deduce y. from the observations. If PP 1 = 5, t * = *— *• Determination of the Principal Points by Rotation of the Lens If the lens is mounted on a stand which can be rotated about a vertical axis we may use the property of the nodal points to locate P and P'. It is convenient to place the lens in a holder which can slide along a scale fixed to the rotating stand. The arrangement of the apparatus is illustrated in fig. 180. A mirror is fixed on one side of the lens normally to the axis and does not rotate with the lens. "t REFRACTION 303 On the other side a pin is set up and moved until its image, by two refractions through the lens and a normal reflection at the mirror, coincides with the object. The pin then lies at a principal focus. The lens is moved along the scale and the pin adjusted so that image and object coincide until a position is found, when slight rotations of the stand fail to cause displacement between the pin and its image. When this is the case the rotation takes place about a vertical axis through the nodal point nearer the pin. The figure explains this, for if P' denote the other nodal point in the symmetrical position and P" the position of this nodal point in a slightly displaced position a ray from F falling on P, first passes along PP'M and is reflected back along its path after incidence on the mirror at M. Fig. 180 When the lens is rotated into the position indicated by the dotted lines the ray, FP, is refracted and emerges from the lens parallel to its original direction, and directed from P', i.e. it takes the course, P'M', and is reflected back to P', emerging once more along PF. The image of the point therefore keeps the same position. The position of the axis thus fixes the nodal point, and the axis is usually clearly indicated on the stand. This finds the point, P, and, by turning the lens, P' may be found in the same way. The Measurement of the Focal Length of an Optical System by means of a Goniometer (Searle's Method). (Proc. Optical " Con- vention," 1912, p. 165.) A simple form of goniometer devised by Dr. G. F. C. Searle, in conjunction with Messrs. W. G. Pye and Co., provides a*very instructive method of determining the focal length of an optical system by means of the properties of the nodal points. The goniometer is illustrated in fig. 181, and consists of a wooden base provided at one end, with a spherical pivot, con- sisting of a ball of phosphor-bronze, and carrying at the other a scale marked in millimetres. A movable arm rotating about the pivot carries an achro- matic lens of focal length about 35 cms., a vertical adjustable frame, across which a vertical wire is tightly stretched, and a fine wire passing across an opening which serves as a scale index. 304 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The ball is adjusted at a distance of 40 cms. from the scale, which can be read to an accuracy of one-tenth of a millimetre, so that a rotation of the arm through one-seventieth of a degree is able to be measured. This is about —5- radian. 4000 Fig. 181 In fig. 182, let HiB and H^B 1 denote the principal planes of an optical system, the points, H^ and H 2 , denoting the first and second principal points. In practice it usually happens that the medium on each side of the system is the same, viz. air. As the general case presents no additional difficulties and can be examined also by this method, we shall write jx x for the refractive index of the medium on the right of the system and i* 3 that of the medium on the left. Then, if U is measured from H t and V from H 2 we have the relation, V U F between the object distance, U, and the image distance, V. F is a constant for the system, and if the value of U is F x for emergent parallel light, i.e. Fj denotes the first focal distance, Fi = - i*iF, and similarly the second focal distance, F 2 , is given by : F 2 = [x 3 F. The usual sign convention is adopted in applying these formulae, directions from the lens towards the source are positive and those in the opposite direction are negative. In fig. 182, F x and F 2 denote the first and second focal points respectively. Nj and N 2 are the nodal points, and with the same medium on either side of the system they coincide respectively with H^ and H 2 . In the general case the first nodal point lies at the distance (F x + F 2 ) from K v and N 2 is at the same distance from H 2 . REFRACTION 305 Thus and similarly : NJF, = H.Fj - H 1 N 1 = F 1 -(F 1 + F 2 ) = -F 2 , N 2 F 2 = H 2 F 2 + H 2 N 2 = F v As drawn in the diagram the value, F t = F^, is positive, since it is measured from H lf while F 2 is negative. Any ray incident on the system in a direction passing through N x gives rise to an emergent ray in a parallel direction passing through N 2 . Fig. 1 8a Thus the ray, ANj, gives rise to CD, and CD if produced would pass through N 2 . If the point, A, lies in the focal plane a divergent pencil from it gives rise to a parallel beam. Two rays, AB and ANj, are drawn and the corresponding ravs are BT. and CI). Since CD is parallel to AN X the angle between the axis and the emergent beam is equal to angle, ANjF v Thus if we measure the distance, AF t , denoting it by d, we can find the length, N^, provided that we measure the wangle between the axis and the parallel beam. If this is denoted by a we have : NjFi = dcot a; or when a is small, N^ = - = F 2 (numerically). The goniometer is a very convenient apparatus for carrying out this measurement. It is first adjusted so that the vertical wire lies in the focal plane of the lens. This may be done by adjusting the wire so that the image of a distant object seen through the lens falls on the wire; or a plane mirror may be placed near the lens on the side opposite to that on which the wire lies and the latter adjusted 3<>6 ADVANCED PRACTICAL PHYSICS FOR STUDENTS until there is no parallax between the wire itself and its iihage. The frame is then clamped. The optical system is now brought up so that its axis lies collinear with that of the goniometer lens when this lens lies so that the horizontal wire index is central. A glass scale is placed in the focal plane of the optical system and may be represented by AF r A denotes one of the marks on the scale. The scale may be adjusted in the focal plane by looking through the goniometer lens and the system and by placing it so that its image is seen coincident with the vertical goniometer wire. The method of parallax is used to obtain an accurate adjustment. When this is made as closely as possible by means of the unaided eye a lens should be used as a magnifying glass to make it still more exact. Note the reading on the glass scale when the apparatus is in the symmetrical position, so that the point corresponding to Fj is observed. Note also the indication of the goniometer scale. Turn the goniometer through some convenient small angle and note the mark of the scale seen coincident with the vertical wire. Again note the goniometer reading. We thus have the values of AF X and of the angle, a, and can consequently deduce F 2 . By turning the system end for end, so that H 2 lies to the right of Hj, we find F x . We can thus locate the positions of N x and N 2 with respect to the outer surfaces of the system. These should be recorded, and since HjNj = F! + F 2 = H 2 N 2 we may also locate the principal points. Record these also. In order to carry out this experiment it is convenient to employ two thin lenses situated at a known distance apart. These form an optical system of the type described, and having the same medium on each side. Hence N x and N 2 are both nodal and principal points, and F! and F 2 are numerically equal but of opposite sign. Find the focal length and the position of the principal points for this case. It is shown in treatises on Optics (see, for example, Houstoun's "Treatise on Light," p. 45), that for two lenses separated by a distance, d, the position of the first nodal point is at a distance from the lens on which light is incident and whose focal length is/i, given by -fid h+h + d> while the second nodal point is at a distance : __M_ fl+ft+d REFRACTION 307 from the second lens, f a , denoting its focal length, while the focal length of the system is : F = A/e . fx+f % + * Measure f 1 ,f 2 , and d, and verify these results, /i aim / 2 may be readily measured by the goniometer by re- placing the optical system by the lenses in turn. Another property of this system, which is of importance in the theory of Optics, concerns the lens equivalent to this system. This lens is such that it produces an image of the same size, but not in the same position, as that produced by the system, when situated at a point at a distance from the first lens equal to — -?-, /a i.e. at the first nodal point. Fig. 182, shows the relative position of the nodal points in the case of two convex lenses. TJie Focal Length of a Microscope Objective The goniometer provides a useful means of measuring a short focal length, such as that of a microscope objective*. The screen, AF 1} is replaced by a micrometer slide provided with a magnifying reading glass. The slide is provided with a small engraved scale with divisions at each tenth of a millimetre. The objective is placed to receive parallel rays emerging from the goniometer lens and focusses them on the scale. Thus an image of the goniometer vertical wire is received and is viewed by the magnifying glass. The position is read for the symmetrical position and again after a slight displacement of the goniometer arm. The image is now focussed at a new point corresponding to A (fig. 182). The distance between the two images is read off, and from a know- ledge of the angle of rotation of the arm, the focal length is calculated as before. The Determination of the Foeal Length and Principal Points of an Optical System by means of the Revolving Table The principle of the method is the same as that described on page 302, for it depends on the same property of the nodal points. The revolving table affords, however, a much more convenient and accurate means of carrying out the experiment. (See figs. 152 and 183.) Each nodal point is determined by the principle that when a small rotation is made about a vertical axis through the second nodal point, the image produced of a distant object is not dis- placed. Thus the position of the axis of rotation of the table locates the second nodal point. 308 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The lens system is mounted on a slider with straight edges, and is placed on the table so that the edge slides along the scale, An index mark on the slider serves to locate positions with respect to the scale. The design secures that there is no side shaking. It is essential in this experiment as in the former (p. 268), to make the axis of the system intersect the axis of revolution. Fig. 183 In order to arrange this the image of a distant object is made to coincide with a pin held in a clamp and the system is moved on the table until, with this focussing exactly made, a slight rotation of the table causes no displacement between the image and pin. Suppose that the axis of the system is represented by N X N 2 (fig. 184), and that S denotes the distant source, while I is the image, S is so far distant that it may be supposed to lie on either of the lines, IN lf N^L, I 1 N V I Nt_N, s I ! L ZZT. * A n! l l' n, n z Fig. 184 Let A denote the point of intersection of the axis of revolution and the horizontal plane. If AN 2 is normal to the direction in which the light is travelling a slight rotation does not displace N a at right angles to the axis. Thus the light incident at the first nodal point still emerges along N a I, and I is not displaced. REFRACTION 3<>9 Nj 1 denotes the displaced position of N x and the dotted line LNj 1 denotes an incident ray from S. Thus with this setting the nodal point, N a , lies in a plane through the axis of revolution and normal to the direction of the rays. The table top is now turned through two right angles, and the lens system moved along the scale until the distant object is focussed on a second pin as before. In general this pin will not coincide with the first, but will be displaced in a direction normal to the direction of the light. This is illustrated in the figure. Nj in the second case become the second nodal point. If I 1 denotes the second pin it is clear that the axis lies on a line drawn parallel to the light and passing midway between the pins. The scale on the table top is undamped and moved through half the distance between I and I*, so as to carry the axis of the system into the correct position. The apparatus is now once more carefully adjusted so that the image does not move for slight rotations. The axis then passes through the nodal point. The system is then turned end for end and the second nodal point found. In order to find the focal length readily, a clamp is fixed to the table which carries a scale (fig. 183). The pin is left in position and the table turned until the scale just touches it and the scale reading is noted. The table is turned so that the other end of the scale just touches the pin and the scale reading is noted for this case also. The distance between the two marks on the scale is determined by subtraction ; one- half of this gives the focal length. A diagram is then made showing to a convenient scale the positions of the cardinal points with respect to the first and final surfaces of the optical system. The apparatus may be employed to examine such a system as that described in the previous experiment with the goniometer. In this case the apparatus is not very convenient for the measure- ment of short focal lengths. It is better to use the goniometer for this purpose. When a distant object is not available in the above experiment it is only necessary to place a plane mirror on the side of the lens system away from the pin and adjust the pin until image and object coincide. The focal length of a thick lens, the radii of the surfaces being r and r 1 , and the thickness t, is given by : F== ¥* >-!).{ nfr 1 - r) - (n - 1) t\ ' Here r and r 1 must be given their proper signs. 310 ADVANCED PRACTICAL PHYSICS FOR STUDENTS It is instructive to determine the value of the refractive index of the glass forming a thick convex lens. Determine r and r 1 by the revolving table method (p. 269), and F by the method just described, while t may be measured by means of callipers. Solve the resulting equation for i*. The experiment is intended to provide practice in working with a thick lens and becoming familiar with the formula for F. The distance between the principal points is given by t Qi - 1) (r 1 - r - *) , (x (r 1 - r) - ((i - 1) t This should be verified by the results of the experiment which locates the position of the nodal points, which are the same as the principal points in this case. The student should verify the two expressions given above. Spherical Aberration with a Thick Lens In the simple theory of lenses it is assumed that all the rays starting out from a point on the axis are brought to a focus after refraction through the lens to another point on the axis. A, Fig. 185 This is only approximately true for rays lying very close to the axis. A ray, PA X (fig. 185), will be made to cross the axis at F, and if PA X makes a small angle with the axis, F will be the focus for all such rays. But a ray, PA 2 , will cut the axis at C 2 , and PA 3 at C 3 , after refraction, these points lying closer to the lens as the incident ray is more inclined to the axis. The distances, FC 2 and FC 3 , are called the longitudinal aberra- tions of the corresponding rays. The distance, FC 2 , depends on the inclination of the emergent rays to the axis. If this distance is x and the tangent of the acute angle at C 2 is m, we may say that x is some function of m. We do not know the way in which x depends on m, but whatever be the form of this dependence we may write x in powers of m, or x = a + bm + cm 2 + dm 9 + . . . , where a, b, c, d, etc., are constants. REFRACTION 311 Since the lens is symmetrical about the axis, FC 2 is the same for positive and negative values of m. Thus if we write — m instead of m in the above equation we have the same value of x, so that x depends only on even powers of m. .'. x = a + cm 2 + em 4 -f . . . When the emergent ray has a very small value of m, the' point, C 2 , is at F, or x vanishes. Thus a vanishes and we have : x = cm 2 -f em 4 -f . . . If m is not very big, m 4 and higher powers of m, will be very small, so that we may write : x = cm 2 . Let F be taken as origin and FP as axis of x. Let v denote distances measured at right angles to FP. The line, B 4 C a , has an inclination to the x axis whose tangent is m, and it passes through the point, C 2 , with co-ordinates (cm 2 , o). So that its equation is : y = m (x — cm 2 ). The emergent rays cross one another at points B 2 , B 3 , B 4 , etc., and the points at which consecutive rays cross lie on a curve known as the caustic of the lens. Suppose that to a ray consecutive to that through C 2 the corresponding tangent is m 1 . Its equation will be : y = m 1 (x — cm 12 )- If we solve these two equations we can find the co-ordinates (x, y) of the point common to both lines. We do not actually require these but only the relation between these two co-ordinates, which will give a locus of the intersections of consecutive rays or the equation to the caustic. Subtracting one equation from the other : x (m 1 —tn)=c (m l3 — m 3 ), x = c (m 12 + mm 1 -f m 2 ) = 3cm 2 , since m and m^are so nearly equal ; m '3c)» C ' \3C J ~ 3 ' (3cY (3c)* \3cJ 3 ' (3c)* * or 2j,cy 2 = 4# 3 . 312 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Thus for values of m, such that m* is negligible the form of 1 the caustic is a semi-cubical parabola. In the figure, DCE 1 and D*CE are the extreme rays from the lens and DEF, D X E X F the two branches of the caustic curve. Fig. i 88 All rays inside the extreme rays touch this curve at points which are nearer to F, the smaller the inclinations of the rays to the axis. The rays are consequently crowded together just inside the caustic curve, and a screen placed between C and DD 1 will show a circular patch of light with a brighter illumination round the circumference. At points between C and EE 1 , a brighter patch appears in the centre which increases to EE 1 where the illuminated area is a minimum. EE 1 is called the circle of least aberration. Beyond EE 1 the first circle diminishes until at F it becomes a point. Set up a large lens on the optical bench with a small hole illuminated by a Bunsen flame containing common salt as a source of light. The light may be concentrated on the hole by means of a small condensing lens. It is best to have the distance from the hole to the large lens about twice the focal length of the lens. Examine the image by means of a micrometer eyepiece, when the appearances described will be seen if the source lies on the axis of the lens. Find the position F, and read off the position of the eyepiece on the bench scale. Move in towards the lens and measure the diameters of the circular rings at a series of distances from F. Plot the results on squared paper and verify the equation of the caustic curve. The radius of the circle of least aberration is r = J cm 3 , where c is constant in the equation and m Q is the value of m for the extreme rays. REFRACTION 313 tn is nearly proportional to the aperture of the lens. Vary the aperture by putting diaphragms over the lens and measure the corresponding diameters of the least circles. Verify that the diameter is proportional to the cube of the aperture. The above value of r is readily calculated by remembering that it is the value of the ordinate of the caustic at the point where it is cut by the extreme ray. It is therefore necessary to solve the equations : v = m (x — cm 2 ) {4) 2ycy 2 = 4a; 3 . The solutions are x = ^cm^, y = ± I cm *. In order to find the value of FL, we note that by equation (4), which is the equation of the line, CL, the value of v corres- ponding to x = o is — cm 3 . This negative sign occurs because we have measured the angle of which m is the tangent from the x axis in a clockwise direction. In the case of the line, CL, the value of m is negative since the angle is obtuse. The lateral aberrations of the extreme ray, i.e. the length of FL, is thus — cm *. But the radius of the circle of least aberration is £cw 3 (nu- merically). FL Thus the ratio 7^=; = 4. VjrJi Verify this result. CHAPTER XII INTERFERENCE, DIFFRACTION AND POLARIZATION Introduction The theory that light consists of waves in the ether leads us to expect, by the principles of superposition, that class of phenomena described by the term 'interference.' Particles of a medium, when simultaneously displaced by the arrival of several disturb- ances, have a resultant displacement obtained by adding together the vectors representing individual displacements. In particular, if a particle is subject to displacements in directly opposite directions, it will be displaced a smaller amount than if it were subject to either separately, and if both the displacements are equal, but oppositely directed, there will be no displacement of the particle. On the other hand, displacements arriving at a particle in the same direction will cause it to be moved a distance equal to the sum of the separate displacements. The arrival of oscillatory disturbances at any point may, therefore, cause larger or smaller displacements of an ether particle than would occur as a result of each separately. The intensity of illumination at any point of a medium is proportional to the square of the amplitude of the vibrations executed by the ether particle at that point. It can thus be seen how it is possible to produce in the ether places of large or small intensity as a result of the arrival of two trains of waves. It may even be possible that there will Fig. 187 be darkness at certain points since it is possible that as a conse- quence of adding vectorially the displacements we get no net effect. This point will be considered in detail. Let Sj and S 2 denote two sources of monochromatic light. From each is thus emitted a train of waves of a particular wave 314 INTERFERENCE, DIFFRACTION, POLARIZATION 315 length, the same for each. In addition, suppose that the distur- bance starting from S x is in the same phase as that starting from S 2 . By this we mean that the ether particles at S x and S a are in exactly the same state of vibration, they are moving in the same direction, and are displaced the same amount from their central positions. These disturbances are propagated with a particular velocity in the ether, and at points equidistant from S t and S 2 the displacements will be in the same phase when they arrive there. In other cases where points are at different dis- tances from S x and S 2 , it may happen that the displacements are in different directions on reaching the point. Draw AB at right angles to the middle point of S^g and let it cut a screen, MjMa, normally at B. We shall inquire as to the illumination at different points on the screen. The disturbances will reach B in the same phase, the displacements will be in the same direction, and will thus unite in increasing the illumination at B, which will always be bright. Of course, we think of the waves as transverse, i.e. the motion of the particles is normal to the direction of propagation. Take any other point, P, distant x n from B. The light from S 2 travels over the path, S 2 P, and that from S x over SjP. Thus those disturbances, which reach P simul- taneously, started at different times from their sources. The vibrations will thus be in different phases on reaching P. If, however, the difference between S 2 P and S X P is a whole number of wave lengths, the disturbance from S a left that point a complete number of periods before the disturbance from S x set out towards P. The displacements are thus in the same direction, and will unite at P to give brightness, just as they do at B. If the difference of path is an odd number of half wave-lengths, the displacements on reaching P will be directed oppositely, and will tend to destroy the disturbance at P. If the amplitudes are equal there will be no displacement, and consequently no illumination at P. In any case there will be a marked falling off in brightness. We ought, therefore, to be able to distinguish on either side of B as we proceed outwards alternate bright and dark places. Suppose the distance AS! = AS 2 = d, while AB = D. Then PMj = PB -MjB = x n — d, and similarly PM 2 = x n -f- d. (Mi and M 2 are the feet of the perpendiculars from S x and S 2 on to the screen.) 316 ADVANCED PRACTICAL PHYSICS FOR STUDENTS .♦.S 2 P={D* + (*„+<*)*}* =D|i+^^j (i) x -\-d in which we neglect higher powers of -—■ — than the square. This means that we suppose that x n and d are of small magni- tude relatively to D. This will evidently be the case if we confine our attention to a few only of the alternations in intensity about B, for the distances apart of adjacent bright points on the screen are of a magnitude comparable with the wave lengths of the light, and this is very small. In the same way S t P = D{l+i2iL^)!j; ( 2 ) Thus for a bright point : 2d and for darkness D x n = «x, (3) 2d 2W + I ^ , , D*« = ^-- X; * (4) or writing 8 = 2d, i.e. SjSg = <5, we have x n • yz= «xor(w + |)x, (5) according as P is bright or dark. In the experiments to be described (5) is of fundamental importance. The first three experiments are three examples of obtaining sources S 2 and S 2 of the kind described. It would be useless to set up two slits and illuminate them by a sodium flame, for we have in such a flame a multitude of sources in different phases. The method adopted is to cause light from a single source to travel to the points of which P is typical by two different routes, and to unite on arrival. We thus have the equivalent of two sources emitting vibrations of the same phase. In making experiments on interference and diffraction accurate measurements have to be made ; for this reason the optical bench is used. It consists of a strong, rigid metal frame, provided with levelling screws on which it stands. The frame consists of two metal rails, one of which is graduated accurately in millimetres. Along the rails slide metal uprights, each of which is attached to a vernier at its lower end so that its position on the bench may be accurately read off. -INTERFERENCE/DIFFRACTION, POLARIZATION 317 The uprights serve to carry a slit, micrometer, microscope, lens, or whatever piece of apparatus is necessary. If it is necessary to move any piece of apparatus transversely across the bench, it is placed in an upright fitted in a support provided with a transverse micrometer screw. Fig. 188 Although it is possible to read off positions of the uprights on the bed of the frame, since the slit or cross-wire may not be exactly above the indicator mark we cannot read off directly the distance from slit to cross-wire. A correction must always be applied. In order to find this correction a stand, carrying a carefully measured rod is placed on the rails, one end of the rod is placed in contact with the slit while the other end is viewed in the micrometer. Let the length of the rod be I, and suppose the distance between the slit and micrometer uprights is I 1 , as observed on the scale. Then to convert the readings as obtained from the upright to the distances required we must add to the observed readings the quantity (I — I 1 ). Determination of the Wave Length of Sodium Light by means of Lloyd's Single Mirror The simplest way of obtaining interference bands is by means of a mirror silvered on the front surface or blackened at the back in order to avoid multiple reflections. The diagram illustrates how the interference is brought about. Light from a slit, S lt travels directly to a screen, PB, and also by the alternative path after reflection at the mirror, MM. For example, a ray may reach P by the direct path, SiP, or by the path, SjCP. The latter ray produces the same effect at P 318 ADVANCED PRACTICAL PHYSICS FOR STUDENTS as would arise from a ray, S 2 P, S 2 denoting the position of the image of S x in the mirror. We thus have the equivalent of two sources, S x and S a , emitting vibrations in the same phase, except for any change of phase that may occur on reflection at MM. MM is mounted vertically on an upright of an optical bench, and the slit, S r , is carried vertically on another upright. 1 P - B S» Fig. 189 MM is mounted as accurately parallel to S x as possible, and the fringes are looked for by the micrometer microscope. They will usually come into view, if not already there, after slightly rotating the mirror, and are rendered distinct by adjusting the width of the slit. The mirror should be parallel to the length of the bench, and we shall see how to make this adjustment accurately shortly. When the fringes appear as definite and bright as possible, it is necessary before proceeding further to make sure that the distance, SjSg, can be found. The method adopted is to use a lens and obtain the magnified and diminished images of S^Sg, but we require a lens whose focal length is less than a quarter of the distance between S X S 2 and PB. In Chapter II, p. 293, we described a method of finding the focal length of a lens in which the object and screen are kept fixed while the lens is moved. In the diagram in connexion with this experiment, suppose that Pj denotes the position of the object, S X S 2 , and lt one of the positions of the lens, while P 2 denotes the position of the image of S a S 2 . By moving the lens to 2 , the image is formed at the same place, but it now differs from the former in magnitude. Let d lt d 2 denote the respective distances between the lines S x and S 2 in the image, while 8 is the actual distance, S X S 2 , between the slits. Then li_L2l ^! _££_*. lhen 8 ~ P^' 8 - OtPj But P 1 1 = P 2 2 and P^ = PaO^ .-. 8* = d x d z (6) Thus, by measuring d x and d z we may deduce <5. INTERFERENCE, DIFFRACTION, POLARIZATION 319 If MM is too close to PB it may be impossible to obtain both images with the lens between the mirror and screen. The mirror must be placed so that this is possible. This is the reason for performing this part of the experiment first since it is useless to measure the distance between consecutive fringes unless we can find the distance, S^. The distance apart of the fringes does not depend on the position of the mirror, but if the mirror has to be replaced in another position it is easy to throw out its adjustment. Both images of Sj and S 2 will be in focus in the plane of the cross-wires only provided that MM is normal to this plane, i.e. to PB. It is important that MM should be normal to PB. We have, therefore, to rotate MM slightly until the two are accurately in focus, the lens being adjusted so that its centre lies on a level with the centre of Sj and opposite the edge of the mirror. On removing the lens, MM should be rotated slightly about a horizontal axis to get the position where the fringes are brightest, when S x and MM are parallel. Set the cross-wire accurately down the Centre of the first bright fringe, and move it always m one direction by means of the screws giving the transverse motion, stopping at every three or four fringes to note the position. From the observations deduce the distance between consecutive fringes. Finally measure the distance from S^ to PB. From (5) we have : N x n = n\ -, and proceeding to the next bright band : D *«+i = (« + 1) x- ; .*. % n +l x n = ~7~ (o) Let s denote the distance between consecutive fringes, and we then have : x = p y/d x d z . As a rule it will be best to make use of the full length of the bench, so that D is large, also <5 should be as small as convenient. At the same time d x or d 2 must not be too small or it will be difficult to make the determination of 6 accurately. In recording the results of this and the two following experi- ments it is a good plan to begin, as mentioned, at one edge of the field, and make out a table as shown below. 320 ADVANCED PRACTICAL PHYSICS FOR STUDENTS We can then readily obtain a series of independent readings for the determination of the separation of the bands. In this experiment the first fringe does not lie at the point corresponding to B, fig. 187. This is because the reflected ray undergoes an abrupt change of phase on reflection at the mirror. This has the effect of displacing all the fringes towards B by a certain distance leaving the separation between successive fringes unaltered. NO. OF 1 BAND MICROMETER READING (a) NO. OF BAND MICROMETER READING (&) SEPARATION FOR 15 BANDS (b) - (a) I 4 7 10 13 16 19 22 25 28 lyiean Separation for 15 Bands Interference by means of FresnePs Double Mirrors Interference fringes may also be formed by the use of two mirrors silvered on the front surface or blackened at the back, very slightly inclined to one another. The diagram illustrates the arrangement. 0M X and 0M t denote the two mirrors which may be mounted on the optical bench and placed accurately vertical. S denotes the source of illumination and may most conveniently be a Bunsen flame coloured by a sodium salt. The fringes are again observed by means of a micrometer, and AB denotes the position of its cross-wires. The screen, CD, protects AB from direct light from S. The source is placed on one side of the bench as the diagram indicates, and may be the slit of the optical bench supported in a clamp. Sj and S 2 are the two virtual images in the mirrors. SxSg and S lie on a circle with centre at O. INTERFERENCE, DIFFRACTION, POLARIZATION 321 The calculation, mode of measurement, and adjustment, are identical with those of the previous experiment. The shaded area of fig. 190 shows where rays from both mirrors interfere, and the section of this region with the plane of the cross- wires of the microscope is the position within which the fringes lie. M; n/ ^ ^~rr rTT7Trr7r777ti B Fig. 190 Fresnel's Biprism The biprism is a prism with one of its angles only a little less than two right angles, and with two equal small base angles. The figure illustrates its action. The biprism is represented by CDEF, and it acts like two prisms placed base to base. Rays from a slit, S, are deviated in each part of the prism and unite on the screen, as in the case of the rays, SGP and SHP. We have virtually two sources, S x and S 2 . Fig. 191 The details of the experiment are very similar to those described in the experiment with Lloyd's mirror. The biprism is held in a stand which can rotate about a hori- zontal axis parallel to the central line of the bench. By this means the edge, D, of the prism can be rotated to bring it parallel with the slit. The adjustments are first made roughly by the eye, and usually the fringes will be observed even with the rough adjustment. Slight rotation of the biprism will, as a rule, improve the appear- 322 ADVANCED PRACTICAL PHYSICS FOR STUDENTS ance of the bands, and still better definition will be obtained by narrowing the slit. The distance, S^, is measured with the aid of a lens as before. In adjusting D parallel to S the following device is of great assistance. If the eye is placed on the side of the prism away from the slit, and moved across the bench, the slit will appear to cross from one side of the prism to the other. As it crosses D, unless the slit is parallel to the edge of the prism, the top or bottom will cross the edge first, while if parallel it will appear to make the transition suddenly. The prism must be rotated until this sudden jump occurs. Be careful after arranging the apparatus to give good fringes, to make the determination of the length, S l S 2 > before measuring the separation of the fringes. The Determination of the Radius of Curvature of the Face of a Convex Lens by means of Newton's Rings Newton's Rings are formed as a result of interference between the incident and reflected rays from a source of monochromatic light on the air film between a plane glass plate and a convex lens in contact with it. G A 1 A The diagram shows the lens, L, and the plate, P, in contact at the point, Q. Two incident rays are drawn, AB and A 1 B 1 . The first is partially reflected and refracted at the points, B, C, D, E, and F. INTERFERENCE, DIFFRACTION, POLARIZATION 323 A ray is drawn which is represented as suffering refraction, except at D, where it is reflected. The neighbouring ray, A 1 B 1 , which follows the course, A 1 B 1 EFG is shown, and these two rays being brought together along EFG will interfere and may produce greater or less illumination than each separately, according to their phase difference. In calculating this it is to be remembered that on account of the reflection at D, at a medium optically denser than that in which the ray travels before reflection, a phase change of half a wave length is imparted to it. The total phase difference between the two united rays is thus : (MD + DE + Jx) as reckoned in path length ; for had the rays both left the lens, the wave front would have been EM, so that the phases are the same at E and M. Each ray, such as AB, has a corresponding ray, such as AjB^ arising from the same point in the source of light with which it can interfere, so that an extended source may be used and a Bunsen flame containing sodium acts very well ; in fact, an extended source is necessary in order to obtain a large area containing rings. The rays entering the eye or an optical instrument will be contained in a small cone, and for the rays in this pencil the change in path difference is very small, so that all the rays from the small region of the lens will be naturally reinforced or caused to interfere. There will be brightness or darkness according as : MD -f DE + |x = »x or (n + £)x, where n is a whole number, i.e. according as MD -f- DE = an odd or even number of half -waves. Let CD fall on the plate at incidence 6. Let t be the thickness of the film at this point. The diagram is drawn with t large, or D a long way from Q, for convenience, but in the formation of the rings the film is very thin, and the part of it with which we are concerned is very close to Q. Just near D we may regard the film as an element with parallel faces separated a distance, t, as in the enlarged element. MD + DE = 2CD - CM 2 CN ~, T . . ~ XT cos 2 = — — - — 2 CN sin = 2 CN . ■ = it cos 0. sin sin Thus the condition for brightness is : 2t cos 6 = (2» + 1) -, 324 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The angle, 0, may be measured by the angle between the normal to the plate and the incident rays before striking the lens, since near Q the lens acts as a parallel plate of glass very approximately. It is usual to cause the rays to fall normally on the lens, when we have : 2t = (n + £)x. If we consider a plane normal to that of the paper through Q and D, we can see that the circumference of a circle with centre, Q, and radius, QD, will pass through points at which the air film has a uniform thickness, and consequently all round this circle rays incident vertically will undergo the same phase change, and thus alternate bright and dark rings are formed about Q. Let the radius of the ring be r n = QD, and let the ring be the (» + i)th from the centre. K The centre itself will be black, for the air film is infinitely thin at this point. If this is not the case at first it is because some dust particles lie between the surfaces, and these should be removed. DQ 2 = DN • DK = t(2R - t), (fig. 193) R == radius of lower surface of the lens, /. DQ 2 == y„ 2 = 2M, approximately, since t 2 may be neglected. Thus for brightness : r w 2 = R(n + £)x. The first ring corresponds to n = ; the second for n = 1, and so on. Thus the radii are proportional to Vi, V3, V57 etc. For the purpose of the experiment a convex spectacle lens of about 100 cms. radius of curvature is suitable, and the light INTERFERENCE, DIFFRACTION, POLARIZATION 325 from a sodium flame is reflected down on to it by means of a sheet of plane glass held at 45 ° to the vertical (see fig. 194). The rings are viewed by means of a travelling microscope. Fig. 194 In order to focus quickly on the rings, remove the lens and focus on the top of the glass plate. On replacing the lens, and adjusting the microscope over the point of contact the rings which lie in the air film should be distinct. A good bright sodium flame is necessary, and often the difficulties disappear if this point is attended to. RING MICROMETER READING (l) MICROMETER READING (R) DIAM. (diam.)* 20 19 18 5 4 3 2 1 ■ 1 (l) denotes the reading on the left of the centre, (r), that on the right. / 326 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Move out the micrometer to about the twentieth ring from the centre, and then, moving back again, turning the screw always one way to avoid any errors due to backlash, set the cross-wire carefully down the centre of each bright ring and observe the micrometer reading. In this way, by observing the rings on both sides of the central dark patch, the various diameters are determined. Make a table as shown above. Draw a curve with the square of the diameter as ordinate*, and the number of the ring as abscissa. The graph should be a straight line. P 2 N 2 = (D„ 2 )*, P X N X = (D n] )*, D tt 2 - 4R*(«+i), P 2 N 2 - P t N x _ (Pg* - (D Wl )« Then if Since N X N S n, 4RX. If sodium light is used, x may be taken as 5890 x io~ 8 cms. In this description we have not taken account of the fact that the rings are seen, not directly, but after refraction through tlae lens. They are formed in the air film in the space between the lens and plate of glass. This difficulty is avoided by placing the plate above the lens and in contact with it, for in that case we view the rings through a plane sheet of glass. Fig. 195 It is, of course, more convenient to place the glass below the lens, for reasons of ease in keeping them steady. The error in this case is not great if a thin lens be used, for then the object — the rings — is at the surface of the lens and consequently at its principal plane. The image is in the second principal plane, and of the same size as the object. For a thin lens these planes and lens surface are nearly coincident. In practice we have an INTERFERENCE, DIFFRACTION, POLARIZATION 327 image of magnification slightly differing from unity, or the diameters in the formula above are to be multiplied by such a fraction. For the purpose of the experiment this factor is omitted. Jamin's Interferometer The apparatus consists essentially of two glass plates, AB and A 1 !? 1 , of the same dimensions and optical character. The plates are very carefully worked and are of the best optical glass. They are mounted parallel to one another, standing on tables on an optical bench at a distance apart of about one metre. The first, AB, is set at 45 to the bench with its surfaces vertical, and it is illuminated by rays from a sodium flame. B Crv ft E flD S l « V Fig. 196 In the diagram a ray, RS, is shown. It is partly reflected and refracted at S, and the refracted beam again partly reflected at T. The divided ray takes the paths, RSS^HPR 1 and RSTUU 1 R 1 , so that it is reunited by the second glass plate. This plate is mounted parallel to the first, and set vertically. It rests on a table and may be slightly rotated about a vertical axis. If both plates are exactly similar and are parallel, the lengths of the two paths will be the same for all rays, but by slightly rotating one of the plates a difference in path may be introduced, differing for different directions so that alternate bright and dark bands will be obtained. The tube, CD, is then placed in the path of one of the divided rays between the plates, the other being allowed to pass clear of CD. CD is a hollow glass tube with ends of plane, optical glass. It is fitted with a tap, E, by means of which it may be exhausted or filled with gas. 328 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The tube is filled with air at the atmospheric pressure, and the fringes found by eye. A telescope provided with cross-wires is then focussed on the fringes. In order to ensure that the fringes arise from interference between rays that have passed, one through CD and the other outside it, we may cover up the end of CD, and note if they disappear as they should if due to this cause. In order to be sure that the rays producing interference do not both pass down CD, it is necessary to intercept the light at the sides of the tube. If they are still present they will arise from rays passing down CD. When the correct fringes are obtained the cross-wire of the telescope is focussed as accurately as possible down the centre of one bright band, after exhausting the tube as much as possible with an air-pump. The pressure should be brought down to i cm. of mercury at least, and a manometer connected up to read the pressures. Air is now allowed to pass very slowly into the tube by the tap, and the fringes watched. They will appear to move across the field of view, and the number passing the cross- wire must be counted. Allow about 5 or 6 to pass, and then stop the inflow and read the manometer. Repeat this, step by step, until the tube is filled with air at atmospheric pressure. Draw a graph showing the relation between the pressure within CD, and the number of fringes that have passed from the initial stage. By an exterpolation deduce the number that pass between the limits of complete exhaustion and the attaining of atmospheric pressure. Let this number be «. This means that the difference of optical path in the tube, when completely exhausted and when filled with air at atmospheric pressure, is wx. If the tube is of length, /, and the refractive indices are y. and Ho, when the pressure is atmospheric and when the tube is exhausted, respectively : l(v- — y-o) = ^x. But Ho = *» wx so that i* = 1 -\--t-* A convenient length for the tube is 30 to 40 cms., when for the exhaustion produced by a good air-pump the value of n is of the order 200. The compensator described in connexion with the experiment with Rayleigh's refractometer may also be used with Jamin's interferometer, and the calculation of the refractive index at normal temperature and pressure may be made with the help of Jamin's interferometer also. INTERFERENCE, DIFFRACTION, POLARIZATION 329 A cylindrical lens with the cylindrical axis vertical is often placed just in front of the point S. This has the effect of widen- ing the source light in a horizontal direction, leaving it unchanged vertically. The source in this case is a slit. The Refractive Index of Air by means of the Rayleigh Refraetometer A diagrammatic representation of the apparatus is given in fig. 197. * 0*1 ■ F JEE 3 B aTc~» SB p CG Fig. 1^7 ABCD is an airtight metal box, divided into two separate chambers, each of which may be connected to a manometer. The pressures in the chambers are varied and measured by means of the manometers. The chambers are closed at the ends by means of parallel plates of good glass. The collimator is provided with a slit at K, and provides a parallel beam of light which falls on a screen, carrying two fine slits, LL, placed in front of the air chambers, so that one slit lies adjacent to the end of one, and the other slit adjacent to the end of the second chamber. These slits are prolonged so as to extend higher than the top of the box, ABCD. Thus, light from the slits passes over the box as well as through it, and finally enters the telescope, T. These two fine slits produce interference fringes in the focal plane of the telescope, and if the pressures in the chambers are equal, the set of fringes in the lower half of the field appear to be continuations of the fringes in the upper half which arise from rays that have passed over the top of the box. In order to deviate the upper set of fringes down, to make comparison with the lower set easy, a prism, P, is provided which intercepts the upper set of rays, and deviates them downwards. When white light is used, two sets of coloured fringes are obtained with a white central fringe. If there is a difference of pressure between the two chambers, there is a displacement of the lower set of fringes owing to the resulting difference of optical path. G consists of two plates of glass, inclined at a small angle, placed to intercept the lower set of rays. When it lies symmetri- cally with respect to the rays striking it, it introduces no additional path difference, but on rotating it the rays through one plate traverse a longer path within the glass than do those in the other. 330 ADVANCED PRACTICAL PHYSICS FOR STUDENTS In this way the lower central band can be moved about, and we may, by a rotation of G, bring back the fringe to its central position after it has been displaced on account of the difference in air density in the two chambers. G carries a pointer moving over a scale, and we may calibrate the scale so that the difference of path introduced by setting G is known. Thus, by altering the pressure within the chamber and then moving G to counteract the displacement of the central fringe, we can read directly from the calibrated scale to how many wave lengths the path difference in the two chambers amounts, and a graph is plotted with the pressure differences as ordinates and the number of wave lengths as abscissae. The calibration is performed by making the pressure in both parts of the chamber equal and illuminating K with mono- chromatic light, e.g. by a sodium flame. The scale is set so that the bright bands of the upper set of fringes lie over those of the lower. The pointer is then moved over the scale until a certain number of bands pass a fixed point in the upper series. The following table will illustrate how this result should be recorded : CALIBRATION OF SCALE IN WAVE LENGTHS WAVE SCALE WAVE SCALE DIFFERENCE FOR 40 LENGTHS READING LENGTHS READING WAVE LENGTHS 1-48 40 4*50 3-02 5 1-85 45 4'8 9 3-04 10 2-24 50 5*24 3*oo 15 2-62 55 5'6i 2-99 20 3-oi 60 5*99 2-98 25 3-38 65 6'37 2-99 30 374 70 6-75 3-oi 35 4-12 75 7-12 Mean for 40 3-00 24-03 wave lengths Mean value of 3-004 1 wave length in scale divi- sions •0752 After the calibration of the scale the collimator slit is again illuminated by white light, and the central fringes arranged one above the other. 334 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The apparatus is shorter since the length of liquid traversed need not be so great as that required for a gas. If a liquid is placed in one of the vessels, and the same liquid containing a solvent in the other, we may determine the effect of the solution on y., or inversely, we may estimate the amount of solvent contained from its refractive index. Fig. 202 In this instrument the movable plate is in the upper beam •instead of the lower, but the method of use is otherwise similar to that of the previous experiment. Two thermometers project into the liquid for recording their temperatures. Examine the changes produced by adding small quantities of a salt to pure water, and placing in one cell the solution so obtained and water in the other. We have, if y. and tx 1 denote the refractive indices of water and of the solution respectively, d the thickness traversed by the light, and x the wave length : where n is the number of displacements of the central fringe. The experiment is performed with white light so as to have a definite central fringe, but x is the wave length of the light used to calibrate the scale. By moving the pointer over the scale until the two central fringes are coincident, the scale reading INTERFERENCE, DIFFRACTION, POLARIZATION 333 the tubes containing the gas, which would normally separate the lower interference bands from the comparison upper bands, produce no image in the field of view on account of the refraction in H. The two sets of fringes stand one immediately above the other. They are focussed by the achromatic lens, Q, and examined by the cylindrical lens, R. This provides a large horizontal magnification of the fringes, which lie close together because Fig. 201 the slits are of necessity rather wide apart (cf. formula 6 of this chapter, d is large in the present case). The lens, however, does not give magnification in a vertical direction, so that the shadow cast by the obstacle is not broadened vertically, and there appears a sharp dividing line between the two sets of bands. The Rayleigh Refractometer for Liquids Fig. 202 shows a Hilger apparatus based on the foregoing principles for comparing the refractive indices of liquids. 332 ADVANCED PRACTICAL PHYSICS FOR STUDENTS If T is the absolute temperature, we have also : —, = constant. pT Thus — - — . T is constant. P If jx is the refractive index of air at N.T.P. {* — i . T _ t*o_— i x 273. P 76 We shall see that y. can be determined by observations in this experiment. Suppose the length of the tubes containing the air to be I, and the refractive indices, jx x and [t it corresponding to pressures. p x and p 2 . Let the wave lengths of monochromatic light be X, and X, respectively. The excess of waves in one tube over those in the other is : ' (t t) ". G£ ~ ■£) = i ( "' - " i) This number is deduced from the scale over which the pointer moves. Suppose it is m. Then : T ■ 76Tx 273l(p2 - Pi) The pressures are measured in centimetres of mercury, and the m ratio, -r — , is deduced from the graph as described, while the length of the tubes, which is usually about 25 cms., may be measured by means of a metre rule B C A D * E Fig. 199 In figs. 199 and 200 a modern form of apparatus is shown diagrammatically. The plates, L,K, correspond to G, and the prism, P, is here denoted by H, and its action shown in the fig. 200. It can be seen how such obstacles as the upper edge of INTERFERENCE, DIFFRACTION, POLARIZATION 331 The pressures in the chambers are varied and measured by the manometers. The central fringes are kept one above the other by means of G, and the pressure differences recorded along with the position of the pointer on the scale. Draw a graph with scale readings as ordinates and pressure differences as abscissae (fig. 198). O 5 lO t5 ZO Pressure Difference. Fig. 198 White light is used in order to provide a definite central fringe. The position of this fringe is independent of colour while all other fringes have positions which depend on the wave length. If monochromatic light is used, all the fringes are alike, and the central fringe is indistinguishable from the others. Since the screw attached to the compensating glasses has been calibrated in wave lengths of sodium light, the observations give the path differences in terms of so many wave lengths of yellow light, and the refractive index deduced will be that for this particular wave length. Fig. 201 gives a general view of the apparatus. From the slope of this graph may be deduced the difference in path in wave-lengths for a difference of pressure of 1 cm. of mercury. Observe the temperature of the air in the tubes by placing a thermometer close to them and noting its indication throughout the course of the experiment. When the pressure is varied by means of the manometers, the changes should take place slowly, and time should be allowed for the air to take up atmospheric temperature. Theory of the Experiment For a gas the relation, u — 1 = a constant, is very approximately true. We assume it in this experiment, (i is the refractive index, and p its density. INTERFERENCE, DIFFRACTION, POLARIZATION 335 will give the number of wave lengths which one ray has fallen behind the other in traversing a path of different nature from that of the other. The calibration is carried out in a preliminary experiment with both liquid cells containing water and with a sodium flame illuminations, as in the last experiment. Michelson's Interferometer The apparatus is illustrated diagrammaticaUy in fig. 203. It consists of two plane mirrors, M x and M 2 , silvered on their front surfaces and mounted vertically on a heavy, firm, rigid stand. The stand consists of a metal bed provided with a large micrometer screw of very fine pitch. Rotation of the screws causes M t to slide along the bed and its position may be read off at the screw-head. A general view of the apparatus is given in fig. 204. M, W The second mirror, M 2 , is fixed at the end of a metal arm mounted at the end of the bed, and at right angles to it. This arm also carries the two sheets of plane optical glass, P a and P 2 , which are equally thick, and are mounted at an angle of 45 to the arm. 336 ADVANCED PRACTICAL PHYSICS FOR STUDENTS In order to produce interference fringes, a source of light, e.g. a sodium flame, is placed at the focus of a lens, L, and a parallel beam of light thrown on to the glass, P x , the direction of the beam being along the arm carrying M 2 . One of the rays of such a beam is shown in the figure as SN. On striking P x it undergoes partial reflection and refraction at N, and the refracted part is again divided at K, aDd later further division takes place at P 2 . Fig. 204 We consider one of the ways in which division can take place, the recombination of two parts, which have started from one ray and have different optical paths, producing interference. A marked difference in this case from that in some of the foregoing is the production of interference between rays differing in phase by very many wave lengths. In the case of Newton's rings or Fresnel's biprism, the path difference amounts to a few waves only. The ray, SN, is refracted in P x and reaches K, where it is partly reflected so that it gives rise to KQT and partly refracted to give KM 2 . These rays are both returned along their paths and reunite to produce KW. In the course of their journey each passes through the glass a distance equal to 3NK, and change in phase is brought about by the difference in the path in air. This may be varied by altering the position of M x . The ray, KW, may be observed by eye, and usually curved INTERFERENCE, DIFFRACTION, POLARIZATION 337 > interference bands similar to those observed in the experiment on Newton's rings will appear. The curves will not be seen closed, only parts of the closed curves can be seen. We use here, as in the experiment on Newton's rings, an extended source of light, for the rays from each point of the source are divided and re-combination takes place between rays which originally belonged to the same ray. Rays falling on P x in the same direction all suffer the same phase change and emerge parallel to KW, so that an optical instrument will focus them together in its focal plane. Other rays in a direction slightly inclined to SN, emerge as a set of parallel rays, slightly inclined to KW. These undergo a different phase change, and also lie in the focal plane displaced from the image due to rays parallel to KW. It is necessary that the mirrors, M x and M 2 , should both be vertical, and in order to allow this adjustment to be made, at the back of M 2 are three screws pressing against springs that cause it to rotate. If a sheet of tin carrying a fine hole is placed in the path of the incident light, four images are seen as a rule, when the eye looks in the direction, WK. The reason for this is best seen by reference to a diagram (fig. 205). The emergent rays are marked p lt p it p s and p lt and it will be noted that p 2 passes through the glass plates three times; so also does p x . Thus if the rays are made to coincide they will be in a condition to annul or reinforce each other according as the path difference is an even or odd number of half wave-lengths. The rays, p a and p x , traverse the plates five times and once respectively. 338 ADVANCED PRACTICAL PHYSICS FOR STUDENTS If a card is placed in front of M x , the rays p x and p s still appear ; and p x will be the brighter since it traverses the plates only three times, so that it can be distinguished. By placing a card in front of M 2 , the rays are cut out except p a and pi, so that by this means the images can be distinguished, and those due to p x and p 2 caused to overlap by adjusting the screws behind M. x and M 2 . When both mirrors are vertical the images coincide in pairs. One constituent of each double image lies behind the other. Those which produce interference lie along the direction, WK. -P Fig. 206 Fig. 206 illustrates the mode of production of the fringes from another point of view. The interferometer acts as if we had a source, S, from which rays could be reflected by two parallel mirrors, M x and M 2 , in which they would produce images, I x and I 2 . The distance, MjM 2 , is equal to the difference of the distances, NMj and NM 2 , of fig. 203. At any point, P, rays, SAP and SBP, would unite, and if their paths differed in length by wx, a bright point would arise. This is equivalent to stating that in this case : PI 2 - PIj = »x. Thus we obtain the particular interference band at all points, P, for which this relation holds, i.e. P lies on a hyperboloid of revolution with SM X M 2 or with WK, of fig. 203 as axis. In a plane perpendicular to that of the figure we have a circular section of the hyperboloid which explains why the fringes are circular as seen on looking along WK. The Determination of the Frequency of Light from a Sodium Flame or any Monochromatic Source A striking feature pi the Michelson Interferometer is the screw which displaces M v In the apparatus illustrated in fig. 204 the screw has a length of 200 mm. and a pitch of 1 mm. The head of the screw is furniihed with a scale divided into one hundred parts, each thus corresponding to one hundredth of a millimetre. The screw may be rotated by the handle seen in front of the apparatus. The sm^ll lever on the right of the front of the apparatus puts into action the slow motion screw, one turn of INTERFERENCE, DIFFRACTION, POLARIZATION 339 which corresponds to one division on the head of the main screw. As the head of the slow motion screw is also divided into one hundred parts, it is possible to record a motion of the mirror of one ten-thousandth of a millimetre. In order to obtain the fringes, set up the sodium flame at S, and place immediately in front of it a sheet of tin with a small hole in it just opposite the bright part of the flame, and adjust the flame and hole to lie on the level of the centre of the mirrors and plates. Mount the lens, L, also so that its centre is at the same height as the hole, and place a piece of plane mirror between L and P a and move L until an image of the hole is thrown back on to the tin close to the hole. The light is then parallel as it leaves L. Remove the plane mirror and look in the direction, KW, when usually four images of the hole appear. Adjust the mirrors so that these images coincide, two by two, as indicated above. Then remove the tin sheet ; allow the light from the flame to fall through L on to the apparatus. As a rule very slight movements of the mirrors will bring the fringes into view if they are not already to be seen. Sometimes a slight motion of M x along the bed of the apparatus helps to discover the fringes. Set up in front of P 2 on the side towards W a sharp pin point to mark the position of the centre of one fringe. Rotate the screw slowly by the slow motion screw, and watch the movement of the fringes across the field of view, counting the number which seem to pass the point, and observing from the scales how far M x has moved. When Mj moves back a distance £x the path difference between the two rays which unite to interfere has been increased by x, so that where a particular fringe originally appeared the neigh- bouring fringe now apparently lies. Thus if Mj moves a distance, I, the number of fringes which 2I appear to move past the point is correspondingly — . If these are couhted, since / can be measured, we can find x. Do this for such monochromatic extended sources as are available. Note also that the intensity of the fringes appears to alternate as the distance of M x varies. We shall make use of this fact in the next experiment. The Determination of the Difference of Wave Length for the Sodium D Lines. Adjust the mirrors Mj and M 2 so that their distances from N are equal, as nearly as can be judged by eye. 340 ADVANCED PRACTICAL PHYSICS FOR STUDENTS This adjustment may be brought about accurately by observing the images of the holes formed as above in the two mirrors, and adjusting M x until there is no parallax between them. In this case they are equally distant from the observer, and the two mirrors are consequently equidistant from N. The light from a sodium flame, though for many purposes considered monochromatic, contains two fairly intense waves whose frequencies are close together. In the present case both sets of fringes which arise from the two waves overlap, but if M x is slowly moved away there is a gradual separation of the two sets, and finally the bright band of the one lies over the dark band of the other. This happens when the distance that M x has moved from the first position contains one more quarter of a wave of the one than of the other, for then a difference of phase, corresponding to one-half wave length has been added to one more than to the other. Or, to put it otherwise, let I denote the distance moved by M lf the additional air path added to each wave incident on M x is thus 2.1. Suppose 2.1 contains n x waves of length x x , and n 2 of lengths x a . Then the difference between n x and n t is one-half. Suppose for the sake of definiteness that n x > n t and conse- quently x x < x t . Then, since n x = —and », = — , Xj x 8 we have : 2l( J = -, \X X Xj/ 2' i i_ i . Xj x a 4' If the two waves had equal intensity the field would become uniformly illuminated, and the fringes would disappear. In this case, since one of the lines is more intense than the other, we get an alternation in distinctness, the brighter fringes still stand out in contrast with the adjacent less bright ones. Note the positions of Mj at the beginning and successively at positions where the fringes become least distinct and again distinct as M x goes further away and the path difference contains one complete wave more of one colour than of the other. Do this for as many cases as possible, and if d denote the distance between the positions of M x in which two successive distinct sets of fringes occur, we have j_ i_ __ i Xj X a 2d' Assume the shorter light in the sodium light to have a length INTERFERENCE, DIFFRACTION, POLARIZATION 341 5890 x 10- 8 cm., and deduce the difference between x x and x 8 in this case. The Production of Coloured Fringes These can be obtained when the mirrors are set for equal paths. Do this as in the last experiment with sodium light and then replace it by a white source. The fringes due to the different colours overlap in this position. With sodium light, as M x is moved, it will be noticed that the circles change the direction of their curvature ; the position required is just when the transition takes place. It is difficult to decide just when this occurs, and it is a good plan to note the two positions of M t in which the fringes are definitely curved in one way and then in the other. Put in white light when M x occupies the position corresponding to one of these directions of curvature, and then slowly move it back to the other position. The coloured fringes will appear during this movement. M, must be moved slowly or the fringes change their places too rapidly to be noted. Fig. 207 The Hilger Wave Length Spectrometer This apparatus, illustrated in fig. 207, consists of a cast-iron stand with two arms at right angles on which are held rigidly the telescope and collimator. 342 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The telescope may be fitted with a high-power eyepiece with adjustable cross-webs or with a shutter eyepiece which can be adjusted to cut out any part of the field except that under particular examination with an adjustable metal pointer. The pointer has a brightly polished fine point and is illuminated by reflecting light from outside by means of the mirror shown in the figure in position above the eyepiece. Thus any point in the field may be taken as a reference point by setting the pointer to it (fig. 208). Fig. 208 In addition the shutter eyepiece may be employed with light filters which impart any desired colour to the bright point. This adds to comfort in reading and consequently to accuracy. A Fig. 209 The vertical collimator slit may also be reduced in length by means of a cross horizontal slit, so that a small rectangular source is obtained. INTERFERENCE, DIFFRACTION, POLARIZATION 343 The principle of the apparatus is based on the constant devia- tion prism which is illustrated in fig. 209. The faces particularly concerned in the deviation of the ray are inclined at the angles marked in the figure, and total reflection occurs at the face AC. The prism is mounted on a turntable in a position marked for it. The mean deviation of the rays is a right angle, and in order to pass through the spectrum the table is turned by means of a screw to which a drum is attached provided with a milled head (fig. 210). On the drum is a scale so that the wave length of any line under observation and appearing in the field of view of the telescope may be read off directly. Fig. 210 Before taking any observations of wave length it is necessary to adjust the prism accurately so that the correct wavelength is indicated when the corresponding line appears at the eyepiece indicator. To make this adjustment, illuminate the slit with light of a standard wave length, set the drum so that the appropriate wave length is indicated at the index of the drum and adjust the prism so that the line appears under the eyepiece index. Clamp the prism in position with the screw provided. Other wave lengths of light illuminating the slit may then be determined by rotating the prism by means of the drum until the line appears at the eyepiece index and reading off the number against the drum index. Fig. 209 indicates the course of a ray in the prism. In the apparatus designed for use with certain accessories — the Lummer-Gehrcke Parallel Plate, the Fabry-Perot Etalon or the Michelson Echelon Grating, the collimator arm is of greater length than that illustrated in the figure to permit of interposing the accessory between the prism and collimator. 344 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The method of fixing the collimator and telescope and of obtaining different parts of the spectrum by rotating the prism is very convenient and accurate ; moreover, the drum can be rotated while looking through the eyepiece, and one is saved the inconvenience of moving round with a rotating telescope. A suitable standardizing wave length is the red line of the helium spectrum, which has a wavelength of frequency 6678-1 Angstrom units (1 Ang. unit = io -10 metre). It is the shorter of the two red helium waves. Or the sodium lines may be used. These are separated by the prism and have wave lengths 5890*2 and 5896*2 Angstrom units respectively. It is a good plan to set the instrument on one of these lines and check the setting by turning to the other and noting if the reading gives the correct wave length. The slit may be illuminated by throwing an image of the source on it. In the case of the sodium lines the source is obtained in the us,ual way, and for a helium line throw the image of the bright part of a helium discharge tube on the slit. A protective metal cover for the prism table is provided. When a photograph of the spectra is required the eyepiece is removed and replaced by a camera with a suitable focus lens. This is shown in fig. 211. It is capable of adjustment by tilting so that the whole spectrum can be photographed and a vertical Fig. 211 displacement enables the same plate to be used giving photo- graphs one above the other. There is, of course, also a shutter for exposure. In addition to the usual vertical slit there is a horizontal adjustable slit attached to a hinge so that it can be swung out of the way or in a position covering the vertical one. INTERFERENCE, DIFFRACTION, POLARIZATION 345 Thus we have crossed slits, and a small rectangular source can be obtained by closing down both the slits as much as is required. The Lummer-Gehrcke Parallel Plate This piece of apparatus 1 s described in the " Annalen der Physik," vol. 10, 1903, p. 457, and a very complete account of its use and theory given. The reader is recommended to refer to the original paper, but we shall give below as much as is necessary for our purpose. The plate can be used in connexion with the Hilger Constant Deviation spectrometer, and is of quartz of refractive index 1*544. It has the following dimensions : Length 130 mm. , Width 15 mm., Thickness \\ mm. (approximately). Like the echelon grating it produces spectra of a high order, and has consequently a high resolving power. The approximate resolving power in the present case is about 200000. It is therefore well adapted for the study of complex lines in the spectrum which in lower orders appear as single lines. It may be used also to measure such small displacements as occur under the influence of magnetic fields, and in the section immediately following the present description we describe how it may be used to determine the value, — , from observations on J m the Zeeman effect. N F3 Q O Fig. 212 The plate itself is shown in fig. 212 and mounted on its stand in position on the spectrometer in fig. 213. In this figure the screws permitting adjustment in the various directions are shown. The action of the plate is illustrated in fig. 214. Here LM X denotes an incident ray of monochromatic light making an angle, i, with the normal to the plate. After refraction the angle is r. The figure shows the production of two beams emerging from the plate on opposite sides. The rays, M x i, M 2 2, M33, etc., and Nil 1 , N 2 2 x , Njtf 1 , etc., on account of their different courses due to successive reflections and refractions, are in different phases when they reach the position denoted by 123 and zH 1 ^ 1 respec- tively. These two traces mark out wave fronts and the emergent beams produce interference bands. The upper beam consists of a system of bands with alternating intensities, the maxima 346 ADVANCED PRACTICAL PHYSICS FOR STUDENTS having an intensity which may be measured by 2 J, while the minima have intensity, J. Thus the effect is the same as is obtained by imparting to the whole field an intensity, J, and drawing bands across it of double the intensity. The transmitted system, however, consists of a set of maxima of intensity, J, with alternating minima of nearly zero intensity. For con- venience in observation the second system is to be preferred, and, by using light at grazing incidence so that i is a right angle and r the critical angle for quartz the sharpness of the lines is increased. Fig. 214 It is not possible to give here the reasoning which leads to these statements ; the discussion is given very clearly in the original paper. These points find application in the Hilger pattern. A slot will be noticed on the right of the carrier of the plate, fig. 213. Just opposite this is a prism of such dimensions that the light along the directions MiNj, M a N 2 , etc., is in the critical direction. The prism lies on the under surface of the plate, so that the beam emerging from above is the transmitted beam. The other beam is absorbed by the black lining of the stand on which the lower surface of the plate rests. In fig. 212 the prism is represented at OQ. The Theory of the Plate Corresponding to every direction, *", of incidence there will be in the wave front, i^^ 1 . . . , a particular variation of phase from point to point on account of the different courses I'UV :,.:i. INTERFERENCE, DIFFRACTION, POLARIZATION 347 taken by the rays. These rays are received by some optical instrument, for example, the telescope of the spectrometer, and focussed in the focal plane. Corresponding to the waves drawn, we shall have a point image in the focal plane. These rays, however, are those lying in the plane represented by the paper. Above and below them lie rays coming from rays above and below LMp and parallel to LM X , whose courses are exactly similar, so that in the focal plane above and below this point image lies a series of points forming a line. We shall work out the intensity corresponding to this line. If we consider a slightly different direction of incidence, i + 8i, the emergent beam is also slightly different in direction, and in its course through the plate. The phase differences in the corresponding wave front will thus be different, and the corresponding line in the focal plane will have a different intensity. A wave travelling along a direction denoted by r will at a time t, be represented by : a sm 2tc (t -{) Here T is the period of vibration in the wave, x the wave-length, and a the amplitude, t is the time measured from some con- venient instant, and r the distance from a convenient origin. In our case the origin will be conveniently the point, N x , and we shall consider the wave motion corresponding to the position, i^^ 1 , at which the time is measured by t. On refraction into the plate there is a diminution of intensity, and since the intensity is proportional to the square of the amplitude of the wave motion we can express this by regarding the refraction as causing a diminution of amplitude, so that amplitude a in the air becomes sa in the quartz, where s is a fraction. On passing out at Nj, the amplitude is again reduced by a fraction, s 1 . The amplitude for the ray Nji 1 is thus We may not suppose that s and s 1 are the same, since in one case the refraction is from air to quartz, and in the other from quartz to air. There is also a change of intensity on reflection at the points M 2 , M 3 , M 4 , etc., and N x , N 2 , N 3 , etc. We shall suppose this to diminish the amplitude each time by a fraction, a. Thus the ray, N^ 1 , has undergone refraction at M x and N 2 , and reflection at Nj and M 2 . The incident amplitude, a, is thus reduced to oHsfy ; and similarly the amplitude of N33 1 is The disturbance which gives rise to the ray, N 1 i 1 , starts out 348 ADVANCED PRACTICAL PHYSICS FOR STUDENTS from N x with an amplitude, ss^, and at i 1 , since it has travelled a distance, r lt we may represent the displacement by : ss^ sin 2tz «-?> Denote the foot of the perpendicular from N 2 on N^i 1 by A, and from N 3 on N 2 2 x by B, and denote the equal distances, N 2 A, N 2 B, by e. Let the distance, MjNj be denoted by <5, and the refractive index of the quartz by [*. Then the distance traversed in quartz is equivalent to the distance, [l6 in air. Thus the point 2 1 is at an equivalent distance [r 2 + 2jx<5) from N lf and since r t = r 1 — e we may denote the displacement at 2 1 by : „ , . ft r x — c + 2fi<5\ aHsH sm 27tf ^ * — J ; and similarly the displacement at 3 1 is : . . ft Y x — 2e + 4tx<5\ o^s 1 ^ sin 2rc( — 5 x ) There is, of course, an indefinitely large number of such terms as the three given above, and since all these rays are focussed at a point in the focal plane of the telescope, the total effect at this point is obtained by adding together all the terms. In practice we shall have a small parallel bundle of rays falling at M lf giving rise to small parallel bundles at N x , N 2 , N 8 , etc. Throughout the bundles there is, however, the same phase, as may be seen by considering a ray parallel to LM X , falling at any angle, i, to the normal. Thus the total effect is merely multiplied by some constant factor on account of the incidence of more light than that we have supposed is represented by the ray LM X . Thus we have a total displacement, Y\ where : Y = SS %sin 2,(| -£)■+ «Wsin a, (I- r - ~ ;+ 2 »> ) , . . ft r x — 2e + 4tid\ -|- ss x a<s* sin 2rc(=r - 1 + ... A typical term may be written : »>..» sin a, j (i - &) + p ■ i^| , where p has the values, o to infinity. * If we write a = 2ir (f ~ ^) ' P = T ^ ~~ ^' we have : Y = Xss l a{o& sin (a - p$)} (7) * In practice the upper limit to the value of P is about 15, the dimensions of the plate permitting about 15 reflections. The terms beyond the fifteenth are email and contribute but little to the value of Y. INTERFERENCE, DIFFRACTION, POLARIZATION 349 The sum of this series is finite since a is a fraction, and has the value sina — o 2 sin (a + ft) 1 — 2a 2 cos ft + o 4 For the proof of the summation the reader may be referred to Hobson's " Plane Trigonometry," §76, p. 91. Thus the resultant displacement is : _ r , sin a — a 2 sin (a + ft) 1 — 2a 2 cos ft + o 4 It will be noted that for any particular direction through the plate the quantities, e and 6, are constant, so that ft is a constant. s, s 1 and a are also constant terms so that it is only a that contains the variable time, t. sin a — a* sin (a + ft) = sin a (1 — o 2 cos ft) — cos a • a 2 sin ft. We may write this = A sin (a. — <f>) ■u a. c 2 sin ft where tan <t> = _— , 1 — o 2 cos ft^ and A — Vi — 2a 2 cos ft 4- 0*/ ===== ' sin (a - 2a 2 COS ft + o 4 Thus Y = . • sin (a — tf>) Vi —2a 2 COS ft + o 4 sra lT-* — r) Vi — 2a 2 COS ft + o 4 This denotes a simple harmonic vibration of amplitude, 55 x a Vi — 2<J 2 cos ft + a 4 ' and it measures the amplitude on the line in the field of view of the telescope due to the reception of the emergent beam along the direction which makes an angle, i, with the normal to the plate. The intensity is proportional to the square of this amplitude, and is therefore a maximum when (1 — 20 2 cos ft+o 4 ) is a minimum, and a minimum when this expression is a maximum. But 1 — 2« 2 cos ft + c 4 = (1 — a 2 ) 2 4- 4a 2 sin 2 Jft. This has a minimum value when sin \ ft has the value zero, and a maximum when sin \ ft has the value unity. Thus lines of maximum intensity correspond to the value : ft = o, 2k, 47c, etc., and lines of minimum intensity correspond to : ft = *> 3*> 5"» etc. But ft = (2(i<J — e). 350 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Thus for a maximum, 2[x<5 — e = n\, and for a minimum, 2y.8 — s = (n + £)x, and n may have any integral value. If d denote the thickness of the plate : 8 — d sec r, and c = MiMa sin * = 2d tan r sin i = 2^ tan r • {/. sin r. Thus 2{A<5 — e = 2yd cos r, and the bright bands lie in directions given by : 2yd cos r = n\. By proceeding further with this discussion, Lummer and Gehrcke have drawn the conclusions, to which we have referred above, concerning the intensity of the bands. Students who prefer it may substitute the following proof which enables the same formula to be derived shortly, but in a way that does not lead to any expression for the intensity of the lines. The path difference between any two of the rays, say Nil 1 and N 8 2 x is equal to : 2(X(5 — e = 2yd cos r. We may take all the rays emerging from the plate in pairs which are separated by. the same distance, equal to N X N 2 . These therefore interfere and will produce darkness or bright- ness, according as this phase difference is equal to wx or (w+|)x, where n is a whole number as before. Thus for bright lines : 2yd cos r = n\. If we refer to fig. 213 and note the position of the plate it is clear that the different orders come out from the plate above one another. As the path difference increases the angle of emergence decreases, so that the higher orders will lie higher in the field of view than the lower in the case of a telescope with an erecting eyepiece. In our case the mean angle, r, is the critical angle and there is grazing incidence. The different orders correspond to angles very slightly differing from grazing incidence. By substituting the value, y. = 1*544, and the value for the critical angle, r = 40°22 ', together with the value of d given above, we find that for X = 5890 tenth metres (10- 10 metre), the order is approximately 18000. In order to set the instrument in position, the spectrometer is first set up in the manner previously described, and the slit illuminated by means of some convenient monochromatic light. Paga 351 INTERFERENCE, DIFFRACTION, POLARIZATION 351 The plate is placed in position and adjusted by the screws until the brightest image is obtained in the eyepiece. Since the orders lie one above another, a vertical slit cannot be used, for the different orders will appear overlapping the image of the slit. Thus the crossed slits must be used with a small rectangular source. The length of the slit need not be very small, for on account of the dimensions of NO, fig. 212, only a fraction of the slit is effective in producing bands. The slit should be small enough to avoid overlapping, but wide enough to produce intense bands. The images may then be viewed or photographed as desired. We have throughout disregarded the possibility of a change of phase on reflection at the two surfaces. The Fabry-Perot Etalon. (" Annates de Chimie et de Physique" 1897, Ser. 7, p. 459.) This apparatus is another means of obtaining a high resolving power, and is in many respects similar to the Lummer-Gehrcke Plate. Fig. 215 shows the apparatus in position on the arm of a constant deviation spectrometer. Fig. 216 shows the two plates, ABCD and FGHE, of which it is composed. These are placed accurately parallel, and are separated by a distance piece consisting of a hollow cylinder of fused silica. This substance has an extremely small coefficient of expansion, so that the distance between the plates may be regarded as independent of the temperature. A B E r D C H G Fig. 216 The faces, BC and EH, are silvered by cathodic deposition in order to increase their reflecting power but so as to leave them partly transparent. Light entering the plates therefore undergoes multiple reflec- tions between the silvered faces and produces also partial transmission through the opposite face. The faces, AD and FG, are inclined to CB and EH in order to avoid interference effects that would occur through multiple reflections and refractions if all were parallel, but AD and FG • 352 ADVANCED PRACTICAL PHYSICS FOR STUDENTS are parallel, so that light incident on one side leaves the other undeviated. Fig. 217 illustrates the action of the plate. The air space between the plates is shown, and the course of a ray as it is multiply reflected between the plates. The refracted portion leaving EH is shown, but the refraction at the other face is omitted for convenience. We consider an incident ray, LM X , from the instant of its arrival at a point, M v within the first wedge and immediately before refraction into the air space. L ^ Y Fig. 217 The wave emerging within the second plate is shown, and XY denotes a wave front. The wave is refracted out at the surface, FG, into the air, but as no further change of phase occurs after the light leaves the air film, we do not need to consider the wave beyond XYZ. Let the wave on starting out from the point, M x , have, in the air space, an amplitude, a, and let its period and wave length be T and x, respectively. Whenever the wave passes from the air space into the second plate, let the amplitude be reduced in the ratio, 0, and on reflection at the partially silvered faces let the amplitude be reduced in the ratio, /. Both these quantities are positive proper fractions. Thus the wave proceeding along M X N X may be denoted by : y = a sin 2n ( - — -J, where r measures the distance from the point, M x , at which the displacement, v, is considered. Let the equivalent path difference between M x and the points, X, Y, and Z, differ successively by an amount, 6. INTERFERENCE,, DIFFRACTION, POLARIZATION 353 Then proceeding as in the theory of Lummer-Gehrcke Plate, we have : 8 = 2e cos i, where e is the distance between the plates, and * the angle of incidence at the surface, EH. Thus the displacement at X may be written : da sin 2n \T x> where r x denotes the equivalent air path between M x and X. The displacement at Y, which after two reflections has been reduced by the ratio, / 2 , and for which the equivalent path is ('1 + <5), is given by : and similarly at Z we have : tf% ana. (i - i±»?). Of course there are many more points such as X, Y, Z ; and if we continue the series of terms until the contributions become negligible, and if we view the rays in a telescope, we have, for the total displacement in the field of view of the telescope, a quantity : Y = da |sin27c (y~^) +/ 2 sin27r (i-^ 1 - a) +/ 4 sin 2k (| — £ - 2AJ + . . .J, (8) where A is written for - •■ x Strictly, A ought to include a quantity due to change of phase on reflection at the air-glass surfaces ; if this is w then A — - + w. In order to compare this with equation (7) above, we may note that the quantity, a, is the same for each, that 6a takes the place of ss^, and that / takes the place of o, while 2* A corresponds to p. If, therefore, we write p = 2tcA, equation (8) may be written : Y = S Oaf* sin (a - p$) (9) where p has the values, o to infinity. * Thus as before, the value of Y is : , sin j 2ir ( — — -^ ] — </> I (10) Vi -2/ 2 cos p+/* ( \T x/ / K ' 2 3 *Strictly p has an upper limit between 10 and 20. See p. 348 354 ADVANCED PRACTICAL PHYSICS FOR STUDENTS where : . . /a sin , x tanft = J ~ a .. . (ii) i — / 2 cos p x ' Thus the intensity of the line appearing in the focal plane of the telescope, corresponding to this particular direction, is propor- tional to the square of the amplitude of this expression, or we may measure the energy by the expression : <W (i -2/ 2 cosp+/*)' This may be written : 2 fl a (i -f 2 )'' 1 + ( T^p sin2TCA Thus the intensity fluctuates between a maximum and mini- mum value, the former of magnitude : fl 2 a 2 (i -/ 2 ) 2 " This may be large provided that the value of / is not far from unity. Denote this by I . Then we have for the intensity : I„ I = I + (Wr sin27cA The minimum of I corresponds to the value for A which makes sin 2 reA = i, and the magnitude is consequently : (h^ < a > Now with / not very different from unity, this may be a very small fraction of I , so that there is a great contrast between the maxima and the minima. Another point may most conveniently be brought out by a numerical example. Suppose / = '87 = ( y^ ) — this value has been chosen for convenience. Then 1 = h 1 -f 48 sin 2 7cA INTERFERENCE, DIFFRACTION, POLARIZATION 355 A maximum occurs when A = m, and the next when A = (m + 1), tn denoting an integer. Consider the case when A = m -f ^, i.e. when we have gone over ■- of the interval between the two values of A. In that case it follows that : I = Io 1+48 sin 2 — ^10 = i I , approximately. This means that the intensity falls off quickly as the maximum position is left, so that we have bright lines in the field of view separated by a comparatively long dark interval. Thus, if there is a second ray in the field of slightly ^different wave-length, its lines will not overlap those of the other ray unless there is very little difference indeed between the two wave-lengths. From the above theory we see the influence of the partial silvering in producing sharp bright lines on a background that is almost black. If a layer of air is used between two plates without any silvering, the reflecting power is small, and we have fringes produced and superposed on a field of uniform illumination with no comparative broad spaces between the fringes. We can explain these points quantitatively by the aid of some numerical examples given in the original paper. It is usual to speak of the reflecting power of a surface and not of the quantity we have denoted by/. But since the intensity of a ray is proportional to the square of the amplitude, and since the amplitude on reflection is reduced by /, we have, if the reflecting power is denoted by R, R=/», so that the expression denoted by (A) above gives for the ratio of the minimum to the maximum intensity : p If R = -042 P = -84, so that for a small reflecting power the maxima and minima have nearly equal intensity. If, however, R = -74, P = -02, and the minima are very feeble. 356 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The way the intensity falls off rapidly as the maximum is left has already been shown. The value for / taken above, corresponds to a value of R = 75, approximately. Fig. 218 shows graphically the difference in the two intensity curves corresponding to displacement in the field of view. The continuous line is the curve for the Fabry-Perot silvered plate, while the dotted curve is for plates of low reflecting power. Displacemcnr in Field of VieuJ Fig. 218 Note in the one case the broad intervals of practically no intensity and contrast with the other in which the intensity falls off slowly, leaving comparatively small dark intervals. Lines due to a second ray of slightly different wave length from that for which the second curve is drawn would produce crests over the dark intervals and leave the field nearly uniformly bright, and it would not be possible to distinguish the separate fringes. The determination of the Ratio, — , for an Electron by means of the m Zeeman Effect When a monochromatic source of light is placed in a magnetic field, and rays are received in a direction parallel to the direction of the field, it is found that the normal frequency is changed, and two or more lines appear symmetrically displaced from the usual position in each direction. The light in each case is circularly polarized ; the component of higher frequency is polarized in the opposite direction of rotation to that for the lower. Reference for the theory of this phenomenon should be made to " The Electron Theory of Matter" (O. W. Richardson). If H denotes the intensity of the magnetic field, and its direction is parallel to that of the light, the increase and decrease in 1 eH. frequency of the two components is of magnitude — • — > and H is measured in ' gausses,' the name given to the electro- magnetic unit of magnetic field. In the experiment we measure the difference in wave length INTERFERENCE, DIFFRACTION, POLARIZATION 357 of the two components, i.e. we measure <$x, corresponding to a change of frequency, *» = — --H. 2-nm Now &v = -g <5x, (numerically), since c = v\ 1 X 2 e so that we measure <5x = — — H . — 2.tz c m tt e c 5\ , ' Hence: -.^.g.- (l2) is measured in electromagnetic units. An electromagnet with adjustable pole pieces and taking a current of three amperes gives good results with an ordinary vacuum tube. A hole is drilled in the pole pieces so that the source may be observed along the direction of the magnetic field. The field is measured by means of a fluxmeter, by which the total flux through an exploring coil placed between the pole pieces is measured. This will be read directly on the fluxmeter. Suppose it is B, Maxwell's. If A is the effective area of the coil and H the field, on placing the coil into the position, where •p H is required, the flux is HA, so that H = -^, and H is in gausses (see experiment on Grassot Fluxmeter, p. 482.) We have now to determine <5X by means of the Lummer plate. First align the apparatus and drilled hole as nearly as possible. Place a sodium flame on the far side of the hole, and obtain the shorter sodium lines as sharply as possible with the drum set at 5890. Place a helium tube or other convenient source in position between the pole pieces, and obtain the yellow line. The yellow line .was used in an experiment and satisfactory photographs were obtained, but it would be preferable to use the blue line so that the exposure is not long, and ordinary photographic plates (special rapid) may be used. We shall, in what follows, consider the yellow line to have been employed, and the wave length is then 5875-6 A. Adjustments will be carried out as described above, and the best possible definition and illumination obtained. On applying the magnetic field it will be observed that the lines broaden, and in some cases actual separation will occur, as in fig. 219. 358 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The state of polarization may be tested by intercepting the light just before it enters the object glass of the telescope by a quarter- wave plate. The plate must, of course, be suitable for the particular wave- length concerned — the usual plate met with in laboratories will be suitable to examine the yellow light. This part of the experi- ment is introduced so that the student may take the opportunity of verifying the character of the vibrations in the lines. Fig. 219 The quarter-wave plate reduces the circular vibrations to two linear vibrations at right angles to one another, and on examining these with a Nicol it will be found that when one line is cut out the other is present in the field. After carefully adjusting the apparatus so that the effect is seen by eye, attach the camera and photograph the spectrum without the magnetic field. By means of the rack and pinion raise the plate, put on the field and take a second photograph below the first. The plates when developed will give lines as in fig. 219. Separation of the lines is seen in the higher orders, e.g. at A and B. Let a denote the line when no field is applied to the source, and A the corresponding line with the field applied Measure the mean displacement of the components of A by means of a microscope, and measure the separation between the two successive orders a and b. Denote these distances by I and L respectively. We proceed to work out a formula, showing how <5x may be . / derived from the ratio j- By the equation for the Lummer-Gehrcke plate : 2,\sd cos r = n\, (13) we see that in proceeding to a neighbouring order (n -f- 1), the angles of emergence and refraction are (*' -j- 51) and (r + <5R), INTERFERENCE, DIFFRACTION, POLARIZATION 359 where: 2d\i cos (y -f <5R) = {n + 1) x (14) From (13) and (14) we have by subtraction : — 2d tan r cos * <5I = x •••(15) In this step we use the relation : sin (*' + 61) = y. sin (r + <5R), and consequently cos idl = y. cos y<5R (16) There is no variation in n since the wave length remains constant ; we pass merely to a new angle of emergence. <5I represents the angle between the rays which produce two consecutive lines as a and b on the photographic plate (fig. 219). Now consider the difference in direction, 5i, for two lines in order, n, of wavelengths, x and x + <5X. , It is an angle, 8i, which corresponds to the displacement between the components of A. Since the plate is fixed in the sjfectrometer and the angles are small : 8i I . . sT=L <M Referring once more to equation (13) we have in the order, n, a wave length, X, corresponding to a direction measured by *, and a wave length, x + 8\, corresponding to * + Si. By differentiating (13) we find : — 2d[i sin rdr = w<5x — 2d cos r -=- <5x (18) and from the equation, sin * = [t sin r, . . . du. fji cos rdr = cos 1 St — sm r j~ <5X (19) In this case it is necessary to take account of the variation of (a since the wave length changes. By eliminating 8r from (18) and (19) it is found that : — 2d tan r cos i 8i = (n 4- J <5x ; (20) \ cosy d\J v ' and on substituting for n from (13) : — 2d tan r cos i 8i — 2 ( - cos r — ^ \ 6\ (21) \X cosy d\J v ' Thus from (15), (17), and (21) : I 6i 2d/y. I d\x \ . , . =-= 7r= — ( —cos r • — - ) <5x (22) L 81 x\x cosy d\J K ' The value of y is the critical value for quartz, and it remains to determine j- for the particular wave length used in the 360 ADVANCED PRACTICAL PHYSICS FOR STUDENTS experiment. The table below showing values of n corresponding to different values of x should be used to draw a graph from which ~ may be obtained by measuring the slope at any point where it is required. From the equations (12) and (22) we find : 2.TZC I W *L e m 2,& (> cosy COS Y di) (23) and the units are electromagnetic If possible take the ratio-y- for more than one order — several orders will usually show sufficient separation for this purpose. REFRACTIVE INDEX OF QUARTZ IN THE VISIBLE SPECTRUM AT io° WAVE LENGTHS (IN (Jt[x) REFRACTIVE INDEX. 396 I-558I 410 I-5565 434 1-5540 486 1-5497 508 1-5482 533 1-5468 589 1-5442 643 1-5423 686 1 -5410 760 1-5392 768 1-5390 Measurement of Wave Length by Diffraction at a Straight Edge The theory of this experiment will be found in Schuster's " Theory of Optics " in Chapter V. In the first few sections of the chapter it is shown that on observing a straight edge illuminated by monochromatic light from a narrow slit a series of alternate bright and dark bands will be seen. If S denotes the slit (fig. 220), EF the straight edge fixed parallel to it, and PQ a plane perpendicular to SE, we shall have the series of bands along PQ. Let Q be the nth bright band from P, where P is the point of intersection of SE and the plane. INTERFERENCE, DIFFRACTION, POLARIZATION 361 Denote PQ by x n , then it can be shown that : (24) Xn=s ^PH4n~i)(p-hq) In this formula p is the distance, EP, and q is equal to SE, the distance between source and slit. F c Q Fig. 220 The bands are not regularly spaced as are interference fringes. To carry out the measurement set up in one of the stands of the optical bench a straight edge parallel to the slit which is illuminated by sodium light. First adjust the parallelism by eye, and finally make a slight rotation by the screw of the holder until the bands are most distinct ; then make the slit as narrow as is compatible with sufficient illumination. Observe the fringes with the travelling microscope, and measure the distance between the first or second bright band and one of the most distant that can be seen plainly. If this is the nth and the first observed is the mth we have : x n -x m = ^(£+g) |v^=l - V^t^i] . . . .(25) The student should take the opportunity of observing the diffraction fringes arising from the light passing a narrow wire, needle point and narrow slit In the case of the narrow wire diffraction bands unequally spaced will be seen outside the geometrical shadow. Within it a series of equally spaced bands will be observed. These may be described as interference fringes due to the two parts of the wave, one on* either side of the wire. The effects of these are equal to those of the two half period zones which lie at the edges of the wire so that they act like two sources at a small distance apart. 362 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Determination of the Wave Length of Light by means of a Plane Diffraction Grating A diffraction grating is made by ruling a large number of equidistant parallel straight lines on glass. The lines are ruled by a diamond point moved by an automatic dividing engine containing a very fine micrometer screw which moves sideways between each stroke. A photographic replica of a plate made in this way is often used in its place. The number of gratings to the inch is marked on the glass. In handling the grating do not touch the faces of the glass, hold it between thumb and finger by the edges. Adjust the collimator and telescope for parallel rays in the usual way and observe the direct image of the slit noting how it lies in the field of view. Set up the grating with its face normal to one side, EF, of the triangle formed by the levelling screws (cf. fig. 161). Throw an image of the slit into the telescope by reflection from one face of the grating and adjust the screws to bring it into the same part of the field as that occupied by the direct image. This makes the faces vertical. In order to set the grating at right angles to the rays adjust the collimator and telescope at right angles to each other, and turn the table until the slit is reflected on the cross-wires of the telescope. Then turn the table a further 45 . It now remains to tilt the grating so that the lines are parallel to the slit. View the first diffracted image, making the slit as narrow as is convenient, and adjust the screw D until the best image is obtained. The lines and slit are then parallel. Find the diffracted images on both sides of the line of direct vision. It will be easy to observe two orders, and if a bright light is used and an image of the brightest part of the flame thrown on the slit by a short focus lens the third may be seen also. If (a + b) is the width of the grating element the formula is : (a + b) sin = wX, for normal incidence, being the angle of diffraction, X the wave length of the incident light, and n the order of the spectrum. Obtain by taking half the angular distance between the corresponding images on each side. (a + b) is deduced from the number of rulings on the grating. Use any source of monochromatic light or light giving well- marked lines as that from a discharge tube containing hydrogen which gives three well-marked lines, red, green, and violet, known as C, F and Hy. INTERFERENCE, DIFFRACTION, POLARIZATION 363 The above formula is obtained by considering rays in pairs passing through adjacent clear spaces of the grating. For example, consider the two spaces AB and CD. We may consider the rays passing through them in pairs, taking together those rays which are symmetrically situated, as for example, QLQ 1 and TMT 1 . These reach the grating in the same phase, and the line ABCDE represents a section in the plane of the figure of the wave front incident on the grating. Fig. 221 Consider a wave front, WW, at a later interval, and suppose that it makes an angle, 0, with the plane of the grating. Draw the rays AP 1 , BR 1 , etc., perpendicular to this wave front and making the angle, 0, with the normal, AO, to the grating. These rays are all received by a telescope and united in the focal plane. Their paths from the grating to the instrument differ, so that they reach the focal plane with different phases. This phase difference is due to the difference of path traversed after leaving the grating. Take the case of the two rays, AP 1 and CS 1 . Their path difference is AN, where N is the foot of the perpendicular from C on AP 1 . Any other pair of rays, e.g. LQ 1 and MT 1 , situated symmetrically in AB and CD have the same path difference. This holds for all pairs of rays symmetrically situated. Thus the phase difference on arrival in the apparatus that receives them is the same for each pair. We shall thus have reinforcement if AN is equal to a whole number of wave lengths. Now AN = AC sin 0. 364 ADVANCED PRACTICAL PHYSICS FOR STUDENTS If a is the width of a space and b that of a line, AC = a + b, (a + b) is called a grating element. Thus a bright line occurs in a direction 0, provided that (a + b) sin = n\. The Plane Reflection Grating This reflection grating consists of a plain polished sheet of metal across which parallel lines are drawn very closely together as in the transmission grating. It is necessary to mount the polished face vertically and this is done as in the previous case. The light does not fall from the collimator normally on the grating, but at a measured angle, *. In order to measure i, the grating is first set with the polished face normal to the rays from the collimator which, together with the telescope, has been adjusted for parallel rays. The table carrying the grating is turned through a definite angle, say io°, and is then fixed. Suppose rays from any direction make with the normal to the grating and are received in the telescope. Consider any three rays, MA, NB, and OC (fig. 222), incident on the grating element ABC. AB is polished while BC is the position of the line where the polish is scratched and where the incident rays are absorbed. A B C D Fig. 222 The path difference between the extreme rays MAQ and OCS is AF—CE, where the dotted lines, AE and CF, denote incident and reflected wave fronts respectively. If AB = a, BC = b, this difference is equal to (a + b) (sin 0~sin *). It is possible to divide up the bundle of rays falling on AB and CD into pairs, one from each bundle, having this same INTERFERENCE, DIFFRACTION, POLARIZATION 365 difference of path. In order to do this it is only necessary to choose rays occupying the same relative positions in the two bundles. If this path difference is a whole numbsr of wave lengths, and the emergent parallel rays AQ, BR, CS, etc., are brought to a focus by a telescope, we shall get a bright image due to reinforce- ment of the rays. We thus have : (a + b) (sin 0*~sin i) = n\, where n may have the values 1, 2, 3, etc. The corresponding values of 0, say, B lt 2 , 3 , etc., give the directions in which the different orders of diffracted images are seen. If the observation is made from a direction on the same side of the normal as i, the formula is : (a + b) (sin i -f sin 0) = nx. Obtain first the reading of the scale marking the position of the telescope when it is directed towards the slit with no grating intervening. As above, set the normal to the grating at some definite angle, *, to the incident rays. Move the telescope round to receive light in a direction corresponding to = i on the opposite side of the normal to *. This corresponds to order zero — it is the position where the ordinary reflected beam is received. Now move round to the next position where a distinct image can be found. This will correspond to n = 1. Carry out this process as long as it is possible to observe images at all. The higher orders get fainter but resolve the lines more than the lower. The number of lines to the inch is given for the particular grating, so that it is possible to deduce (a + 6), which must be expressed in centimetres. From the values of *, and n, it is then possible to evaluate X. It should be possible to obtain separation of the sodium lines in the second order, and the wave length for each constituent should be calculated. Use three or four different values of i and compare the value of x obtained, finally take the mean of the series. Resolving Power of a Telescope Theory Let a parallel beam of monochromatic light fall on a slit, AB, and let us examine the intensity of the light along a direction CE, inclined at an angle, 0, to the normal to the slit. Let the light in this direction be brought to a focu$» at E. From A draw ADF perpendicular to the beam so that z. BAF is 0. The rays issuing from the slit on arrival at E will have different phases on account of the varying length of their paths from AB to E. We may measure these differences by such lengths as PM, varying in amount from zero at A to a maximum of BF for the extreme ray from B. 366 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Divide the slit, AB, into a very large number of equal parts, such as PQ. If we make each of these small enough, all the rays in the element of which PQ is a section, may be supposed to be in the same phase on arrival at E. Suppose that the rays from each small element differ in phase by an amount, 3, from the rays of the adjacent element. Each element will be supposed to contribute an amplitude, A, to the whole beam so that if there were n slits, and all had the same phase on reaching E, the amplitude would be«A. Let the disturbance at A be denoted by ' A sin pt. This is an expression which represents the displacement in any simple harmonic vibration, and rays of light afford an example Fig. 223 of this type of vibration. The rays from the next element, differing in phase by 6 must be represented by A sin (pt + <5), from the next by A sin (pt + 28), and so on for all the elements of number n. Thus the total effect at E is obtained by adding up these separate elements. Let T measure this effect so that : T = A {sinpt + sin (pt + d) + ... + sin (pt + n -1 <5)}. BF The maximum phase difference is measured by 2n -— radians. INTERFERENCE, DIFFRACTION, POLARIZATION 367 Let <p denote this phase difference ; it is also measured by n — 1 8. The above series for T can be expressed by the more convenient formula : T = A sin (pt + - — — 3 J sin - <5 . 6 sm- 2 as is shown in textbooks on Trigonometry. When 6 is very small, sin \d = £<5 ; n A sm — = «Asta l£sin( ^ + W) in which |<5 is neglected because of its smallness. When 9 is zero there is no difference of phase, and the direction CE is normal to AB. In that case the amplitude at E is nk, and we denote this by B. Thus the amplitude in any direction, EF, is , *? , the factor sin {pt + \<p) denoting the oscillating character of the disturbance. The intensity is proportional to the square of the amplitude, or the intensity at E is measured by : 4_ — ' sm t L. rrw - • 1- sin 2 £? . This is a maximum when — ^~ is a maximum. 9 2 If this function be examined for its maxima and minima by means of the differential coefficient, it will be found that the maxima occur at points where the values of - are : o, i-437r, 2-4675, 3-4775, etc., and the minima at rc> 27r, 375, 47c, etc. At the latter values the intensity is zero. The graph drawn in fig. 224 shows the relation between intensity and values of \<p. When the light is examined at H the intensity corresponds to the ordinate at I, and on moving round we come to a place, E, say,where there is darkness corresponding to the zero ordinate at 75. In this case : 1 1 BF i.e. BF^= X. 368 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Thus a minimum occurs in the direction in which BF == X. Now the angular distance between CE and CH is the same as angle BAF, and since this is small it is measured by or x (breadth of incident beam). 1 AB ^ M bAstt Fig. 224 If another beam were incident on AB in the direction along CE, there would be a maximum for this beam at E, and it would overlap the minimum of the former. The intensity curves might then be represented on the same diagram as in fig. 225. The curve on the left represents the intensity curve for the second direction while the upper dotted curve shows the two compounded. Fig. 225 The resultant has two pronounced maxima with an appreciable dip between. It will thus be possible by the aid of this falling off in intensity between two bright regions to distinguish the two beams or they will be 'resolved.' Moreover, the angle between the two directions is ECH or X ~ (width of beam). This is taken as a limiting case, and the resolving power is measured by this ratio. If the angle between the incident beams is less than this the two maxima cease to be distinguishable as two and blend into one. In order to verify this theory a telescope is fitted with an adjustable slit which is placed as close as possible to the object INTERFERENCE, DIFFRACTION, POLARIZATION 369 glass. The width of the slit is carefully measured by means of a micrometer microscope. A suitable object for this experiment may be made by coating a sheet of plane glass with tin-foil and cutting two fine parallel lines in the foil with a razor blade at a distance of two or three millimetres apart. When these are illuminated with a sodium flame they provide two bright slit-sources. This object should be placed at different distances from the object glass of the telescope, the aperture of which may be varied by means of an adjustable slit placed immediately in front. A certain minimum width of this slit will be found for which the two lines appear as separate lines. This width varies with the distance from the object glass to the two slits, and for smaller widths the lines appear as one. A table is made of the minimum widths of the slit and the corresponding distances. The angle subtended by the two fine lines at the object glass is measured by== > where d is the width of the slit and D the distance. The theory above described shows that the value of this angle is-> where x is the wave length of the light, and a is the width of the aperture. The object of the experiment is to compare the theoretical and practical resolving powers, the former determined by - and the latter by ~* An examination of the practical resolving powers measured in the Wheatstone Laboratory of King's College, London, during the past session has shown that the values obtained were about 20 per cent, greater than the theoretical value. Repeat the experiment with two point sources and circular apertures of different diameters. In this case the field of view contains a bright small central circle with concentric alternate bright and dark rings. If two sources close together are such that the bright centre due to one falls on the first dark ring of the other, the two sources just cease to be distinguishable as separate. Sir G. Airy has shown that if d is the diameter of the aperture, the angle subtended when separation ceases, e = i-22-j. d The point sources should be two fine holes in a sheet of tin. They should be illuminated by monochromatic light and placed at such a distance from the object glass that they just cease to be seen as two holes. 24 370 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Place the telescope at different distances from the holes and focus it on them. Adjust the aperture diameter until the holes just cease to be distinguishable as two separate sources of light. Measure the distance between the centres of the holes by means of a micrometer microscope, and the distance between the plane of the holes and the aperture by a metre rule. From these measurements deduce 6 and compare it with the theoretical value, x • v i'22-j, in each case. a Polarization by Reflection. Verification of Brewster's Law When light is reflected from surfaces the reflected beam is partially polarized, that is to say, that the transverse vibrations constituting the light have, on the whole, a greater component along a particular direction than in any other. Ordinary light is supposed to consist of a transverse vibration which changes its direction in space, though of course always in the wave front, so rapidly that on the average in any appreci- able interval of time the component in one direction is the same as that in any other. The reflected light has lost this property and is polarized so that it has a greater component in one direction. This direction is normal to the plane of incidence. The transmitted light has a greater component in the plane of incidence. 8 S M Fig. 226 In the diagram (fig. 226), a ray, SA, is represented as being reflected at a glass sheet, MM, so that AB is polarized. In order to test the polarization the ray is received in an analyser, which in some instruments consists of a Nicol prism. This- is a prism of Iceland spar which is cut into two along a diagonal plane and cemented together with Canada balsam. Iceland spar has the property of dividing a ray of light into two rays refracted in different directions and one polarized perpendicularly to the other. The layer of Canada balsam serves to reflect one INTERFERENCE, DIFFRACTION, POLARIZATION 371 of these rays to the side of the prism where it is absorbed by. the blackened walls of a case, and the other ray is transmitted. In this way the emergent beam is made to consist of light vibrations all in one direction. Incident light with its vibrations in this direction passes through the Nicol, while if the vibrations are perpendicular to this direction the light is unable to get through. Light with vibrations in any intermediate direction has only the components parallel to the direction of trans- mission passed on. The vibrations transmitted are parallel to the shorter diagonal of the end of the prism. On examining AB with this prism it will be found that as the prism is rotated there is a change of intensity in the trans- mitted light. This means that AB consists of vibrations with the components in one direction greater than in another or it is partially polarized. On altering the inclination of MM the alternations can be varied in extent, and in one position the change between brightness and darkness is a maximum. The angle of incidence when this occurs should be noted by the help of the scale of angles attached to MM. Theoretically the light should be completely polarized in one position, so that for one particular S Fig. 227 setting of the Nicol darkness should be complete. In practice it will be found that the light is not quite all cut out, though with care a position will be found when, this is very nearly true. The apparatus is represented diagrammatically in fig. 227. Brewster's law is that for complete polarization : tan t = {*, 372 ADVANCED PRACTICAL PHYSICS FOR STUDENTS where * is the angle of incidence and ^ is the refractive index of the reflecting material. This law should be verified. It will be found best to allow light from a window on the opposite side of the laboratory to fall on MM, and to adjust for the maximum effect, and afterwards to set up a sodium flame as at S, and make the exact adjustment. In another form of apparatus the Nicol is replaced by a second glass plate which can be rotated about the direction AB as well as inclined at different angles to the horizontal. When the mirrors are parallel the polarized light in AB is readily reflected by the second mirror, while on rotating it through a right angle from this position about AB this polarized light will not be reflected. We shall thus obtain alternations in intensity on rotating the mirror about the vertical axis, and there will be a maximum effect for a particular inclination of MM, in which case tan i. =\j. Rotation of the Plane of Polarization. Laurent's Saccharimeter The essential parts of the saccharimeter are two Nicol prisms, N x and N 2 , illustrated in fig. 228, one of which serves to polarize a beam of light passing through it while the other analyses the transmitted beam and detects its plane of polarization. These Nicols are spoken of, respectively, as the polarizer and analyser. 6 » t^y o L N, H Fig. 228 r\7 N 2 When N x has reduced the light vibrations to a particular direction, viz. parallel to the short diagonal at the end of the prism, all the light transmitted by N 4 can pass through N 2 if N 2 is oriented exactly in the same way as Nj, i.e. if its shorter diagonal lies parallel to that of N lf and its length lies parallel to that of N ; . We are here neglecting the diminution of intensity due to absorption, which always goes on, since actual bodies are not perfectly transparent. We mean that no light is cut out in this case on account of polarizing effects of N 2 . In this position the Nicols are said to be parallel. If, however, N t is turned from this position through a right angle no light from N! can get through N 2 , since N 2 is now so oriented that the light vibrations falling on it are in a direction perpendicular to its short diagonal, and such vibrations are not transmitted. In this position the Nicols are said to be crossed. Certain substances like quartz, and solutions like that of sugar, INTERFERENCE, DIFFRACTION, POLARIZATION 373 possess tiie property of rotating the plane of vibration of light as it passes through them, so that if N x and N 2 are crossed when the active substance is not placed between them (in which case no light will get through N 8 ),on inserting the active material, on account of the change in direction of the vibration, some light will pass through N 2 . It is found that a rotation of N 2 in one direction or the other will bring it into a position when the light is once more stopped. This shows that the light is still polarized, but its vibrations have changed direction in traversing the medium. We ought, therefore, to be able to measure the amount of this rotation by measuring the angle through which N 2 is turned ; but, unfor- tunately, N 2 can be turned through an appreciable angle when the light is cut out without any apparent return of the light. This lack of sensitivity is overcome in the saccharimeter by a special device. Just in front of the polarizing Nicol on the side towards the analyser is placed a semicircular sheet of quartz cut parallel to the optic axis. The complete circle is made up by a semi- circle of glass of such thickness that it absorbs the same amount of light as the quartz. The position of this circle is at H, and it covers the open end of N x completely. F A O ~ Q Fig. 229 When the light falls on the quartz it is separated into two components, polarized normally to one another, which travel through the quartz with different velocities. Let one component be represented by OI and the other by OQ just as the light reaches the quartz plate, these are the components of a vibration along OR ; It is here supposed that the ether particle is at O but just moving in the direction OR, so that its component directions are OQ and OI. As the disturbance passes through the plate 374 ADVANCED PRACTICAL PHYSICS FOR STUDENTS there will be a gradual change of phase between these components on account of the differing velocities of transmission. After a time the disturbance will reach a point in the plate where one component displacement is along 01 while the other component is along OQ 1 . These combine to give a displacement OR 1 . The quartz plate is cut so that, as the disturbance just leaves the plate on the other side, this difference of phase exists between the components. The difference is one-half of a period, and the plate is called a half-wave plate or half shade. Of course the light which traverses the glass side proceeds undisturbed, and its oscillations are still along the direction, ED, shown in the upper part of the diagram, parallel to OR. If N 2 receives this light when its short diagonal is at right angles to OR 1 this component is not transmitted, and no light from the quartz side gets into the eye, while part of the com- ponent OR gets through and the glass side appears illuminated. Generally, light passes in from both sides of the plate, but both sides are not equally illuminated. When both sides present the same illumination the principal plane of the Nicol is either along AB or normal to it, for it is clear that in either of these positions the components transmitted are the same for both sides. It happens that the eye can readily detect a change from the equality of illumination in both halves of the field, parti- cularly if both halves are equally dark, i.e. when the Nicol is so placed that the smaller components are transmitted. If the Nicol is set for equal iUumination on both sides, and an active substance is interposed, it will be necessary to rotate the Nicol to find once more the position of equal intensities. The amount of rotation measures the angle of rotation of the plane of polarization. S Fig. 230 Another common method of bringing about an increase in sensitivity is to use the biquartz. This consists of two semi- circular discs of quartz fitted together to form a complete circle. One of these rotates the plane of polarization of the incident light in a clockwise, and the other in a counter-clockwise direction. The amount of rotation per unit thickness varies with the colour INTERFERENCE, DIFFRACTION, POLARIZATION 375 of the light. For a particular thickness the rays from a sodium flame will be turned in opposite directions through a right angle, so that if the short diagonal of the Nicol lies parallel to this direction the yellow rays get through, and if the diagonal is perpendicular to this direction these rays are cut out. When white light is used it is robbed of the yellow constituent when the Nicol lies in this latter position, and the colour observed is greyish and is called the tint of passage. It is easy to detect a slight change from this uniform colour, for an appreciable change takes place to a partly blue and partly red field, one colour belonging to each side. Let LMNS denote one end of the analysing Nicol, and let UV denote the direction of vibration of the light. This light will be cut out if MS— the shorter diagonal— lies normally to UV. A small rotation of MS counter-clockwise will bring it into a position to cut out the light and so will a larger rotation in the opposite direction, the sum of these rotations being 180 . It is thus not easy to decide which way the plane has been turned. But if two lengths of the rotating substance be used, one slightly longer than the other, the rotation for the longer must be greater than for the shorter. The direction of rotation of MS which shows a larger angle in the case of the longer is the direction in which the rotation has taken place. Tubes of glass with carefully worked end-pieces are used to carry the solution to be examined. The ends are held in position by metal caps screwed against them. It is necessary to have rubber washers between the glass ends and the tube to avoid strain when screwing up ; for a strained end will produce rotation. Find the amount of rotation for a solution of sugar in water and deduce its specific rotation. This quantity is denned to be the amount of rotation produced by one decimetre of solution divided by the weight of dissolved substance In unit volume. Let w grammes be dissolved in ioo ex. and suppose a length, / cms., produces rotation, 0. The specific rotation is : 6 w e iot -. = iooo T' I ioo lie Repeat for various strengths of solution, and for different lengths of tubes. When several tubes are obtainable it is interesting to observe the effect of causing transmission through different lengths of a solution of a particular strength. By this means it may be verified that the amount of rotation 376 ADVANCED PRACTICAL PHYSICS FOR STUDENTS is proportional to the distance traversed by the light in the solution. Another instructive experiment is to make solutions of different known concentrations, which may be measured by the number of grammes of substance dissolved in ioo c. c. of solvent, and to measure the amount of rotation in traversing a particular distance through the solution. A curve showing the relation between the rotation and concentration should be plotted. The Lippich Polarizing System In modern polarimeters the half shade is replaced by a more convenient method of dividing the field. In the most recent polarimeter the field is divided into three parts (fig. 231), the two outer similarly illuminated for all positions of the analysing Nicol and the central portion which may be differently illumin- ated from the neighbouring regions and which has to be matched with the outer parts of the field. Fig. 231 l n l /// M 1 j Fig. 232 The mode of producing the divided field is illustrated in fig. 232. N is the polarizing Nicol and LL are two small Nicols fixed in position and mounted in a brass cylinder in front of N. The directions of vibration of the light emerging from the cylinder and falling on the optically active substance are repre- sented by the arrows, I, n, I. The central portion passes through the Nicol, N, only, while the outer parts pass through N and L. When the analyser, M, is turned so that it transmits vibrations along a direction bisecting an angle between n and I the whole field is uniformly illuminated. For all other positions of M the field is not uniform. Thus, to measure the amount of rotation of any substance placed at A, the analyser is first put into the position correspond- ing to uniformity of field. The substance to be examined is then put into position and M again rotated until the field is once more uniform. The angle of rotation measures the rotation due to the active substance. INTERFERENCE, DIFFRACTION, POLARIZATION 377 The optical system by which the field is examined is not shown in the diagram. It lies to the right of M and is focussed on the plane through the right-hand ends of the Nicols, LL. This system has the advantage that it is suitable for the examination of all wave lengths, whereas the half shade has to be constructed for one wave length only. Fig. 231 shows at OCO how the field is divided into three parts. The Half Shadow Angle The various devices employed to enable accurate observations to be made in polarimetry, which have been described, produce two beams of polarized light with vibrations in directions inclined to one another. In the Lippich system we have denoted the two directions by I and n. In fig. 233 these directions are denoted by B/ and Bw respec- tively, and the angle between them is 20. This angle is called the ' half shadow angle,' and the magnitude of this angle has an important bearing on the question of sensitivity. E Fig. 233 Suppose that BC bisects the half shadow angle, and that DBE denotes the direction of vibration of the light which traverses the analyser. When this lies at right angles to BC the intensities of the two beams are equal as seen through the analyser, for the components of the displacements transmitted, viz. BF and BG, are equal, and the intensities are in the ratio : BF 2 : BG 2 . Suppose the analyser is turned through an angle, a. Then the transmitted components are : BZ cos ZBE 1 and Bw cos nED 1 , or BZ sin (0 — a) and Bn sin (0 + a). Thus the ratio of the intensities is : sin 2 (0 — a) : sin 2 (0 + a). In photometric work it is assumed that the eye can detect a difference of intensity of one per cent. Thus, if we regard as a given angle, we may say that the 378 ADVANCED PRACTICAL PHYSICS FOR STUDENTS change in setting of the analyser of the amount a will just be detected when sin 2 (0 — a) = «99 sin 2 (0 + a). From this equation it follows that when a is a small angle, so that we may write : sin a = a, and cos a = I, a = -0025 tan 0. If a has a small value the apparatus is sensitive, and it would appear that the sensitivity is improved by making as small as possible. But as gets small difficulties arise on account of the fact that the light is never plane polarized, it is always elliptically polarized in practice. may not be indefinitely diminished. Polarimeters are usually fitted with a small movable arm projecting from the tube which carries the polarizer. This arm carries an index mark which moves over a scale. By means of it the polarizing Nicol can be rotated so that the half shadow angle can be adjusted within limits. A modern form of the apparatus is illustrated in fig. 234. The sensitivity is thus to some extent under the control of the observer, who will discover as he becomes familiar with his instrument the best adjustment for sensitivity which suits him. The reader may be referred for a more detailed and complete account of all these questions to the article on " Polarimetry " in the " Dictionary of Applied Physics." Soleil's Compensator Sometimes the saccharimeter is fitted with a piece of apparatus consisting of two quartz wedges. This is known as Soleil's Compensator. Fig. 235 illustrates the apparatus. The wedges are ABC and DEF, and these are mounted in metal holders which can be moved by means of a rack and pinion, so that the wedges are translated in either direction parallel to AC or DE. Thus if a ray of light is passed perpendicularly to AC it is possible to place varying thicknesses of quartz in its path. The quartz wedges are cut so that the optic axis lies perpendicular to AC and DE, and polarized light passing through them suffers rotation of its plane of polarization. The amount of rotation can be varied by moving the wedges by means of the rack and pinion. The quartz wedge is placed just in front of the analyser. An index mark moves along a scale as the wedges are displaced, so that a record can be made corresponding to each thickness of quartz interposed. i'aae 378 INTERFERENCE, DIFFRACTION, POLARIZATION 379 The analyser is first rotated until with the quartz wedges occupying a convenient zero position the field is equally dark on both sides, supposing that the polarimeter is fitted with a half shade. The quartz wedges are then moved a small amount by the rack and pinion, and the amount of rotation of N 2 necessary to restore the uniformly dark field of view is recorded. Fig. 235 By making a number of observations a curve can be plotted which shows the rotation corresponding to the various dispositions of the wedges. When light has undergone a rotation before passing through the wedges, this rotation may be counteracted by interposing the correct thickness of quartz. Thus suppose the Nicols and quartz occupy the zero position described above, and that an optically active material is inter- posed, the rotation resulting may be counteracted by displacing the quartz wedges either so as to increase the thickness traversed or to diminish it. When the appearance in the analyser is the same as that of the zero position, we know that the wedges have caused a rotation equal in magnitude, but opposite in direction to that of the active substance. Thus by observing the record opposite the index mark we can deduce the amount of rotation due to the substance. In plotting the graph, rotations in one direction will lie on one side of the origin up to 180 °, while those in the other direction will lie on the other side up to 180 . CHAPTER XIII PHOTOMETRY Introduction The light emitted from a small source is absorbed very little by the air through which it passes, so that we may say that any surface, surrounding the source completely, will receive the same total amount of light. Let this total amount be denoted by M. = luii Fig. 236 Imagine a cone with its apex at a small source of light and let its solid angle be w. All surfaces receive the same amount of light on the parts lying within this cone. If w is small, say 6w, the cone may be regarded as denning a particular direction and the intensity will be regarded as the same for all rays within this cone. If L denote the total amount of light emitted within this cone per second, we write : L = Kdw. This equation defines K, which is sometimes called the 'candle power ' of the source. In general, K is dependent upon the direction, but when K is the same for all directions M = 4ttK, where M is the total amount of light emitted per second by the source. Let the small cone cut a surface in the element, <5S, and let the mean direction of this cone make an angle, 0, with the normal to &S. 380 Then aS = PHOTOMETRY 381 r 2 8w cos0 , T K cos ^ and L = — -—- — <5S. r 2 The intensity of illumination is defined to be the amount of light falling on unit area per second, i e I — — — ^ cos ® ~ ss 7* The intensity, I, varies inversely as r* and directly as cos 0. When I is the same for two similar surfaces they appear to the eye to be equally bright, and on this principle the use of photo- meters depends. The efficiency of a source of light may be denned as the ratio of K to the material or energy consumed per second. The average candle power, in the case when K is not constant, divided by the amount consumed per unit time is called the mean spherical efficiency. This term is employed because the average M value of K is denned as — , and this denotes the average amount of light falling on 1 sq. cm. of a unit sphere placed with its centre at the source. The efficiency of a candle is : K -^- weight of wax consumed per second, of a gas flame : K -r- cubic feet of gas consumed per minute, and of an electric lamp : K 4- watts supplied to it. The watts are measured by the product, volts x amperes, one watt denoting the rate of working when a current of one ampere falls through a potential difference of one volt. The unit in which K is measured is the candle power of a standard candle in a horizontal direction when the flame is 50 mms. high. If another source is used and compared with the standard the two are adjusted to the same height and placed at such distances, r 2 , and r it from a conveniently placed screen that each makes it appear equally bright. In this case by what has been said above concerning intensity we have : t _ Kjcos T K 2 cos Ii - -JT- - I. - -jt— 382 ADVANCED PRACTICAL PHYSICS FOR STUDENTS since the angles are the same, power its value is unity, and, If K, is the standard candle It is assumed that the student is familiar with the simple forms of photometer such as Rumford's and Bunsen's. It is difficult to make accurate comparisons of the illuminating powers of sources with these instruments. We shall be concerned chiefly with the more accurate types of instruments in this chapter. Even with these care and practice are required, but a skilled observer can obtain accurate results. The Efficiency of Sources of Light Take a gas-burner provided with an indicator registering the quantity of gas supplied, and adjust the flame until it stands at the same level as a candle flame and the grease spot of a Bunsen photometer. The candle may be taken as the standard with the value of K unity, and it must be shielded from draughts and must burn steadily. Determine K for the flame corresponding to different rates of supply of gas, by adjusting the distances between the sources, until the photometer screen appears alike on both sides. Plot a curve, showing the relation between the efficiency and the supply per hour. It will be found that the efficiency increases with the supply up to a maximum and then diminishes. It is most economical to adjust the supply to the value appropriate to the maximum. • r v " • A R 1" ■ 1 Fig. 237 We may similarly measure the efficiency of an electric lamp. Arrange the lamp, L, on the same level as the grease spot and candle flame, as before, and measure its candle power when the current is supplied at different voltages. Fig. 237 illustrates the arrangement of apparatus. V is a voltmeter joined to the terminals of L, and A measures the current in amperes. The resistance, R, is adjustable and is PHOTOMETRY 383 used to vary the current supplied to the lamp.. The power is obtained by connexion to the main through a plug. A con- venient voltage is 100 volts. Care must be taken not to short-circuit the mains. Plot a curve, showing the variation of candle power, with rate of supply of energy as measured in watts. The Flicker Photometer One form of this instrument is illustrated in figs. 238 and 239. The essential part consists of a white wheel, W, of which the edge is about 1 cm. wide, and is cut to the shape of a ridge running spirally. This is shown at RR, in fig. 239. The wheel is mounted in a box, black on the inside, provided with a rod, so that it may be supported in the carrier of an optical bench. J Si s, Fig. 238 The wheel is provided with a central axis and is rotated by a spring within the box, which is wound up by the key, K. The two sources to be compared are placed one on each side of the apparatus, as at S! and S 2 . These are carried in stands on the bench, so that the distances from the wheel are measured accurately and their heights are adjusted with the aid of two lenses fixed on the box. The small doors EF and GH, are closed, and the lenses used to throw an image of the sources on marked points, P 1 and P a , on the doors. When the images fall on these points the heights of the sources are correctly adjusted. 384 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The edge of the wheel is observed by means of an eyepiece, J, and usually unequal parts of the edge are seen on the two sides. When the wheel rotates the widths of these vary, and unless the illumination is equal on both sides a flickering effect is observed. If the illumination is the same on each side the ridge character disappears. S x and S 2 are moved to or from the apparatus until the flicker effect ceases, when the illuminations are equal, and if I x and I 2 represent the illuminating powers of the sources, and r x and r 2> their distances from the wheel, we have : \* The distances should be measured from the middle of the wheel. It is not easy to decide when the "flicker ceases, and the degree of accuracy obtainable is not very high. Compare in this way a standard candle and a lamp. The Lummer-Brodhun Photometer This photometer is one of the most accurate. It is illustrated in fig. 240, and consists of a box, LMNO, containing the prisms, P lf P„A, and B, by means of which light is reflected and trans- mitted into the telescope, T, placed at 45 to the sides of the box, with its object glass in one corner. The two sources to be compared are at S x and S 2 , from which light falls on the slab, DD, which consists of magnesium car- bonate. From the diffuse sides of the slab rays are scattered and absorbed by the sides of the box, except those that cut the sides of V x and P 2 normally. These are reflected by the hypot- enuse faces into the two right-angled prisms, A and B. These prisms are the principal part of the apparatus. The hypotenuse of A is rounded off, except for a circular central portion which is placed in optical contact with the PHOTOMETRY 385 iarger face of B. The reason for this is, that rays of light may pass from A to B at this junction just as if the prisms formed one solid medium. Rays falling on other parts of the hypotenuse faces are totally reflected. In this way rays from P x pass on through B, forming the central bundle of rays in the beam emerging from the right and entering T. The rays outside this circle are reflected and are absorbed by D Fig. 240 the sides of the box. In the same way, rays from P 2 are reflected outside the circle, while those falling on the circle are transmitted. Thus the field of view of the telescope is illuminated by a central circle of rays, originating at S 2 , while the outer rays come from S x . Generally these two parts will be of different brightness, and on moving S x or S 2 , the two parts may be made equally bright. The eye can judge this easily and readily appreciate a slight deviation from equality. It is on this fact that the sensitiveness depends. From DD both sets of rays follow similar paths and light is absorbed equally. When the field of T is uniform, the slab is 25 386 ADVANCED PRACTICAL PHYSICS FOR STUDENTS illuminated equally on both sides. If Lj and L 2 denote the illuminating powers respectively, and d x and d 2 denote the respective distances from DD, we have : L x L a df d 2 Thus any two sources may be compared with accuracy. Compare two sources as in the last experiment. In order to eliminate errors arising from inequality of the reflecting power of the two surfaces of the slab and a possible difference in optical paths, DP X A and DP 2 B, it is usual to mount the box, LNOM, on a central horizontal axis lying perpendicu- larly to SiSa. Readings are first taken with the telescope as in the figure, and then with the box turned through two right angles so that the telescope lies on the left. The Nutting Photometer and its use for the Determination of the Absorption of a Solution This form of photometer is the most accurate instrument of its kind. It is based upon principles described by P. G. Nutting, but has been modified by Messrs. Hilger and Co., whose apparatus is here described. It is used in combination with the Constant Deviation Spectrometer (pp. 341 to 345), and the apparatus is illustrated in fig. 241, in the position ready for use. The essential features of the photometer are illustrated ki fig. 242. The box, A, is of aluminium, blackened inside, provided with two small windows, Q and Q 1 , by means of which light is admitted along the two directions, QR and Q X S. Light from a suitable source is deviated by two prisms, P and P 1 , carried in the plate, C. On the plate a number is inscribed, indicating the distance that the source must be placed from it, in order that the necessary deviation may be produced. This distance is usually 19 cms. Behind the window, Q 1 , lies a Nicol prism, Nj, which polarizes the light entering at Q 1 ; but behind Q in the form of apparatus shown no Nicol prism is placed. In another form a Nicol lies behind Q also, and its purpose is to counteract as much as possible the elliptic polarization that occurs on reflection at the surfaces marked R and S. It is found that this is reduced to a minimum by a particular orienta- tion of the Nicol. The prism, RS, is composed of three slabs, the two outer are alike and are cut at the ends, R and S, at an inclination of 45 ° to the length and to the incident light. Pm& 3^5 PHOTOMETRY 387 Thus the beam, QRST, is totally reflected at R and S, by these two end faces. Between these slabs lies a central one of the same composition and thickness cut at R at 45 like the others, but cut square at the other end, and projecting as the diagram Fig. 242 indicates. Thus the central portion of the beam, QR, is totally reflected at R, and transmitted out at the other end, and is ab- sorbed by the blackened walls of the box. The beam, Q X S, is totally reflected at the end, S, by.the upper and lower slabs, and the light absorbed by the walls of the box, but the central portion is transmitted in the direction, ST, by the central slab. Thus a tripartite field is produced and may be observed from the end of the tube, F ; the outer portions are illuminated by light which has entered at Q, and the central portion by light from Q 1 . The instrument is constructed and the prism and Nicol, N 1 , chosen to cause the light within the box to suffer approximately the same absorption along the two paths, QRS and Q X S. The tube, F, carries a second Nicol which is not shown, which acts as an analyser and which can be rotated by means of the divided metal circle, G. This circle carries two scales, one marked in degrees and the other giving ' densities ,' a term which will be explained below. When the instrument has no zero error the degree scale reads zero when both Nicols are parallel. In this case the light already polarized by N x is transmitted by N 2 , while the unpolarized outer portions of the field are polarized by N 2 in the same way as the central portion by N r Thus when there is no absorbing medium between the source and one of the windows, the field appears under these circum- stances of the same intensity in the outer and inner portions. F also carries a condensing lens which may be adjusted by the rod, R, projecting downwards from the tube. The purpose of this lens is to converge the light so that on exit all the light may enter the pupil of a normal eye. This is an important condition with which accurate photometric apparatus must comply in order that intensity comparisons may be of any use. 388 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The rod carries a scale past an index mark in the slot, B, on which are engraved two sets of numbers which have to be as- sociated in pairs. The upper scale records the distance of the source from the front of A in cms., and the number below this record on the lower scale denotes the maximum breadth of the source which is permissible if the photometric condition is to be satisfied. Beyond the circle, G, the tube carries a lens system which focusses the light on to the slit of the spectrometer, and if it is desired to examine the field of view directly by eye an additional eyepiece is fitted into the tube. The apparatus requires careful adjustment which may be carried out as follows : Remove the prism from the spectrometer, illuminate the colli- mator slit and place on the prism table a piece of plane mirror or a right-angled totally reflecting prism, and adjust it until an image of the slit lies on the cross-wire of the telescope. Now place the source to be used with the photometer at the distance from the collimator slit at which it is to be situated during the experiment, having regard to the photometric condition which limits the distance to some extent on account of the size of the source. Adjust the source until an image of it lies in the centre of the field of view of the telescope, the collimator slit being now wide open and the eyepiece removed from the telescope, so that the source appears to lie at the centre of the object glass. Slide the photometer into position with the end of the tube, F, as nearly as possible 1*4 cms. from the slit, and with the window, Q 1 , directly between the collimator slit and source. Cover up the window, Q. Adjust the photometer by means of the three screws on which it stands, and by rotation about a vertical axis until the image of the source lies once more in the centre of the objective of the telescope. Place the plate, C, into position at a distance of 19 cms. from the source. The distance between P and P 1 is 3-8 cms., so that by moving the source a distance 1*9 cms., it can be brought to lie opposite the middle point of PP 1 and it should then be in the correct position. - This may be judged first by observing if bright circular patches of light lie symmetrically round the windows, Q and Q 1 . Make sure that, with the rod, R, adjusted so that the index lies opposite the mark denoting the distance between the front of A and the source, the width denoted by the lower reading is greater than that of the source. If this is not the case the source has either to be displaced* farther from A or diminished in size. Place the constant deviation prism and eyepiece of the telescope PHOTOMETRY 389 in position and adjust the spectrometer correctly for sodium light as indicated on p. 343. In doing this a strongly coloured Bunsen flame may be placed just closer to C than the light source. Make the line as sharp as possible by rotating slightly the milled head at the end of F. Now remove the Bunsen flame and open the slit to let in a convenient quantity of light. It will be probable that the Fig. 243 central part of the field of view is slightly displaced with respect to the outer parts. This is exaggerated in the upper part of fig. 243. This may be corrected by a further small rotation of the photometer about a vertical axis. The three parts of the field should be separated by fine dark lines. If these are too wide they may be made narrow by adjusting the base screws of the apparatus. Correction of Zero Error In practice it is usually found that when the apparatus is set at zero the three parts of the field are not uniformly bright, and that the error is not the same for different wave lengths. In order to correct for this the readings of the apparatus are recorded when the field is uniform for different wave lengths. A shutter eyepiece is fixed to the telescope to cut down the light except over a narrow central strip, the wave length for which is recorded on the drum. When the central portion is the brighter, rotation of the analyser cuts it down, and the readings on both sides of the zero are observed and the mean taken for a series of different wave lengths and a curve plotted, showing the relation between error and wave length. 390 ADVANCED PRACTICAL PHYSICS FOR STUDENTS These readings are observed on the degree scale or density scale as may be required. If the central part of the field is the darker it is not possible by merely turning the analyser to bring about uniformity. In this case a weak absorber, e.g. a plate of glass, is put into the path of the beam just before it enters Q. The thickness is chosen so that in the zero position the central field is just stronger, when the correction may be made as before. This should not often occur because the instrument is made so that the central portion is brighter than the outer, so that as the apparatus deteriorates with time the central portion may still be the brighter and the interposition of the weak absorber may be avoided. The substance of which the absorption is required is placed in the path of the light before it enters Q. In the case of a liquid it is necessary first to measure the absorption of the vessel containing it. The observations consist of noting the readings on the scale, G, when the field is uniformly bright for both directions of rotation of G. Let I denote the intensity of the light entering at Q 1 , and let I denote the intensity of that which enters at Q. The density scale records the values log ~, i.e. the logarithm of the ratio of the intensity of the light entering the medium to that transmitted by it. If this number be divided by the thickness of the material traversed a quantity known as the ' extinction coefficient ' is obtained. Let a denote the amplitude of the polarized light transmitted by Ni and suppose that the analyser is turned through an angle, B, from the position in which it is parallel to N r The amplitude of the vibrations transmitted by the analyser is therefore a cos 0. When the field is uniformly matched the intensity is I, and hence Io_ a 2 . I a 2 cos 2 ' .-. log ^-= log sec 2 6. This shows the relation between the density scale and scale of degrees. A solution which gives a characteristic absorption curve is one of eosin in alcohol. Eosin may be obtained from a bacterio- logical laboratory, and the solution must be very dilute, or so much absorption takes place that the transmitted light is very feeble. PHOTOMETRY 391 Carry out the determination described above and plot on the same curve the values of the ' density,' log —, and the cor- responding wave lengths, for the zero error, for the vessel alone and for the vessel and solution. From these it is possible to obtain the value, log^ for the solution alone. Fig. 244 shows the result of an experiment with eosin. Some- times the variations occur very rapidly and a curve is obtained like that drawn in diagram 243. Unless frequent observations are made in the neighbourhood of AB the variations at C and D may be missed. Whenever the curve shows any sign of change the neighbourhood where this occurs must be examined with care. AlosoY^Hon Curue for a u>cak Solution of Eostn >n AlcoKo\. the. o|p|per cwtye. \s ■&><• tElo&n -v- Glass etc. "The \oujer curve. i» •¥<* /toro covrecC\or\ <wA cjVass cxbsor^Hon . ■4400 *=\BOO 5EOO B600 6000 WaveWr^Yhe x \o cms Fig. 244 It may not be necessary to determine exactly the value of log y for the solution, and in this case it is sufficient to make the zero correction and that for the containing vessel together, the whole appearing as a combined correction curve. CHAPTER XIV SOUND To Find the Frequency of a Note by means of the Siren In this instrument a musical note is produced by puffs of air following one another in rapid and regular succession. The series of puffs is produced by blowing air through a number of holes in a rapidly rotating plate. The diagram (fig. 245) illustrates the instrument. It consists of a cylindrical metal chamber provided with a tap through which air can be blown. In the upper end is cut a series of holes lying regularly spaced on a circle with its centre on the axis of the Fig. 245 cylinder. Above this lies a circular metal disc provided with a similar series of holes which fit above the former. The disc is mounted so that it can rotate about the axis of the cylinder and so alternately cover and expose the lower series of holes. 392 SOUND 393 The two series of holes slant in opposite directions as the figure shows, and when a current of air is blown into the chamber and the disc given a slight rotation, the puffs of air on escaping produce a pressure which drives the disc. By adjusting the influx of air the regularity and speed can be controlled so that notes of varying frequencies can be produced. If N is the number of revolutions made by the disc per second, and the number of holes is n, the frequency is N». In order to measure the number of revolutions the disc is provided with a metal bar, provided with a screw at one end, which works two dials, one registering units and tens, and the other hundreds of revolutions. This form of apparatus is due to Cagniard de la Tour ; but it has the disadvantage that the speed can only be increased with greater air pressure and a consequently louder note. It is also difficult to keep the speed uniform. It is preferable to drive the disc with an electric motor, of which the speed may be regulated by including a resistance in the supply circuit, and the holes should be cut normally to the disc in order to avoid air pressure in the direction of the rotation. The siren gives a large number of harmonics, and it is necessary carefully to single out the fundamental note. Let it be required to find the pitch of a given note, as, for example, that produced by an open organ pipe. Carefully adjust the speed of the siren until beats are heard between the note it gives and that of the organ pipe by blowing at a particular pressure from a bellows connected to the chamber. The blower should then endeavour by a slight change of pressure to produce from the siren a note giving no beats. To some extent the frequency may be controlled by the tap, but it is important to keep a steady pressure on the bellows. At the same time a second observer should measure the speed of revolution by observing the number of revolutions recorded on the dials in a definite time (30 or 60 seconds). Verify the result of the determination of frequency by measur- ing the length of the pipe and its diameter. For an open organ pipe emitting the fundamental the wave length is approximately twice the length. The correction necessary to obtain a more accurate result is to add «6 radius for the open end and 2*8 radius for the flute mouthpiece. If the length of the pipe be I, and the radius of the pipe, r, the half wave length is given by : - = I + 3'4^ or X = 2(1 +■$•#). 394 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The velocity of sound in air for ordinary temperature may be taken as V = 33,300 cms. per second, or make the accurate correction for temperature by formula (A), p. 416. V Thus the frequency is -•• Determine the frequencies of several organ pipes theoretically and experimentally, and draw up a table recording the speed of rotation of the siren disc, and the observed and calculated frequencies in each case. The Tonometer In Scheibler's tonometer a number of tuning forks are arranged in ascending order of frequency, each of which gives the same number of beats with its neighbour. The forks thus form a series in which the frequency increases by equal steps, and they are arranged so that the highest frequency is twice that of the lowest. In Appunn's tonometer the forks are replaced by reeds set in vibration by a blast of air from bellows of large capacity, and the apparatus has the appearance of a small harmonium provided with a series of stops by means of which any note may be sustained. This form of apparatus is not so accurate as the original one, for Lord Rayleigh has shown that the frequency of a vibrating reed is to some extent affected by the vibrations of its neighbours. As it is necessary to vibrate two successive notes in the experiment we have no longer a constant register of frequency as in Scheibler's instrument. It is first necessary to find the absolute frequency for each note on the instrument. Suppose there are (k + 1) notes, and con- sequently k intervals between them, and that the frequencies are N x , N 2 , . . , N* +1 , beginning from the lowest. If the number of beats be observed between all the successive notes, and be denoted by n x , w 2 , . . . n k , respectively, we have the following relations : N a -N 1 =» 1 , N 3 — N 2 = n 2 , N* +1 — N* = n k . Then adding both sides : N* +x -N 1 = # 1 + » l + ...+» 4 . SOUND 395 But N* +1 = 2N X ; /. Nj = n x + n 2 +. . . + n k . We thus find N x by counting the successive numbers of beats, and we can then deduce from the equation the frequencies of all the other notes of the series. In practice, of course, the numbers of the beats will vary between the successive notes in different parts of the scale to a slight extent. In order to determine the frequency of the lowest note, count the number of beats at five or six different parts of the range and deduce the average difference of frequency. Let this be denoted by n. Then the frequency of the lowest note is k x n. After this determination has been made the frequency of any given note coming within the range may be determined. For example, suppose the frequency of a fork is required. Find by trial the note nearest to it in pitch and count the number of beats when the notes are sounded together. If this number be denoted by x, and N/ is the frequency of the note nearest the unknown, the frequency of the latter is N, ± x. The frequency of the note next above N, is N, + n, and on sounding this and the unknown note together the frequency of the beats will be less than n if its pitch is higher than that of N/, and greater than n if its pitch is lower. This enables a distinction to be made between ± x. In making this last part of the experiment it is advisable not to rely on the accuracy of the average number of beats for each interval, but to measure separately the frequency of the beats 6i the notes immediately above and below N/. It will then be easy to decide on the exact position of the unknown pitch above or below N/. The Determination of the Frequency of a Tuning Fork by the Method of the Falling Plate A smoked glass plate, P, is suspended vertically by a piece of thin string or thread over two nails, QQ, the thread being attached to the upper edge of the plate by means of sealing wax or by any other convenient method. The fork is held in a clamp, H, and carries a light style of bristle or thin aluminium wire attached to one prong by as little wax as possible. The style is just in contact with the plate so that when the plate falls it removes some of the soot and leaves a trace. In order to prevent breakage a padded wooden stand, AB, is placed just below the plate. The fork is stroked gently by means of a violin bow and the /*' 396 ADVANCED PRACTICAL PHYSICS FOR STUDENTS plate is allowed to fall straight down by burning the thread between the nails, QQ. A wavy line is traced by the style similar to that shown in fig. 247, but with more waves, and usually of small amplitude. A point, O, is chosen just clear of the indistinct portion drawn when the plate was moving down in the first stage of the motion, and consequently before its velocity had sufficiently increased Fig. 247 to open out the waves, and from it is counted a number, n, of complete waves to the point S. Again, n waves are counted to the point S r Let the spaces, OS and SS X be of lengths, s and s lt l respectively. Then the time taken to fall over these two lengths is the same, let us say t. If N denotes the frequency of the vibrations : n = N*. SOUND 397 Let u denote the velocity of the plate at the instant correspond- ing to the mark, O. Then by the equation for space described under acceleration, g, s = ut + \gP, and s -f- s x = zut + 2gt 2 , for the time of description of s + s t is 2t. Hence Sl — s = gt 2 , or -4 N = n • J- g vs x — s "1 " The distances, s and s v are measured carefully by means of a travelling microscope, and the value of N obtained is that for the fork vibrating with the load consisting of the style and wax and affected by the friction of the style against the plate. This should be allowed for by taking a second fork of nearly the same frequency as that under examination, before attaching the style, but of slightly higher pitch. Carefully load this fork by adding wax until no beats are heard when both sound together. Then when the first fork is loaded and has the style touching the plate as in the experiment, again sound the two together and count the beats. The number of these per second gives the number of periods lost per second on account of the loading and friction. This number added to the value of N, determined in the experiment, gives the corrected frequency. In marking off the points O, S, and S v be careful to choose them at corresponding points of the waves. If O is at the summit of a crest, S and S x must lie in a similar position n and 2w waves later, respectively. The plate may be conveniently blacked by holding it just over a turpentine flame which gives a good deposit of soot ; the flame of a paraffin lamp gives also a satisfactory deposit. Chronographie Methods of Determining the Frequency of a Fork In both the methods to be described under this title a fork is set in vibration and, electrically maintained, with a style attached to one of the prongs lightly touching the blackened surface of a cylinder which can be rotated about its axis. On rotating the cylinder the track made by the style appears as a wavy line which can be opened out so that each vibration is distinctly separated from its neighbours by adjusting the speed of rotation. 398 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The drum may be rotated by the handle, H, or by a falling weight not shown in the diagram, which rotates the drum at a convenient speed. In this case the rotation is regulated by a governor, and by releasing a catch, the handle, H, can be employed for rapid movement of the drum. This also is not shown in the figure. The drum moves along an axis cut with a screw thread, so that the track drawn by the style forms a wavy helix round it. Fig. 248 It is then necessary to have some record of time with which to compare the vibrations. The two methods differ in the way the time is obtained. In the first a stand is mounted conveniently near to the fork, as illustrated in fig. 248, which carries a pointer actuated by an electromagnet, M. The pointer, P, is carried on a lever one end of which consists of a strip of iron or steel, which is attracted by the electromagnet core when a current flows through the exciting coils. When the current is off the lever is held in a position with the strip a short distance from the magnet by means of a spring. Thus, as the cylinder rotates, a line is drawn round the surface. On passing the current in the coils the pointer moves and makes a kink in the line. If the current is put on at regular and known intervals, a time record along the side of the waves is produced on the drum. Thus by counting the number of waves between consecutive kinks the frequency of the fork may be deduced. The regular intervals are obtained by connecting the wires, AA, to a battery, and completing the circuit by means of the mercury cup, M, and the pendulum, P, as shown in fig. 249. Mercury is poured into a hole cut in wax, W, so that it stands just above the wax surface, and the wax so placed that a strip of wire hanging down from the pendulum just touches the surface as it passes its lowest position. The circuit is completed twice in each complete period, and at each instant the pointer makes a record on the drum. It is improbable that the interval between successive records will be one half a period, since this would require exact coincidence of the point of contact with the mercury and the lowest point of SOUND 399 the swing. The time between alternate records will, however, measure the time of a complete period. Thus, in counting up the vibrations, find the mean number between alternate records. Obtain a long helix, begin at the first stroke of the pointer, and count the number of vibrations up to some later odd numbered stroke. Find the mean number per complete period. Repeat this, beginning with the second stroke and ending at some later even numbered stroke, and find the mean number again. The two values should agree, but if there is a slight variation take the average value of the two results. . Fig. 249 To obtain a blackened surface, take *a sheet of smooth white paper, and wrap it round the drum, one layer thick, holding it in position by means of gummed paper. To blacken it, rotate it over a turpentine flame, or coat it with camphor smoke. The coating of soot should not be very thick, the style will then remove the soot and leave a white wavy trace. The paper may be smoked again when once used and the track covered up. The time of a complete period is measured in the usual way, by timing the pendulum. The frequency obtained is, of course, subject to a correction similar to that of the last experiment, and this should be determined and applied in the manner described. Fig. 250 represents a convenient method of carrying out the determination in an alternative way. The cylinder is held vertically, and is rotated by a handle or string round a drum as shown. The fork is maintained electrically as before, but the axle of the drum is connected to one secondary terminal of a small 400 ADVANCED PRACTICAL PHYSICS FOR STUDENTS induction coil, while the second terminal is connected to the fork. The primary circuit is completed through a pendulum and mercury cup as before. Each time the primary circuit is com- pleted the induction coil is excited and a spark passes to the drum from the style, knocking off a little soot and leaving a white dot to record the instant of closing of the primary circuit. The same procedure and precautions are adopted as before. A thin, flexible copper wire will be found suitable for suspension of the pendulum, and with a small induction coil there is no incon- venience on account of shocks obtained when the apparatus is touched. Fig. 250 This method was employed by A. M. Meyer. It should be noted that uniformity of motion of the drum is not necessary. The methods are both inferior to those in which an optical method is employed for determining frequency as in Rayleigh's method. The Frequency of a Tuning Fork by the Stroboscopie Method The fork is fitted with two very light plates fastened at the extremity of the prongs, one to each, and so that one may vibrate freely past the other. These plates may be of thin cardboard or aluminium, so that the loading affects the fork to the smallest possible extent, and they are stuck on to the fork by means of a little wax. A disc is taken provided with a number of dots placed at equal intervals round circles concentric with the disc. Each circle has its own interval length. SOUND 401 The fork is maintained electrically (p. 138), and the disc placed behind it with its dotted surface brightly illuminated. Each plate attached to the fork is provided with a slot, and when the fork is at rest the slots lie directly behind one another. Thus the disc can be seen through them. The slots and one of the circles of dots are so placed that the dots can be seen by looking through the slot, as the disc slowly rotates. If the fork is vibrating it is possible to see through the slots twice in each complete period, and thus 2,n times per second, where n denotes the frequency. The disc is caused to rotate uniformly by means of an electric motor provided with a resistance in circuit to vary the speed. The speed is gradually increased by adjusting the resistance until when the fork is vibrating the dots in one of the circles appear to be at rest. In this case a dot moves up as the disc rotates, so that each time the slots are in line a dot is just in line with them. The eye sees apparently one stationary dot, and the effect of rotation is lost by looking through the slot. In order to count the number of rotations per second made by the disc, a counting arrangement is attached and the time taken over a definite number of rotations, when the dots remain apparently steady, by means of a stop-watch. Let there be N rotations per second in the case when the dots belong to a circle containing p of them. In this case the time taken by a dot to take the place of the one preceding it is : ~-r second. This is equal to one half the period of the fork. Hence — = — — N^> 2 n ' or n = - Np. Now further increase the speed of rotation until once more the dots appear steady. In this case the speed of the disc is such as to cause a dot to take the place of the dot two intervals in front. The time taken to do this is ^n, where N 1 is the N 1 ^ number of rotations per second From this we may calculate the value of n once more. On further increasing the speed until a dot takes the place of another three intervals in front we can obtain a third calculation. Repeat this for several series of dots. With the fork loaded it is necessary always to correct for the loss of pitch due to loading as before. 26 402 ADVANCED PRACTICAL PHYSICS FOR STUDENTS We may, however, avoid this by making one prong of the fork bright over a small area, and by rotating a disc provided with several series of concentric holes in front of it. If the fork is well illuminated and the disc carefully mounted, it will happen that for some particular speeds the fork appears stationary when seen through the holes. The experimental details and the mode of deduction of the fre- quency of the fork from the observations are similar to the former. The observations in stroboscopic experiments can nowadays be made much more conveniently by the use of a neon lamp than by the method described above. This lamp has the property that it lights up immediately a voltage is supplied to it without any appreciable lag, and is extinguished immediately when the voltage is taken away. The lamp consists of a flat piece of aluminium with a rod of the same metal lying a short distance from it and parallel to it in a bulb containing neon at low pressure. When included in the secondary circuit of an induction coil, it lights up each time the current in the primary is broken. There is no effect at ' make ,' because in the construction of the usual type of induction coil the effect at make is suppressed and that at break of the primary circuit intensified. If the primary coil is put into the circuit which actuates the fork the primary current is made and broken once per vibration of the fork and the lamp flashes out once per complete vibration. Of course, the make and break attached to the primary must not be allowed to work, the fork takes its place. All that is necessary is to make the connexions in the usual way and to screw the platinum-iridium point close up to the clapper to prevent separation and to place some object, e.g. a small block of wood, between the soft iron on the clapper and the end of the armature to prevent oscillation as the current fluctuates. The lamp is used to illuminate the rotating disc provided with a series of dots on a white background and the speed adjusted until one row of dots appears stationary. When this is the case one dot just moves up to take the place of a dot somewhere in front of it during the interval of darkness between the flashes of the lamp, i.e. during the period of vibration of the fork. The calculation of frequency is made as before. It is a great advantage to be able to avoid loading or marking the fork and to have the disc in any convenient position where it may easily be observed. Instead of a series of concentric circles with regularly spaced dots, it is more convenient to draw on a circular disc a series of concentric regular polygons. A small triangle is drawn just SOUND 403 about the centre of the disc, about this a square, then a pentagon, and so on. c The triangle may be coloured white, the space between it and the square blackened, the space between the square and pentagon coloured white, and so on alternately. When the/speed is adjusted exactly one of these figures appears stationary and the frequency is easily calculated. By varying the speed the figures may be made to appear stationary in turn and several determinations of frequency made. The Composition of Two Simple Harmonic Vibrations in the Same Direction (Beats) An apparatus which will combine graphically two simple harmonic vibrations in the same direction has been invented by Koenig. It consists of a large fork mounted on a stand and provided with a clamp by means of which a strip of glass can be held horizontally and fastened to one of the prongs (fig. 251). B Fig. 251 AB denotes the strip and C a weight, attached to the other prong for the purpose of balancing. The glass is coated with a thin layer of lamp-black, by means of a smoky flame. A second fork is mounted above the first and carries a light style adjusted so that it just touches the glass plate. This fork is fixed to a sliding base by means of which the style can be drawn along the smoked plate. If both style and plate are vibrating a curve can thus be traced which represents the motion of the upper fork relative to the lower. Thus if at any instant the lower fork is displaced a distance, y, from the standard position, and the upper is displaced a distance, y\ the displacement of the style over the plate is {y 1 — y). The forks are made to vibrate with nearly equal amplitudes. This may be done by bowing or by using a strip of metal which is wide enough to open out the forks to a convenient extent ; the metal is then quickly removed. The slider is drawn along, not too quickly, and the trace obtained examined. > When the forks have nearly the same frequency this will con- sist of a wavy line with waves of varying amplitude. In this case, with the amplitudes of the forks nearly equal, the smallest waves will have almost zero amplitude, and the fluctuation expresses graphically what the ear recognizes as beats. 4 o 4 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The time between consecutive minima measures the interval between the beats. If the glass plate is long enough three or four intervals extending between consecutive minima will be obtained. The speed of motion of the slider must be adjusted so that the individual vibrations are drawn out to an extent which enables them to be counted easily. It is best to count the number of intervals on the plate and the total number of vibrations between the first and last minimum points. This enables the mean number of vibrations between consecutive beats to be obtained. Let this number be x. We shall suppose that the two vibrations have frequencies n and (n + m) per second, so that they may be represented by : y 1 = a sin 2% (n + m) t and y = asm. (2nnt + a), where a is included to take account of any difference in phase that may exist when the vibrations begin. The resultant displacement recorded on the plate is : Y = (yi — y) = a {sin 2-n (n + m) t — sin (2nnt + a) } = 2a cos \2rtln -\ — \t +- isin (271— t — -V In the cases when beats are heard, m is much smaller than n or in + -\ so that the simplest way of regarding this expression is to consider it as a S.H.M. of amplitude : 2a sin (2tc • \mt — |a) = A (say), and then Y = A cosj 2* \n + ~J t + £<x • represents a S.H.M. of amplitude, A, and frequency, \n + ~J. The amplitude, A, attains a maximum value, 2a, and sinks to zero alternately. It is zero for the values of t given by : imt — |a = o, re, 2tc, etc., i.e. for values of t = ^ — ^ + -. ^ + - , etc., i.e. at intervals of— sees., or m times per second. m Thus the number of beats is m per second, the same number as the difference between the frequencies of the two notes. The frequency of the note heard is In + — V SOUND 405 (tn\ n + — J vibra- tions per second and m beats. The number of vibrations between two beats is : H» + -)• m\ 2 / But this is counted on the plate and found to be x. Thus — In -\ — ) = x, tn\ 2/ or n — m (x — \). The beats can be timed by means of a stop-watch. When the notes are sounding, count them for as long as possible, and take the time of the interval. If the number of beats counted is N and the stop-watch is started on the first and stopped at the Nth, the interval between the beats is : T N^T where T is the time interval recorded on the stop-watch. Thus -is known, and the above equation gives n. The M frequencies of the notes are thus, n and (n + m). The method must be regarded as an illustration of the phenomenon of beats — it is not an accurate method for the determination of frequency. B rf .P Fig. 252 The Composition ol Two Simple Harmonic Vibrations Perpen- dicular to one another. (Lissajou's Figures) Let the co-ordinates of a point, P (fig. 252), be (x,y) and let P move so that x = a sin pt y = b sin (pH + a). The motion of P then consists of two simple harmonic motions along two perpendicular directions. 406 ADVANCED PRACTICAL PHYSICS FOR STUDENTS When t is zero, x is also zero ; but y has the value b sin a. Thus the two S.H.M.'s are in different phases. From the equations it follows that : .1 . x t = — arc sin- p a and t = — arc sin \ — — • p 1 o p 1 Hence between x and y there exists for all values of t the relation : i . x I . y a - arc sin - = -— arc sin \ — -:• /> a p 1 b p 1 This represents the curve on which P lies. When the relation between p and p 1 is simple, as for example : p =p\ p = 2p\ 2p = 3P 1 , etc., the point, P, describes the curve completely in a short time, and afterwards retraces it. The case, pj= p\ is very simple, for we then have : x = a sin pt, y — b sin (pt + a). In the general case this represents an ellipse with its centre at the origin. If a = o, then : x y a = b' and P describes a straight line. 1Z If a = —, P describes the curve : 2 a 2 ^ 6» This is an ellipse whose principal axes lie along OX and OY. In the case when the amplitudes of vibration are equal along both directions, a = b and the locus of P is the circle : x* + y 2 = a*. Thus when the periodic times are equal, in general P describes an ellipse ; special cases of this general case are the straight line and circle. All these curves are therefore appropriate to the case of equal frequencies. If OB is equal to b sin a, B represents the position of P at the time t = o, or at instants later by a complete period than this initial time. SOUND 407 If the frequencies are not quite equal, let us say that the frequency along OY is a little the greater, then when another period is complete the displacement along OY will be a little different from that in the first case, i.e. P will lie at some point, B 1 . If OB 1 = b sin a 1 , a 1 is slightly different from a, and there has been a slight change in phase on the part of the OY motion. Thus a will continually vary, and will pass through all the values from o to 2n. This will cause a continual change in the shape of the curve described by P. It will sometimes be a straight line and sometimes an ellipse. If it happens that the two amplitudes are equal, we shall have a circle sometimes. The closer p 1 is to p, the slower will this change take place, so that by watching the movement of P we can test the closeness of the two frequencies. If the figure is maintained steady, without change, the frequencies are equal. Tfre same argument can be applied to the cases where one of the other simple relations exists between p and p 1 . When the shape of the curves corresponding to these relations is known the approximate ratio of the frequencies can be recog- nized and the exactness of the ratio tested by observing the rate of change of shape throughout the series. We can deduce the ratio of the frequencies by examining one of the curves, e.g. fig. 253. It cuts the Y axis in four points and the X axis in three, so that the frequencies are in the ratio, 4 : 3, for the vibrating point makes four vibrations parallel to OX in the same time that it makes three parallel to OY. Fig. 253 This principle may be employed to investigate the way the period of vibration of a rod, fixed at one end, varies with the length of the rod. The apparatus consists of a vertical flat rod or spring with a lens fixed at its upper end and a horizontal spring carrying a screen with a small aperture (fig. 254). 408 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Light from a source, S, is focUssed by a lens, L 1? on to the aperture, A, while the lens, L 2 , carried by the vertical spring, Vr focusses an image of the aperture on the screen so that we have a bright point, P. Motion of the horizontal spring alone, causes P to trace a vertical line, and represents the S.H.M. of A. Motion of the vertical spring alone, produces a horizontal S.H.M. on the screen. When both springs vibrate together, the path of P represents the combination of the twb motions. The vertical spring is of fixed length, but the horizontal spring can be clamped at various points, so that the vibrating length can be adjusted. Make the first adjustment so that P describes an ellipse, straight line, or circle without change of form. In this case the periods are the same. Now change the length until another steady curve is obtained without change of form. Draw it carefully and deduce the ratio of the periods of vibration. Make several determinations of the ratio of frequencies and corresponding lengths, and draw a curve showing the relation between the length and frequency, taking the vertical spring as a standard. Fig. 255, shows the curves described by P for a few frequency ratios which will serve for reference. The Vibration Microscope The essential features of this apparatus are the same as those described in the last experiment. The two vibrating springs are replaced, one by a fork and the other by a vibrator, the frequency oi which is to be compared with that of the fork. The second vibrator is sometimes another fork or violin string. SOUND 409 A bright source of light such as a speck of chalk is attached to the second vibrator, while the fork carries, attached to one prong, a lens which forms the object glass of a small microscope. If the lens alone vibrates, on looking through the eyepiece the motion of the chalk is simple harmonic, on account of the vibra- '/. 2 *> %. Fig. 255 tions of the lens. If the other vibrator is in motion and the lens is at rest, the motion observed is, of course, that of the vibrator alone. These two motions are arranged to take place in two perpendicular directions so that a figure of the type described in the last experiment is observed. It is steady if the frequencies are exactly adjusted, but goes through the appropriate series if the frequencies are not identical. The rate of progress through the series may be observed and determined by means of a stop- watch. After one completion of the cycle there has been a gain of a whole vibration by one vibrator over the other. If the frequencies be N and N 1 , and the time for completion of the series is t seconds, then N~ N l = 410 ADVANCED PRACTICAL PHYSICS FOR STUDENTS for N2 and N 1 ^ are the numbers of vibrations made respectively, and these differ by one. If the fork be slightly loaded we can find which is the greater of the frequencies by again observing the rate of progress through the cycle. If this time is shorter than before, the time of gaining a period is less than before and the frequency of the fork is the smaller. We shall describe how the apparatus may be used to find out the character of the vibrations performed by a stretched string in the manner in which Helmholtz used the apparatus to examine the vibrations of a violin string. P L P T Fig. 256 8 The string, SS, is stretched below the prongs of the fork, PP, to one of which the lens, L, is attached. The part of the string to be examined is slightly blacked by ink, rubbed with wax, when dry, and powdered with starch or chalk. A few white particles will remain sticking to the string, and one of them is illuminated by a lamp and focussed by the microscope and its movements observed. The tension of the string is adjusted until the figure apparently described by the chalk, as seen in the microscope, remains steady. The vibrations of the fork are electrically maintained while the string may be bowed or plucked. The frequency of the fork is known, so that, as its motion is simple harmonic, we can find the displacement due to it at any of the instants during the vibration. The string vibrates at right angles to the direction of vibration of the fork, so that Fig. 257 from the curve obtained we can subtract the vibrations of the lens and draw a curve showing the displacement of the string at different times. The white speck is first obtained in the centre of the field, and its mean position represented by the origin, O (fig. 257). SOUND 411 The curve is drawn to a convenient scale accurately from measurements observed by the scale in the eyepiece. If the microscope is not furnished with a scale, throw an image of an illuminated scale to lie coincident with the string and view the white speck and image together. In the figure, OA denotes the amplitude of vibration of the fork, its motion being assumed to take place along AOA 1 . If its frequency is n, the motion is given by : x — OA sin 2nnt. Thus for any time, t, we can find the displacement, ON, along the x-aods. At this instant the displacement along the other direction, i.e. due to the motion of the string, is NL. Thus by observing several values of the ordinates and the times corres- ponding, we can plot a second curve showing the time-displace- ment for the string. Its shape will indicate the character or quality of the note emitted. Fig. 258 shows the observed curve and the time-displacement diagram for a stretched string when bowed. The first curve was observed while the string was being bowed. The bow is drawn slowly and regularly across the string. Slight fluctuations are liable to occur during this process, but they appear as slight variations of a figure remaining on the whole permanent. The dimensions of this figure were obtained. Fig. 258 The deduction of the diagram for the string is made in the following way : Draw on squared paper the figure observed (fig. 258), and draw the extreme vertical tangents, AP and A 1 ? 1 . We are assuming that the vibrations of the fork are executed along AA 1 . With the middle point of AA 1 as origin, describe a circle on AA 1 as diameter, and divide the circumference into a convenient number of equal parts. In the diagram the number is twelve. If a perpendicular be drawn from any point on the circum- ference of this circle on to AA 1 , the displacement of the foot of this perpendicular from O will represent the displacement of the prong of the fork from its mean position. Along the line, A*A, produced, beginning at o, twelve equal 412 ADVANCED PRACTICAL PHYSICS FOR STUDENTS intervals are marked off, as shown, to represent intervals of time corresponding to the points on the circle. Beginning at A, draw ordinates to the curve passing through the points marked on the circle. For convenience in drawing only one of these, that through the point, 8, is drawn. Let this cut the curve at the point, Q, and the line, AA 1 , in N. Then NQ denotes the dis- placement of the string from its central position in magnitude and direction. This displacement is plotted at the point, 8, on AA 1 produced, and we thus obtain a point, q, on the displacement diagram for the string. This process is carried out for all the points on the circle, and the diagram plotted. It will be noted that the ordinate, NQ, cuts the curve in a second point, B. There is no doubt as to which point is to be taken when actually drawing the curve, for we begin at A and pass along one branch of the curve and back along the other. In this discussion we have associated the upper part of the curve from P to B and then to P 1 , with the times corresponding to the points from o to 6. Transverse Vibrations of Strings. (Melde's Experiment) The object of the experiment is to verify the laws of vibration of a string under tension. In such a case a disturbance travels along the string with a velocity, v, given by : \ m T denotes the tension expressed in absolute units, i.e. poundals or dynes, and m is the mass per unit length. When the string is fixed at both ends, there is a node at each end in its fundamental mode of vibration, with a loop or antinode in the middle. The corresponding wave length is twice that of the string. If this wave length is denoted by \ Xt we have : ni== t \S' «! denoting the frequency of this note. This mode is illustrated in fig. 259. The string may also vibrate to produce the overtones or harmonics as illustrated in figs. 260 and 261. In these cases, if the frequencies are n 2 and n z , and the cor- responding wave lengths x 2 and x 3 , we have : SOUND 413 n. x 2 \w' and Wo = and if / denote the length of the string : Thus n x : n 2 : n s : Fig. 259 In the experiment the string is set in resonant vibration by impulses having the same frequency as one of its modes of vibration. For this purpose ordinary string is unsuitable. It is not uniform and does not divide into equal segments; but it will be found that a length of fishing line is satisfactory, as a rule it is sufficiently Uniform. One end of the cord is attached to the prong of a tuning fork, by tying it to a small wire hook soldered on to the prong, or to a small screw which is held in a hole bored in the prong. The other end passes over a small pulley and carries a weight which produces and measures the tension. The vibrations of the fork are electrically maintained (see p. 138) and by properly adjusting the length and tension the string can be made to break up into stationary undulation with well-defined nodes. The fork may be placed so that the motion of the prongs is in the direction perpendicular to the string (fig. 262), or along it (fig. 263). 414 ADVANCED PRACTICAL PHYSICS FOR STUDENTS In the former case the frequency is the same as that of the fork, in the latter it is half as great. For when in the second case the prong is in the extreme position on the left the string is slack in the first vibration, and when in Fig. 263 the extreme position on the right it is horizontal and tight. The inertia of the string carries it onward so that when the prong returns to the extreme left position, and thus completes one vibration, the string completes a half vibration. The student should examine the interesting effects produced when the prong moves in a direction between these two and thus produces in the string a combination of the two modes of vibration. We shall consider the former case only in this description of the experiment. The frequency of the fork is denoted by N, and when the string is in resonant vibration, this is also the frequency of the mode of vibration of the string, corresponding, let us say, to a wave length, x. Then : N = £ J.2, x \ m £ =_!__ constant. Thus by varying the tension and consequently the wave X 2 length, we should find ~ constant. To determine X, measure the distance between the first well- defined node on the right of the string and the last on the left. Let this be d and suppose there are k loops between, then, _ 2 d X_ k ' Note that the ends of the string at the fork and the pulley should not be taken. There is a certain amount of movement at these two points. x" Draw up a table showing values of X, T, and „• SOUND 415 KUNDrS TUBE (a) The Determination of the Velocity of Longitudinal Waves along Rods Kundt's apparatus consists of a glass tube about a metre long, and of diameter about 3 cms., provided with an adjustable piston near one end. The tube is supported horizontally on a table by resting it on two wooden V-shaped stands. Near the other end of the tube is a second piston, Q, attached to the end of a metal rod, DQ (fig. 264). This rod is clamped at its middle point, C. A _B c D — * 3TFT I"1 r~7 IS~P z Fig. 264 For the purpose of the experiment the tube must be quite dry, and a light powder, such as lycopodium powder, is placed in a line at the bottom of the tuj>e extending along its length between the two pistons. A convenient way of inserting the powder, is to spread it along a metre rule, place the rule in the tube and turn it upside down. If the tube is not dry it must be warmed above a Bunsen flame, and a current of air blown through it. The metal bar, which may be of brass with a diameter of about •5 cm., can be set into longitudinal vibration by stroking it along CD with a piece of wash-leather and powdered resin. In the fundamental mode of vibration the ends, D and Q, are antinodes, and the fixed point, C, is a node. The wave length is twice the length of the rod. x = 2l. If the frequency of the note is n, and the velocity of the waves, v, n\ = v. If the distance between the pistons is L, the air between them will have a fundamental wave length, 2L, and overtones with corresponding wave lengths: L, f L, I L, etc The corresponding frequencies are : v y v_ 2L' L' |L ' etC " where V denotes the velocity of sound in air. If one of these frequencies is the same as that of the rod, the air will be set into strong resonant vibration, and will move the light powder. This will settle down at and near the nodes where 416 ADVANCED PRACTICAL PHYSICS FOR STUDENTS the air is least in motion. As many as possible of these should be used to find the average distance between two nodes. Choose as carefully as possible the position of a node at one end of the tube and locate the node nearest the other end. Measure the length between these two points, and divide by the number of spaces, such as NM (fig. 265). -ijiSis-.iiijjii-.ijjjjii. N M Fig. 265 The^pattern will be somewhat similar to that in this figure, and the longest line of each set marks the nodal position. It will probably happen that there is not strong vibration of the air at first, but by slowly moving the piston, P, forward or backward, the length between the pistons may be adjusted so that resonance occurs. Twice the distance between the nodes is the wave length of the sound in air. The velocity of sound at o° in air is 33,060 cms. per second, and at a temperature, t : v, = 33060(1 +^y (^ Thus the frequency, n x — — ', where x 1 is the wave length in A air. Since resonance occurs, n = n 1 ; 1 _ Y« **• 2/ ~~ x 1 ' or v =^V t - In carrying out the experiment it is a good plan for one observer to continue stroking the rod, while the other carefully adjusts the piston until the powder moves violently and settles down into the pattern of fig. 265. (b) The Veloeity of Torsional Vibrations in a Rod If instead of stroking the rod longitudinally with the resined cloth, it is held near the end, D, and the cloth turned so that it slips over the surface in a direction that would cause the rod to rotate round its axis, a note is emitted of different frequency from that given when the rod is in longitudinal vibration. This note corresponds to torsional vibrations and will set the air in resonant motion as before. SOUND 417 Find in this way the velocity of these waves. It is not easy to obtain a loud note by this method — the force applied should not be great, but, with a little practice, it should be possible to produce the note. (c) The Determination of Young's Modulus and the Modulus of Rigidity These constants may be determined from a knowledge of the velocity of longitudinal and torsional waves in the bar. The formula for the former is : and for the latter : «-£. *->£■ where E is Young's Modulus, n the modulus of rigidity, and p the density of the material of which the bar is made. Its value may be taken from a table of physical constants. When ex- pressed in C.G.S. units, E and n are expressed in dynes per sq. cm. Find the values of E and n from the determination of the velocities in the previous experiments. {£) The Velocity of Sound in Carbonie Acid Gas Kundt's tube may also be used to determine the velocity of sound in gases. Suppose the tube to be filled with a gas in which the velocity is V and the wave length, \. Then the frequency is Vj V. \ X 1 ' • Yi _ h. — distance between nodes in the gas V ~~ X 1 distance between nodes in air. The procedure is, therefore, first to obtain resonance between the rod and air column, and to find the mean distance between the nodes, then to drive out the air and fill the tube with the gas, and again obtain resonance, and measure the distance between the nodes in this case. \ a is then found from the last equation. The gas must be quite dry or the powder will stick to the glass and fail to respond to the motion of the gas when resounding. It may be necessary to pass it through drying tubes before filling the tube. A slight modification of the apparatus is necessary for this purpose (fig. 266). 27 4x8 ADVANCED PRACTICAL PHYSICS FOR STUDENTS In the former case the adjustable piston may fit loosely, but for the present purpose it must be both adjustable and gas-tight. It may consist of a cork round the outside of which a rubber band or piece of cloth is stretched. f*T* Q Fig. 266 The cork carries a tube to admit the gas and the piston may be adjusted by means of this tube. The metal rod passes through a tightly fitting cork, also pro- vided with a tube which can be opened and closed by means of a stop-cock. We shall suppose that the velocity of sound in carbon dioxide is to be measured. Connect the source of gas to the inlet tube, I, and open the exit tube, E, to the air. Allow the gas to flow in steadily so that it will fall to the bottom of the tube and the air will flow over at E, also, so as not to disturb too much the powder which we assume has been spread out along the bottom of the tube. The experiment should be performed close to an open window or in a draught cupboard to prevent escape of the gas into the room. Continue the passage of gas long enough to ensure that the air is driven out and then close the inlet and outlet tubes by means of stop-cocks, and proceed as in the last experiment. Care must be taken that the temperature of the gas is the same as that of the air, unless the temperature of the gas is measured in some way. If the gas is delivered from a cylinder it will be colder than the air and it must be allowed to acquire the air temperature before closing the stop-cocks. Otherwise we shall be measuring the velocity in the air at one temperature, and that in the gas at another. Reduce the velocity to that at zero. To do this, note that : ^ = ^ = ratio of nodal distances, where the affixes t and o denote temperatures. In order to determine when the air is all driven out from the tube, collect a little of the gas issuing from E, in an inverted glass cylinder over mercury, and introduce on to top of the mercury column a little of a solution of caustic soda or potash. By noting how much of the gas is absorbed it can be seen if any air is left. The C0 8 is all absorbed by the solution and no gas should be left. SOUND 419 (e) To Calculate the Ratio of the Specific Heats of a Gas From the result of the last experiment we may determine the constant, r, for carbon dioxide, i.e. the ratio of the specific heat at constant pressure to that at constant volume. For the velocity of sound in a gas at o°, is given by the formula : Po where p is the pressure and p the density of the gas at o° C. The value of p may be determined by the barometer since the tube has been filled at atmospheric pressure. p must be expressed in dynes per sq. cm. To make the cal- culation, take the density of mercury as 13*60 gm. per c.c, and P = '001974 gm. per c.c. The measurement of the distances between nodes may be performed simply by the ordinary use of a metre scale ; but a slight addition to the apparatus will add to the accuracy. A metre rule is fixed just below the tube and parallel to it, and sliding over the rule or along one of its edges is a wooden base, B, carrying a metal disc, D, with a hole, H, at the centre, and a frame, F, with cross-wires, C (fig. 267). Fig. 267 J H serves as an eyepiece and HC is aligned on the nodes marked out by the powder. An index on the base indicates the position of the stand. If the apparatus is aligned consecutively on the nodes, and the 420 ADVANCED PRACTICAL PHYSICS FOR STUDENTS positions of the index recorded, a table may be made out as follows : NO. OF NODE READING AT INDEX NO. OF NODE READING AT INDEX LENGTH OF 5 HALF WAVES I 2 3 4 5 6 7 8 9 10 Average Length of 5 Half Waves . . Mean Half Wave Length ... Mean Wave Length Chladni's Figures The nodes of a vibrating stretched string are points of its length where, theoretically, there is no motion. On either side of the node the string moves simultaneously in opposite directions. In the case of a vibrating plate there exist nodal lines, i.e. lines in the plate where there is no motion. They divide the plate into segments so that the parts on either side of the nodal line at any instant are moving in opposite directions. The point of support of a plate is necessarily on a nodal line. Round and square plates are usually supported centrally in the experiment, but interesting results arise when they are supported elsewhere. By touching any point on the edge or surface of the plate with the finger nail, the plate is prevented from vibrating at the point, and if it lies on a nodal line the corresponding mode of vibration may be excited. In the case of a circular plate clamped at the centre the nodal lines are radial, and the fundamental vibration gives two per- pendicular diameters. These may be obtained by sprinlding white sand over the surface of the plate, touching two points on SOUND 421 the edge separated by one quarter of the circumference, and bowing vertically across the edge at a point one-eighth of the circumference from one of the fingers. A square plate clamped at the centre gives a large variety of figures. Some of these are shown in the figure in order to assist in producing them. 111 J V ^ > r 13) . <l3l Fig. 268 ( 14) Some of the nodal points on the edge and surface of the plate should be touched, and the plate bowed along the edge at one of the points midway between the nodes. The fundamental vibra- tion gives two perpendicular lines through the centre parallel to the edges of the plate. The simpler figures are easy to obtain, but skilful manipulation of the bow is required to produce the more complicated ones. 422 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The student should obtain as many as possible, and clamp the plate at other points than the centre to investigate other modes of vibration. The Relation between Pitch and Volume In the ease of a Narrow- necked Resonator Consider the case of a bottle containing air closed by a piston, P, without friction in the neck of the bottle. Jil B Fig. 269 Let v denote the volume of the bottle below the piston, m the mass of the piston, and A its area of cross-section. Let the piston be originally in the position of equilibrium and let the pressure outside be p 9 , while that inside is p. Then we have : pA = ptA + mg. If now the piston be displaced downwards a distance, x, so quickly that the change may be regarded as adiabatic, a new pressure, p 1 , will be generated, such that : p 1 (v — A*)* = pv\ We shall suppose that x is small, so that we may write : The total force downward is now : mg + p<A - p x A == (p - p x ) A v Thus the equation of motion is : d % x pyA'' x. m W = x. SOUND 4*3 This is a simple harmonic motion and the time of vibration about the position of equilibrium is : 2w J mv (see p. 26) In the calculation we have assumed that the pressure is the same throughout the gas during the oscillations. This is not true since time is required for the transmission of the pressure. The other assumption concerning the adiabatic character of the compression is very approximately true, es- pecially as the neck is narrow and heat will not easily escape from the bottle or be transmitted to it. The piston will thus behave only roughly according to the formula, and will have an approximate period of oscillation given by the above value, i.e. it will have a frequency : T 27T \ £yA s mv Thus we have approximately : rih) = constant. It is the object of this experiment to verify to what extent the gas behaves according to this formula. The piston, in practice, is the layer of air in the neck of the bottle and by pouring water into it, resonance is produced be- tween it and a series of forks of known frequency. A medicine bottle will be found convenient and must first be calibrated by pouring in water from a measuring flask to various depths and measuring the height of the water surface above the bottom of the bottle. This should be done for a series of intervals up to the base of the neck. Make a table thus : HEIGHT OF VOLUME VOL. OF AIR ABOVE WATER POURED IN HEIGHT IN COL. I. *1 Vl v n -v x K v* f„-t> 2 K *>s v n -v 3 K t>n 424 ADVANCED PRACTICAL PHYSICS FOR STUDENTS v n denotes the capacity of the bottle. Draw a curve showing the relation between the heights of the water and the volumes of air above it, i.e. draw a curve with the values h as abscissae and the volumes in column 3 as ordinates. By means of this curve we can deduce the volume of air in the bottle corresponding to various heights of water. Forks of pitch varying from 256 to 512 should be used and a curve plotted with volumes as ordinates and values of — as abscissae. The resulting curve should be a straight line through the origin if the theory is correct, but in practice it will be found not to pass through the origin although the relation is very nearly linear. The curve will obey closely the law : n % (v -f- c) = constant. The value of c is a correction to be applied to v. This may be regarded as a neck correction, and the ratio of t to the volume of the neck should be recorded. The agreement of pitch between the fork and bottle should be tested by blowing across the neck and noting beats between the note obtained and that of the fork. The Interference of Sound Waves. Determination of a highly pitched Note by means of a Sensitive Flame When a continuous note is produced in front of a smooth wall the reflected and incident trains of waves produce stationary undulation in front of the wall, and consequently there exist nodes and antinodes in the air. The experiment consists in locating the positions of consecutive nodes from the wall out- wards. Thus the wave length is determined, for the distance between two consecutive nodes is half a wave length. By noting the air temperature and deducing the velocity of sound in air appropriate to it (eq. (A), §10 of this chapter), we may thus deduce the frequency of the note by the relation : V =wx. In order to locate the nodes a sensitive flame is used. This flame is produced by supplying the gas under pressure to a pin- hole burner. The requisite pressure can be obtained by leading the gas from the main into a large gas bag and placing a board on the top of the inflated bag to carry a weight. From the bag a pipe leads the compressed gas to the burner. When the bag is full, turn off the main tap and put weights on the board so that a tall flame from one to two feet high is pro- duced and there is no flaring. It will be found that in the sensitive state on making slight noises as, for example, by jingling SOUND 425 keys in the neighbourhood of the jet, the character of the flame changes. It flares and shortens, recovering its former state when the sound ceases. This is caused by the motion of the air as the sound wave passes. Start with the flame close to the wall and sound a note from a highly pitched whistle, or other suitable source, and move the flame slowly outward from the wall. It will be found that at certain points the flaring ceases and the flame increases in length. At these points the air is still, and the points are at nodes of the stationary wave motion. If a smooth wall is not conveniently situated, set up a large sheet of glass or smooth board at the end of a table, and move the flame outward from the surface along the table. The flame will probably need adjusting before it will respond readily. To do this use the tap leading from the bag to the burner, and also vary the weights producing the pressure. It appears to be necessary to use a rather long gas tube to convey the gas from the bag to the jet. In obtaining the most sensitive flame it is to be noted that the orientation of the flame is important and the burner should be turned about a vertical axis, so that different sides of the flame are presented towards the direction of the sound. The flame appears to have different degrees of sensitiveness on different sides. CHAPTER XV MISCELLANEOUS MAGNETIC EXPERIMENTS Measurement of the Pole Strength of a Bar Magnet, using a Grassot Fluxmeter The search coil of the fluxmeter (see p. 482) is placed on the bar magnet as shown in the figure so that it encircles the mid- point of the magnet. At this stage the reading of the fluxmeter is noted. Then if the bar magnet is uniformly magnetized, the coil, when withdrawn, cuts all the lines due to the pole past which it moves. The fluxmeter indicates the flux change in units which are specified ; in the case of the instrument described on p. 482 each division corresponds to a change of flux equal to 10000 maxwells. If there are x divisions change during the withdrawal of the search coil, and there are n turns of wire in the search coil, then since from a pole of strength, m, there are 471m lines, we have : iooooa; ioooo* or m = « The experiment should be repeated, using search coils having different values for n. This gives quite constant values for m as seen in the following experimental results. Coil A : n = 100. Initial reading of Fluxmeter 3 Final reading of Fluxmeter 46 Deflection, first experiment 49 Deflection, second experiment 50 Mean deflection 49-5 40-5 x 10000 m = -^-^ = 394-1. 47c x 100 J ^ Coil B : n = 8. Deflection, 4 divisions. 10000 x 4 m = -^8 " 3 ^4 mean value of pole strength, 394*25. 426 MISCELLANEOUS MAGNETIC EXPERIMENTS 427 Magnet was 1*58 cms. by -75 cm., i.e. 1*23 sq. cms. in cross- section, i.e. intensity of magnetization, assumed uniform, is : 394^5 _ 1-23 320-5. Distribution of Magnetism along a Bar Magnet This may be determined by using the fluxmeter in a manner very similar to that of the last experiment. The magnet is marked off in centimetres along its length, and the search coil is placed at the mid-point, around the magnet. The coil is then advanced in centimetre steps and the deflection of the instrument noted, i.e. for o to 1 cm., 1 to 2 cms., 2 to 3 cms., etc. The deflection in each case is proportional to the magnetization in the space moved over by the coil. The variation of magnetization along the length is seen by plotting deflection against distance from the centre. Gauss's Proof of the Law of Force The most satisfactory proof that the force between two magnetic poles varies inversely as the square of the distance between them was first given by Gauss. The method consists of a comparison of the magnetic force at a point on the axis of a magnet with the force at a point on a line drawn at right angles to it, at its mid-point. Let us first calculate the value of the force at two such points, assuming that the force between poles varies inversely as the l\ S, m£ Xk ■* ~JP Fig. 270 «th power, so that the force between two poles of strengths, m and m', r cms. apart is : mm' F = r" ' End on' position ('A ' position of Gauss) Let NS be the magnet (fig. 270) and P a point along the axis produced, such that the distance from P to O, the centre of the 428 ADVANCED PRACTICAL PHYSICS FOR STUDENTS magnet, is r cms. If the length of the magnet is 2.1 cms., we have the magnetic force at P, which is the force on unit pole placed at P, is : m m t, .= r A . (r - l) n (r + iy This is the net repulsion due to N and S on the unit pole at P, and is equal to : F A =*m F =-^ ^ ~ \ ~r) If we use a bar magnet such that / is small compared with r, we have : expanding the two expressions. Neglecting the powers of - higher than the first, this expression becomes : _ 2mnl _ Mw . ** A r n+l Jn+i' W where M is the magnetic moment of the magnet. ' Broadside on ' Position (' B ' Position of Gauss) Again in fig. 270 let Q be a distance r cms. from O, measured along OQ, which is drawn at right angles to the magnet at the mid-point. Due to the pole at N, there is a repulsion at Q along NQ equal to m , {\/(y a + / 2 )}' and a similar attraction due to S, along QS. The resultant of these forces may be obtained by considering MISCELLANEOUS MAGNETIC EXPERIMENTS 429 the isosceles triangle, QNS, as triangle of forces. The sides the resultant QS and QN each representing a force ^ 2ml + IH*' NSis i.e. F R = M {r> M f a + / 2 n+> yn+lK-j- 0' rf .(2) or when I is small compared with r. F --* Comparing equations, (1) and (2) we see that for such a short magnet the magnetic force at the two points, P and Q, are as n : 1. The numerical value oin is obtained by comparing the magnetic force at two such positions. A magnetometer is employed. This consists of a small magnet fastened in a light frame which carries a small mirror, the whole being suspended by a thin silk fibre in a cylindrical brass case as shown in fig. 271. Fig. 272 shows the usual form of small suspended magnet. Fig. 271 Fig. 272 The magnetometer magnet is allowed to come to rest, care being taken that all the torsion is removed from the silk suspension. Two boxwood scales, C and D, are arranged as shown in fig- 273, one in the magnetic meridian, and one at right angles to it. A lamp, L, and scale, S, are arranged in the usual way, as shown, so that an image of the lamp is reflected and focussed on the scale S. * The distance of S from the magnetometer mirror should be about one metre. When the magnetometer needle is arranged to swing freely in the centre of the case by means of the levelling screws shown (fig. 271), the position of the needle with respect to the ends of the boxwood scales can be very readily obtained, if the radius of the cylindrical case is measured. 430 ADVANCED PRACTICAL PHYSICS FOR STUDENTS If now a very small bar magnet is placed at any point, such as B, fig. 273, in an E. and W. position, a magnetic field will be set up at M, corresponding to the ' broadside on ' position. The distance, r, between the magnet centres being obtainable by § 1 1 1 1 1 1 1 1 1 1 u 1 1 ifcgfr A direct measurement on the boxwood scale. The deflection of the magnetometer is found by observation of the movement of the reflected beam of light. The magnet is then placed, again E. and W., at an equal distance from M, on the scale D (A, fig. 273), and the deflection again obtained. It is desirable to eliminate the zero reading of the magneto- meter ; the double deflection, left and right, may easily be obtained by reversing the magnet at the two positions A and B. The experiment is repeated for various values of r. For example, with a magnet 4 cms. long the values of the deflection at 50, 60, 70, 80 cms. may be obtained and tabulated as under. DISTANCE OF MAG- ' END ON ' (A) POSITION ' BROADSIDE ON ' (B) POSITION NET FROM CENTRE MAGNET- OMETER IN CMS. VALUE A - = n B SCALE READING MAG. DOUBLE DEFLEC- TION A SCALE READING MAGNET DOUBLE READING B DIRECT REVERS- ED DIRECT REVERS- ED 50 cms. I7'6 — l6'7 34-3 8.2 • -9.O 17.2 2.0 60 10-3 — IO'I 20-4 4-6 -5*1 97 2.1 70 6-5 - 6-2 127 2-8 -3'3 6.1 2-08 80 4.4 - 4-i 8-2 i-8 -2-3 4-i 2-07 90 3'3 - 2-9 6-2 i"° —2*2 3-2 i-94 mean 2-04 MISCELLANEOUS MAGNETIC EXPERIMENTS 431 The deflected position of the magnetometer needle depends on the relative strengths of the horizontal component of the earth's field, H, and the field due to the deflecting magnet. If 6 is the angular deflection produced, we have, H sin = F cos 0, or F = H tan 0. So that for the same value for r F\ _ n __ tan t F B ~ 1 "" tan 0, ' where X is the deflection when the magnet is at A, and 2 when the magnet is at B, in the ' broadside on ' position. If the distance from the scale S is fixed and the corresponding deflections are d 1 and d 2 , tan t _ d x tan 0, ~ d 2 ' d x i.e. n = -—- This ratio is tabulated in the last column of the table, and a mean value obtained. The Variation of Residual Magnetism with Temperature The object of this experiment is to investigate the behaviour of a magnetized carbon steel rod when subjected to temperature changes. Fig. 274 A rod of carbon steel of about 8 cms. in length and -5 cms. in diameter is magnetized between the poles of an electromagnet. The magnet so formed is set up inside a copper or brass tube, which is of slightly larger diameter, as seen in the upper portion of fig. 274. 432 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The magnet is drilled with a sma^lhole which receives a thermo- junction, as shown. This thermo-junction is calibrated as described on page 545 and serves to register the temperature of the magnet. Some simple heating device is arranged to vary the temperature of the magnet. In the diagram a second brass tube provided on the upper surface with a series of holes is shown, fitted on the end of a brass Bunsen burner. If a brass Bunsen burner is not available a simple heater may be made by taking such a second tube as the one shown, perforated with a series of holes, having one end closed and at the other a device for admitting an air-gas mixture similar to that of the common form of ' gas ring.' The magnet is set up at right angles to the meridian, and at a distance d cms. from a mirror magnetometer, of the form shown in fig. 271 ; the distance, d, should be such that at room tempera- tures the magnet produces a full scale deflection. The magnitude of the deflection is noted, and the temperature is obtained from the thermo-junction balance point on the potentiometer. The temperature of the rod is gradually raised by means of the heater, and the magnetometer deflection and the thermo- junction balance point are noted for every 20 C. rise, until the temperature is about i8o°C. At this point the observations are made much more frequently as the critical part of the varia- tion is being observed. At about 250 C. the readings may be again taken at about every io° C. or 20 C. intervals, and observa- tions continued until the temperature is at the maximum obtainable value for the heater employed. The results are tabulated and the value of the temperature obtained for each point ; the tangents of the corresponding angular deflections of the magnetometer are also tabulated. The magnet is then allowed to cool slowly, and the observations are continued until finally the specimen is once more at room temperature. The magnetometer is at a constant distance from the magnet, and in a fixed control field, H, hence the moment of the magnet is proportional to the tangent of the angle of deflection. This quantity is plotted against temperature, and a curve somewhat similar to that of fig. 275 is obtained. The firm line shows the relation between tan a and t° C. as the temperature rises, and the broken line the relationship as the temperature decreases. For carbon steel in general, it will be found that at about 200 C. the magnetic moment is reduced to zero. An increase in temperature causes a reversal of the polarity as shown in the curve. This negative moment is a maximum at about 210 C. and then decreases to zero at about 8oo°C. In general such a high MISCELLANEOUS MAGNETIC EXPERIMENTS 433 temperature will not be available with the heater described, but the most interesting part of the curve may be obtained. On cooling, the negative moment is a maximum at D, at a temperature about io° C. or 20 C. above the original position, and on regaining room temperature will have a small positive moment, corre- sponding to E in fig. 275. ioo C Fig. 275 The above experiment which demonstrates an interesting feature of the magnetization of such an annealed carbon steel rod is due to Prof. S. W. J. Smith and was described by him in the " Proceedings of the Physical Society of London," No. xxiv, 15 Aug., 1 91 2. Reference should be made to that paper, which gives an explanation of the observed results in terms of the magnetization of the iron and iron carbide molecules which compose the bar. Briefly, the iron carbide molecules are set in line on magnetiza- tion and exert a demagnetizing effect on the iron molecules which are reversed in the internal field due to the carbide. At about 210 the carbide is totally ineffective, but the reversed iron molecules being more retentive have a maximum external effect. As the temperature rises, the iron molecules approach their neutral temperature, and finally at about 800 entirely lose their magnetic effect. (See the original account in the place stated above.) The Increase in Length 0! a Bar on Magnetization To investigate any change in length produced in a rod of iron when magnetized, the following apparatus is set up. Fig. 276 shows a section of a brass case wrapped with two solenoids and provided with a water jacket, WW, through which water at a steady temperature may be circulated via tubes, T : the two coils, C, are wound together so that their magnetic effects 28 434 ADVANCED PRACTICAL PHYSICS FOR STUDENTS are identical at the axis of the cylinder. The free ends of each coil are connected to separate terminals, P, Q, R and S. The number of turns and mean diameter of the two coils are as nearly as possible the same, so that if a current were sent through from T w IZL W zdT PQRS Fig. 276 P to Q, the magnetic effect is the same as when the same current is sent from R to S through the second coil, or if the two coils are connected in opposition there would be no magnetic field along the axis. The method of winding and the resulting magnetic field should be tested at the outset by sending a current through the coils, and noting the magnetic effect on a compass needle held near the end of the solenoid. At the top of the brass case is a circular brass end-plate to which are soldered rigidly three pins, S, which terminate in points on which a sheet of plane glass, G, may rest. The iron specimen to be investigated is first thoroughly de- magnetized by heating, etc., or by reversals of a diminishing current through a solenoid surrounding it (as described on pages MISCELLANEOUS MAGNETIC EXPERIMENTS 435 449-451) . The specimen should be a cylindrical rod with plane ends made to fit the cavity provided along the axis of the solenoids. The end effects of magnetizing solenoid are overcome by first placing inside the apparatus a cylindrical length of brass, B, with plane ends, the lower end resting firmly on the thick brass base of the instrument. Above the iron rod is placed a second brass cylinder, A, also with plane ends ; the upper end projects slightly above the level of the upper surface of the end plate. A long focus lens is fastened firmly to this brass cylinder by means of soft wax or plasticine. The length of the supports, S, is such that a small gap is left between the upper face of the lens and the lower surface of the glass sheet, G. Above the parallel walled sheet of glass, G, is placed a second sheet, inclined at 45 to the normal. This reflects light from a sodium flame towards the lens and plane surface. The reflected beams are viewed by a microscope M, and the Newton rings formed are observed in this way. The microscope contains a cross-hair which is used as a point of reference. When a current is sent through the solenoids arranged in series, and to add their magnetic fields, the rod may be gradually magnetized by increasing the current strength. If any change in length occurs, there will be a movement of the Newton ring system, which may be observed in the microscope. It will be seen (by reference to page 323), that if the rod changes in length by - where X is the wave length of sodium 4 light, that one particular part of the field which was formerly dark will now be bright or vice versa. The direction of move- ment of the fringes will show whether the rod increases or decreases in length. Now, due to the current circulating in the coils, a certain amount of heat will be developed. Thus for a rod of 50 cms. length a rise in temperature of about -03 will cause an increase in length more than sufficient to move fringes corresponding to a change in size of the gap equal to -. It therefore becomes necessary to ensure that the rod is maintained constant in temperature throughout. Water at constant temperature is circulated through the jacket, WW, for some time before com- mencing observations and continuously during the experiment. Under these circumstances tap water from a supply removed from any hot water pipes will be admissible. As a further check on the observations, to ensure that the results are not spurious due to the heating, each observation is preceded by one in which the current to be used is sent through 436 ADVANCED PRACTICAL PHYSICS FOR STUDENTS the twp coils arranged in opposition, i.e. the heating which may affect the dimensions of the bar will be present, and any movement of the fringes is due to that cause, since there is no net magnetic effect. The coils are rapidly changed over, and the same current causes a magnetization of the bar. The true effect on the length of the bar due to magnetization may therefore be obtained To facilitate the rapid change in the direction of the current, a Pohl commutator may be employed : the ends of one coil, say, R and S, are connected to one pair of the terminals of the commutator ; the other coil is connected in series with a battery and variable resistance to the middle terminals. Thus by throwing over the movable arm, the coils may be changed from a condition of producing fields in opposition to one in which their magnetic fields are in the same direction. Carry out observations using currents up to a maximum value which is determined by the current-carrying capacity of the coils, CC. • Plot the results showing change in length in terms of the wave length of sodium light, against the field strength measured in Gauss. CHAPTER XVI TERRESTRIAL MAGNETISM Determination o! the Horizontal Components of the Earth's Magnetic Field (H) The method due to Gauss, described below, is usually employed to determine the horizontal component of the earth's magnetic field ; it may also be used to measure any magnetic field which is uniform over a sufficiently large volume. The method involves two experiments. In the first a magnet of known moment of inertia is suspended freely in the earth's field, at the place where H is to be found, and from observation of the time of swing, the product MH where M is the moment of the magnet, may be calculated. In the second experiment the field due to the same magnet is compared with the earth's field by means of a magnetometer. Determination of MH Let the magnet be suspended in a light stirrup and perform oscillations whose periodic time is T seconds, it being supposed that the suspension has no initial twist. Then if I is the moment of inertia of the magnet about the axis of suspension, and i the moment of inertia of the suspending frame about the same axis, and if t be the restoring couple per unit angular displacement due to torsion of the suspending fibre, we have : = 2*J '(MH + t)' {1) for (I + i) is the moment of inertia of the system, and (MH + t) is the total restoring couple per unit angular displacement, when small displacements are considered, whence : MH = A, (say) (ia) M To find jr A magnetometer is set up in the place at which MH was found, and the magnet placed with its centre d cms. from the centre of the magnetometer needle, east or west of it and lying east and west. If be the deflection of the magnetometer, and F the value of the field due to the magnet, we have : 437 438 ADVANCED PRACTICAL PHYSICS FOR STUDENTS F = H tan 0, and F = ^ 2 _j 2)2 > where il is the distance between the poles of the magnet. Alternatively, if the magnet were placed so as to be in the € broadside on ' position, i.e. E. and W. with its centre in the meridian through the centre of the magnetometer needle, we have F lf the field produced at the magnetometer, in this case is M ~ (d t * + /*)*' where d t is the distance between the centre of the magnet and the magnetometer in this case, and I is the distance between the poles of the magnet. Whence for the ' end on ' position Htan6 = (i2 _ /2 p (2) or for the ' broadside on ' position M Htan^ = p^47iji (3) M in either case the values of g may be found. Let this value be B. Whence from the two experiments : H -£ In performing these experiments special forms of magnetometer, etc., are advisable. All iron is removed from the neighbourhood at the outset, and care taken to maintain the magnetic conditions the same throughout the two experiments. The magnet used may very well be a true cylindrical bar magnet of not more than a few centimetres in length. Let the length as determined to & of a mm. be 2l x cms., and the radius, measured with a micrometer screw, be r cms, the mass of the magnet, weighed to at least one in a thousand, being m grammes. I, its moment of inertia about an axis through the centre of gravity and normal to its length, is —ftr+i) To find T, use is made of a light stirrup suspended by a fibre from a torsion head. A brass cylinder of the same mass as the magnet is first placed in the stirrup, and all the twist is then removed from the fibre. The torsion head is then turned so TERRESTRIAL MAGNETISM 439 that the brass rod is in the magnetic meridian. It is then replaced by the magnet. The frame is provided with a small mirror, otherwise one end of the magnet is polished to act as a mirror, and a lamp and T Fig. 277 scale is arranged so that a beam of light reflected from the mirror on to the scale enables the movement of the suspended system to be observed. Set the magnet swinging through a very small angle, and obtain the time taken for 100 complete swings. Repeat this and take a mean value, from which T may be obtained.* To find i in equation (1) weigh the stirrup and estimate the moment of inertia from the dimensions. The restoring couple due to the torsion, t, may be obtained by rotating the torsion head through, say, a complete turn. Note the resulting angular deflection of the magnet, y> radians, whence (2* — y>) t = MHy, MH V or T = 7 r- (27C — y>) It will be found that, if the fibre is thin and the stirrup a very light one, t and i are negligible compared with MH and I. Including these corrections we have : T = 2* I+* ^H(x +_£_)' * Alternatively, the method of timing set out on page 118 may be used. 440 ADVANCED PRACTICAL PHYSICS FOR STUDENTS whence MH 4* 2 (I + i) T 2 ^r~ 27t — \p The magnetometer experiment is performed, using a special form of magnetometer (Kew type magnetometer). This consists oi several small magnets rigidly fastened in a metal frame, which also carnes a small mirror. The whole is encased in a brass case, provided with suitably arranged glass windows so that a beam of light may be reflected from the mirror on to a scale S cms. away from it. The magnetometer frame carries a long bar which has four pegs, Pj, P 2 , P 3 , p 4) so arranged that when a small carriage is placed on any of them, the cylindrical magnet, which the carriage supports, lies with its axis in the same plane as the magneto- meter needle. Figs. 278 and 279 show diagrammatically the arrangement of the magnetometer. Dl ^a. -1 J S "i 1 1 1 1 ri 1 1 1 1 Fig. 278 M ?4. _J Fig. 279 The magnetometer and bar are first levelled (using a spirit-level) by means of the levelling screws on the base of the instrument. Pj and P 4 , P 2 and P 3 are fixed, in the construction of the instrument, equidistant from the needle when the system is level. Measure P X P 4 = D x cms., say ; measure P 2 P 3 = D 2 cms. The scale is then adjusted so that the reflected spot of light is at its centre when the magnetometer is under the influence of the earth's field alone. The arm which carries the pegs is set first of all in the direction of the magnetic meridian, and the small carriage in which the cylindrical magnet is carried is placed on V x and turned round so that the magnet is at right angles to the meridian. The true position for the magnet at right angles to the meridian is obtained by noting the deflection of the magnetometer. This will be a maximum when the normal position is acquired. TERRESTRIAL MAGNETISM 441 In this manner the deflection in cms. on the scale may be measured. The magnet is then rotated through 180 until a maximum deflection in the opposite direction is obtained in the magnetometer. Let these deflections be <5 X and <V cms., and the corresponding deflections when the magnet is supported on the other pegs be 6 2 , 6 2 ', etc. Obtain the mean value of 6 t d t ', <5 4 6 J — 6 say, that is the mean deflection for distance £D X — d lt ; also take the mean of #2 <V> ^3 <V — sl > sa Y> corresponding to a distance £D 2 = d 2 . If q> and <p' are the corresponding angular deflections of the magnetometer, we have : St St tan 29 = ^- ; tan 29' = ~-» Hence we may calculate q> and q>' from the observed deflections. The cylindrical magnet was placed as described above so that the field at the magnetometer was that due to the ' broadside on ' position ; thus from equation (3) above H tan * = w+w M i.e. g = (df + l*f • tan g> = (d 2 * + If tan 9'. H Now (i» + jy = rf»(i+!j!-. . .), neglecting the fourth and higher powers of -5. i.e. H = d, 3 I+ !(j,)^ tan "' M ■•'^.fCr i )fw. M H H Hence / may be eliminated, giving ( * - * A =d 2_ d , V^tany d 2 tan <p'J x 2 ' M Hence =5. is determined, and the value of H may be calculated using this and the 1 previously obtained value of MH. The bar carrying the four pegs may be alternatively arranged at right angles to the meridian, and the carriage containing the magnet again placed on the pegs with the magnet at right angles to the meridian, producing a field of force at the magnetometer corresponding to the ' end on ' position. The process described above is repeated, and the average deflections obtained enable 442 ADVANCED PRACTICAL PHYSICS FOR STUDENTS the calculation of the mean deflections, x and 2 , for the distances, d x and d v Hence again : TTi „ 2M I H tan X = ^r-T r x D x 3 2£ 2 ) d^ ' ' '] TT A „ 2M Htan 2 = ==r-. X D 2 S ( T _ i 2 ) ' to the same order as before. Once more eliminating I, half the distance between the poles of the magnet, we have 2 .M(_I j * ^ =^-^, H V*! tan ©j d % tan 2 / M whence ^ may be again calculated. This second method gives larger deflections and is therefore preferable to the first. Fig. 280 The Dip Circle— Measurement of the Angle of Dip The dip circle consists of a long magnet supported on a hori- zontal axis which passes (approximately) through its centre of gravity and is at right angles to its length. The magnet is sup- TERRESTRIAL MAGNETISM 443 ported in wheel bearings or on agate edges, so that it may turn freely and with a minimum of friction. On the same base as the support for the knife-edges is a vertical graduated circle which enables the position of the ends of the needle to be obtained. The whole structure is supported on a circular table, which is capable of rotation around a vertical axis. This rotation may be measured on a horizontal circular scale about which the table may rotate about its axis, and, by means of a suitable vernier, may be measured to at least one minute (see fig. 280). The position of the ends of the magnet may be read on the vertical scale by means of microscopes carrying verniers, which move round the scale. If the plane of the vertical scale be turned in the direction of the meridian, so that the horizontal axis of sup- port of the magnet is at right angles to it, the magnet sets in the direction of the earth's lines of magnetic force, and the angle included between the horizontal and the position of the needle is the angle of dip. ' However, there are many sources of error . The axis of rotation / . of the magnet may not quite coincide with the centre of the scale. One end, therefore, of the magnet would read too small, and the other too large a deflection. 2 The axis of rotation may not be truly through the centre of " gravity, in which case a couple is exerted, tending to turn the magnet so that the centre of gravity comes vertically under the axis of support. ^ The axis of the magnet does not usually coincide with the geometric axis of the needle, so that the centres of the ends of the magnet do not give a correct reading for the magnet when set along the earth's line. The possible errors are fully set out in any textbook of magnetism. A consideration of the observations taken and described below will show that these errors may be eliminated. To determine the angle of dip, the dip circle is set up and by means of the levelling screws on the base of the instrument, using a spirit-level, the 'horizontal ' scale is made truly horizontal. The plane of the meridian must then be found, so that the magnet may set freely. To do this, the instrument is first turned so that the upper, S, pole is at the 90 mark, and the reading of the vernier on the horizontal scale is noted. • It will in general be found that the lower, north, pole of the magnet is not quite at the lower 90 scale reading. A slight turn of the screw which rotates the instrument about the vertical axis is then made until the north pole is at the 90 reading, and the horizontal vernier scale is once more read. The circle is next turned through 180 , and the two more observations are made. The needle is now reversed in its bearings and the four 444 ADVANCED PRACTICAL PHYSICS FOR STUDENTS observations repeated. The mean value of the horizontal scale readings for these eight positions is taken. This is the mean position for the plane of rotation at right angles to the meridian. The case is then moved through an angle of 90 from this mean position so that the plane of the needle now coincides with the magnetic meridian. The following 16 readings are then obtained : (1) Read the position of each end of the magnet. (2) Rotate the instrument through 180 , and once more read both ends of the magnet. (3) Reverse the needle in its bearings and read both ends on the graduated circle. (4) Turn the instrument through 180 and again read both ends. (5) Remagnetize the needle and repeat the above process. The needle is remagnetized so that its poles are intercharged. The north pole for the observations, 1 to 4, above now becomes a south pole. Under these new conditions eight more values are obtained. The mean value of the 16 readings gives the true value of the inclination. The observations may be con- veniently tabulated as under. MARKED SIDE OF MAGNET FACING MARKED POLE DIPPING (NEEDLE READINGS) magnet remagnetized (needle readings) UPPER POLE LOWER POLE MEAN UPPER POLE LOWER POLE MEAN East West a P a' P' Needle reversed in bearings. East West Y <5 Y' d' mean q+ P + r + 8 4 == A Angle of Dip = mean A +B a' + P' + Y' + <r = B Note. — It may be necessary to obtain corresponding 16 values for the determination of the zero instead of the 8 described on pages 443-4' CHAPTER XVII PERMEABILITY OF IRON AND STEEL Measurement o! Permeability by the Magnetometer Method Consider a long thin specimen of iron wire placed in a magnetic fiield of uniform strength H. Due to the fields the soft iron specimen becomes magnetized, having poles of strength, ± m. Let , 2/ be the distance between the magnetic poles, s be the cross-section of the wire. The number of lines coming out from the north pole end is 47cm, and an equal number enter the south. Due to the poles developed, there are a number of lines of magnetic force passing from the south to the north in the specimen, equal to ^nm. This is equivalent to - — per sq. cm. of cross-section of the iron, assuming that the lines enter and leave at the ends only. In addition to the above there are H lines per sq. cm., due to the field, making the total number per sq. cm. of H + - — • This is the induction, B, in the specimen. Thus N The magnetic moment of the wire is 2,1m, its volume is s • 2/. Hence the intensity of magnetization, which is the magnetic moment per unit volume, is 2.1m _ m _ T 2.1s ~ s ~ Hence B = H + 4* 1 ( x ) The permeability (v) for any value^of H is defined by the ratio — • We therefore have : [L = 1 + 4« jj The ratio -=y is usually defined as the susceptibility or coefficient XX of induced magnetization, so we have, putting k for this quantity, jx = 1 -f- 4nk. 445 446 ADVANCED PRACTICAL PHYSICS FOR STUDENTS We have imagined the length of the magnet to be large. Other- wise there is an effect due to the poles at the end of the specimen to be considered, in expressing the value of H. For, due to the north and south poles developed in the specimen, there will be a magnetic field in the opposite direction to H, reducing the value to H 1 , say. This demagnetizing field can be neglected, however, if 2.1 is made large compared with s. A suitable length is obtained when 2X is greater than 200 times the diameter of the wire. If the specimen is arranged horizontally or vertically near a small magnet, suspended by an unspun silk fibre, the magnet will be deflected by the induced magnetism in the specimen. From a knowledge of the deflection and of H, the value of the constants, B, \k, I, k, may be calculated. To find the above constants for a given wire specimen, the type of reflecting magnetometer described on page 429 is used. The inducing magnetic field is obtained from a solenoid, through which a current of known magnitude may be passed, and the deflection of the magnetometer may be observed. Iron Wire Specimen arranged Vertically The arrangement of the apparatus is seen in fig. 281. L is a lamp and scale, M the magnetometer which is clamped to the wooden base shown, at a convenient distance from the specimen which is placed in a vertical solenoid, S. The wooden base is arranged in an east and west direction so that the magnetic effect of the specimen may be measured. Fig. 281 Before introducing the specimen, the solenoid is connected in series with a small coil, C, and several accumulators, a variable resistance, an ammeter reading to 3 amperes, and a reversing switch. The coil, C, is so arranged that the magnetic effect produced by it on the magnetometer is the opposite sign to that produced by the solenoid, S. PERMEABILITY OF IRON AND STEEL 447 The position of C is adjusted so that whatever is the value of the current in the circuit there is no movement of the needle. In that way the whole movement of the magnetometer will be due to the magnetism induced in the iron wire. The current is then switched off and, having noted the position of the reflected spot of light on the scale, the unmagnetized specimen is introduced. If there is any movement of the spot of light it may mean that there is a certain amount of magnetiza- tion in the specimen. It should be demagnetized by first of all heating to red heat and then hammering vigorously. Having once more replaced it in position in the solenoid, the current should be adjusted to about one ampere. The direction of the current should then be constantly reversed, whilst at the same time its magnitude should be reduced to zero. In this way any small residual magnetization may be neutralized. A preliminary experiment on a second sample of the same iron wire enables a value of the maximum current to be determined. The distance MS is adjusted so that a full scale deflection is obtained when the sample is saturated. The original specimen is then introduced so that its lower end is level with the axis of the magnetometer magnet. Starting with the soft iron unmagnetized specimen, the zero position of the magnetometer on the scale is observed. The value of the current is then raised to, say, '2 ampere and the deflection noted. Then without breaking the circuit the current strength is increased by increments of m 2 to a final value of about 3 amperes, a limit which depends upon the thickness of the wire in the solenoid winding, and also upon the result of the preliminary experiment, whichever limit is the smaller fixes the maximum current. Having increased the current to a suitable maximum value, it is then decreased by similar steps, again taking care that the circuit is not broken in the process, until zero current circulates. The current is then reversed and increased to the same maximum in the other direction. The current is again brought back in steps to zero, and having once more reversed the current it is raised to the maximum in the original direction. Tabulating these results a connexion is found between the value of the current in amperes and the magnitude of the deflec- tions or the value of the scale reading, which, when plotted, will show the general form of the curve below. The results are then converted to the corresponding C.G.S. values for B and H. In fig. 283 let n s represent the magnetized specimen which will produce at P the position of the magnetometer, a field 448 ADVANCED PRACTICAL PHYSICS FOR STUDENTS strength, F, at right angles to the Earth's horizontal component, and which causes the deflection, 0, say. We have F=H'tan0, where H y is the magnitude of the Earth's horizontal component. Fig. 282 Let m be the pole strength induced in the specimen, I be the length in cms., r the distance from the lower end of specimen to the magnetometer in cms. I I \ I I 1, \ \ \ u» r - Fig. 283 \ Then F = — — ^r-. cos SPn r* SP 2 m m r 2 r 2 + /* (r 2 +l 2 )* /i r \ V 2 (r 2 +/ 2 )v PERMEABILITY OF IRON AND STEEL 449 Whence : H'tane m = /L I \ ' \y2 ( y 2 + l^ij and I = -, B = H + 47rl. Further, using a solenoid of n turns per cm. having a current of c amperes, the uniform field H = - — is effective on 10 the specimen. Hence, for each reading in the table of results, I, B and H, also (x and k, may be calculated. If the curve is drawn for I and H, the area enclosed is equal to the work done on the specimen in the cycle (i.e., the heat developed). The values of H and B are in C.G.S. units, i.e. Gausses (see table on page 633). (2) The B-H Curve for a Sample of Iron (using a Ballistic Galvanometer This method is specially applicable to the determination of the B-H curve for a specimen in the form of an anchor ring, or a very short hollow cylinder. For such a specimen, magnetized by a magnetic force of, say, H gausses, no free poles will be developed, and therefore no demagnetizing field will be set up. The value of the induction, B, will therefore correspond to H, and not some smaller field, H', as in the previous case. The experimental arrangements of this method, as shown in fig. 284, are such that a variable field, H, may be set up, by passing a current through a primary winding, P, which is closely wound on the anchor ring, and some means of measuring B in the specimen. The method employed to measure B is to wrap a few turns of wire, S, round the anchor ring and primary winding and measure the quantity of electricity which passes through a ballistic galvanometer, BG, due to the change in the induction in the specimen for a known change in H. Since the ballistic galvanometer must be standardized, the secondary circuit is completed through a second small coil of a mutual inductance, M, used in the standardizing experiment. Thus the ballistic galvanometer is in a fixed resistance circuit in all measurements. The current from an accumulator may be regulated by resist- ances R x and R 2 and measured by an ammeter, A. By means of the Pohl commutator the current may be sent directly or reversed through the primary windings, P, when K 2 is to the left, or through the standardizing mutual inductance, M, when the key K 2 is closed on the right. 29 450 ADVANCED PRACTICAL PHYSICS FOR STUDENTS By closing the key K lt the resistance R 2 may be cut out. The value of H may be calculated, as explained later, from the value of the current strength as obtained from the ammeter ; and B may be calculated from the observed throw of the ballistic E -p; Fig. 284 galvanometer, BG. Since the galvanometer circuit is composed of a comparatively low resistance, a moving coil instrument may be too heavily damped to be serviceable (see page 481). A moving needle instrument is used if this is the case. As a preliminary experiment, the key K 2 is closed to the left, and R x and R 2 decreased, until, on closing the Pohl commutator, the galvanometer gives a full scale deflection from the zero, which should be the central graduation of the scale. The current required to do this is noted and is used as a maximum value in the main experiment. Usually from 2 to 3 amperes will suffice, the sensitivity of the galvanometer being adjusted to measure the throw produced. In general, the past history of the iron anchor ring will be such that residual magnetism in the specimen is almost certain. This must now be reduced to zero. To do this the galvanometer circuit is broken and the resistances R x and R 2 reduced to a minimum. The current passing in P is then reversed many times and R x and R 2 gradually increased until finally the current which is reversed is very small. It should be noted that R x and R 2 are such that changes in resistance may be brought about without breaking the circuit. PERMEABILITY OF IRON AND STEEL 451 Another method of demagnetizing the specimen is to pass an alternating current through P and a liquid resistance in series. The alternating current is gradually reduced to zero by with- drawing the electrodes of the resistance. The galvanometer is once more put in circuit with S, etc., by closing the key in the circuit (not shown in the diagram) : K x is closed, and R x given the value corresponding to the maxi- mum current as determined by the preliminary experiment. The commutator is then closed to the right, and the throw of the ballistic galvanometer, <5, noted. The reading of the ammeter is also noted. The throw will correspond to an induction, Bj, and the ammeter reading to a magnetizing force, H x , represented by some point such as S in fig. 285. We now use this as a point of reference. The galvanometer circuit is again broken, and the commutator is reversed rapidly some 20 to 25 times, and finally left on the right, i.e. the iron is taken several times round the cycle represented in fig. 285 and is said to be in the 'cyclic state.' BG is again put in circuit. When all is steady, R 2 is given a small value and K x is opened. The magnetizing field is thereby decreased and the throw of the galvanometer, d 2 , noted. This throw corresponds to a decrease in induction, Bj — B 2 . The value of the current corresponding to H a is noted on the ammeter. Fig. 285 Kj is now closed, the galvanometer key opened, the Pohl commutator reversed 20 to 25 times, and finally left to the right. The galvanometer is put in circuit ; R 2 is given a larger value ; K x is then opened and the throw due to the change in the induc- tion, say, Bi — B 3 , is noted ; the ammeter is again read. This process is repeated with the commutator to the right, until R 2 is infinite and consequently the current and H zero, i.e. the relation of B to H represented by the path, SA, of fig. 285, is investigated. After each measurement, the iron, by the reversal of the maximum current, is returned to the state represented by S, which therefore becomes the reference point. 452 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The key, K lf being closed, etc., the commutator is reversed some twenty times once more and finally left to the right ; R 2 is given a large value and the galvanometer is again put in circuit. The commutator is then thrown over to the left, and at the same time K x is opened, i.e. the current is reversed and at the same time made of small value. This gives a point on the part AB of the curve. The same starting point is taken (S), the drop in induction measured, and the change in H from a maximum position to a small negative value obtained. This process is repeated in many steps until finally R 2 = o, i.e., until the change in field, H, and induction, B, corresponds to a reversal of the maximum positive to maximum negative values. At this stage we may assume that the curve is symmetrical, and obtain the full hysteresis loop having obtained SABS 1 experimentally, or we may proceed to find SfA^S by experi- ment, using S 1 now as our reference point. The process is as before except that the commutator is now left on the left-hand side. K x is opened in turn to R 2 having values from o to infinity. CURRENT C IN E.M. UNITS FIELD H IN E.M. UNITS THROW OF GALVO. (<5) CHANGE OF INDUCTION FROM S OR S 1 INDUCTION (B) ASSUMING SYMMETRY (I) (2) (3) (4) (5) FOR CURRENT CHANGE Drop from S •24 to -24 = 2-o cms. 10000 •22 •24 to -22 = 1-5 cms. 500 9500 •20 •24 to «20 = i*8 cms. 700 9300 •18 •24 to- 18 = 2-4 cms. 1000 9000 . . . etc. +•02 •0 — 02 — 20 — 22 —24 •24 to — 24 = 20000, say — 10000 PERMEABILITY OF IRON AND STEEL 453 At this stage (point A 1 ) R 2 is given values from infinity to o. and Kj opened as the commutator is thrown over to the right- hand side. It should be found that the value of the throw for the reversal, when the full current is passing through the primary is identical for the change from S to S 1 and S 1 to S, and that either value is approximately twice the first recorded throw, <5 X . The results may be tabulated as shown. The original direction of the current in the coil, P, is taken as positive and the current value may be recorded in column (1) and the corresponding throw in column (3). To Calculate H Imagine a unit magnetic pole to be taken round the axial circle of the solenoid. If H is the value of the magnetic field strength due to a current, C, in the solenoid, the work done on the pole for such a complete circular path is 2i*r • H, where r is the mean radius of the anchor ring. If there are N turns in the winding, the unit pole is linked with each winding in the complete path, therefore doing work 47rC for each or a total of 4nNC, e.g. 471NC = 2TtrK H = ?^,* <D r where C is the current strength in E.M. units (not amperes). The value of the current in E.M. units (not amperes) shown in column (1) may be converted to gausses (lines per sq. cm.) by multiplying by the factor — as seen in equation (1). To calculate the induction, B, corresponding to the observed value of 0, an auxiliary experiment is necessary. The two-way switch, K 2 , is closed on the right, so that a current maybe sent through the long straight solenoid, M (about 40 cms. long and 4 cms. in diameter), which has m 1 turns per cm. Inside M is the second small solenoid which has been in series with S and the galvanometer throughout the preceding experiment. For a current of C E.M. units flowing through M, since the solenoid is long, the field strength at the centre is ^imfi gausses. *The result of (i) may be obtained from the general formula giving H inside a solenoid of n x turns per unit length, i.e. H = 4irtt]C. N In our case n x — -—-,> H r 3NC or 454 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Let the radius of the inner coil be r 2 cms., and N a the total number of turns in this coil, then the total flux in the inner coil is : 4izm 1 C(izr 2 e N 2 ) maxwells. If now the current be reversed in M the change in induction in the central coil is : 8n*m 1 Cr t , N 2 lines. Let R be the total resistance in the galvanometer circuit. dN Since the electromotive force is numerically ~rr# at r> <*N f ,, fdN Q = **n£*S mC (2) For an instrument of the moving-needle type : e = *G sm 2 ~( I+ S> Q = K 1 sini( I+ ^). T H where K t = — . ^-and is constant. 7C (jr Now for small deflections, if d is the scale deflection in cms. a sin oc d oc sin - and x is constant in the present case, even using a moving coil instrument, for the circuit is constant, i.e. Q = K 2 i; where K 2 is a constant (i.e. Qccd for either type of galvanometer) substituting in (2) above ^ = iL^a, or Kd = Stt^jN^C, where K = K 2 R == a constant. The last equation, which connects the deflection, d, with the total change in flux, enables the calculation of the value of B corresponding to each field H. The change in the number of lines, threading the entire circuit, corresponding to 1 scale division deflection on the galvanometer scale is : K ^ d PERMEABILITY OF IRON AND STEEL 455 The deflections 8 V <5 2 , etc., in column (3) therefore correspond, to a change in flux equal to Kd lt K<5 2 , etc. This is a change in the total flux in the space of the secondary winding S. If r x is the radius of cross-section of the iron anchor ring, approxi- mately that of the secondary winding, S, and m is the number of turns in the secondary, and (Bj — B 2 ) in the change in induction to be entered in column (4), we have, since B is the number of lines per square cm. of cross-section : (B x - B 2 )m • Ttf-j 2 = K<5, Bl-B « = ^'* ( a m r x 2 ) The term in the brackets is evaluated and used as a reduction factor, converting 6 to induction for column (4). To obtain column (5), assume a symmetrical form for the curve, i.e. assume that the induction for the maximum positive and negative current values are equally placed from the zero induction line, i.e. equal to half the induction for a full current reversal, which is the mean of the extreme values in (4). The method of calculation may be understood from the numbers shown in the suggested table. Note The B-H curve could also be obtained for an anchor ring as above, using a fluxmeter in place of the ballistic galvanometer. The calculation of H is as before. When the induction is large the number of turns in the secondary coil can be diminished, so that the fluxmeter reading is not excessive. The calculation of B from the observed deflection, 0, for any current change in the primary, is more direct than for the ballistic galvanometer. Unit scale deflection on the fluxmeter corresponds to a flux of 10000 maxwells. So that if the secondary coil used is of m turns, and the radius of each turn of the secondary is r lt the induction change (Bj — B 2 ) corresponding to the deflection 0, is given by : WX(Bi — B 2 ) Xtt/! 2 = 100000, „ -. 100000 i>i — Jt>2 = r« The experiment is carried out in a manner similar to that described above for the galvanometer as flux measurer. CHAPTER XVIII AMMETERS, VOLTMETERS AND GALVANOMETERS The most convenient form of current measurer is the ampere meter or ammeter, which indicates directly, on a graduated scale, the strength of the current passing through the circuit in which it is placed. As a result of the method of construction the range of useful- ness of the instrument is limited, in the usual form, to measure- ment of current not less than a milliampere. Recently, however, by improving the suspension, etc., the two-pivot instrument has been manufactured capable of measuring one microampere, for example, a Weston ammeter is made with an open scale division equal to one microampere. Similar conditions hold for the voltmeter. Fig. 286 The Voltmeter This instrument consists essentially of a coil of thin copper wire, C, which is supported on an axis pivoted on jewelled pivots, P, P 1 , and is free to move in the cylindrical gap between the soft iron pole piece, DD\ of an aged steel permanent magnet, NS. 456 AMMETERS, VOLTMETERS AND GALVANOMETERS 457 The field is strengthened and made approximately radial by the insertion of a soft iron cylinder; I, in the space inside the coil. This cylinder, as Seen in the lower fig. 286, is fixed to a brass bar, B, which forms a bridge between the pole pieces. The air gap is thus reduced and the coil moves freely in this space. To establish a restoring couple, the two hair-springs, H, H 1 , are attached to the axis, and to points on the framework which are insulated from each other. These springs also serve as leads for the current, to and from the coil. Attached near the upper spring is a light counterpoised pointer which moves over a scale G. The centre of gravity of the whole system is arranged to coincide with the axis of suspension. When in use the instrument is placed in parallel with the points whose potential difference is to be measured. The internal resistance of the instrument must, therefore, be large in order to avoid any appreciable rearrangement of current and potential drop in the circuit. The current passing through the voltmeter is therefore very small for such a high internal resistance, and therefore the heating of the coil is not very great. The internal resistance is not made up entirely of that of the copper coil. In most forms the greater part of the internal resistance consists of a resistance in series with it (fig. 287). The chief reason for this is to avoid any error due to heating in the moving coil. Such heating may be due to (a) atmospheric rise in temperature, ( p) the Joule (C 2 R) effect. An increase in resistance of the moving part would occur in either case unless the temperature coefficient of the wire were small. Manganin has a small temperature coefficient, but a higher specific resistance, i.e. for the same coil resistance a less radiating surface is available for manganin than copper. The effect of (a) in raising the temperature of the coil may be best eliminated by making the moving coil resistance fairly low, and hence the percentage change of the whole is reduced. Hence the usual compromise is a copper moving coil of com- paratively small resistance and a series manganin coil of high resistance. This series resistance is constructed of thicker wire than would be possible for the moving coil. Having in this manner secured the best approximation to constancy for the internal resistance, it will be seen that the small current, C, passing through the instrument is proportional to the potential difference between the terminals. Now the deflection produced is proportional to the current for a large range of deflection, when the field is radial, and hence the deflec- 458 ADVANCED PRACTICAL PHYSICS FOR STUDENTS tion produced is proportional to the potential difference between the terminals. The instrument is made ' dead beat ' by winding the moving coil on a copper frame. Seia Rfesisra^ce To >->pPer >b \ouo«r v P ?AVWVVWWVWWWV-1 B -3 Volts Fig. 287 H -»50 Volte The same instrument may be used to measure different ranges of potential. This will be seen from fig. 287. Thus if a potential of 3 volts, when applied to PL, produces a full scale deflection in the instrument, whose internal resistance (AB + coil) is 345*5 ohms, it will be seen that the current in the coil is -00865 ampere. Now, if a higher potential, say, 150 volts, is to be the new value corresponding to a full scale deflection, this may be con- nected to H and P so that a bigger series resistance, BC, is included, such that the total internal resistance, R, is given by •00865 = ^, or R is 17270 ohms. The same current will flow through the coil and hence the deflection will be again a full scale deflection. The graduations on the scale will therefore subdivide the o to 150 into equal increments, and each division corresponds to fifty times the value which corresponds to the lower voltage applied to PL. For the lower range voltmeter, measuring potential of the order of a millivolt, the value of the internal resistance is smaller, for the deflection is proportional to BC, where B is the magnetic flux in the gap and C the current. B is constant and so C to produce the deflection, when a low potential is applied, is obtained by decreasing R. AMMETERS, VOLTMETERS AND GALVANOMETERS 459 In such a moving-coil instrument the direction of deflection depends upon the directions of the current. The higher potential should be connected to P which is marked +. An accidental reversal of this order is apt to strain the needle. Ammeters To measure the current in a circuit, the measuring instrument used should be of low resistance, unless some account be taken (as in galvanometers) of the resistance introduced in this way. For example, if a small resistance of known magnitude, r, be included in the circuit, the current strength, c, may be found if the value of the potential drop, V, along r, be determined by V means of a milli voltmeter, for c = — . This arrangement of a T voltmeter shunted with a low resistance is utilized in the am- meter. The fixed-range ammeter usually contains the shunt inside the case. The value of the shunt resistance is small, and therefore the resistance of the whole instrument is of the same order. The moving coil of the ammeter is often, also, provided with a series resistance as in the voltmeter to minimize temperature variations as described above. Many of the better forms of ammeter are not provided with fixed shunts but require an external one. The value of the shunt resistance determines the range of the instrument. For bigger ranges the smaller is the resistance of the shunt. Suppose, as before, the maximum scale reading is obtained for a current, c, through the coil ; this is proportional to the potential difference between the ends of the shunt. It will be obvious, that if the external current, C, be doubled the drop of potential along the shunt will be doubled, so that if a shunt of half the original resistance replace the first, the potential drop will be equal to that which is required to send a current, c, through the coil and series resistance of the instrument, and so produce a full scale deflection. The shunts are made of manganin, which has a low temperature coefficient. The dimensions required in using a definite manganin strip may be calculated. If it is found that to produce no appreciable heating the shunt width has to be excessive, it is usual to construct the shunt of several strips in parallel. The instrument should never be used without the appropriate shunt for the current to be measured. The ammeters and voltmeters described above have the advantage of being direct reading on a calibrated scale ; they are robust and do not require any adjustment. But as indicated at the outset, the general type of instrument is not sufficiently sensitive to measure currents of less than a milliampere or 460 ADVANCED PRACTICAL PHYSICS FOR STUDENTS potential less than one millivolt ; but by more delicate con- struction and general improvement they can be made to measure to one micro-ampere and micro-volt. In such a case the extra sensitivity entails^ very precise work, and makes the cost of the instrument somewhat higher than for the ordinary range (i.e. one millivolt or ampere). Some forms of instrument are available which combine the voltmeter and ammeter. The necessary shunts and series resistance are contained inside the case, and by connecting to the proper terminals, the instrument may be used either as an ammeter, or as a voltmeter of several ranges (see, for example, fig. 288). Unipivot Instruments To increase the sensitivity of the above types of instruments, a modification of the support of the moving part was introduced by R. W. Paul. The pivot friction was reduced very considerably by the use of the one-pivot method of suspension, and at the same time all the advantages of the form of double-pivot sus- pension were retained, so that a sensitivity corresponding to one subdivision per micro-ampere is obtainable for the uni-pivot instrument. The construction is shown in figs. 289 and 290. A circular coil is suspended about a spherical core of soft iron between the poles of a permanent magnet. Fig. 290 Fig. 290 shows the detail of the coil support. A vertical spindle carries a light counterpoised pointer, and rests on a polished jewel at the bottom of a cylindrical hole drilled in the soft iron sphere. 1-'JG. 288 Fag* 4(10 AMMETERS, VOLTMETERS AND GALVANOMETERS 461 The cylindrical spring at the upper end has a very, slight lifting effect on the coil, and produces a restoring couple when the coil is deflected ; it also serves as a lead for the current to the coil.. The current leaves the coil by the flexible wire shown at the lower extremity. The centre of gravity of the moving part is at the point of support. A simple device is included to raise the point off the jewel when the instrument is not in use. This is shown under the coil in fig. 289, which gives the general appearance of the instru- ment when one pole piece and the magnet are removed. For many purposes it is necessary to be sure that the ammeter or voltmeter used is reliable to a fair degree of accuracy. The instrument may be calibrated in the laboratory (e.g. by potentio- meter and standard cell), but it is here suggested that each laboratory be provided with one form of ammeter or voltmeter (or both) which has been tested at the National Physical Laboratory, and is provided with a correction certificate. Such instruments should be retained at laboratory standard, and the working instruments checked against such standard instruments. Galvanometers When smaller currents are to be measured, use must be made of some form of galvanometer, an instrument which is not as robust as the above, it must be levelled before use, and further the current must be calculated from the observed deflection produced by it. Thus it is not as convenient and simple to use as the ammeter, but has a sensitivity which is impossible to attain in the latter. The increase in sensitivity is largely produced by a more sensitive method of suspension. The friction of the pivot is entirely removed by the use of a fine suspension of silk or phosphor-bronze. The suspension carries a small concave mirror by means of which small movements of the moving part may be magnified. The two common methods of producing such magnification are by use of (1) a lamp and scale, (2) scale and telescope. (1) Lamp and Scale Method In this method a beam of light from an incandescent lamp or Nerst filament is directed by means of a lens on to the concave mirror, which reflects it to a scale some distance away.* The greater the distance, D, between mirror and scale, the greater *The scale should be placed a distance away from the mirror equal to its radius of curvature. The condensing lens, over which is stretched a vertical wire, acts as an illuminated object whose image, a circular patch of light with vertical black line, is used to measure deflection. i _,. ^ 462 ADVANCED PRACTICAL PHYSICS FOR STUDENTS the magnification produced. When the mirror rotates through an angle, 0, the reflected beam moves through twice that angle, causing a movement of the spot of light, say, d cms. on the scale. Hence tan 20 =■=> To measure such deflection it is essential that the mirror should produce a clear image. For this reason, with the size and character of the mirror available on such a suspension, the usual maximum value of D is one metre. (2) The Telescope and Scale Method A scale is set up horizontally at about one metre from the galvanometer mirror, and a telescope, usually under the central graduation of the scale, is turned towards the mirror. When the mirror, which should be a plane one for this method, is parallel to the scale, the latter may be seen in the telescope. The reading of the scale in coincidence with the cross-hair in the eyepiece of the telescope is noted. When the mirror moves a second scale reading will coincide with the cross-hairs. The difference for an angular rotation, 0, of the moving system may be accurately measured. If this is equal to d cms. then, once more, tan 2 = ^- It is advisable, for simplicity of reading, to use either a telescope with an erecting lens or prism or to use a scale provided with ' inverted ' graduations, so that the scale appears the right way up. Sensitivity For many reasons it will be apparent that the instrument cannot be calibrated once for all. The scale distance is variable , the suspension may of necessity be replaced, and so on ; there is, therefore, no permanent direct reading scale as in case of ammeters and voltmeters. The usual method of converting scale readings to the corres- ponding currents is to obtain the ' sensitivity ' of the instrument. The sensitivity of a galvanometer may be defined in many ways. In addition to being a ' reduction factor,' converting scale divisions to current, as indicated above, the sensitivity also gives an indication of the possibilities of the instrument. How- ever for a comparison of two instruments another factor will be discussed later. Current sensitivity may be defined in either of the two following ways : (a) The number of mms. deflection produced on a scale one metre away by one micro-ampere, i.e. by io- e ampere, or ( p) the number of micro-amperes required to produce one mm. deflection on a scale one metre from the galvanometer mirror. The second definition is perhaps from some points of view AMMETERS, VOLTMETERS AND GALVANOMETERS 463 the better, but (a) gives a value which varies directly with the property measured. It must be understood that a sensitivity of 1000 mms. per micro-ampere means only that a very small current produces a deflection which would correspond to 1000 mms. deflection when a micro-ampere passes, assuming the deflection to be still proportional to the angular movement of the coil. It is purely a theoretical mode of expression. The deflections measured never actually exceed from 5° to 8°. Volt sensitivity may be similarly defined, substituting micro- volt for micro-ampere in the above. It will be seen that for a fixed current strength the current sensitivity is greater for a bigger coil resistance, i.e. a bigger number of terms. Volt sensitivity, which deals with a fixed potential applied to the galvanometer, is the greater the smaller the value of the resistance of the coils. Determination of Current Sensitivity The galvanometer whose resistance, G, has been determined, is connected in series with a megohm (10 6 ohms) and a cell of known electromotive force, E. The deflection on the scale one metre away is noted = d x mms., say. E x 10 * Now the current passing is-; — micro-amperes; from which the number of mms. deflection produced by one micro- ampere may be calculated. An alternative method is shown in fig. 291. Fig. 291 A steady accumulator whose electromotive force, E volts, is known accurately — either by a potentiometer comparison with a standard cell or by measurement with a calibrated voltmeter — is connected in series with a high resistance, R ohms, through a commutator, C, to a low resistance, S, which is in parallel with the galvanometer of resistance, G. 464 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The mean deflection d, say for both positions of the com- mutator is obtained. The current causing the deflection is readily calculated, for SG the effective resistance of the shunt and galvanometer is u . - ; the main circuit current is therefore equal to* ^Wo W where B is the battery resistance which, if an accumulator is used, is negligible compared with R. Hence neglecting B in (i), the current through the galvanometer is E S C« = cT — x /o _i_ r\ X io 6 micro-amperes (S+G) E'Sxio 6 micro-amperes (2) R(S + G) + SG . . . . d Hence the sensitivity is ^- mms. per micro-ampere. Thus the sensitivity having been obtained the deflection produced in any case may be converted to the corresponding current values so long as the suspension remains the same and the scale is at the same distance, one metre, from the galvano- meter mirror. STEADY CURRENT MEASUREMENT Galvanometers for the measurement of steady direct current may be subdivided into two general classes according to the nature of the moving part: {a) moving needle, (b) moving coil. (a) Moving-needle Galvanometers The student will be familiar with the simple tangent galvano- meter. In this form a small magnet is suspended at the centre of a coil of wire of n turns. When a current of C, expressed in electromagnetic units passes through the coil, placed in the magnetic meridian, the magnetic field set up causes a deflection, 0, such that the restoring couple due to the horizontal component of the earth field, H, balances the couple due to the magnetic field of the current, and we have : Hr C = — tone, (3) where r is the mean radius of the coil. In general the assumptions upon which the formula is developed ■of* * N.B. — If S is very small, c.f. R, the factor „ , ^ may be omitted as this S + G ES 106 is a little less than S. If further S is small, c.f. G, (2) becomes j^ s ^tg) AMMETERS, VOLTMETERS AND GALVANOMETERS 465 are not justified, and corrections should be applied to allow for the width of the coil, etc. ; such corrections are but seldom used, and the galvanometer is not used as a small current measurer. A modification, known as the Helmholtz galvanometer, consists of two similar coils placed on a common axis at a distance apart equal to the radius of either. A short magnet is supported at the mid-point between the centres of the coils. As with the tangent galvanometer, the plane of the coils is placed in the magnetic meridian, then using the same notation as before, C^-^tanS; (4) 327m n is tne number of turns in each coil. These two forms are such that the absolute value of the current may be simply calculated, but they have no claim to great sensitivity. Sensitive Current Detectors The result expressed in (3) for a simple tangent galvanometer shows that, for this form of instrument, the deflection for a given current may be increased by (a) decreasing H, the control field, (p) decreasing r, (y) increasing n. There is a limit to methods ((J) and (y) which is prescribed by the practical problem presented. Further, as r is decreased and n increased, the conditions upon which (3) was developed no longer hold. (a) may be best carried out by the use of an external control magnet which neutralizes the value of the horizontal component of the Earth's field. Further, by using an astatic pair of magnets or groups of magnets the control effect may be reduced without interfering with the magnitude of the deflecting couple. The above conditions are embodied in the Thompson (or Kelvin) and Broca galvanometers. The Thompson Galvanometer The Thompson galvanometer is illustrated in fig. 292. The control magnets, EE, enable the galvanometer to be used in any position independently of the Earth's field, the suspended astatic system is shown at the right of the diagram. Each set of magnets is placed at the centre of a pair of coils, FF. The magnets are mounted on a light rod, the whole being supported from a torsion head by an unspun silk or a quartz fibre. 39 466 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The magnitude and direction of the control field may be varied by alteration of the positions of EE along the vertical rod shown. Sets of coils of different resistances are usually supplied. Using high-resistance coils a sensitivity of about 600 mms. per micro-ampere may be attained, i.e. the instrument will detect currents of the order of io~ 9 ampere. The movement of the magnet is detected, using the small mirror, B. Fig. 292 The Broea Galvanometer This instrument is shown in fig. 293. The moving ' astatic pair ' is made up of two vertical magnets having consequent poles as shown. The control field is due to the magnet, B, which moves in a ball socket to any desired position. The coils, EE, may be of any suitable resistance, say, 10, 100 or 1000 ohms as required for sensitivity in the circuit. They are connected in series, one at each side of the spaCe occupied by the centre magnet, and arranged to produce a field in the same direction. As in the Thompson galvanometer, an aluminium vane, G, moves between two parallel plates which may be adjusted by the rods terminating in the metal knobs, CC. The damping of the system may be altered by an adjustment of the distance between these plates. Currents of the order of io -10 ampere may be detected, AMMETERS, VOLTMETERS AND GALVANOMETERS 467 The Thompson and Broca galvanometers are most advan- tageously employed as very sensitive current detectors whose sensitivity may be rapidly adjusted over a wide range by the control magnets. Both forms require levelling before using. Fig. 293 (b) Moving-coil Galvanometer The moving-coil galvanometer is constructed in a manner very similar to the ammeter already described. A phosphor-bronze strip, F, fig. 294, acts as suspension to the coil, C, which is free to move in the gap between the pole pieces of a permanent magnet, one of which is not shown in fig. 294, and a soft iron cylinder, I, which is screwed to a brass plate as seen in the lower figure. In this case the magnetic field in the air gap is approxi- mately radial, as shown by the broken lines in the lower figure (fig. 294). The current enters the coil via the phosphor-bronze strip, F, and leaves at the under end by means of a helix of phosphor- bronze, S. The control is mainly due to the twist of F in such a case. If B is the magnetic flux in the air gap, A the area of one turn, n the number of turns, and t the restoring couple per unit angular twist of the suspension, F, we have, for the case of 468 ADVANCED PRACTICAL PHYSICS FOR STUDENTS a radial field, when a current, c, passes, causing a deflection, 0, BnAc = t6, (5) Gc = t6 G = BnA and is the couple on the coil for unit current and is called the galvanometer constant, or c = kQ, when k is a constant and equal to =^-i . BwA era Fig. 394 For small deflections, 6 = tan 0, i e. © is proportional to d, the deflection on the scale. From (5) above it will be seen that to increase the sensitivity, t must be made as small as possible, and B, n and A as large as possible. To decrease t we may decrease the cross-section of the strip or increase its length. For a circular wire it was shown on page 102 R* Ttt 7' The most profitable method of decreasing t is therefore to reduce the cross-section, i.e. use a fine suspension. This is limited by the fact that very fine suspensions are also very fragile and somewhat difficult to use. From this point of view Fig. 295 Pagt 469 AMMETERS, VOLTMETERS AND GALVANOMETERS 469 phosphor-bronze is the most satisfactory material and is almost universally employed . The usual type of moving coil is supported by a fibre of the greatest convenient length. The increase of B, n, and A must be considered together as the terms are interdependent. Assuming the magnet to be fully saturated, the value of B in the air gap depends on the size of the gap. The smaller the air space, the larger is B. The effect of increasing nK is to increase the size of the coil, and consequently the air gap, and hence decrease B. In practice it is usual to use a standard size of air gap which allows a frame of such size to be wound with wire, and move freely in the available space, that the best compromise between these two effects is obtained for a maximum value to the product B-ff'A. An example of galvanometers of this type is shown in fig. 295- Such instruments may have a sensitivity as great as 1500 mms. per micro-ampere, i.e. the instrument will measure currents of the order io -10 ampere. Adjustment The moving-coil instrument must be adjusted before use. The coil is first released, then the instrument is levelled by means of the levelling screws, so that the coil does not touch either the pole pieces or the iron core, and is thus able to swing freely. The instrument may be used in any position, and the coil is turned by means of a torsion head, which carries the suspension, until the plane of the coil is approximately parallel to the sides of the magnet. It is inadvisable to make this adjustment, unless the reflected beam does not fall on the scale, for there is a danger of breaking the suspension. Onwood Moving-coil Instruments The Onwood galvanometer differs somewhat from the above general types in that it requires no levelling, is not so fragile, and is more convenient to move. The difference is in the method of suspension as seen in fig. 296. NS is the permanent magnet with a soft iron core, d, which is drilled to the centre. From this point, h, is a short suspension c, which is of sufficient length to support the coil, a, from the point, g, clear of the fixed parts. A small mass, e, at the end of the rod, /, keeps the system vertical. 470 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The current is led through the frame to the iron core, and thence through the suspension, and the tube, b, to the coil which it leaves by means of a flexible ligament at the base of the coil, but not shown in the diagram. Fig. 296 Since the coil is supported at the centre it will be equidistant from pole pieces and iron core alike for all positions of the instrument not too far removed from the horizontal, i.e. the instrument does not require levelling. A small spring, not shown, at the upper end of the suspension protects the latter against sudden shocks. The galvanometer is a very small one and the sensitivity claimed ranges from 20 to 500 trims, per micro-ampere according to construction details. THE CHOICE OF A GALVANOMETER FOR THE MEASUREMENT OF A STEADY DIRECT CURRENT To select the galvanometer most suitable for a particular experiment many factors must be considered. To decide firstly between the two general types described above, their relative advantages and disadvantages are discussed below. Moving-needle Type (1) The value of the control field is affected by the proximity of external magnetic fields. The instrument may register a deflection when no current circulates through the coils when AMMETERS, VOLTMETERS AND GALVANOMETERS 471 magnets are moved in the laboratory, or dynamos set in motion, or even by a passing electric tramway car. Thus there is some uncertainty unless such stray fields are eliminated. (2) The moving magnet may have its moment altered by the field set up when a current passes through the coil. (3) Unless the instrument is made ' dead beat ' the needle takes a long time to come to rest and is therefore somewhat troublesome to use. This may be overcome to some extent by using an external damping circuit, consisting of a solenoid through which a current may pass when the circuit is closed. This solenoid is placed near the case of the instrument and the circuit closed momentarily as the needle swings, in such a way as to produce a field which opposes its motion. (4) The damping of the instrument is independent of the resistance of the circuit in which it is placed. (5) The needle may be supported by means of a quartz sus- pension which has the property of returning after deflection to its original position. However, the slight torsion in the fibre is usually neglected, and the expression for the current is therefore not strictly accurate. Moving-coil Type (1) The moving-coil galvanometer has a large permanent magnet, and is therefore practically unaffected by stray external fields. (2) No demagnetizing effect is possible on the moving part. (3) The movement of the coil may be very rapidly arrested, even when the instrument is not ' dead beat,' by connecting the ends of the coil together through a very low resistance as the coil passes its rest position. (4) The electromagnetic damping varies with the value of the resistance of the external circuit. If used in series with a low resistance this damping is very high, and the coil may take several minutes to attain the full deflection, a fact which is very often overlooked when the instrument is used. (5) The zero-keeping quality of the suspension depends upon the degree of sensitivity attained. The torsion at the fibre is taken into account in the expressions for the current. Having decided, from the conditions of the experiment, the type of instrument most suitable, the next problem is, What is the most satisfactory galvanometer resistance ? The Order of Resistance of a Moving-coil Galvanometer which is most sensitive in a Given Circuit In general terms we may state that if the current is to be of a fixed value, independently of the galvanometer, considering all 472 ADVANCED PRACTICAL PHYSICS FOR STUDENTS factors, the sensitivity will be proportional, approximately, to VG, i.e. will increase with increasing resistance. If a fixed potential difference is to be measured the sensitivity will obviously be greater the smaller the resistance ; approxi- mately, the sensitivity is proportional to —j^, i.e. high resistance i VG for detecting current, and low for detecting small potential differences. The problem usually presented is, given an external resistance, R ohms, what is the best value for G, the galvanometer resistance for measuring the current due to a fixed electromotive force. We saw (pages 467-8, equation (5)) that the couple due to the current c is nABc, and if E is the electromotive force in the circuit -ETG (6) Further, it was shown on that page that in a galvanometer there is but a limited space between the pole pieces to obtain a maximum sensitivity. Fig. 297 shows a cross-section of the frame which will just move freely in this space. Let the cross- section of the whole of the windings be a sq. cms. and assume that the windings entirely fill the space with copper. If p be the mean perimeter of the coil windings, we have length of wire used = np, cross-section of the wire = — r n Therefore G = — *- = ■ * • a a n where o is the specific resistance of the wire, say, copper, i.e. ft = VT (7) ypa Hence the couple due to the current = BwAc T> A 155 _E from (6) and (7). AMMETERS, VOLTMETERS AND GALVANOMETERS 473 The condition for the couple to be a maximum and therefore produce a maximum effect is that BAEVoT. VG Vp^ R + G be a maximum. We have seen that a and p are constant, so that the condition is p , „ is to be a maximum, K -f- (j •p or that —7= -f- y^Q be a minimum, i.e. R = G. That is, under the circumstances stated, the maximum sensitivity is obtained when the galvanometer resistance is equal to the total eocternal resistance. It should be noted that the galvanometer resistance in this discussion refers to the copper coil resistance only. The resistance of the suspension should be included in the value R. This being so, it is apparent that there is a lower limit beyond which the resistance of a coil may not be reduced with any advantage. If the galvanometer is to be chosen as a detector or measurer of small differences of potential the most suitable instrument will be one of low resistance (e.g. for thermo-couple work). Other qualities of galvanometers to be considered when making a selection of galvanometers are : — (1) Damping The damping of the moving part in a galvanometer apart from external artificial agency may be considered due to two separate causes. (a) The damping due to the viscosity of the air. This is present in moving coil and needle alike, and is approximately proportional to the angular velocity of the system. It is always present, but is usually small. (b) Electromagnetic damping. In the case of a moving magnet the amount of damping due to this cause is very slight when the magnet is in a non-metallic case, e.g. when the coils are wound on wood or ebonite. This is the reason for the long and troublesome wait which occurs before the needle returns to its rest position. This may be reduced as described under. However, in either case the amount of damping is obviously independent of the external circuit. In the case of the moving-coil instrument, the suspended coil, when closed by an external circuit, is moving in a strong magnetic 474 ADVANCED PRACTICAL PHYSICS FOR STUDENTS field. Under such circumstances the electromotive force induced in the circuit sets up a reverse current in the closed circuit, which is therefore brought to rest. The value of the damping current depends on the magnitude of the external resistance, and may become very great for a low series resistance. For many purposes it is necessary or convenient to have a galvanometer such that the moving part very rapidly returns to the zero position after being deflected. A galvanometer having this property is said to be ' dead beat.' In both types of instrument this may be brought about by increasing the electromagnetic damping. In the moving-needle type this is accomplished by encasing the needle in a copper, or similar metal, case. The movement of the needle sets up eddy currents in the copper and the magnet is rapidly brought to rest. The moving coil may be rendered dead beat by winding it on a metal ' former ' or frame. This constitutes a closed metallic circuit, and the desired result is obtained. Alternatively, if the galvanometer is wound, for ballistic purposes, on a non-metallic frame, e.g. bamboo, the same result is obtained by facing the coil with a thin sheet of copper foil (cut into a ' picture frame ' which is the same size as the edge of the coil). (2) Period and Constancy of the Zero A galvanometer which is very sensitive has a long time of swing ; and also, due to a very fine suspension, some trouble may arise due to the ' creep ' of the zero. However, the type of instrument used in experiments in this book will not be of the extremely sensitive order at which this trouble arises. The main cure for the trouble lies in the selection of suspension, and that really involves the selection of an instrument maker who will take the trouble to minimize this fault. Beyond this the correc- tion of residual effect must be solved by the ingenuity of the experimenter as applied to the particular experiment involved. Quartz for moving magnets and phosphor-bronze for moving coils cause least trouble in this respect. For average work 5 to 10 seconds per complete swing will be best value for direct steady current measurement or for use in ' null ' methods. MEASUREMENT OF QUANTITY OF ELECTRICITY The Ballistic Galvanometer A galvanometer suitable for measuring a quantity of electricity is called a ballistic galvanometer, and has the following essential features : AMMETERS, VOLTMETERS AND GALVANOMETERS 475 (1) The periodic time of swing, T, of the moving part is fairly large. (2) Damping of the moving part is very small. The first condition is fulfilled by making the moment of inertia of the needle or coil which forms the moving part as large as practicable and by reducing the controlling forces, for — £ where I is the moment of inertia of moving part, and t is the restoring couple per unit angular displacement. Thus by increasing I and decreasing the restoring forces, T, the time for one complete swing, is increased. The second factor, damping, is reduced in a way which depends on the instrument (needle or coil). As seen when considering damping (page 473) the electro- magnetic damping only may be reduced. The air damping is usually small. (3) A third condition is, that when used to measure a quantity of electricity, the whole of the transient current shall pass before the needle or coil moves from the zero position. Should there arise a case in which the quantity of electricity to be measured takes longer time to traverse the instrument, due, for example, to inductance in the circuit, the time of swing of the needle must be increased, by loading it, so that this third condition is fulfilled. As indicated above, the galvanometer may be of the moving- needle or moving-coil type. We shall develop a relation between the throw or angular deflection in either type, and the quantity of electricity which passes. Moving-needle Type This type of ballistic galvanometer consists of a needle sus- pended by a fine quartz or unspun silk fibre, at the centre of two coils, through which the quantity of electricity, Q, passes. The control in this case is either the Earth's field or a control magnet. The needle in its zero position is arranged at right angles to the axis of the coils, so that when a current passes a field is set up at right angles to the control field. Let G be the galvanometer constant, i.e. the field due to the coils for unit current circulating through them, H the value of the control field strength, I the moment of inertia of the magnet about the axis of suspension, M the magnetic moment of the magnet. 476 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Suppose a current of strength, c, to pass through the coils for a very small interval of time. Since the third condition above holds, the needle will be at right angles to a field of strength Gc, and will experience a turning moment, GcM. This couple will produce an angular acceleration in the needle. Hence (page 53) IB .= GcM, 16 = GM fcdt = GMQ (8) At the end of the swing, the needle having turned through an angle O , the kinetic energy of the moving needle, \1Q % , has been reduced to zero in doing work against the magnetic force, Hra (m being the pole strength of the needle) at each pole. The work done is 2Hm f cos O J = Kml (1 — cos O ) = 2MHsin 2 -^> 2 where I is the distance between the poles. Thus 2MHsin 2 -^ = -I0 2 , 2 2 l0 2 = 4MHsin 2 ^ (9) Hence squaring (8) and dividing by (9) T Q 2 M 2 G 2 , x 4HM sin 2 ^ We have also for the period, T, of the suspended needle in the control field, H, t T'MH , ^ I= -^- («) Equating (10) and (11) : Q 2 MG 2 = T 2 MH „ . ' O "" 4T5 2 4H sin 2 -^ ^ ^ 2 Hence : ^ T H • e o Q = «'G' sm i •••• < I2 > AMMETERS, VOLTMETERS AND GALVANOMETERS 477 In developing this result we have assumed that the original kinetic energy of the needle is wholly used in moving the magnet against the field, H, through an angle, O . If, however, there are any frictional forces, i.e. the needle is slightly damped, some energy will be required to overcome this force, and the result will be that the observed angle of swing, say, a, is not truly of the magnitude given in the undamped case. The true value for Q is therefore dependent not on the observed value, a, of the angle, but on some slightly bigger angle O . It will be shown that O = a(i +-)> where xis the logarithmic decrement of the suspended system, o-"-B(-+i)] ™ Moving-coil Ballistic Galvanometer There are several disadvantages in the moving-needle instru- ment as in the case of measurement of steady direct current, e.g. (1) and (2), pages 470-1* These defects are overcome in the moving-coil type of ballistic galvanometer. The electromagnetic damping is reduced by using a coil wound on a bamboo frame ; T is increased by using a fine phos- phor-bronze strip suspension. To obtain the connexion between Q, a quantity of electricity discharged through the galvanometer, and O , the first throw, we may first of all assume that there is no damping. Lei: G be the galvanometer constant, i.e. the couple acting on the coil when unit current passes through it. t the restoring couple due to the suspension, for unit angular displacement. I the moment of inertia of the suspended system about the axis of suspension. As for the moving magnet galvanometer (p. 475) let c be the value of the current at a time t, then Gc = I0 (13) Integrating Gfc . it = GQ = 10 (14) Considering the kinetic energy of the coil, we have, if is the original angular velocity given to the coil by the impulse due to the discharge of a quantity of electricity Q, K.E = £I0 2 478 ADVANCED PRACTICAL PHYSICS FOR STUDENTS This energy is used in twisting the suspension through 0. At any angular displacement 0, the restoring couple is t0 ; to twist through a further angle dd, the work done is t0 . dd ; i.e. the total work done in deflecting the coil is f. -cd.de =^ (15) 2 Thus we have t0 o 2 _ 10* 2 ~~ 2 I0 2 = T0 O * (l6) Squaring equation (14) and dividing by (16) we have I = ^H- (17) Again the time of swing of the coil is given by T = 2* -yjl- or * = 7T Substituting this value in 17 T 2 t = G 2 Q2 4* 2 ~" t0 o 2 Q -f £-t (l8) or if a is the observed first swing, and x is the logarithmic decre- ment, o-H-K'+i) ;•■«*■» Suppose now we send a steady current of known magnitude, c, through the galvanometer and observe the steady deflection, <p, Gc ■-« T9» AMMETERS, VOLTMETERS AND GALVANOMETERS 479 or _ = - Hence Q=l.t.*( I+ ?) (i 9 ) 76 -K <p 2\ 2/ v It should be noted that the preceding paragraph gives a very ready method of finding the ballistic reduction factor for the galvanometer. We see that ^ , , , T x steady current Q = ka where k — — J „ — -, — 2re X steady deflection Subsequent deflections may be converted to quantity by multi- plying these deflections by k. See also page 481, equation (23a) for the moving magnet type. To Correct the observed first swing (a) for damping in either form of galvanometer. The observed first deflection, a, is reduced by damping forces which are proportional to the angular velocity. The equation of motion, therefore, must be modified from the simple form : IB + F0 = o, where F may be t or MH, according to the type of instrument, to include a term proportional to the angular velocity. Let this term be K0, then the equation of motion becomes : 10 + K0 + F0 =0, an equation similar to that already dealt with on page 27. _K The amplitude of the system is therefore O e 3l ', using the notation of page 161. _K T Thus the first observed swing, <x x = 6 Q e *i 4. K T ' Logarithmic decrement, x = -^ • - , 2X 2 i.e. a x = O e~\, or O = a i<^ etc ) In a ballistic galvanometer K is small and although T may be KT rge, I is also large, and -^ is small, i.e. X may be neglected in comparison with unity, KT large, I is also large, and -^ is small, i.e. X 2 and higher terms 4 1 4 8o ADVANCED PRACTICAL PHYSICS FOR STUDENTS i.e. . ».-«(i+i)- x may be obtained by one of the methods of pages 162 to 164, and the correcting factor fi -f - ) obtained. The galvanometer must be in the circuit of the experiment when x is obtained, so that the damping is the same. Another way of correcting for damping in the moving part does not involve an independent measurement of x. Suppose that in addition to noting the first swing, a lr we observe the next swing on the same side of the zero, a 3 , we have : K T _K 3T _k 5T ai = o e~*i'*; cc 2 =d e *"*; a 3 =0 e *i"4,-etc; or & = ^ = e + § T (20) a 2 a 3 K T ( + * T )-* i.e. a. x =0 e-* 4 = © |^ aI ; or by (ao) a ' = 9 »t-^' ie - *•— (£)' (3I) Thus, suppose there is an error of one per cent in the deter- mination of an angle, f — J is liable to two per cent error or (— ) may be in error by one-half per cent. The correction for damping obtained in this way will be sufficiently correct for many experiments. Of course a x , a 2 , and <x 3 are not measured as angles. The corresponding scale deflections, <5 X , <5 a , and <5 3 , are measured. Now — = tan 2(Xl for small deflections, d 3 tan 2<x 3 i.e. since under these circumstances the value of tan 2a is approximately the same as 2a, _£i __ «i. <5 3 a 3 So that (19) for the moving-coil galvanometer becomes Q_t.i.«*(«!y (22) * re <p 2 \a 3 y and (i2«) AMMETERS, VOLTMETERS AND GALVANOMETERS 481 o-Sff-f-®'! <*3) In the above method the damping under the conditions of the experiment is obtained for the correction factor. Equation (23) may be modified to conform with (22). For if a steady current, c, produces a deflection q>, cG = H tan y, i.e. Q=7 t w s +Cl)l (23fl) To Reduce the Excessive Damping in a Moving-coil Ballistic Galvano- meter used in a Low-resistance Circuit It was shown in the discussion of damping that the moving- coil galvanometer has a large amount of electromagnetic damping when in a closed circuit of low resistances. This would make the use of a moving-coil ballistic galvanometer inadmissible in many experiments unless some means were taken to reduce the damping. The quantity of electricity passing through the instrument is, in general, due to some current change. If, therefore, the galvanometer is in circuit whilst such change takes place, and is then allowed to swing freely when it has received the impulse due to that cause, the damping is almost eliminated. '////A i=i , o '///A ^ B WA S JZZL E Fig. 298 By use of a compound key, such as that shown in fig. 298, this may be brought about. The three brass strips mounted on an ebonite block (shaded) are connected to separate terminals, as is the stop S. The battery circuit is connected to A and S, and the galvanometer to C and B. When the key is depressed,' C and B, A and S are connected, but B and A remain insulated by the ebonite stops shown (shaded). When the key is released A and S are broken ; impulse, due to induction or whatever cause is operative, is given to the galvanometer, and then C and B are disconnected by the upward move of the key. If the time interval between the break of the battery and galvanometer circuits is small compared with the period, T, of the coil of the galvanometer, the latter will swing an amount which is almost independent of electromagnetic damping due to the low resistance circuit. 3* 482 ADVANCED PRACTICAL PHYSICS FOR STUDENTS If such a key is used for a moving-coil instrument the damping correction used should be obtained from observations of the coil when swinging freely in open circuit. The Grassot Fluxmeter The Grassot Fluxmeter is an instrument which performs the same function as a ballistic galvanometer, as for example in experiments on pages 426, 557, 582, 357, etc. However, the instrument is specially designed for measuring magnetic field strengths directly. Fig. 299 It is a suspended coil instrument which depends entirely on the electromagnetic damping for control. The coil, D, is supported by a single cocoon silk fibre, which has a negligible torsional control, from a flat spiral, E (fig. 299), to eliminate the effect of shocks. The current enters and leaves the coil by the silver strip coils, H, as shown in the figure. The usual iron core, B, and permanent magnet pole pieces, N, S, supply a constant magnetic field in which the coil swings. The light frame, FF, carries a pointer and a concave mirror, not shown in the figure. The points, T T ', correspond to terminals on the case of the instrument. In general use a search coil, C, of known mean area and number of turns is connected to T,T'. The search coil is placed into the magnetic field to be measured and the induced electromotive force causes a current to flow in the closed circuit and so produces a deflection which may be measured either by a lamp and scale arrangement, using the Fie, 300 Page 483 AMMETERS, VOLTMETERS AND GALVANOMETERS 483 concave mirror, or, if sufficiently large, by direct reading of the pointer over a scale. The instrument therefore produces a deflection which is proportional to the total quantity of electricity which passes through it. Fig. 300 shows the general appearance of the instrument in use to measure the magnetic distribution along a magnet. The method of calibration of the graduated scale and general possibilities of the instrument will be apparent from a con- sideration of the theory of the instrument. Let R = the resistance of the circuit, i.e. of the search coil, C, and leads, and the suspended coil, D, L = the self-induction of the whole circuit, I = the moment of inertia of the coil, D, about the axis of suspension, E = the E.M.F. induced in C ] c = the current in the circuit I at any instant, v> = the angular velocity of the coil J K = induced E.M.F. set up in D for unit angular velocity, G — the galvanometer constant, i.e.. the couple exerted on the coil due to unit current passing through it, A = the couple due to air resistance for unit angular velocity (see damping, page 473.) Expressing Ohm's Law for the circuit, we have : cR = E - IT- - Ko>, .(24) i.e. -i(*- L s-*-> The equation of motion of the coil is : t d * e r A , ^ 1 dtf =Gc ~ A< * -to) Substituting the above value for c and remembering that dd Ci>= — dt' T ^6> _ GE _ LG dc _ /GK \ dt R R dt \R +AJ G> - Now L, G, R, K, and A are constants ; further, if the coil starts from its rest position when no current passes, and becomes 484 ADVANCED PRACTICAL PHYSICS FOR STUDENTS deflected through an angle, a, by the quantity discharged through it, the coil is at rest at the end of the swing when no current again passes through it. Therefore, integrating the last equation over the whole swing with respect to t, we have ■[•I-S/"-¥['T.-<?«)[-]:- /M,?(!f + A).. fEdt = (^ + K) a (26) Thus it appears from (26) above that the value of the deflection, <x; is determined by / JLdt, since the other terms are constant. For example, if the search coil is placed in a magnetic field, thereby setting up an induced E.M.F. in the circuit, the total deflection is independent of the speed of insertion or withdrawal of the search coil. AR Now in general the value of A is small and the term -^~ may be neglected in comparison with K. And hence we have : Edt= Koc, i.e. or / approximately. With this approximation the deflection a is independent of the value of R, i.e. the search coil may be replaced by another, provided that R does not become very large for then the approxi- mation is not justifiable. For the purpose of this account the noteworthy feature is that the deflection is, as for the ballistic galvanometer, proportional /■p g • dt, which passes through the coils, i.e. a is proportional to the quantity. The standard form of fluxmeter has a scale, graduated in maxwells ; this is graduated experimentally. The deflections corresponding to known flux change define points on the scale, and since a is proportional to f Edt, which itself is proportional to B, the number of lines cut, the subdivision of the scale between such fixed points is a matter of dividing angles into equal parts. The experimental arrangements for the instrument in measur- ing magnetic flux and quantity of electricity are described in the chapters where such measurements find a place. AMMETERS, VOLTMETERS AND GALVANOMETERS 485 For many of the experiments for which the instrument is used the graduated scale is too coarse. Use is then made of the mirror, using a lamp and scale arrangement as in the ordinary galvanometer. The deflection on the scale is proportional to the quantity of electricity discharged through the instrument. A full scale deflection is produced in the instrument by a definite flux change in a search coil ; the position of the reflected spot of light is noted, as is the movement of the pointer on the grad- uated scale. By repetition, the corresponding values are obtained. From such observation the value of the flux change, corresponding to 1 mm. scale deflection (at, say, 1 metre), is deduced from the graduated scale deflection. MEASUREMENT OF VARYING CURRENT The preceding account has included the method of measuring (1) steady direct current, (2) quantity of electricity. We now consider briefly methods available for the measure- ment of varying currents. Such currents may be subdivided into (a) currents of short duration at regular or irregular intervals, (b) alternating currents. (a) To detect currents of this kind, and also to measure the time interval between such short duration currents use may be made of The Einthoven Galvanometer The principle of this instrument may be understood from fig. 301. A 'string/ CC, of fine platinum or tungsten wire is supported vertically between the poles of an electromagnet. If a current be sent down the string, the latter will be deflected m a direction parallel to the face of the pole pieces. In the diagram, the direction of movement is shown by the arrow, a, for a field in the direction, SN. To observe the movement, the pole pieces of the magnet are drilled, as shown by the broken lines, in a direction parallel to that of the magnetic field. Light from a bright point source is concentrated by a condenser, CF, placed in one hole, and the movement of the illuminated string is magnified by a telescope, DE, placed in the other. The " string ' is usually attached to the ends of two small springs which keep it stretched. The tension may be altered to any desired amount by a micrometer adjustment at one end. The string and tension-varying device are supported on a frame which may be removed bodily from the gap between the pole pieces. * 486 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The deflection produced for a given current depends upon the tension on the string and the strength of the magnetic field. The former may be varied as indicated above, whilst the latter may be adjusted by regulation of the current flowing through Fig. 301 the coils of the electromagnet. The sensitivity of the instrument may be therefore varied over a wide range quite simply. At a fixed sensitivity the deflection is proportional to the current. The usual method of detecting deflections is to use a photo- graphic arrangement. The beam of light emerging from ED if allowed to fall on a screen forms a shadow image of the string. This is reflected on to a cylindrical lens which forms an image of the central portion of the line, i.e. if the string is still and a photographic plate is moved vertically at the focus of the lens a straight white line is produced on the plate when developed. If a current passes, the string moves, and a shift of the shadow image results. The point focus of the cylindrical lens is thereby deflected. This results in a lateral displacement of the lme image on the photographic plate. The magnitude of the displace- ment gives a measure of the current strength. The natural period of the string is small and it rapidly returns to the rest position when the current ceases to flow (not more than a few hundredths of a second is required). Thus if a succession of small currents pass in the circuit, the instrument detects them, even when but a few hundredths of a second interval occurs between successive currents, whereas an ordinary galvanometer would not distinguish the break between them. To measure the interval of time between successive impulses, or the duration of one of them, a time scale is imprinted on the record of the current by means of a 'time marker' as shown in fig. 302.' This consists of a device for intercepting the light at Fig. 302 Page 4&G AMMETERS, VOLTMETERS AND GALVANOMETERS 487 regular intervals and thereby making transverse white lines across the photographic plate on which the current is recorded. A small motor is made, as shown in the figure with a soft iron armature of, say, 10 teeth. Intermittent current is supplied to the electromagnets by connecting them in series with the circuit of an electrically maintained tuning fork, the impulses given to the motor are therefore regular. Suppose the fork vibrates 50 times per second, the synchronous motor will rotate 5 times per second. This drives a circular disc provided with projecting arms as shown in the figure. These arms are usually allowed to move across the beam of light illuminating the apparatus. The figure shows five such spokes, the fifth being broader than the others. A spoke intercepts the light 25 times per second and makes a time scale on the photographic plate of ■£ g second. By making 20 spokes to the wheel T hs second graduations may be obtained. The width of a division may be varied by varying the speed of movement of the plate. For a continuous record a cinematograph film may be used. Of course a number of strings may be used, and the image focussed simultaneously on the same film by placing a 45 right- angled prism in the path of each image. The whole is therefore concentrated on to the slit in the camera box. The arrangement of the parts is seen in fig. 303 which shows a plan of the apparatus. prisms ~_ Fig. 303 (b) Alternating Current Two types of instrument may be used for the measurement of alternating current : (1) The oscillograph which gives the wave form of the current. (2) Ammeters or dynamometers which give the effective current. The Duddell Oscillograph Fig. 304 shows the essential features of this form of instrument. A thin phosphor-bronze strip, ss, is supported over a small ivory bobbin, P, and fastened at the lower ends. The tension on the strip may be adjusted by regulation of the tension on the spring suspension of P. 488 ADVANCED PRACTICAL PHYSICS FOR STUDENTS ■ If a current is passed through the loop, ss, the two strips will suffer a deflection in opposite direction, and consequently rotate a mirror, M, which is attached to both. If the current direction is reversed, the direction of rotation is also reversed. Fig. 304 Thus, for alternating current the mirror would rotate back- wards and forwards, provided that the natural period of the loop is small compared with that of the alternating supply. The image of a source of light reflected by the mirror on to a scale or screen by M would therefore be drawn out into a line for such an alternating current. If the image were focussed on to a photographic plate in a camera, and the plate were moved at constant speed in a direction normal to the beam of light and the direction of vibration of the image, the trace on the plate would be approximately a sine curve. The form of the current time curve may then be investi- gated from the record. To see the wave form on a screen the photographic arrange- ment could be dispensed with, and a mirror made to rotate and reflect the first beam on to a screen. The linear patch of light is again converted to wave form by thus adding a constant velocity normal to that produced by M. The amplitude of the curve gives an indication of the maximum current strength, which is approximately proportional to it. This instrument can also be employed for many of the purposes to which the Einthoven galvanometer may be used. 'For a description of the other forms of alternating current measurers, attracted iron ammeters and dynamometers, the student is referred to any text-book of electrical engineering ; e.g., T. F. Wall : " Electrical Engineering." Methuen. CHAPTER XIX RESISTANCE MEASUREMENTS The Wheatstone Bridge The student will be familiar with the Wheatstone net as shown in Fig. 305. When the bridge is balanced the relation PR Q = s W holds. For maximum sensitivity, using a fixed galvanometer and battery and measuring a resistance R, it has been shown* that P, Q and S should be chosen so that Q 2 = BG, P 2 = RG R + B S 2 = RB R + G R + G R + B If choice of galvanometer is practicable, it should have a resistance comparable with the other arms. When P ^= Q =cr R ^= S maximum sensitivity is obtained when G — P. Measurement of the Resistance of a Galvanometer This may be done in several ways, of which one or two are given below. Fig. 305 Kelvin's Method In this method the galvanometer acts as its own detector of balance in a Wheatstone net. The galvanometer is placed in the arm, DC (fig. 305), and the galvanometer of that figure is replaced by a single-way key, so that B and D may be connected together when the key is depressed. When the battery circuit is completed, a steady current flows through the galvanometer. P, Q and R are adjusted until on joining * See Gray : "Absolute Measurements," Vol. I, p. 331 [1 489 490 ADVANCED PRACTICAL PHYSICS FOR STUDENTS B to D throughthe key, no change is produced in that deflection, P R when ^- = -p, where G is the resistance of the galvanometer. W ** The usual difficulty with a sensitive galvanometer is that the steady current is too large. The galvanometer may not be shunted in this experiment, but the E.M.F. applied may be reduced, e.g. the cell may be connected through a high resistance, and leads from the end of a small fraction of the resistance may be taken to A and C instead of the battery directly applied. This, however, decreases the sensitivity of the method. For a moving-magnet instrument it is better to apply the cell directly, and to reduce the steady deflection to zero by adjustment of the control magnet, or, if that is not sufficiently strong, by the adjustment of an external bar magnet. The sensitivity is thereby retained. For a moving-coil instrument no such adjustment is available, and the simplest course to follow is to clamp the coil of the instru- ment and find its resistance using another galvanometer as detector in the usual way. See also page 643. The Carey Foster Bridge The Carey Foster Bridge, shown in fig. 306, is a modification of the metre bridge. *lt is provided, as seen, with four gaps, PR R 1 Q « 4 D D l E E 1 F F 1 nafe 6 6i g— 3r-arer FlG. 306 CC 1 , DD 1 , EE 1 and FF 1 , which may be closed by the insertion of resistances. E 4 Suppose the gaps be closed with resistance Y in CC 1 , R in DD 1 , R 1 in EE 1 , Z in FF 1 , as shown in fig. 307. The battery, E, and gal- vanometer, G, are arranged at points, A C and B D, which corre- spond to the same points in the theoretical net diagram, fig. 305. RESISTANCE MEASUREMENTS 491 If the point D is chosen such that no current passes through the galvanometer, we have, from (1) : R = Y+^+SxP , , R 1 Z + r t + (ioo-*,)p' - K) where P is the resistance per cm. of bridge wire, x x the length, SD, DT = (100 — x x ) if ST is one metre, r x is the value of the resistance at the soldered junction, S, r 2 the resistance at T. If the simple metre bridge were used to compare R and R 1 , i.e. Y = Z = o, we should have a balance at, say, l x cms., such that, neglecting r x and r 2 for the moment, ?L — h? R 1 (100 -l x ) P ' Now suppose Y = vp and Z = zp, equation (2) becomes : ?L= (y + *i)p R 1 {{z + (ioo-x l )}?' if r x and r 2 be neglected. It is obvious from a comparison of the two results that the Carey Foster bridge functions as though the length of the wire were increased, i.e. the same error in obtaining a balance point corresponds therefore to a less percentage error in the Carey Foster bridge determination. Comparison of the British Association Ohm and the Legal Ohm The comparison of two resistances, very nearly equal, serves to show a common use for this bridge. In the following method it will be seen that the end resistance is eliminated. The resistances R and R 1 are made approximately equal to the values of Y and Z. Connecting as in the fig. 307 with, say, Z, a B.A. ohm, and Y, a legal ohm, whilst R and R 1 are each, say, 1 legal ohm, we should obtain a balance at a point, D, x x cms. from S, so that : 5. = Y + '1 + *iP ( o\ 1 Ri Z + r 2 + (100 - x x ) p' u; If now Y and Z are interchanged, Z being connected in the gap occupied in fig. 307 by Y, and Y replacing Z, a balance for such an arrangement could be obtained at a distance x 2 from S, from which we have : J* _ Z + r x + x iP () R 1 Y + r 2 + (100 -*.)?' ■•" w 492 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Equation (3) may be rewritten : R _ Y +?!+*!! R + R 1 Y + Z + r x + r % + ioop' and (4) similarly becomes : R _ Z + r x + x 2 p R + R 1 Y + Z + r x + r 2 + loop ' Equating numerators of these equations we have : Y -f r x + #jP = Z + r x + # 2 P> Y-Z = (*.-*,)p (5) i.e. the difference between Y and Z is equal to the resistance of the bridge wire between the two points of balance. It is independent of the value of r x and r 2 , and of the total length of the bridge wire. To obtain the value of the B.A. ohm in terms of the legal ohm, we have, Y being the legal ohm : B.A. ohm = {1 — (x t — *j)p} legal ohm. The Value of the resistance of the bridge wire between the points of balance may be obtained by calibrating the bridge wire (see page 495), or, if the wire is uniform, the following simple method will serve. Resistance of Unit Length of Bridge Wire (p) (1) The bridge connexions of the main experiment above re- main as before . R and R x are approximately equal, and may very well be the same as above. Y and Z are replaced respectively by a fraction of an ohm, say, Y 1 , and a stout strip of copper of negligible resistance, say Z 1 = o. Following the same procedure as before, a balance is obtained at a distance x x x from S, when Y 1 and Z 1 are in the positions of Y and Z in fig. 307. When Y 1 and Z 1 are interchanged the balance will move to another point, x 2 x from S 1 . Then by equation (5) : Y*-Z 1 = (V-*i 1 )p. or since Z 1 = o, and Y is known, Xn X-i '2 "*1 Hence, using this value of p, the difference between Y and Z in the first case may be evaluated. This method may obviously be applied to any similar case, and a comparison between two nearly equal resistances obtained. RESISTANCE MEASUREMENTS 493 Experimental Details The resistances and Y, Z, Ri and R 2 are connected by means of stout copper strips. These will have practically zero resistance, and the small difference to be measured will be truly the difference in the resistance of the coils. It is also essential, of course, that all connexions be very tightly screwed, for the same reason. In performing the second part of the experiment, i.e. to find p, two or three sets cf observations should be made ; the exact number will depend on the total resistance of the bridge wise. For example, Y 1 should be made -i, *2, -3, -4 ohm successively, and p calculated in each case, from which a mean value is obtained. It may be found that, -4 ohm acting as Y 1 , no balance is obtain- able. In that case the total resistance of the bridge wire is less than «4 ohm. The following set of observations shows the order of the result obtained in such an experiment. Using Z = 1 B.A. ohm, Y = 1 legal ohm, x 2 = 51-2 cms., x x = 48-2 cms. The wire was not calibrated. EXPERIMENTS TO DETERMINE P Y r in ohms. •1 •2 •3 •4 x x x in cms. 36-35 2376 11*2 no balance x 2 x in cms. 6375 76-34 88-9 p ohms per cm. _ Y 1 •00365 •0038 •00386 x% x^ Mean value of p = -00377. Hence since Y - Z = (# 2 - x x ) P , Legal ohm — B.A. ohm = (51-2 — 48-2) (-00377), B.A. ohm = 1 — 3 x -00377 = -989 legal ohm. (2) Another equally simple method for finding the resistance per cm. of bridge wire may be used. Suppose Y, Z, R and R 1 (fig. 307) are all 1 ohm coils (of the same kind, e.g. B.A. ohms). R and R 1 are set up as before. To introduce a small difference between Y and Z, one of them is shunted with, say, 10 ohms, i.e. the net result is ££ ohm. If now the process described above is carried out, interchanging the 1 ohm and effective ££ ohm, we have two balance positions, l x and l 2 cms., say, and Y -Z = {l x -l 2 ) 9 , i.e. Hence p is determined. 5- <*i-^ P. 494 ADVANCED PRACTICAL PHYSICS FOR STUDENTS To Construct a Resistance Coil of Known Magnitude To construct, for example, a i ohm coil, a wire with small temperature coefficient between io° and 20° C. is selected, say manganin, and the resistance per cm. of the specimen available is obtained by finding the resistance of about ioo cms. The length of wire required to have a resistance not less than 1*1 ohms is calculated, and cut off. The insulation covering is removed from the ends and, using a non-corrosive flux (say, resin or ' fluxite '), the two ends are soldered to two stout copper wires which are soldered to flat copper forks, A and B (fig. 308). Fig. 308 A and B and the rods are then fastened to opposite sides of a wooden bobbin by means of terminals as shown, and the wire wrapped, as in the diagram, in a non-inductive manner round the bobbin. The middle of the wire T is freed from the silk cover. The resistance between A and B is obtained by the Carey Foster method. The middle, T, is twisted with pliers, cutting out the end loop until, compared with a standard 1 ohm coil by the method of page 491, the balance is in the centre of the bridge. T is then soldered in position, and the value of this copy of the standard ohm when completed is compared with the true standard as already described. CALIBRATION OF A BRIDGE WIRE The simple metre bridge, briefly referred to on page 489, also the Carey Foster Bridge, the potentiometer and similar instru- ments, depend upon measurement to a point of balance on a stretched wire. For simplicity it is often assumed that the wire is of uniform cross-section, and that its resistance per cm. is consequently constant throughout the length. Further, the soldered end and the thick copper connecting strip is assumed to be of zero resistance. ' In a practical measurement it is better to make no such assumptions, but to determine the variation due to these causes by a preliminary calibration of the bridge. RESISTANCE MEASUREMENTS 495 To Determine the End Correction of the Bridge Wire For this determination the outer gaps of the bridge (CC 1 , FF 1 in fig. 306) are closed by short clean thick copper strips of negligible resistance, and in the inner gaps are two unequal resistances, say a 10 and a 1 ohm coil (R and R 1 ). A balance is obtained at, say, x x cms. from A. The 10 ohm and 1 ohm coils are interchanged, and a second balance is obtained at x 2 cms. Now, if the ' end resistance ' at A is equivalent to l x cms. of the bridge wire, and if the end resistance at B is equiva- lent to l 2 cms. of bridge wire, we have : R. _ x x + h R 1 100 — x x + /, ' and R1 = *■ + '» R 100 — x 2 + 1% from which l t and l 2 may be calculated. Calibration of the Wire This may be done in many ways, of which we will consider two. The object of these experiments is to find, at different points along the wire, lengths having the same resistance, usually equal to that of a gauge employed. Knowing the total resistance, S, of the wire, the mean value of the resistance of such a length is readily calculated, and hence the correction to be applied at each segment taken, to reduce to the mean value, may be deter- mined. Alternatively, having obtained the lengths which have the fixed resistance, the method of analysis used in the calibration of a tube may be applied (page 43). Suppose we wish to test every 5 cms. of the wire, and this will be quite sufficient for most cases, the following methods may be used. (1) Carey Foster's Method ^ We saw on page 492, that if two resistances, R and R 1 , are fixed in the inner gaps of the Carey Foster Bridge, and resistances, Y and Z, are balanced, and then interchanged and once more balanced, that : Y — Z = (% 2 — *i)p> where x t — x x is the difference in the balance points, and p the mean resistance per cm. between these balance points, i.e. the difference between the resistance of Y and Z is equal to the resistance of the wire included between the balance points, the end resistances being eliminated by this method. If now we make Y-rZ equal to the approximate resistance of 5 cms. of the bridge wire, and so arrange R and R 1 that the 496 ADVANCED PRACTICAL PHYSICS FOR STUDENTS values of x 2 and x t are made in turn to pass along the whole wire, the calibration is completed. This is done in the following manner. The value of S, the total resistance of the bridge wire, is found either by using another measuring device or the same bridge,* Knowing S, the mean value of 5 cms. of the bridge wire may be found. A length of wire, preferably of the same material as the bridge, is then taken ; its resistance is measured and a length cut off 2 cms. in excess of that required to be of the same resistance as 5 cms. of the bridge wire. This is then soldered to two stout copper lugs as shown in fig- 309- The two cms. excess being soldered to the lugs. This gauge is used as resistance Y. Z is composed of a thick copper connecting strip of practically zero resistance (fig. 310). B 1 1 Q) ^B «r— Fig. 309 Fig. 310 Carrying out the process of balancing Y and Z, inter- changing, etc., we have, if x 1 and x 2 are the two values of SD, as before, Y — Z = resistance of wire between x 2 and x v By altering R and R 1 the part of the wire compared with Y — Z may be varied. This process is best done by using another bridge wire as (R + R 1 ) seen in fig. 310. PQ is the second bridge wire connected with thick copper strips to the bridge as shown. The galvanometer, G, is joined to B and D by means of variable contacts : the gauge is placed in the left- hand side and the zero resistance in the right, the point D is chosen near the end of the wire and B adjusted until the galvano- meter when closed in the circuit shows no deflection. Y and Z are interchanged and, leaving B fixed (i.e. R and R 1 are fixed), D is adjusted to some position, D', where a balance is again obtained. , The resistance of the length, DD ', is then equal to the resistance of the gauge. Keeping the contact at P 1 fixed, * The value of S may be found using the same bridge. Referring to'fig. 307, the gap, Z, is closed with a stout copper strip of negligible resistance. The galvanometer is permanently corrected between points S (x=o) and B. R, R 1 and Y are adjusted for balance, the whole bridge wire making the fourth arm in R 1 a Wheatstone net. When balanced we have S «■ ^r * Y. This gives S with- out allowing for the end corrections. RESISTANCE MEASUREMENTS 497 the gauge and the strip are returned to their original positions and B is adjusted to some position, B 1 , where balance is obtained (i.e. a new ratio R : R 1 is obtained). Y and Z are interchanged, and the contact D' is moved until again a balance is obtained, i.e. it is moved a further distance having a resistance equal to that of the gauge. This method is carried out until the whole wire, SP, is covered in approximate 5 cm. steps. The length of wire (x 2 — #1) for the different steps are noted for each part of the scale, and tabulated as under. REGION OF LENGTH OF WIRE DIFFERENCE CORRECTIONS TO WIRE ON METRE OF SAME RESIST- FROM REDUCE TO SCALE ANCE AS GAUGE MEAN LENGTH OF MEAN p 0- 5 4-8 + •15 +0-15 5-10 5*1 — 15 0-00 10-15 5'2 —25 —0-25 15-20 4'9 + •05 — 0-20 20-25 5-o —05 -0-25 etc. etc. etc. etc. 90-95 - -05 95-100 o*o Mean length = 4*95, say. The correction is obtained by finding the mean of column (2), and finding what must be added or subtracted from the first value in column (2) to give the mean value For the 5-10 cm. range the correction is the algebraic sum of the first 0-5 cm. range, and the 5-10 cm., and so on. A correction curve is drawn which will convert any length of the wire to the equivalent length of a uniform wire of the same length and resistance. Thus, if in use in an experiment the bridge balances at 65 cms., and from the correction curve the correction, —0*2 say, is obtained, if the wire were uniform and of same length and resistance, 64*8 cms. would be the balance point, and the resistance ratios of the two segments of the wire is 648 : 35 '2. It will be seen that in this method the interchanges which are frequently made must be assumed to cause no variation in resistance. The gauge and the straight copper connecting strip are thoroughly cleaned and always screwed tightly in the gaps to avoid, as far as possible, such changes. ' , This method may be used to calibrate wire, PQ, since the contact, B, is moved a distance corresponding to equal increments 32 498 ADVANCED PRACTICAL PHYSICS FOR STUDENTS of resistance. It would therefore serve well for the calibration of a potentiometer wire. Direct Calibration by a Potentiometer Method The wire, ST, of the bridge is connected, as shown in fig. 311, in series with an adjustable resistance, R 2 , and a 2 volt accumu- lator which is in good condition and well charged. A similar accumulator is connected in series with a second adjustable resistance, R x , and a length of, say, 20 cms. of stretched wire of the same material and approximate cross-section as the bridge wire. G is a high-resistance galvanometer. If A B 3 IE 13 "n2k C D {Elr 1^— ^ K Fig. 311 C is soldered to the wire, M?and with A and B in contact with ST, say, between the o and 6 cm. readings, and about 5 cms. apart, D is adjusted until no deflection is given in the galvano- meter. D is then soldered to M. It should be noted that convenient resistances for R x and R 2 are of the order 20 to 50 ohms. The adjustment of these resis- tances is made until the currents are about equal. Leaving B in contact with ST, reverse the current in M by means of the commutator, K, and move A to the other side of B; adjust till again no deflection is produced. Repeat this process over the length of ST. If the latter is uniform, then of course the length AB will be the same. Variations of the cross-section will be made apparent by the different values of AB. Tabulate the length of AB as, before, and deduce the correction curve for the wire. This method may also be used for any stretched wire, e.g. potentiometer, etc. * Alternatively, the potentiometer may be made up of two resistance boxes, Rj and R/, in place of the wire M and box R x . RESISTANCE MEASUREMENTS 499 DETERMINATIONS OF THE VALUE OF A HIGH OR LOW RESISTANCE Using the ordinary Post Office Box method of finding resistance is a very ready method, but for resistances of a million (10 6 ) ohms or more it is unreliable. The ordinary Post Office Box enables a magnification of 1000 to 10 only to be obtained, and in the adjustable arm 10000 ohms is the maximum resistance available, so, for a resistance greater than 10 e ohms a special method is required. Equal limitations for very small resistances make it necessary to employ special methods in this case. Some of these special methods are described under. Low Resistance (1) The Direct Deflection Method To find the resistance, r, of a low resistance wire, AB, the following simple method gives a fair approximation. AB is joined in series with a known low resistance, R, and an accumu- lator (2 volt), and an adjustable, fairly large resistance, S. Fig. 312 Having by means of S adjusted a small current, c, through the circuit, there will be a drop of potential, cr, across AB, and cR between the ends of R. If a high-resistance galvanometer is used, the current in the main circuit is least disturbed, and the resulting current in the galvanometer is proportional to the potential cr or cR applied to it. If this causes a small deflection, d x cms., using the usual lamp and scale method, cr oc 0, approxi- mately, and 0j oc d x approximately. The same galvanometer will have deflection i a cms. (angle 0^ when connected to the ends of R. Again cR oc 2 and 0„ oc d 2 approximately i.e. B x d x cr ~e t ~~ d 2 ~ cR r ~R r = ^-R d % Hence r = -=*.- R „. (6) 5oo ADVANCED PRACTICAL PHYSICS FOR STUDENTS This method of observing deflections may be used to find the specific resistance of copper. A length of copper wire is soldered to two terminals about i metre apart on a wooden base, and is connected in series with •i or «oi ohm, and a constant source of potential such as a steady 2-volt accumulator, and a resistance box of o to ioo ohms. (It is not advisable to have less than 20 to 30 ohms in the circuit.) The ends of the wire AB (r) and of the ^ ohm (R) are con- nected to a double pole two-way switch (indicated by the broken lines in fig. 312), which is connected to a high-resistance galvano- meter, G, the deflections of which may be observed by the usual lamp and scale method. The deflection given when the potential difference between the ends of r and R are applied is measured by taking the reading of the deflected spot of light in each case. The battery is then reversed and the reading is obtained on the other side; half the difference in readings giving d x and d 2 . S is adjusted and the experiment repeated for 2 or 3 values of the current, and the mean value of r is obtained. Hence, putting R = t& or y^r in (6), r is evaluated. (2) Potentiometer Method A steady lead accumulator E x is set up in the potentoimeter circuit AB. Fig. 312a, The two small resistances to be compared, R and r, are RESISTANCE MEASUREMENTS 5oi joined in series with a third variable resistance S, which is adjusted so that the current is the maximum compatible with the capacity of the lead accumulator E 2 . A galvanometer G, is connected as shown, and a double pole two-way switch is used. This should be a mercury cu|> key, using well amalgamated copper connecting strips. A balance is obtained at H when cups 1 and 4, 2 and 5 are joined together. A second balance is obtained at H' when 2 and 4, 5 and 3, are joined together. The P.D. between C and D is cR where c is the steady current in the circuit CFS ; in the same way the P.D. between D and F is cr. So that if AH = l x and AH' = l 2 , cR l x R l x — = -i or — = -1 cr l 2 r l 2 The length l x is obtained, then l 2 , and finally l x is checked ; if any difference is found the mean of l x and l x ' is compared with 1 2, in the usual way. (3) The Kelvin Bridge To obtain a more accurate comparison of two small resistances than may be obtained by the foregoing experiments, the Kelvin Bridge method is used. Fig. 313 shows the general arrangement of this network. r x and r 2 are equal resistances, as are r 8 and r 4 . These are arranged as shown, so that the total resistance, D to N = that from N to G and resistances in the arm EK = resistance in KF. The two resistances to be compared, if in the form of wires, are arranged as at AB and BC ; B being a mercury cup or other low-resistance junction. The two resistances, r x and r s , are connected to AB at points D and E, and the second low resistance is connected to r 2 and r x at F and G, the latter being an adjustable point. 502 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The two pairs of equal resistances (about i or 2 ohms each) should be made of wire of the same material, or having similar temperature coefficient, and should be placed in close proximity to ensure a common temperature throughout the experiment. Further, when the current from the battery is sent through the circuits, its duration should be short to eliminate excessive heating in these resistances, and in the specimen AB and in BC ; the latter, presumably, will have different coefficients, and the comparison might therefore be upset if such temperature variation are introduced. The galvanometer between K and N should be of low resistance. The points E and F should be near B and the point of contact, G, is adjusted until on closing the battery and galvanometer key no deflection is produced in the latter; then if the resistance DE = r, FG = R, we have 1 = r -l = *J. R r t r 4 Thus it is apparent that r v r* r z , and r 4 are not of necessity equal, but the relation r 1 = ^ w r 2 *4 must be satisfied. This condition is satisfied in the construction of the bridge which, like the Post Office Box, may be used with equal ratio arms or with factors 1 : 10, 1 : 100, 10 : 1, 100 : 1, in the usual form. To establish the relation,^-= - = -, let us put c = the current ■K ^2 ?3 in DE (f), Cj the current in EKF (when in the balanced condition no current passes through the galvanometer), c t the current through DNG. Then, since the current c — c t passes along EBF, the current in FG (R) is c. The above is the balanced condition of the net of conductors, i.e. when the potential at K is the same as the potential at N, i.e. when cr + c x r x = ctf* _ r _z; c x r 2 +cR c % r A r 4 i.e. r A {cr + c^) = r z {c x Yi 4- cR), or c(*r 4 - r 3 R) = c x {r# 3 - r x u). But the condition satisfied by the bridge, by construction, is that £ = £2 or r x u = r 2 r 9 , r t r A i.e. c(rr A — r s R) = o ; and since c is not equal to o : rr k = y,R; RESISTANCE MEASUREMENTS 503 or 4 = ~ = - fr om (7) (8) With such an arrangement of resistances as already described the comparison of two nearly equal low resistances such as two wires, can be made ; then if a, I, and s, represent the radius, length and specific resistance of r, and similar terms apply to R, we have : 7 o /» 2 y o / 1 2 2 = -* ,whence — may be calculated. To obtain the specific resistance of a metal in the form of a thick wire or rod, we may carry out the above process using a copper wire as a second low resistance, and assume s for copper (s for drawn copper at i8°C. =178 x io- fl ). Alternatively the copper rod may be replaced by *oi ohm (r), and the unknown rod placed in BC (R). The length, FG, to balance may be found as above ; whence s for the material : •01 r t sl/a* r % When a definite length of wire, or fixed small resistance, is to be measured it becomes advisable to have a variable value for r. This may be a variable length of copper wire of known cross- section for very small resistances (R), or a series of »oi ohm coils, and a calibrated wire in series with the coils for larger resistances. This latter method is usually used in the ready-made Kelvin Bridge now on the market. Of course it must be realised that when the values of R, the unknown resistance, is very small, the current passing through it must be correspondingly large if any appreciable potential is to be attained between the ends, and if the measurement is to be at all satisfactory, the reason for a sensitive /oze/-resistance galvanometer will also be apparent. As examples of the types of Kelvin Bridge on the market, consider figs. 314 and 315. For measuring the specific resistance of a wire of low resistance, the special clamping contact arrangement shown at the back of fig. 314 is used. For other forms of low resistance, the ter- minals X of that figure and fig. 315 are used directly. Such an apparatus has a wider range and is more adjustable than the simpler form already described but is, in principle, identical with the bridge illustrated in fig. 313. To emphasize this, fig. 315 has been drawn showing the internal arrangements of fig. 314, lettered to agree with fig. 313. 504 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The ' ratio arms ' may be adjusted as indicated to give various ratios, viz. -oi, «i, i, 10, ioo. The coils which make up ?xt r* ?z, r if are accurately adjusted to ensure the condition of (7) above. As shown in the figure, r is the unknown resistance, and R may be one or more of the four coils each of -02 ohms, together with a part, F'G', of the graduated and calibrated wire, G'G". The length of the wire is about 450 mms. and each mm. has a resistance of '00005 ohm, i.e. 40 cms. of the wire add -02 ohms to the value taken from CG' (-04 in the diagram = GG'). The wire is graduated in fractions of an ohm. A comparison of figs. 315 and 313 will show the general arrange- ments. For example, suppose an unknown resistance r were balanced w j 1 en — was made — «oi, by -04 ohm from the coils, and 290 units of the slide wire, as shown approximately in fig. 315, t — R f — 1 = (-04 -f 2qo x -00005) * 01 \ r, / = (-04 + -0145) -oi = "000545 ohms. This instrument will measure resistances of 10 to -ooooi ohm. Fig. 317 shows a second form of Kelvin Bridge made from the potentiometer illustrated in fig. 338, and a ' double ratio ' box (fig. 316). This enables a range of measurement from 1-5 ohms to -ooooi ohm, about. The ' double ratio * box contains resistances (fig 317), LN, NN', N'N" and N"M, which are respectively \, ^, tttt and 1 TTJI Of the total resistance of LM. A similar arrangement holds for I'm. 314 Fig. 3 r6 Pw 504 RESISTANCE MEASUREMENTS 505 the parts of HI, so that the resistances LN : NM = HK : KI (1 : 1) or LN' : N'M' = HK' : K'l (= 10 : 1), etc. The scheme of connexions is lettered to conform with the letters of fig. 313. A comparison with that figure shows the reason for this scheme : R, the unknown resistance, is equal to the resistance between E and D when the galvanometer is connected to NK as in the case taken, or is ^ ED when the galvanometer is connected at N'K', etc. The coils to which D is tapped are each «i ohm and the resistance of the slide wire is *ooi ohm per small division. Thus, using N* K* as galvanometer terminals, a resistance of -ooooi may be measured. <2X r. » 6 vd C.3 4-C-* T* VO I* 1-4 B« Fig. 317 High-resistance Measurement As already stated, a special method is required to determine the resistance of greater value than 1 megohm (i.e. 10 6 ohms). One of the simplest ways of evaluating resistances of this order is the method of substitution. Fig. 291, page 463, shows the scheme of connexions. A steady accumulator, E, is connected in series with the unknown resistance, R, and connected to a commutator, C, whence the current may be sent in either direction through a galvanometer, G, which is provided with a variable shunt, S. Let the steady deflection due to the current be d cms. on a scale one metre away (corresponding to a movement of the suspended system of 0°). A known adjustable resistance is substituted in place of R, and if on adjusting the known resistance a steady deflection d cms. is obtained again, the known resistance is of the same value as the unknown. If no variable known resistances of the order of the unknown resistance are available, the galvanometer is shunted so that one-thousandth part of the current is allowed to pass through it. When a large known resistance (e.g. 10000 to 20000 ohms from two P.O. boxes in series) is placed in series with it and the steady accumulator, a deflection of the same order as that given when the unknown resistance is placed in the circuit may be obtained. 506 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Under these circumstances, suppose that d x is the deflection caused by the battery when in series with the unknown resistance, R ohms, and d 2 when in series with r, the known resistance. Then if E is the E.M.F. of the accumulator, B the resistance of the battery, and G the resistance of the galvanometer (if the latter is a moving-coil galvanometer with a radial field), we have, where k is the galvanometer constant, Mx = B + G + R' kd* = E S . (■» SG , I S + G' . *i |fr + B)(S+G) + SG} S + G '•<*■ {(S + G)(B+G + R)} S M S + G (,+B)(S+G)+SG W s + ^ _ () S(B + G + R) G + R w when, as is obviously the case, B may be neglected in comparison with r or R. If further, G is negligible compared with r or R : * m dx d % R » i.e., if Trnrry part of the current goes through the shunted galvano- * S + G meter, — ^ — = iooo, d x _ iooor 1,e - T 2 -~R The experiment may be carried out, using an adjustable known resistance, and shunting the galvanometer with a known shunt. The known resistance is adjusted until d 2 = d v i.e. equal deflections are obtained. Then, if G is small compared with R, R = iooor, (10) if the shunt is -^. Alternatively, if the known resistance is not sufficiently adjustable to cause equal deflections, the values of d x and d 2 , both of the same order, are noted, then, neglecting G in com- parison with R, we have : rf, S, + G * But according to p. 473 G should be as high as possible to make G as near equal to R as possible for maximum sensitivity when the experiment is per- formed without the shunt S. RESISTANCE MEASUREMENTS 507 Again, if the shunt takes ^^ of the current R =-r • iooor. d x In both these cases, if G is not negligible compared with R, then the corresponding formulae for the two cases is given by equation (9) above. Determine by the above methods the value of the resistance of a leaky condenser. Another suitable high resistance to be measured by these methods is made by taking a sheet of ebonite, say 20 cms. by 5 cms. Two holes are drilled about 15 cms. apart, and the surface of the ebonite blackened, round the holes, with a black lead pencil. Two terminals are screwed down in the holes, and a fine black lead pencil line is ruled between them. The sheet is covered with a thin protecting second sheet of ebonite to prevent any accidental change in the dimensions of the line. The apparatus should be set up as in fig. 291, and the values of R obtained. For example, using a thin black lead pencil line, &x = i«95 cms., R = ? no shunt ; ^2 = 9*3° cms., t = 10000 ohms, shunt fa, Hence : R = -2-2- x 10000 x 1000 = 47-7 x 10 « ohms. 1.95 Using a thicker line on ebonite, i.e. smaller resistance, the first method (equation (10)) : deflection 10 cms. when R was in circuit, no shunt, deflection 10 cms. when 3640 ohms replaced R, shunt fa. Thus, R = 3640 x 1000 = 3-64 x 10 6 ohms. THE VARIATION OF RESISTANCE WITH TEMPERATURE The value of a resistance of a specimen of wire, in most cases, increases with temperature. The relation between R , the resistance at o° C, and R f , the resistance at another temperature, t° C, is : R, = R (i +at + &*), (11) where a and p are constants. The constant, p, is small, and therefore over small ranges of temperature the resistance is practically a linear function of the temperature, i.e. R t = R (i + «t) (12) expresses the relation, for small temperature ranges, between 508 ADVANCED PRACTICAL PHYSICS FOR STUDENTS resistance and temperature, (t should not exceed ioo° C. for (12) to be valid). From (12) we have : ^a ^ R * ~ R °, R t and a may be called the ' coefficient of increase of resistance with temperature ' for the limited range taken. To measure such resistance changes and determine the value of a, which is practically constant for all pure metals, we may make use of the Carey Foster bridge, since these resistance changes are small. This method is specially advantageous for such a determination of small differences, as has already been shown. The theory of the method is almost identical with that already given, so that we may deal with the experiment as such. Determination of the Resistance of a Wire at Temperature from 0° C. to 100° C. A length of thin platinum wire of about 1 ohm resistance is mounted on a mica frame in the non-inductive manner shown in fig. 318. The ends of this wire are soldered to two thick copper wires whose resistance is negligible compared with that of the platinum. This arrangement is mounted rigidly in a glass tube, with mica or rubber supports as shown at M. The free ends of the copper wires are soldered to two long leads, and two identical leads, cut from the same wire, are joined (D) together. The four free ends of the leads are soldered to copper connecting strips, shown in fig. 322, P, P 1 , C, and C 1 . These ensure good contacts with terminals under which they are fixed. The pair, P, P 1 , are the ends of the leads from the platinum wire ; C, C 1 are the compensating leads. - The inner gaps in the Carey Foster bridge are occupied by two equal resistances of the same order as that of the platinum wire ; in the case taken these resistances, R t and R 2 , are both made 1 ohm (fig. 319). P and P 1 are connected in one of the outer gaps and the com- pensating leads in series with a resistance box, S, fill the fourth. It must be remembered that R 1} R 2 and S should be connected to the bridge by means of copper strips, and every source of uncertain resistance, such as a bad or dirty connexion, must be eliminated. All the resistances should be ' non-inductive/ for in this method it will be seen that the galvanometer is permanently connected in the circuit, and the battery takes the position in the sliding contact. RESISTANCE MEASUREMENTS '509 This is essential, for we must balance the resistance of the platinum at the temperature which is fixed by the surroundings, so that the current should not pass for any appreciable time and cause a heating in the spiral. fr Ur y sn»«r Copper" Leads Wame Platinum Wire Fig. 318 Fig. 319 S is a resistance box in which are all values from *i to :eo or more ohms. The function of the compensating leads will be apparent. They are in opposite arms to the bridge to PP 1 , and so, since the material and length of the leads are identical with those of the leads to the platinum, the value of the resistance of the last- named leads is eliminated ; and since the two pairs of leads are, in the main, side by side, variation in the resistance of this part of the circuit is also balanced. In obtaining a balance at a point, A, x cms. from the end of the wire, the value of the resistance, S, is so adjusted that A is near the middle of the wire. Also the balance is obtained by having no immediate deflection of the galvanometer when contact is made with the bridge wire. If the current continues to flow, a change in the resistance of the platinum wire, due to the heating by the current, will cause a deflection in the galvanometer. Suppose the glass tube containing the platinum wire specimen is immersed in a constant temperature bath and a balance is obtained at x cms. from the end of the bridge, we have, if R, is the resistance of the platinum at this temperature t°, Rj ^ R« + r + x ? R, r -f S + (100 — #)V 510 ADVANCED PRACTICAL PHYSICS FOR STUDENTS where r is the resistance of either pair of leads, p is the resistance of i cm. of bridge wire. Since R x = R 2 = i ohm, R t -f r + x 9 = S +r + (ioo -*)p, R t = S + (ioo — 2X) P (13) Hence, if p be known, R f may be calculated. The value of R is obtained by surrounding the platinum, etc., with melting ice. After twenty minutes or half an hour the whole of the immersed tube will have attained the temperature of o°C. ; the balance is obtained near the centre of the bridge by making S have a value S . A is x cms. from the end. If after five minutes the balance is still at x , we may safely assume that the resistance is at o° C. The value of P may be obtained at this stage, by varying S to S ' and finding the new balance at x '. S ' is again ad- justed to a third value, S ", and the balance is now at x ". The platinum wire is meanwhile at o° C. and therefore its resistance remains R„, i.e. R = S + (100 - 2*0) p, Ro = S ' + (IOO "2* ')p, R = S " + (I00-2« *) P . These taken in pairs yield three values of p, and further give by any one, preferably the value of S which makes the balance in the neighbourhood of the centre of the wire, a value of R . The glass tube and contents are then placed in a hypsometer, and a balance is obtained once more, when the temperature of the wire has acquired that of the steam which is made to pass round it in the hypsometer. The value of the resistance at, say, 40 C, 6o° C, 8o° C. about, is obtained in the same way by immersing the container in a water bath maintained at steady temperatures near these points. In each case time is allowed for the platinum to acquire the temperature of the bath. The values of R obtained are plotted against temperature. The result will be approximately a straight line. From the slope of the straight line drawn through the observed points, calculate a as defined above. In the theoretical account of the balance given above it was assumed that there were no end corrections to the bridge wire. If l x and l 2 are the equivalent lengths obtained in the bridge calibration (see page 495), equation (13) would become R t + r + x? + h? = S + r + (100 - x) 9 + l 2 ?, ' R t = S + (100 - 2*)p + (h - h)?- Since l x and l t are known the effect produced may be allowed in this way throughout the determinations. RESISTANCE MEASUREMENTS 5ii r\ The values of x and (100 — x) should be corrected from a calibration curve as obtained in the manner given on pages 495-8. The Platinum Resistance Thermometer. Callendar-Grifflths Bridge The platinum resistance thermometer is a temperature measur- ing instrument which depends for its action on the variation of the resistance of a wire with temperature, as investigated in the last experiment. The temperature of any enclosure is calculated from the observed value of the resistance of a calibrated specimen of platinum wire. The instrument consists of a length of fine platinum wire, wound in a non-inductive manner on a mica frame. The ends of this thin wire are soldered to two thick leads which are con- nected at the other ends to two terminals on the cap of the porcelain tube which contains them. Arranged parallel to these thick leads are a second pair, of identical material and size, which are likewise soldered to two more terminals on the cap. The other ends of this second pair are joined together as shown in fig. 320. The terminals are usually marked in a distinc- tive manner, P, P, C, C ; C, C being the ends of the compensating leads. We have already seen (page 507) that the re- sistance, R f , of a given specimen of wire, at a temperature t° C, may be expressed in terms of the resistance at o° C, R , and two constants a and in the following manner : R, = R (i + <*t + ftf 2 ) (n) Thus, if the resistance of the platinum wire were measured at o° and two other known temperatures, we should have two equations from which a and p could be calculated, and the wire would be standardized, so that if R t were measured at an unknown temperature the latter could be calculated from equation (11) above ; or having R , a, and p, a calibration curve could be drawn showing the relation between R and t, and hence the temperature corresponding to any resistance could be obtained. Alternatively we may define a new scale of temperature which expresses the relationship rm. 3*u between temperature so defined and the value of the resistance in a very simple manner. 512 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Let R and R 100 be the resistance of the platinum wire at o° C. and ioo° C. ; R 100 — R being the increase in resistance for ioo° rise in temperature, and is called the ' fundamental interval.' The platinum scale makes the size of the degree such that each degree rise in temperature on this scale corresponds to an equal increase of resistance of the specimen of wire and is equal to Rioo Rq ioo The platinum scale so defined coincides therefore with the gas scale at o° and ioo°, but will differ at other points since equation (n) above expresses the true relation between R and t on the gas scale. Let R be the resistance of the wire at any temperature, t° C, or t P on the platinum scale ; by definition : tp ==ioo p ~ p • ( x 4) ■•Moo ss -o The difference between t p and t has been found to be given by t t - U = k- (14a) 100 j 100 where k = 1-5 for pure platinum, but its value for any particular specimen may be obtained by obtaining the resistance R , R 100 , and R t at o° C, ioo° C. and a third standard temperature, say, the boiling point of sulphur corresponding to a known t° C. The values of R , R 100 , etc., may most conveniently be deter- mined by using the Callendar-Grimths Bridge. The Callendar-Grifflths Bridge The Callendar-Grimths bridge is a compact form of the Carey Foster arrangement described previously in considering the variations of resistance with temperature. Fig. 321 shows the general appearance of a modern example of this type of bridge. Reference to fig. 322 may make the principle of construction clear. R x and R 2 are equal resistances. EF an adjustable resistance capable of giving 5, 10, 20, 40, 80, 160, 320, 640, 1280 arbitrary units of resistance. ML is a straight stretched wire chosen by reason of its uniformity ; T a second parallel wire of the same material, which may be connected to ML by means of a short length of the same material wire. Thus, the possibility of thermo-electrical effect when the galvanometer is connected to the wire is eliminated. Fig. 321 Pag* 512 RESISTANCE MEASUREMENTS 513 Suppose the thermometer be balanced at a temperature t p , against S units from EF, when the contact is made at O, the centre of ML, then using the same notation as before (page 509). R t + r + Resistance LO = S -f r + Resistance OM. or R, = S. If now at a second slightly different temperature t\ the Lcadi Fig. 322 thermometer has resistance R v and is balanced by moving the contact 1 unit of length to the right, Rj+rH- resistance of (OL— 1) =S+r-f resistance of (OM+i), i-e. R 2 =S -f-2 resistance of 1 cm. of LM. If the resistance per cm. of ML is half an arbitrary unit, we have : Rx = S + 1 arbitrary units. The scale on which measurements of MO, etc., are made is usually inscribed in arbitrary units equivalent to the movement of the contact position. If the balance is to the right of the mid-point of LM, the number of units must be added to S ; if to the left the number is taken from S. The length of the wire, LM, is usually sufficient to allow about 15 units each side of the centre. To arrange this simple relation between the length of the wire and the arbitrary units, a coil, s, of suitable resistance is shunted across the bridge wire. The coil is adjusted to give the desired value of resistance per cm. to the bridge wire. The arbitrary unit of resistance often chosen is deduced from the value of the fundamental interval of the thermometer (i.e. the change in resistance for a change in temperature from o° to ioo° C). 33 514 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The platinum thermometer is constructed with such a resistance that its fundamental interval is I ohm and one hundredth part of this, i.e. *oi ohm, is taken as the unit. The coils in S have, therefore, resistances of -05, *i, etc. The resistance per cm. of bridge wire in the example taken would therefore be £(-oi) or -005, the effect of 1 cm. change in balance being 2 x -005 = •01, i.e. a change of balance of 1 cm. corresponds to a change in resistance of the platinum wire of one unit. The form of Callendar-Grimths bridge illustrated in fig. 321, is provided with a scale for the slide wire bridge which is gradu- ated from o to 15 units, i.e. the readings are continuous from one end of the wire to the other. The balance points give directly the number of units to be added to S to give the equivalent resistance of the thermometer. The half, MO, of the wire is obviously in this case within the instrument in the form of a coil. A more recent feature of the bridge is the use of mercury cup contacts instead of the usual plug contacts in the adjustable arm, S. Fig. 323 shows an enlarged view of one of the contacts. Fig. 323 The coil is connected to two mercury cups inside a tightly fitting cover. A plug, D, when inserted in the hole, G, corresponding to this resistance, strikes a thin circular sheet, E, which is covered with baize and is ordinarily held tightly against the hole by means of the spring, A, and a spiral spring, F, thus keeping dirt and dust from the mercury. When D is allowed to depress this arrange- ment, the copper connecting strip, B, which is amalgamated at the ends, dips into the cups and the resistance is thereby cut out. RESISTANCE MEASUREMENTS 515 The total value of the resistance, S, is therefore the sum of the numbers opposite the holes without plugs. The makers suggest that when not in use all plugs should be in to maintain the amalgamation of the connecting strip, B. The balance point on the bridge wire may be maintained by a rough movement of the slider, followed by a fine adjustment by means of the small lever attachment. The position of the oalance is read on a vernier which enables one-tenth of the small scale division to be measured, i.e. if graduated in lengths corresponding to a unit and subdivided into tenths, one may read to y^ of the unit. Reference to fig. 324 may make clear the internal wiring of the bridge. The lettering in this diagram corresponds to that of fig. 322. A two-volt accumulator or Daniel cell is connected to BB 1 and, say, a Broca galvanometer to GG 1 . PP 1 and CC 1 are gaps for the thermometer and compensating leads. It will be noticed that Rj and R a are contained in the bridge. Fig. 324 The point A in fig. 324 is capable of slight adjustment in many forms of the bridge ; so that, if in error, Rj may be made equal to R 2 . Before using the bridge with the thermometer it is advisable to check some of the points of construction which have already been described. (a) See that the zero of the scale is truly the mid-point ot the bridge. This is readily done by inserting equal resistances, e.g. two thick copper strips, in the gap provided for the thermo- meter and compensating leads. With S = o, find the balance point on the wire. This is the mid-point and should coincide with the zero graduation ; if this is not so, then probably the values of R and R 1 may not be truly the same ; this may be further checked by using two practically equal resistances of 5i6 ADVANCED PRACTICAL PHYSICS FOR STUDENTS 5 or 10 ohms in the gaps PP 1 and CC 1 . A balance is obtained. The coils are then interchanged and a second balance obtained. The mean of the two readings should be the zero of the scale. This method cannot be very well used in the second form of bridge wire described above. However, for the purpose of comparing resistances the slight error in equality will not affect the comparison. (b) Verify the relation between arbitrary units of resistance in S and the resistance per unit length of bridge wire, and see that the resistances in S are consistent within themselves. The gap CC 1 is closed by means of a copper connecting strip, and an external resistance or resistances are connected by copper strips to the gap PP 1 . The coil 1280 is unplugged and balanced against an external resistance of, say, i2«9 ohms. Suppose x x be the reading on the bridge at balance. Replace the 1280 coil by the rest of the coils, i.e. 640 + 320 + .... 10 + 5, and again balance these coils (nominal value 1275 units) at a point, x x x , using the same external re- sistance. Let l x be the resistance corresponding to a change in balance x x x — x x . Then coil 1280 — sum of the rest =l x (15) Carry out this test using the 640 coil : balance against an external resistance making the point of balance at x 2 , say. Cut out the 640 coil and, again leaving the external resistance the same, balance the rest of the coils (320 -f 160 + 80 + 40 + 20 + 10 + 5) at a second point x 2 x cms. Let the resistance corres- ponding to the change x 2 x — x 2 = h> then : Resistance (coil 640 — the rest) — l 2 (16) Carry out this test with each coil in turn. Finally : Resistance (coil 5 — o) = / 9 . Nine equations are obtained in this way for the bridge described. The difference between the first and second equations, where each term represents the corresponding resistance is : coil 1280 — coil (640 + 320 ... +5) — {coil 640 — coil (320 + . . . +5)} =li—h> i.e. coil 1280 — 2 coil 640 = l x — 1 2 .. coil 1280 , l 2 — l x . v or coil 640 = + 2 (17) In the same way, taking the third equation from the second : .. coil 640 , l z — 1 2 ,_™ coil 320 = — — — + ± — — » (1-0) RESISTANCE MEASUREMENTS 517 and so on. Substituting in (18) the value of coil 640 in terms of 1280 from equation (17) : coil 1280 . l z — h coil 320 = • + - l z — 1 2 (19) 442 This process is carried on throughout the range. We may express each coil in terms of the largest one. Equation (17) and (19) give the corrections to be applied to each coil to make them consistent with the largest. For a perfect set of coils l x = / 2 = • • • = h = h = tne length of wire having 5 units of resistance. It is important to note that the unit often taken is -oi ohm, EXTERNAL RESISTANCE COIL BRIDGE WIRE READING x units re- sistance DIFFERENCE I CORREC- TION 1280 640 +320 +...+5 64O 320 +160 +...+5 320 160+80+...+5 160 80+40+. ..+5 80 (+1280) 40+20+10+5 (+1280) 40 (+1286) 20+IO+5(+I28o) 20 ( + I28o) 10+5 (+1280) 10 (+1280) 5 (+1280) 5 (+1280) (+1280) 1 so that for safety the 1280 coil should be out as a permanent addition when testing the 80, 40, 20, 10 and 5 unit coils. The results may be tabulated as shown above, the last column giving the correction to each coil. The above calibration gives the mean value per unit length of bridge wire. 518 ADVANCED PRACTICAL PHYSICS FOR STUDENTS For some purposes, and especially when the bridge has had considerable use, it may be necessary to calibrate the bridge wire. This may be done by a slight modification of the method given on page 498. But a new bridge wire should be uniform to within 0-3 per cent, and therefore will give readings correct to the second decimal place in degrees on the platinum scale. Calibration and Use of the Platinum Thermometer Having tested the bridge and calibrated the coil and wire as indicated above, the thermometer is placed in melting ice and allowed to remain there until the resistance remains constant. The value of this resistance is noted. The thermometer is next placed in a hypsometer, in which water is boiled, and again the steady resistance, at the tempera- ture of the steam, is obtained (after being in the steam for, say, 20 to 30 minutes). The temperature of the steam corresponding to the atmospheric pressure is determined from tables (= b° C. say) then, if R ft is the resistance obtained in the steam, and R in ice at o° C, R& — Rp b is the change in resistance per degree centigrade. The thermometer is next immersed in the vapour from boiling sulphur (see below for method), and from the balance of the bridge, when steady, the value of R, the resistance at the tempera- ture of the sulphur vapour, is obtained. The boiling point of the sulphur in degrees on the platinum scale is from (14) : _ b(R - R ) tp ~ K 6 - R * Now the value of the boiling point of the sulphur at 76 cms. of mercury pressure is 4447 C, and the value at any other pressure, j^mms., is : 2=4447 + -0904(/> — 760) — «oooo52(£ — 760) 2 . The value of t, the boiling point at the pressure which obtains during the experiment is calculated and hence, substituting t p and t in (14a), the value of k may be calculated. It will be found to be very near 1-5 in most cases. The thermo- meter is now fully calibrated and may be used, for example, to find the B.P. of aniline first in degrees platinum from which, by (14a), the corresponding value in degrees centigrade may be found. To Obtain the Boiling Point of Sulphur The apparatus of fig. 325 is convenient. This is readily made by taking a length of iron tube about two inches diameter, and RESISTANCE MEASUREMENTS 519 about four or five inches longer than the thermometer. The lower end of the tube is closed by brazing on a circular disc of iron. The lower end of the tube is covered completely with asbestos paper and then wound with nichrome wire of suitable gauge and length to produce sufficient heating. For example, the one Fig. 325 shown in the figure was made by winding with nichrome wire of -092 cm. diameter and six metres in length. This was satisfactory for use with a series resistance on 100-volt mains. The length of the same gauge wire may be varied to suit the potential of the mains available. It might be found advisable, in addition to the thick layer of asbestos around the nichrome winding, to surround the boiler with dry sand. The sulphur vapour condenses on the upper walls of the tube and runs back. The boiling point under the pressure which obtains during the experiment, may be obtained from the formula given on page 518. The Variation of the Resistance 0! a Bismuth Spiral in a Magnetic Field In this experiment a bismuth spiral such as that shown in fig. 326 is placed in a variable magnetic field and the resistance Fig. 326 is determined for each value of the field strength. The measure- ments involved are therefore (1) the strength of the field, (2) the resistance of the spiral. The magnetic field should be of as large a range as possible, and may be obtained by using a large du Bois magnet as shown in 520 ADVANCED PRACTICAL PHYSICS FOR STUDENTS fig. 327. Current from the electricity mains is passed through a series of frame resistances, R, to the magnet, and an incandes- cent lamp, L, in parallel. The current passing through the windings of the magnet may be measured on a calibrated ammeter, A, and may be reversed by the commutator, C. The ammeter is placed at as large a distance from the magnet as is convenient. SHIHS Fig. 327 By varying the frame resistance, the current passing may be of values from, say, to 5 amperes. The bismuth spiral is placed centrally between the poles of the magnet with the plane of the spiral normal to the magnetic field of the magnet and coinciding with the meridian. The resistance of the spiral is measured either by a Post Office Box, or by the Carey Foster method. In the latter case the scheme of connexions shown in fig. 319 may be used, where the gap, PP 1 , is closed by the spiral and the resistance, S, may be of any value from o to 50 ohms. R x and R 2 may be given convenient values ; for a spiral of resistance 15 to 20 ohms make R!=R 2 =20 ohms, say. In either case the determination of the resistance, R 1} say, of the spiral will present no difficulties. Its magnitude is determined for values of the field corresponding to currents in the electromagnet windings ranging from o to 5 amperes, by, say, '3 ampere steps. The initial value of the resistance will depend upon the past history of the specimen, but will increase with increasing magnetic fields. (The order of this increase will be 10 per cent of the original value, depending upon the field change which a current variation of o to 5 amperes creates.) RESISTANCE MEASUREMENTS 521 The values of the current and resistance are tabulated. The next part of the experiment is an estimation of the mag- netic field corresponding to the currents used. This may be carried out in either of the following ways : (1) Fluxmeter Determination of the Field Strength (H) The fluxmeter is set up, away from the electromagnet, and a search coil of the same area as the bismuth spiral is made. This is connected to the fluxmeter and introduced between the poles of the magnet. The deflection of the suspended system is noted, either directly or on a scale one metre away (for the weaker field this method is essential). The scale readings are converted to maxwells\>y comparison with the graduated scale on the instrument as described on page 485. If a is the area of the search coil (per turn) and there are n turns, the total flux recorded in maxwells (m maxwells, say) is m = nail, or H = — gausses. na ° As explained in dealing with the theory of the fluxmeter (page 482) the total deflection is independent of the speed of insertion of the search coil. The field is found in this way for each of the current values used in the resistance determination. (2) Ballistic Galvanometer Method of Finding H In this method a similar search coil is made and connected to a ballistic galvanometer. This forms a low-resistance circuit, so that a moving-needle type would be least damped by such a low-resistance circuit. This, however, should be removed to a very great distance from the magnet and ammeter for obvious reasons. It will generally be better to use a moving-coil instrument, also removed from the magnet, though not necessarily as far away as the needle type. In such a case a key such as described on page 481 should be used so that the circuit may be broken immediately after the passage of the discharge. Under such circumstances, when the current is passed through the windings of the electromagnet, the search coil being in the gap between the pole pieces of the magnet, replacing the bismuth coil, a transient E.M.F. is set up in the galvanometer circuit through which a quantity of electricity, Q, will be discharged. The quantity, Q, as shown earlier, may be expressed by equation (19), page 479. * n <p 2 \a 3 / 522 ADVANCED PRACTICAL PHYSICS FOR STUDENTS where T is the undamped periodic time (when the key described is used), a x is the first deflection, a 3 is the second deflection on the same side, c is a small steady current, <P is the corresponding steady deflection. The correction for damping (—V will only be applicable for the moving-coil instrument when the circuit is broken, as described, as soon as the impulse is given to the galvanometer. The value of <p corresponding to a current, c, may be obtained by passing a current from a 2 volt accumulator through one megohm, and the galvanometer shunted by a small resistance. The deflection is observed and the current, c, calculated, as on page 465. T, a lf a 3 are observed in the usual way. Now suppose that the field strength to be measured is H gausses, and let c 1 be the current passing through the coil at any instant, t, after the circuit is made. During a small interval of time, it, let ^N be the number of lines threading the circuit, then cZN dE = -=- (numerically), , rfN or re 1 = -jt, at where r is the total resistance of the galvanometer circuit (galvano- meter, leads and search coil), i.e. i.e. fredt = fdN, when a is the area of one turn (= itR 2 ), and n is the number of turns, in the search coil, ~ Han i.e. Q = — * it (Tc i.e. H ;N(-) 4 - [jc tp 2an) The bracket term is constant during the experiment and may be evaluated and used as a reduction factor throughout. The observations described above under distinct headings may be carried out successively. The search coil is placed in the gap between the pole pieces ; a lf a 3 are observed. The bismuth * Alternatively, if the search coil is placed in the gap between the magnets 2 anil and the current reversed in the windings, Q = — — • RESISTANCE MEASUREMENTS 523 spiral is then introduced and its resistance measured. The search coil is replaced and the current cut on. The values, "fci 1 * as 1 * are again noted. A new value of the current strength is obtained by an adjustment of the frame resistance, and the current switched on. The throws are observed and then the bismuth spiral resistance is again measured. The check value for the throws is again obtained when the current is switched off, and so on. With either method of working compile a table showing the relation between the magnetic field strength and the resistance, and plot a curve showing this relation. The form of the curve will suggest that the relation between R and H is of the form : R H = R + d¥L + 6H 8 , where R H is the resistance in a field of strength, H, R the resistance in zero field, a, b are constants. By substituting values for two points, find a and b. Calculate the expected value of R for another magnetic field, in the range of the experiment, as far removed from the first two points as convenient. The observed value will be found in close agreement with this value. Determination of the Absolute Resistance of a Metal Rod. (Lorentz's Method) This method of measuring resistance in absolute units is one in which a steady drop of potential at the ends of a rod (as determined by Ohm's Law) is balanced against one set up by induction. The measurements are reduced to those of length and time. Fig. 328 shows the details ; the sketch shows a copper disc, mounted on a horizontal axis about which it may be rotated, either by hand or by a motor. A metal brush, C, makes contact at the rim of the disc, and a second contact to the disc is made at the axle. At an adjustable distance, d, from the plane of the disc is a coil, D, of N turns of wire, of about the same radius as that of the disc (a cms). By regulating the position of the movable base, S, the distance, d, may be made any length within the limits of the length of the base. Fig. 328 also shows, in a diagrammatic form, the connexions used in the experiment. AB is the metal rod, say, of brass. A single accumulator is connected in series with the coil, D, and the resistance, AB. 524 ADVANCED PRACTICAL PHYSICS FOR STUDENTS If R is the resistance of the rod in absolute E.M. units, the drop in potential at the ends of AB is cR, when c is the current in like units. The rod is also connected, by copper wires, in series with the disc and a galvanometer. Fig. 328 When the current from the accumulator, E, circulates through the coil, a definite number of lines of magnetic force cut the disc. If M is the coefficient of mutual inductance of the disc and the coil, there is a flux Mc lines through the disc. If, there- fore, the latter is made to rotate, an E.M.F. will be set up in the disc circuit, which will depend on the direction of rotation of the disc for its direction. The direction of rotation is chosen such that the E.M.F. set up by induction opposes the E.M.F., Re, due to the steady current of the coil circuit. By adjusting the speed of rotation, or by altering the distance, d (usually by both these methods), we may balance the potential due to the two causes. Suppose there are n revolutions per second when this balance occurs, as shown by no deflection in the sensitive galvanometer G. In the disc circuit we have an E.M.F. due to the induction equal to (Mc)n, since Mc lines are cut by any radius for one revolution, there are n revolutions per second. ~M.cn is the number of lines cut per second, and opposing this is an E.M.F., Re. For balance Men = Re, or R = nM. The resistance, R, is therefore determined if M, the coefficient of mutual induction, is known, and n the number of revolutions per second is counted. Maxwell's formula for the coefficient of mutual inductance in the case of two circuits of one turn each, of radius, a, and separated by a distance, d, is : = 4*«(log.^( I+i ^)-( 2+ -g 5 ) m RESISTANCE MEASUREMENTS 525 The coefficient, M, above is thus Nw, 1 - e - R = wwN, or R= 4 ^N« J 2-303 -10^.(1+^) _( 2 ■ ^ 16a 2 , , The resistance in ohms, where the ohm is defined as io 9 E.M.U. is, of course, R x 10-'. Experimental Details To avoid the complication of the Earth's field in the induction of the E.M.F. in the disc, the plane of latter is turned so that it is in the magnetic meridian. *" The Broca, or other sensitive moving-magnet galvanometer, works very well in this experiment. If no motor is available, a fair result is obtained by rotating the disc by hand. The speed of rotation which may be best maintained steady is found by a trial experiment, and the coil and disc placed at a convenient distance, d. When all is ready the disc is rotated, until the spot of light from the galvanometer is brought back to the zero reading. At this stage the rotation is maintained constant. The number of revolutions made is counted and timed, with a stop-clock, over as long a period as the light spot may be kept at zero. From this a knowledge of n may be obtained. d and a are measured in cms., and the value of R calculated. If the length and cross-section of the rod are measurable, the value of the specific resistance may be also calculated. The following is an experimental result for a determination made with such an apparatus. 400 revolutions of the disc in 172-5 seconds, .*. n — 2-325 revolutions per second, a = n-8 cms., d = 5-0 cms., N = 100 cms., R = 100 x 2-325 x 148-1 1 2-303 x 1-2761 ( 1 -j- — ^~ ) ( \ 2227-8 / V ^ 2227-8/ ( = 35372 absolute E.M. units. Specific Resistance : Length of rod = 167 cms., area of cross-section = ^(.64)2, Specific Resistance = 35372* 1-20, 16-7 = 2732 • E.M. units (or -000002732 ohm). CHAPTER XX RESISTANCE OF ELECTROLYTES Resistance of a Battery. (Mance's Method) The resistance of a battery may be found by a modification of the Wheatstone net as shown in fig. 329. The resistances, P,Q,R, and the battery of resistance, B, form the network. The connecting wire and key, K, take the place of the battery in the usual form of the net. K Fig. 329 It will be seen that for all values of P, Q, R, a current will pass through the galvanometer, G (i.e. in circuit, DCB). The resistances, P, Q and R are adjusted until, on depressing K, no change is produced in the steady deflection ; under such circumstances we have : P R „ OR — = — or B = — Q B P The main difficulty in such a determination is that the steady current in the galvanometer causes a deflection which is too large to keep the reflected beam of light on the scale ; or, if on the scale, the galvanometer is insensitive in detecting change in deflections, as seen by the fact that over a large range of resistance in the arms, R or Q, the above condition seems equally well fulfilled. 526 RESISTANCE OF ELECTROLYTES 527 This is best overcome by placing a condenser of, say, one-third microfarad in series with the galvanometer (shown in broken lines in the figure). The steady current in the galvanometer is therefore eliminated. When the key, K, is closed and the bridge is unbalanced, a throw of the galvanometer will be recorded. When balanced, no kick will be given by the galvanometer when K is closed. Another method which may be employed to overcome the steady deflection, using a moving magnet galvanometer, is to alter the control field in such a way as to reduce the deflection to zero, and proceed as first described. This method, if possible with the control magnet on the instrument, gives good results as the galvanometer is sensitive to the variations produced by closing K, when in the zero position. If the control magnet is too weak to restore the moving magnet to the original zero position, it may be restored by an external magnet. Using a Kelvin (Thompson) galvanometer, the same result may be brought about by sending the current through one pair of coils only. The second pair of coils are connected to an independent circuit composed of an adjustable resistance and cell, and the current sent in such a direction as to produce a field in the opposite direction. The magnitude of this current is adjusted until the spot of light is at the zero position. The effect of tapping the key, K, will then be seen in the resulting deflection. The control magnet, in such a case, may be adjusted to give good sensitivity to the instrument. Conductivity of Salt Solutions (Electrolytes) When a direct current is passed through an electrolyte, the resulting polarization causes an increase in the resistance of the electrolyte. Further, for continued passage of current through such a liquid, the resulting decomposition also causes a change in the resistance, due to alteration of the concentration of the solution. The usual special methods adopted to measure such resistances are designed to overcome these difficulties, and are either a potentiometer method, or one making use of an alternating current. The second is the one most generally employed. If the alternating current is small, and the area of the electrodes large, the polarization effect is reduced to a negligible amount. The more completely is this brought about when very rapidly alter- nating current is used. Thus, whereas it is impossible to obtain reliable values for 528 ADVANCED PRACTICAL PHYSICS FOR STUDENTS the resistance of salt solutions by the Wheatstone bridge method in the ordinary way, by using alternating current in conjunction with a Wheatstone net, the value of the resistance of the electro- lyte may be found. Of course in such an arrangement an ordinary galvanometer is useless as detector ; a telephone replaces it in the usual modification. It is not advisable to introduce any inductance or capacity into the net, so for this reason the wire bridge is preferable to the post office box type, as the one or ten metres of wire have less self-inductance than the coils of the post office box. The scheme of connexions is shown in fig. 330. AB is a stretched wire of one or ten metres in length, R an adjustable resistance, S a vessel containing the solution, T a telephone, C a small induction coil. \ Fig. 330 The alternating current is supplied by a small induction coil which will give but small potential. The induction coil is driven by a cell and the secondary is connected to the ends, A and B, of the bridge. If possible an induction coil without an iron core should be used ; the coil with the greater number of turns should be used as primary, and the current supplied to it should be reversed by a rotating commutator driven by a small motor. The telephone, T, should be a head-piece receiver. J The vessel, S, shown in fig. 330, should be surrounded by a water bath, and be provided with platinum electrodes. The temperature of the water jacket is maintained constant, as the resistance of the solution varies rapidly with temperature. The platinum electrodes are coated with finely divided platinum to increase their effective area, and to decrease the back electro- motive force due to polarization. If this has not already been RESISTANCE OF ELECTROLYTES 529 done, the electrode should be immersed in a solution of platinum chloride* and a smaller current passed first in one direction, and then the reverse. To prevent an absorption of salt from the solution, the platinized electrodes are then raised to dull red heat. The solution of known concentration is placed in S, and R is adjusted so that a point P is obtained near the centre of the bridge, such that the sound in the telephone is entirely cut out or, as more often happens, until the sound is reduced to a minimum, when the usual Wheatstone result may be applied, RAP S ~PB' It is usual to express the results in terms of the specific con- ductivity of the solution, i.e. the reciprocal of the resistance of one centimetre, one square cm. in cross-section. If r is the resistance in ohms between the electrodes, s the specific conductivity, then sec— , r A or s = — , r where A is a constant depending on the dimensions of the vessel used A may be obtained by finding r for a liquid of known specific conductivity. We may take potassium chloride as a standardizing solution, using ihe data in the table shown on page 530, where the specific equivalent conductivity is the quotient of the specific conductivity and the number of gramme molecules of the salt per litre. It will be seen that for dilute solution this quantity is nearly independent of the concentration. Make up a solution of KG containing a definite number of gramme molecules per litre, and find the resistance, r, of the solution when filling the vessel, S. Dilute this solution so that the solution contains, say, half this number of gramme molecules. The value of the specific conductivity for these two strengths may be taken from the above table, and the mean value of A calculated. Then, using this calibrated vessel, find the resistance of several solutions of NaCl from a concentration of, say, 29*25 grammes * The Platinum chloride solution to use is made by taking 1 part platinum chloride, 30 parts water, '008 part lead acetate. 34 530 ADVANCED PRACTICAL PHYSICS FOR STUDENTS per litre P^\ to -2925 grammes per litre, by diluting the concen- trated solution first made. Plot a curve showing the relation between specific conductivity and concentration, and specific equivalent conductivity and concentration. The variation of the resistance of one of the solutions with temperature may also be investigated by heating the water baths surrounding the cell, S, and the value of the temperature co- efficient may be calculated. Using a container for the solution, of more measurable dimen- PARTS BY WEIGHT GRAMME SPECIFIC SPECIFIC OF KCI IN IOO MOL. PER CONDUCTIVITY EQUIVALENT PARTS OF SOLUTION LITRE AT l8°C. CONDUCTIVITY •OO746 •001 0-125 x 10- 3 125 xio- 8 •O746 •01 1-206 „ 120-6 743 •I 11-300 „ 113-0 I-48 •2 22-00 „ IIO-O 3^4 *5 50-60 „ ioi-6 7'43 1-0 99-10 „ 99-1 137 2*0 188-8 94-4 197 3'0 274-2 91-4 sions, the specific conductivity, and hence equivalent conductivity may be calculated in the way usually employed in solids. For example, in fig. 331, a uniform tube connects two bottles in dh X j Fig. 331 each pf which is immersed an electrode ; the column of liquid conveying the current is practically one coinciding with the tube, i.e. of length I cms. and cross-section equal to that of the internal cross-section of the tube. The resistance is measured as above ; the specific resistance is then directly calculated in the usual way, and the experiment proceeds as already described. CHAPTER XXr MEASUREMENT OF POTENTIAL Standard Cells 'the two standard cells in common use are the Weston or Cadmium cell and the Clark cell ; of these, the former is better for general service. The two cells have each a constant electromotive force at one temperature, and also a definite temperature coefficient. The Weston cell is shown in diagram form in fig. 332. Two tubes are arranged as shown, each being provided with an external lead which is in contact with the bottom layers. These layers consist of pure mercury, M, and an amalgam of pure mercury and cadmium, A, respectively. Above the pure mercury is a layer of a paste of mercurous sulphate, P, made as described later (shown by horizontal shading). Above this and the cadmium amalgam is a layer in each tube of pure cadmium sulphate crystals, CC. Finally, a layer of a saturated solution of pure cadmium sulphate occupies the upper parts of the tubes. To make a Cadmium Cell The following is a method of construction for the Weston cell ; for a more permanent and exact cell the specification given in the Report of the British Association Meeting, 1905, on page 98, by F. E. Smith, should be consulted. 531 532 ADVANCED PRACTICAL PHYSICS FOR STUDENTS (a) Mercury is at first obtained in as pure condition as possible commercially. It is then passed through a dilute solution of nitric acid, drop by drop. To bring this about take a tube of about three-quarters to one inch in diameter, and about 60 to 100 cms." long ; draw out the end to a smaller diameter and bend this smaller tube so as to leave a short U-tube at the end ; the shorter end is bent over as seen in fig. 333. Fig. 333 A second length of the wide tube is drawn out at one end to a fine capillary of such diameter, that when the tube is filled with mercury the latter will just emerge as very small drops. This tube is inserted at the upper end of the first, which is filled with dilute nitric acid. The mercury is collected in the U-tube and passes over into a collecting vessel. The process is repeated and the partially purified mercury is next distilled under reduced pressure, air being bubbled through it during this process. To carry out this distillation Fig. 334 the mercury is placed in a round-bottomed flask, provided with a side tube which leads through a condenser to a second round- bottomed flask, itself connected to a good water filter pump, as shown in fig. 334. MEASUREMENT OF POTENTIAL 533 Into the first flask, through a tightly fitting cork, a narrow- glass tube passes under the mercury. A clip regulates the inflow of air. When the pump has reduced the pressure inside the system, the mercury is heated on a sand bath, and the condensed mercury is collected in the second flask. The air is allowed to enter through the mercury as a slow succession of bubbles. This oxidizes such impurities as zinc and minimizes the risk of their distillation with the mercury. (b) The amalgam is made by dissolving pure cadmium in pure inercury, to make a 12 to 13 per cent cadmium amalgam (i.e. about in the proportion of 1 gramme of cadmium to 7 grammes of pure mercury). (c) The paste is made in a mortar by grinding together pure mercurous sulphate, cadmium sulphate and purified mercury in the proportion of 8:4:1. The mixture is made into a thin paste by the addition of a solution of cadmium sulphate. Before filling the cell the platinum wires must be amalgamated. In the form of tube shown in fig. 332, this is done by the electro- lysis of mercuric nitrate in the glass tubes, using the wires as electrodes and reversing the current. Another form of glass container for the cell consists of a double tube with open upper ends. Corks are selected to fit the open ends tightly, and through holes in the corks a narrow glass tube may be inserted in each side tube, to carry the wire through the cell content to the mercury or amalgam. Such platinum wires could well be amalgamated by heating to red heat and dipping in mercury. Having amalgamated the platinum wires, the tubes are next carefully filled as shown in fig. 332. If the open-tube type is used, the corks should be finally coated with marine glue or a mixture of beeswax and resin. The E.M.F. of the cell so formed will be found constant with constant temperature. On no account must a current of any appreciable magnitude be taken from the cell. The International Conference on Electrical Units and Standards, 1908, adopted the following formula as giving most accurately the E.M.F. of the cell : E t = 1-0184 — 4'o6 x 10- 5 (t — 20) — 9-5 x 10- ' (t — 20) 2 + I0- 8 (tf-20) 3 VOlt, . . « M ... (I) where t is expressed in degrees centigrade. The temperature coefficient is therefore small. The Clark Cell This is made in a manner identical with that described above, With the exception that cadmium is replaced in this case 534 ADVANCED PRACTICAL PHYSICS FOR STUDENTS by zinc ; cadmium sulphate by zinc sulphate, etc. Proceeding as above, again using pure salts and mercury, the standard cell so constructed has an electromotive force expressed by the formula : E t = 1-4328 - 1-19 x 10- 3 (t - 15) — 7 x 10- 6 (*-i5) 2 . .(2) Thus this cell has a larger temperature coefficient than the Weston, a fact which explains the more general use of the former. When using either form of cell in potentiometer work the device of fig. 337 (i.e. a large series resistance) is a useful one as a safeguard against damage to the cell when the ' balance point ' is not approximately found. Comparison of Electromotive Force The potentiometer method of comparing two electromotive forces is the most satisfactory one. It is assumed that the reader is familiar with the direct comparison of two E.M.F.'s, using a stretched wire potentiometer. B In that method a steady accumulator is arranged as at E in fig- 335 > and E x and E 2 whose E.M.F.'s are to be compared are in turn placed in series with the galvanometer, and a point of balance obtained at l x and Z,*cms. respectively ; then, if the wire is of uniform resistance per cm., E 2 l 2 Using one of the methods given on page 495, tt seq., the wire, AB, could be calibrated and then, more accurately, if l x x and Z 2 * are the corrected lengths corresponding to l x and l a For an absolute value of the E.M.F. of either cell a third balance could be obtained at, say, l 3 for a cadmium cell whose E.M.F. is known at the temperature of the experiment and is * l t is the mean of, say, three observations, and l 2 the mean of two observations, taken alternately ; this eliminates the effect of the variation of E.M.F. of E. MEASUREMENT OF POTENTIAL 535 approximately 1-0184 volts. Whence E x and E 2 may be obtained in volts. For making such a comparison, the accuracy of the deter- mination depends on the accuracy of obtaining the balance poiftt. If instead of using a one-metre potentiometer, a wire of ten metres (i.e. ten wires in series, each one metre long), be used, then each cm. of wire has a potential drop equal to one-tenth the drop in the simpler potentiometer, i.e. a movement of 1 mm. in the single wire bridge would correspond to 1 cm. movement in the ten-wire instrument ; hence by using the ten-meter potentiometer the true balance point may be more nearly estimated. It is often convenient to make such a potentiometer direct reading. To do this we arrange that between the ends of a definite length there is a fixed potential difference, say, 10- s volt. Using a ten-metre potentiometer the most convenient length to employ to correspond to io _a volt is 5 mms. This is brought about as follows : A standard cadmium cell (E.M.F. = 1-0184, assuming tempera- ture is 20 C.) is connected to A,* and the jockey makes contact with the wire at 1018-4 units of length from the end. We have chosen the unit for this purpose as 5 mms. i.e. P is fixed at 509-2 cms. from A as in fig. 336. R is now adjusted E 1 . — I— 1 — B S.C. G Fig. 336 until the galvanometer gives no deflection, i.e. the current in the wire AB, due to E, is such as to cause io -3 volt drop per 5 mm. of the wire (assumed uniform). Leaving R fixed, any other E.M.F. may be found by balancing on the potentiometer at, say, / cms. from A or 2/ x 5 mms., i.e. 2.1 x 10- 3 volts is the value of the balanced E.M.F. The above assumes that the accumulator E remains steady. This should be checked at intervals, if a series of comparisons are to be made, by reinserting the standard cell, SC, and adjusting R to bring a balance at 509-2 cms. * A resistance of about io 4 ohms should be in series with S.C. until balance is almost complete. See fig. 337 and page 537. .536 ADVANCED PRACTICAL PHYSICS FOR STUDENTS For many purposes it is convenient to replace the wire by- variable standard resistances when comparing potentials. A suitable arrangement of apparatus using such a method is seen in fig. 337. R 2 and R 2 are two resistance boxes, each having a E x l |U? K 2 Fig. 337 resistance up to, say, 10,000 ohms (two post office boxes do very well). E is a steady accumulator which is connected in series with R x and R 2 . The cells to be compared are connected to A and P through a galvanometer, G. For direct comparison the resistance (Rj + R 2 ) is kept constant, say 10,000 ohms, and R x and R 2 are varied until no deflection is obtained when K 2 is closed. The value of R x is noted. The process is repeated with the second cell, say, a cadmium cell, a balance being obtained for a resistance R^ in the box between AP. Then, as before ~r = ^j> Ji 2 Kx 1 E R for the drop in potential between AP due to E is = ^ — 5 i R-i + J^2 + -fc> where B is the resistance of the cell E. When a balance is obtained similarly and Hence E. = R x + R 2 + B R1 1 E, R 2 i + R, 1 + B Rj 4. R 2 == Rji -j- R 2 i = 10,000 ohms. 5s E. 5l Rx 1 ' Another way of using the above form of potentiometer is similar to the direct reading method of using the wire potentio- meter. MEASUREMENT OF POTENTIAL 537 One of the cells to be balanced against E is a cadmium cell. If the temperature of the experiment is not far removed from 20 C, the electromotive force of such a cell is 1-0184. R x is given the value 1018-4 ohms (a fraction ohm box may be included in series with R a ) . Knowing the approximate electro- motive force of the cell, E, the value of R 2 may be estimated such that the potential drop in AP ^ 1-0184 volts. The standard cell is placed in series with the galvanometer G ; K 3 being open, r, of about 10 4 ohms is in series with SC, to avoid damaging it during the preliminary balancing. R 2 is now adjusted until no deflection is noted in G. K s is closed and the final balance verified. The total value of R x + R 2 under these conditions is noted and maintained constant throughout the comparison. Now 1018-4 ohms have a drop of potential of 1-0184 volts, i.e. each ohm corresponds to a potential drop of io~ 3 volt. Therefore, when a second cell (E 1 ) is introduced, if R t has a new value, R x * ohms at balance (R^ + R 2 X being equal to R, and R 2 as obtained in the first test), E 1 is R x x x io~ 3 volts. The above methods of making the potentiometer direct reading are only suggested for those cases where several comparisons are to be made, for under such circumstances subsequent calculation is eliminated. The experimental arrangements described above are most sensitive for comparison of electromotive forces of the order of one volt. If now a small difference of potential is to be deter- mined it will be apparent that these arrangements are not sufficiently sensitive. In the case of a stretched wire, the sensitivity increases with increase in length ; therefore, to measure a potential difference of the order of, say, io~* volts with fair accuracy the potentiometer wire would require an extension of several metres of wire, or, what is more convenient, the inclusion in the circuit of a resistance several times that of the wire (r). For example, if a potentiometer wire of one metre were of one ohm resistance, and the accumulator E had an E.M.F. of 2 volts, a potential difference of the order of 10 milli- volts would balance at a distance of 5 mm. from the end of the wire. This could very easily be estimated at 4 mm., or 20 per cent in error. Further, under such circumstances the calibration of the wire would be of greater importance. If on the other hand 99 ohms were placed in series with the wire, the drop of potential in the wire would be 1/100 of 2 volts, and the- true balance point would therefore be at 50 cms. An error in estimating the balance point of 1 mm. would only be •2 per cent. An example of the application of this method is seen in fig. 343, page 542. 538 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Direct Reading Potentiometers ■> As described above, the potentiometer may be made ' direct reading.' Several forms of potentiometer are now on the market which are constructed on the above principles and are calibrated directly in potential. Of the simpler forms of compact manufactured instruments, which measures potential of the order of a volt to one millivolt, we will describe the form illustrated in fig. 33,8. The internal wiring is indicated by white lines drawn on the case, and the instrument is an application of the form shown in fig. 336. The 2 volt accumulator which is connected to the terminals EF, sends a current through an adjustable rheostat, a series of coils provided with tappings to the studs shown, and through the slide wire. This constitutes the main circuit. When the current is adjusted to the correct amount, the fall in potential along each of the resistances, connected to the studs, is *i volt, and along the slide wire, «I2 volt. The wire is divided into 120 parts, each corresponding in adjustment to a fall of potential of 10- * volts. To adjust the current to this strength a standard cell is inserted in the gap marked ' potential,' and a galvanometer of low resis- tance in the gap marked ' galv.' The two adjustable contacts are set at points which correspond to the true E.M.F. of the cell at the temperature of the room, e.g. if the E.M.F. of the cell is 1-018, the sliding contact at the back of the instrument is set at i«o and the contact on the wire made at •018. The rheostat is then adjusted so that when the key is closed no current is indicated in the galvanometer. If the rheostat is not sufficiently large to bring this about the battery should be connected to E and G, not EF. This introduces more resistance in the circuit as indicated in the figure, and, using a normal 2-volt accumulator, the balance for the standard cell will be attained. To obtain the value of an unknown electromotive force, it is connected to replace the standard cell. Leaving the rheostat in the balanced position, the sliding contacts are adjusted until a balance is obtained, as indicated by no current in the galvano- meter. The value of the electromotive force is then directly obtained on the calibrated scales. The width of the small divisions is sufficiently large to allow eye estimation to £ of a division, but is not of suitable range for the measurement of the corresponding £ of a millivolt. Thus, whereas the instrument would measure a potential of 1-0182 with a good degree of accuracy, it cannot be used to measure •0002 volt with any certainty, nor is it reasonable to expect PtlRfi 5 3 B Fig. 339 Pg£B 5J9 MEASUREMENT OF POTENTIAL 539 such measurement with an instrument having a range o to 1*5 volts.* For such measurements, as, say, for a thermo-junction, an instrument having a range of to 50 millivolts is more suitable. Potentiometers of such a range are manufactured by many firms, e.g. Nalder, Cambridge and Paul, Crompton, Gambrell, etc. Figs. 339 and 340 show the general appearance and internal arrangements of such an instrument. The current to main circuit is supplied by a 2 volt accumulator, B. As seen, this circuit consists of adjustable resistances, Rj and R 2 , fixed resistance E and F, MVC, D x and D 2 , and the two stretched wires SS and W. The range of the instrument is 30 millivolts (o to 30, or 30 to 60, or 60 to 90) . The resistance MVC is made up of 29 similar coils of such a resistance that when the current is adjusted as described below, the potential difference between the ends of each coil is 1 millivolt. Also the wire, VV, is of a resistance such that for this adjustment the drop of potential along its length is 1*2 milli- volts. By subdivision, a value of '005 millivolts may be obtained. Further, the value of the resistance, F, and the slide wire, SS, is such that the graduation along SS gives the potential between M and N when adjusted. To standardize the potentiometer, the standard cell, SC, is connected to the galvanometer, G, by means of the double pole switch. The point, N, is chosen equal to the potential of the standard cell, R x and R 2 are then adjusted until the low resistance galvanometer shows no deflection, i.e. the drop of potential along MN corresponds to the graduation value. Under such circum- stances the potential difference per coil of MVC is one millivolt, etc. To measure an unknown electromotive force between .0 to 30 millivolts, plugs are inserted at the points shown in fig. 340, and the unknown potential connected to X. The commutator is thrown over so that XX are connected to the galvanometer, and a balance is obtained by varying the point of contact, P, on MVC and Q on VV. If D 2 is of zero resistance the unknown potential corresponds to that between P and Q and is therefore obtained directly from the scale. Suppose a bigger potential, say, 30 to 60 is to be measured, D a is now made of such resistance that there is a potential drop equal to 30 millivolts along its length, and therefore for the * A modification of this form of instrument, having a range suitable for thermo-junction work, is now to be obtained. 540 ADVANCED PRACTICAL PHYSICS FOR STUDENTS balanced position the potential is 30 + the readings of MVC and VV. Similarly for 60 to 90 millivolts. D 2 is increased so that for the steady current in the main circuit the potential difference between the ends of D 9 = 60 millivolts. r-0 Fig. 340 To maintain the current in the main circuit at a fixed value, the resistance of D x must be decreased by the same amount as the increase in resistance of D 2 . This is done in the manner shown in fig. 340. Thermo-Electrieity— Thermo-Junetions When a circuit is composed of two dissimilar metals and the junctions of these metals are maintained at different temperatures an E.M.F. is set up in the circuit. This electromotive force varies with the difference in temperature between the junctions, and when one junction is maintained at o° C. is given by E f = at + U\ where a and b are constants and t expresses the temperature of the hot junction in degrees centigrade. The direction of the electromotive force depends on the metals. It is customary to express E t for any metal with respect to a standard metal which is taken as one of the pair. The usual choice of standard metal is lead. In drawing the curve giving the relation between the E.M.F. and temperature, the E.M.F. is taken as positive when the current tends to flow from lead to the metal at the hot junction. MEASUREMENT OF POTENTIAL 54i Thus fig. 341 shows the form of these curves for Pb/Fe and Pb/Cu. At a temperature t° C. AB represents the E.M.F. developed in a Pb/Cu junction and, according to the above rule, the electro- motive force is from the lead to the metal at the hot junction. Similarly, AC is the magnitude of the electromotive force developed from lead to iron at the hot junction. + c y 1 ^r EMF f B A? O L^ A Temperature — Pb Fig. 341 The law of intermediate metals may now be applied to deter- mine the value of the E.M.F. developed at a copper-iron junction at a temperature t° C, the other junction being maintained at o°C, for, according to this law the E.M.F. developed at the temperature t°, for a Cu/Fe junction, is expressed by E.M.F. Cu/Fe = E.M.F. (Cu/Pb + Pb/Fe) = - AB + AC . , = + BC, i.e. there is an electro-motive force of magnitude BC from copper to iron at the hot junction since the BC is positive. Reference to fig. 342 will show this from another point of view. If junctions, X and Y, are maintained at temperature, t°, Z being maintained at o° C, the arrows show the direction of the E.M.F. 's 542 ADVANCED PRACTICAL PHYSICS FOR STUDENTS of magnitude AC at X counter-clockwise and AB at Y clock- wise — a net result BC counter-clockwise. This, according to the Law of Intermediate Metals, is the value of the E.M.F. if XY are brought together, the lead being removed. The result is an E.M.F., BC, in counter-clockwise direction, i.e. from the copper to the iron at the hot junction. Experimental Determination of the Thermo-Electromotive Force — Temperature Diagram The magnitude of the thermo-electric E.M.F. is of the order of a few millivolts. It is best investigated by means of a potentio- meter of the form described on page 537. An instructive result is obtained using the three metals, copper, iron and lead. As will be seen from the above, it is only necessary to obtain the E.M.F. for two pairs of the metals, the third pair may be estimated, making use of the law of intermediate metals. A uniform wire potentiometer of, say, 10 metres is employed (one of one or two metres, if the other form is not available). If the best results are to be obtained a preliminary calibration of the wire is advised. A steady accumulator, C, is joined in series with R lf R 2 , and the potentiometer wire (R x and R 2 should contain resistances up to 10,000 ohms). A preliminary experiment gives the value of r the resistance of the ten metres of wire. Her ice Fig. 343 A thermo-junction is constructed, using copper and iron wire. Care must, of course, be taken that the wires are in contact at the junction only. For this purpose a suitable form of junction is seen in fig. 344. The one wire, B, passes down MEASUREMENT OF POTENTIAL 543 a thin glass tube, G, the other, A, is joined to it at J. To ensure good contact, J is dipped into mercury at the bottom of the test tube, T. This does not affect the E.M.F. If one or both of the metals are affected chemically by mercury, it must be dis- pensed with, and special care paid to the welding of the junctions. j . Three such junctions are made. One, the hot / / A junction, using iron and copper wires. The free ends of the copper and iron wires are each joined to connecting leads as illustrated. These two junc- tions are maintained at the same temperature, o° C. in the experiment described. The use of such junc- tions is to eliminate thermal electromotive force between the metal of the connecting leads and the metals of the thermo-junction proper. In the par- ticular case taken, the copper to copper connecting lead is not essential generally, but is a safeguard against the possibility of impurity in one wire, and is necessary for the Pb/Fe junction. If the junctions to the connecting wires are main- tained at o° C. by surrounding with ice, and the ' hot junction ' is placed in a water bath at a tempera- Fig. 344 ture t o c ^ a;a £jip_ yrfft be developed in the direction A to L via the junctions. C is therefore connected with the positive pole to B. The maximum E.M.F. developed in the above couple is of the order of 1500 micro-volts : the rise in potential along AB should therefore be arranged not very much in excess of this. A voltmeter gives the approximate potential available from C, and r being known, the value of R x + R a to cause such a drop may be calculated, potential difference in AB r E.M.F. of C r + Ri + R»' the accumulator resistance being negligibly small. Having fixed R x and R 2 , the point, L, is made to coincide with A, and the standard cell, say a cadmium cell, is put in series with the galvanometer ; R 2 is adjusted, keeping R x + R 2 at the value determined above, until no deflection is given in G when contact is made at A. If the room temperature is approximately 20 C. the E.M.F. of the cadmium cell is 1-0184. Hence the drop of potential along AB is : £- x 1-0184, whence the drop per cm. of the wire is : ^- x 1-0184 X I0 " 3 volt • — — — ~ (3) 544 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The thermo-junction is now placed in circuit and the hot junction is raised to about 95 C. in a water bath : the length of wire required for balance is obtained. The water is allowed to cool and balance points obtained for intervals of temperature of about 5 C. The hot junction is placed in a hypsometer and again the E.M.F. is balanced : when the junction is at the temperature of the steam a steady balance point is obtained and noted. The hot junction test tube is next placed in a boiling tube containing mercury. This is heated slowly. Balance points are obtained at intervals and the temperature noted on a special mercury thermometer which reads to 360 C. The process is carried on until the mercury boils. The results are tabulated — temperature and lengths for balance. The lengths are converted to E.M.F. by multiplying by the reduction factor given in (3) above. The process is repeated, using a lead wire in place of the iron, and the results are plotted as in fig. 341, E.M.F. in micro-volts (io - 6 volt) against the temperature of the hot junction. From these results obtain the corresponding curve for lead — iron. It will be found that at a temperature of about 240 ° C. the Fe-Cu junction will give a maximum E.M.F. This tempera- ture is called the neutral point. The same results could be obtained directly, using a direct reading potentiometer. With a ten-metre instrument r would probably be just greater than 10 ohms. If a shunt, S, were placed between A and B (fig. 343) and S given a suitable value which can be calculated from a knowledge of r, the wire and shunt may be made to have exactly 10 ohms resistance. As before, suppose the E.M.F. of the cadmium cell were 1-0184 at tne temperature of the experiment (any other value can be treated in the same manner). Then to obtain a drop of 10- 6 volt per cm. of bridge wire R 2 is given the value (1018*4 — 10) 1008-4 ohms. The cadmium cell is placed in series with the galvanometer, and the sliding contact, L, is made at B (fig. 343). Ri is adjusted until a balance is obtained. Then the drop in potential from A to B = 5 — of 1-0184 or I0 ~ 8 volt, hence 1010-4 the drop per cm. is io -6 volt. The lengths for balance in the thermo-couple experiment now give the potential in micro-volts directly. The same process applies in the case of a single-metre poten- tiometer. The value of r is found, and the shunt value, S, to make the value for the two in parallel 1 ohm, is calculated and R 2 made (1000 E — 1). In that case each millimetre of the wire corresponds to a potential drop of io~ 6 volt. MEASUREMENT OF POTENTIAL 545 If no standard cell is available the value of the potential per cm. of wire may be calculated from a knowledge of r, R x and R 2 and E, the E.M.F. of the cell, C, as determined by means of a high-resistance voltmeter. Consider the curves obtained. If we take any two fixed temperatures and determine from the curve the value of the E.M.F. developed, in micro-volts, we shall be provided with two equations having two unknown constants, for we have seen that the relation between the E.M.F. developed E, and the temperature t° C. is : E = at + bt*. So by choosing two such pairs of temperatures and finding E on the Fb-Cu and Fb-Fe curves, the values of a and b for iron and copper may be obtained. dF Hence since ' -=- = a-\-2bt, at we know the value of the thermo-electric power, -=--, at any dt temperature. The thermo-electric power lines for Cu and Fe against the standard, lead, may thus be drawn and the point of intersection, which gives the neutral temperature for iron-copper, may be ascertained. This will be found, of course, to agree with the neutral temperature as found directly from the E.M.F. temperature curve for the iron-copper junction. The Use of a Thermo-Eleetrie Couple as a Thermometer It will be seen from the curves obtained (fig. 341), that a couple such as copper-iron is not a suitable one to use as a thermometer. The neutral point is too near o° C. for such use. The ideal couple for such purpose is one with a neutral point well removed from o 6 C, and which will therefore give an E.M.F. approximately proportional to the temperature difference. In many experiments in this book such a couple is required. A couple of copper and eureka or constantan serves well for this purpose. When the thermo-junction is used in this way the form of potentiometer described on page ^ 539, fig. 339 is a great convenience, but the form of direct reading potentiometer of ten metres of wire (page 544) when once adjusted serves quite well. In either case the electromotive force-temperature curve of the junction should be obtained as described up to, say, 240 C. The value of the temperature corresponding to any other electro- motive force may then be obtained from the standardizing curve so obtained. 35 546 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Lippmann's Capillary Electrometer The essential feature of the Lippmann Capillary Electrometer is a very fine capillary tube drawn from a thin glass tube, mounted so that a column of mercury of variable height may be placed over the meniscus of a liquid which rises in the capillary. Fig. 345 Fi g- 345 shows how this is usually arranged : C is the capillary attached to a vertical glass tube, AB, by means of a short length of pressure tube at B. The side tube is connected by pressure tubing to an adjustable reservoir, R, full of mercury. A scale in millimetres is placed behind either AB or the reservoir R, so that changes in level of the mercury may be measured. The capillary tube, which dips into sulphuric acid, should be of sufficiently small diameter to prevent the acid from being driven from the tube by the mercury above it. At the bottom of the beaker, E, which contains the acid, is a layer of mercury, into which a wire, passing down the centre of a glass tube, D, may be placed. A second copper wire dips into the mercury above the meniscus, either by means of a platinum wire lead fused into the tube AB, or by inserting a wire into the mercury in AB or R. The two leads are connected to a potentiometer which consists of two Post Office boxes in series, R x and R 2 , so that, maintaining R t + R 2 at 10,000 ohms, and adjusting R x and R 2 , any fraction of the potential of the cell, E, may be applied to the junction of the upper mercury and the acid. MEASUREMENT OF POTENTIAL 547 If the meniscus is focussed and made to coincide with the horizontal cross-hair in the focal plane of a high-power microscope, it will be found that when a potential is applied in one direction, the meniscus descends, and when applied in the other direction the meniscus ascends. The effective value of the surface tension of the acid mercury surface is altered by the applied voltage. Suppose the reservoir, R, is lowered so that only a small pressure is applied to the surface, and the meniscus is focussed on the cross-hair of the observing microscope (which should be of high magnifying power), the observed level does not corres- pond to the true surface tension level. At the junction of two liquids there is a contact difference of potential set up, and the value of the effective surface tension depends on this contact potential. In whichever direction this contact potential acts, the result is a decrease in the effect surface tension value. For two given liquids, i.e. sulphuric acid and mercury, the value is fixed in direction and magnitude. We may regard the surface of separation of the liquids as a double layer functioning like a condenser, and having an energy per unit area of £cV 2 , where c is the capacity and V the value of the contact potential. Thus, if T is the effective surface tension, and T the value of the surface tension if no potential exists, we may write : T=T -£cV2 (4) so that whatever sign be given to V the value of T < T . If a potential be applied to the surface of separation in the same direction as the contact potential, then V increases and T becomes less, causing the column of acid to descend to a new equilibrium position, whereas, if the applied potential be of the opposite sign to the contact potential, we obtain a larger value for T, and consequently the acid rises in the tube. S.-4A - Fig. 346 The capillary when drawn out from the glass tube is always slightly conical, and the state of things showing two positions is seen in fig. 346. If the applied E.M.F. opposing the contact potential increases 548 ADVANCED PRACTICAL PHYSICS FOR STUDENTS T becomes larger, until, when the applied potential is further increased, we have the double layer effect again coming into play, due to a net potential difference of the opposite sign. For each value of the applied opposing E.M.F. the meniscus will take up a definite position, and it is obvious that when the contact potential is just neutralized, the meniscus will be at the highest point. The conical shape of the capillary tube, with resultant change of diameter and of focus, renders unreliable the observations of the level of the meniscus. To be sure that the diameter of the tube at which the surface of separation lies is the same, the meniscus is always observed at one point. For example, as the value of T increases the level is brought back to the previous one by raising the reservoir R and increasing the pressure on the surface. The microscope used for observing the level of the meniscus should be high-power and the reservoir readily adjustable, so that the level of the mercury in R or AB may be read for each value of the applied potential. This process should be repeated for all values between o and 2 volts, or until electrolysis interferes with observation. The values of the applied E.M.F. should be plotted as abscissae, and the pressure in cms. of mercury as ordinates. The form of the curve as obtained from the results under, is seen in fig. 347. The maximum of the applied pressure^corre- Fig. 347 sponds to a maximum, T, or if the value of T may be represented in some such form as (4) the maximum T is T when the term depending on the contact potential is zero., i.e. it corresponds to an applied potential equal to the contact potential. From a MEASUREMENT OF POTENTIAL 549 knowledge of the direction of the applied potential we may say at once which of the liquid is electro-positive and which electro- negative to the other. » t . In performing the experiment the first thing to do, having assembled the rest of the apparatus, is to draw out a suitable capillary. After one or two attempts a suitable one will be obtained, which is of sufficiently small dimensions to support the pressure. It should be noted that the capillary tube should be drawn from a clean tube. This may be obtained by boiling the tube in nitric acid and rinsing in tap water, or by leaving the tubes to be drawn in a solution of potassium bichromate and sulphuric acid for 12 to 24 hours and then rinsing in tap water. The mercury also should be cleaned, preferably redistilled. The following is a record of an experiment where the above precautions were observed. ? Rj + R 2 = 10000 ohms. Ri APPLIED POTENTIAL APPLIED PRESSURE ohms volts cms. Hg o-o 227 1000 •2 25'3 2000 *4 26-8 3000 4000 •6 •8 27-2 267 5000 1*0 25'9 25-8 6000 1-2 7000 i-4 25'4 8000 i-6 24-9 9000 1-8 247 Contact potential from curve = -57 (fig. 347)- The negative terminal was connected to the upper mercury, i.e. in the experi- ment the mercury was electro-positive to the sulphuric acid. CHAPTER XXII MEASUREMENT OF CAPACITY AND INDUCTANCE COMPARISON OF CAPACITIES OF CONDENSERS (1) Deflection Method Suppose Ci and C 2 are the condensers, of capacity K x and K 2 and that to each is imparted a difference of potential equal to E volts. Let £>! and Q 2 be the respective charges on the plates. I hen we have : K —Sdl If _ Q*. Kl - E ' K » ~ "E ' or El „ 2l / T , In the experiment described below the quantities, Q x and Q 2 , are measured by discharging the condensers in turn through a moving-coil ballistic galvanometer. In such a case, if B x and 2 are the angles of the first throw of the ballistic galvanometer, corresponding to movement of the spot of light, d x and d g cms., from the zero on the scale, we have (see page 479) : q, - 1 . 1- ?tf 1 + *Y Wl 7T G 2\ 2/ Q,_J .i.i»( I+ i\ *" 7T G 2 \ 2/ or $i = -i, If the deflections are small : e t _ 2 tan At _ tan 20 t _ rfi 2 ~ 2 tan 2 ~ tan 20 2 d a ' whence Ml = -i ^ _ - 1 , from (1) Q 2 *2 K, KT^iT- — (2) 550 MEASUREMENT OF CAPACITY AND INDUCTANCE 551 If the deflections are not small this approximation cannot be used. The values of X and 2 may easily be obtained, for tan 20! == y 1 ' where L is the distance from the scale to the mirror of the galvanometer ; hence 20 x and B v It often happens that with a steady source of potential, say, a steady 2 volt accumulator, that the throw 6 X and a is too large, i.e. E is too great. Under such circumstances, instead of using E directly we may take any fraction of E by a potentiometer method. Connect E in series with two resistance boxes, R x and R 2 , making R t -f R a = 10000 ohms. From the ends of R t connect up the leads which in the diagram (fig. 348) are shown directly connected with E. By adjusting R lf making R x + R 2 = 10000, we may obtain a suitable fraction of E, for the potential drop through the resistance, R 1( is Ri E. These resistances are then Ri + R t kept constant and the throws are obtained. The scheme of connexions is shown in fig. 348. K x and K a are condenser keys. K is a single-way tapping key, useful in bring- ing the galvanometer to rest after observing the swings. BG a ballistic galvanometer of the moving-coil type. -\ <L_S E Fig. 348 K x is depressed and the condenser, C lt is charged. Kj is now raised. The charge, Q lt is thus sent through BG, and X noted. As soon as the galvanometer is brought to rest, the key K 2 is depressed for the same short time and then raised, a being noted. - During the interval of the two depressions of the keys, the potential difference between the poles of the accumulator will remain sensibly constant. Of course the experiment could be done using one condenser key, and placing C x and C 2 in turn in the single circuit. The observations are repeated, say, four times with Cj and three times with C 2 , and the mean values taken. 552 ADVANCED PRACTICAL PHYSICS FOR STUDENTS (2) Null Method, (de Sauty's Method) For this method the two condensers, C t and C 2 , are arranged as two arms in a Wheatstone net ; R x and R 2 , two adjustable high resistances making the bridge complete. Fig. 349 gives the scheme of connexions. G is a high-resistance galvanometer and E a battery of cells having an E.M.F. of several volts (say, 4 or 6 Leclanche cells). FIG. 349 The key, K, may very well be an ordinary condenser key, so that when depressed the condensers are charged, and when raised the condensers are discharged. The values of R t and R 2 are so arranged, that when the key K is moved up and down, charging and discharging the condensers, there is no move- ment of the galvanometer coil, i.e. in both processes the potential at B is the same as at D. Let E 1 be the potential at A, and v x be the potential at B at a time, t, v 2 be the potential at D at a time, t. The drop in potential along R x is (E 1 — v x ) and along R 2 is {E 1 - v 2 ). Thus, the currents in R x and R 2 are ^-^ and ^"^ K t K 2 In a small interval of time, dt, the quantity of electricity passing along R t is ( ~ Vl \ dt, and along R 2 is ( E * ~ ^2 In charging the condensers the quantity, / III!- 1 • dt, flows to C, J Ri and f ~ v * -dt to C r dt. MEASUREMENT OF CAPACITY AND INDUCTANCE 553 Now when the condensers are charged, since the potential difference is E (that of the cells) we have : /jri v T5 L * &» and similarly K 9 E / Ri E 1 - v, R< dt, where K x and K 2 are the capacities of the two condensers ; or K^E = /(Ei - v x )dt, K 2 R 2 E = /(E 1 - v 2 )dt, when no flow occurs through the galvanometer, i.e. v x = throughout the charging, i.e. /(E 1 - vjdt = /(Ei - v z )dt ; or, KiRiE = K 2 R 2 E, ie Ki-^1 K 2 -R x In performing the experiment R x and R 2 may be obtained sufficiently high by using a Post Office Box for each, and following the scheme of connexions shown. (3) Fig. 350 (3) Method of Mixtures Comparison of the capacity of two condensers may be made in one or two ways using the 'mixture' method. In this particu- lar method the condensers are arranged to have equal charges *and the potential required to do this is measured or compared by a potentiometer method. The theory of the method presents no special points which require separate treatment. 554 ADVANCED PRACTICAL PHYSICS FOR STUDENTS A battery of cells (two accumulators) is connected as in fig- 350 to send a current, c, through two high resistances, Ri and R,, which are adjustable and connected together at a point, B. The potentials between A and B, B and C, are proportional to Ri and R 2 . B is connected, as shown, to the two condensers whose other plates are connected to the central cups of a cleaned ebonite Pohl commutator from which the cross-connexions have been removed (or a slab of clean paraffin wax with six holes full of mercury will serve). A galvanometer, G, is connected to the cups, X, X 1 , of the commutator. A and C are joined to Z and Z 1 . When connexion is made (by the rocker of the Pohl commutator) between Y and Z, Y 1 and Z\ the condensers, C t and C 2 , are charged to the potential differences of AB and BC, respectively. If Qi and Q % are the charges, and K t and K 2 the capacities, we have : K «=fir K - = fc- -(4) c being the current through R x and R 2 . When the switch is thrown over so that Y and X, Y 1 and X 1 are connected, the condensers are discharged through G. Suppose the current flow from A to B to C in the potentiometer circuit, then the inner plate of C x is at a lower potential than the outer, when charged, whereas for C a the inner plate is at a higher potential than the outer. So when the condensers discharge through G, C 2 will send its charge in opposition to C v The above process is repeated, altering Rj and R 2 between observations, until finally, a value of R x and R 2 is obtained, such that on discharging, no current passes through the galvano- meter. Under such circumstances, Q t = Q 2 . From equations (4) above we have : Q 1 = K 1 R 1 c, Q 2 = K 2 R 2 c. In the adjusted position, since Q x = Q 2 , KjR^ = K^, K, R 2 or — - = — =• K 2 Rj Determination of the Absolute Capacity of a Condenser The value of the capacity of a condenser may be determined in any system of units if a measured potential, E, in these units, MEASUREMENT OF CAPACITY AND INDUCTANCE 555 applied to the condenser imparts Q units of charge in the same units ; for the capacity, K, of the condenser is denned by Q =K-E, or K=2 (5) For example, if Q is expressed in coulombs and E in volts, K» the capacity, obtained in the equation above is expressed in farads. This can be transformed to micro-farads since this unit is 10- 6 of the farad. The usual method of finding K experimentally is to apply a known potential (in volts) to the condenser and measure the charge, Q, by discharging the condenser through a ballistic galvanometer. The moving-coil ballistic galvanometer is the best form to use, as in most capacity experiments, whence if a! be the first observed throw of the instrument, due to the discharge, we have (see page 479) : or, if the method of page 480 is used, the value of x is not obtained so fully. If ax and a 8 are the first and second displacements of the spot of light on the same side of the zero, we have : it 9 2 \a 3 / when <p is the angular deflection produced by a steady current c. L®J K ISP K: ' ^ ■ K* £5 »1 R< E Fig. 351 For small displacements of <5, d lt and d 3 cms. on the scale corres- ponding to <p, a, lf and a 3 , we have : O = - .-.^±( dl djf dj (6) 556 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Fig- 35 1 shows the scheme of connexions for such an experiment. K is the condenser whose capacity is to be determined ; R x and R 2 are resistance boxes, introducing a high resistance to the circuit ; AB a resistance box of decimals of an ohm, r say ; K 1 a single-way switch ; K 3 a tapping key, whereby the galvano- meter may be brought to rest ; K 2 a condenser key ; and BG a ballistic galvanometer of the moving-coil type. c- The circuit, R 1} R 2 , and the battery, E, serve in the first part of the experiment as a potentiometer.* R x and R 2 have a constant sum of about, say, ioooo ohms. By adjusting these two values, keeping (R x + R 2 ) constant the potential difference between B and C may be made any desired fraction of the potential, E, of the battery. This adjustment is carried out, and the deflection produced in the galvanometer when jthe condenser is afterwards discharged through it is noted. Since R t + R a is high and r is never greater than I ohm, the potential between BC = = — -~-*'E=y volts, say. Ri + K 2 When K 2 is depressed, the condenser is charged, the potential applied being V. The charge, Q, will flow through the galvano- meter causing a deflection, d x cms., when K 2 is raised. This process is repeated, say three times, and a mean value of & x obtained for the fixed potential, V. For each discharge, the reading corresponding to the second deflection, d 3 , on the same side as d x , is measured. This provides the data for the damping correction for the galvanometer under the identical conditions under which oc 2 is measured. The time of swing, T, is obtained by timing, say, 20 swings of the needle with a stop-watch. It is most convenient to use the position of rest of the spot of light as a reference point of such counting and timing of swings. To standardize the galvanometer a steady current of known magnitude must now be sent through it and the scale deflection, 8, measured. The key, K lt is now closed. The current from the battery E (the same 2 volt accumulator throughout), now passes through R 2 and R x , and then through the galvanometer, shunted by the small resistance, r. The resistance of the galvanometer and shunt is : rG G +r where G is the resistance of the galvanometer, t t G may be obtained by including a resistance between B and BG, and adjust- ing till the deflection is reduced to half the value, when G is obviously equal to this resistance. MEASUREMENT OF CAPACITY AND INDUCTANCE 557 Hence, if E is the potential difference in the circuit, the current in the main circuit, *, is : E R 1+ R,+B+Jg; where B is the resistance of the battery. This may be neglected when the latter is an accumulator, and — ^is > 1 ohm, and is also negligible compared with R + R 2 - Of this current, the value of the part, c, through the galvano- meter is : Er = (r + G)(R 1 +R 2 ) We have also seen that the potential applied to the capacity is V = 5i — • E. R x + R 2 Hence, substituting this value of V and Q from (6), K=2 V E-r 2* (r + G) (R, + R 2 ) d JL(<h\* Ri Ri + R 2 ^ILftA* ( r V 27T d \dj XRfi) if r is negligible, cf. G. E is assumed constant throughout the experiment. Z Fig. 352 Measurement of Capacity, using a Fluxmeter A simple method of estimating the capacity of a condenser is illustrated in fig. 352. 558 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The condenser, C, of capacity, K x , is connected to two condenser keys, Si and S 2 . The lower studs of the keys are connected to the mains of the electricity supply (i.e. to 100-200 volts). When the keys have been depressed the condenser is charged to that potential, V. On releasing the keys the charge on the condenser is discharged through a low resistance, r, which is shunted across the fluxmeter, F (the position shown in the diagram). r is made small, about ^ to ^ ohm is suitable, and the shunt may therefore be regarded as taking the whole of the discharge, say, Q units. The potential difference at the ends of r varies from V, to zero. i.e. the current through r at any instant when the potential is E E is — and the total quantity of electricity passing through r is therefore : Q = f— ■ Now, if the resistance of the fluxmeter, usually about 30 ohms, is large compared with r, the instrument gives a deflection which is a measure of /E • dt (see page 484). Let this deflection be x divisions or x x 10 4 maxwells, o — x x iq4 W — f If now r is expressed in ohms, the units must be made consistent, x x 10 * i.e. a; xio* maxwells should now be written ^ — practical units to obtain Q in coulombs, i.e. Q (in coulombs) = If the potential applied to the condenser is measured by means of a voltmeter, the capacity in farads may be readily obtained. The Effect of Inductance and Capacity in a Circuit Conveying an Alternating Current In many of the subsequent methods of finding the self -induct- ance, mutual inductance, or capacity in a circuit, alternating current is employed and a system of non-inductive resistances, capacities, inductances are arranged to produce a balanced Wheatstone net, the usual galvanometer being replaced by a telephone as detector. In such circumstances it is not a difficult matter to find the relation between the values of the resistance, etc., when such a balance is obtained. The majority of cases may be solved by precisely the same method. MEASUREMENT OF CAPACITY AND INDUCTANCE 559 Consider an alternating potential E cos pt to be applied to a circuit, of resistance R, self-inductance L, and containing a capacity K (fig) 353), (p is, of course, given by 27m = pt where n is the frequency of the alternation). 'Ofl'OWOO N — — I B At/^ E co* yt Fig. 353 Now E^ = E (cos pt -h i sin pt), where i = V^i a* 10 ^ tne rea ^ ^ axt °* E^*' expresses, therefore, the alternating potential applied. If c is the resulting current at any instant the net applied potential = Re, Lie Q i.e. Re =E cos^--^--^, when Q is the charge on the condenser at any time, t. Further, we have : c ~ dt' dc It £2. dt* ** dt* + dt + K y Hence : L ^ + R 3? + ^« = real of E^*', If Q = A-e*< , i.e. § = #Ae*< , ^ - - p*A& , we have, where A is a constant, - L£ 2 Ae*' + R# Ae*< + g Ae* = E e*'; or and A = E ( Rip-Lp*+± Q Epg^ R#-I4> a +^ 560 ADVANCED PRACTICAL PHYSICS FOR STUDENTS or i.e. c = dQ ipE^ 1 dt Rip -Lp* + ± E^ c = R + i( L p-^) Ea e»<\R-i(Lp-±)} (7) R8 + ( L ^- ^) . E e» f R if i R2 +(^-^) a r^i Ra +(^-i) 2 } R 1 7 FTm* = cos * : R ' + ( L *-^)I R 2 + (*-b) Fjl = sm a, we have c = - E n e*' The real part of this is] ^ E cos (pt — a) "> ,+ ( i *-i5) , r (8) i.e. the current lags behind the applied E.M.F. unless a = o, i.e. cos a = I, MEASUREMENT OF CAPACITY AND INDUCTANCE 561 i.e. unless R = \ R 2 + (*-*)'}' or Lp = =jr-, or p =- . F Kp' r VK-L Thus we see that in the general case taken the method gives the value of the current at any time. From equation (7) we see that the maximum current C is equal to the applied electromotive force divided by ( R + Lip — — Y By analogy with the simple Ohm's Law equation ( R + Lip — J may be termed the ' quasi-resistance ' of the circuit, and be used in the same way as ordinary non-inductance resistance values, when alternating electromotive force is applied to a circuit. For example, consider a Wheatstone net as illustrated in fig. 360. If a direct current were applied at AC and the net balanced we have : Resistance AB _ Resistance AD Resistance BC ~~ Resistance DC Now for alternating current applied using the quasi-resistances in the corresponding arms : i r s + Lip R _ * K^ which gives on rearranging and equating real and imaginary quantities, thus : (real) r x R = M or L = K^R, Kjp (imaginary) ]0 = 1T^ or K i r i = K * r » a result which agrees with the fuller treatment given on page 575. The above method of investigation assumes that a true sine E.M.F. is applied. Using a ' hummer ' or the secondary of a small induction coil does not produce such a simple form of E.M.F., but would be the resultant of a series of such simple wave forms. Such an E.M.F. would be satisfactory in those cases where the end result, as in the case taken above, is independent of the frequency or p. 36 562 ADVANCED PRACTICAL PHYSICS FOR STUDENTS In cases where p is involved in the result, a timed telephone, which responds to one frequency only should be used. As an alternative, a vibration galvanometer such as that designed by A. Campbell (" Physical Society of London Proceedings," 1907, page 626). As such methods are not very much used, they are not included in this account. MEASUREMENT OF SELF-INDUCTION The measurement of the coefficient of self-induction of a coil may be carried out in several ways, of which the following methods are representative. Some of these methods are very tedious as they involve a double balance of a bridge, for steady and variable currents. Perhaps the most unsatisfactory methods from this point of view are (1) the direct comparison with a standard self -inductance, (2) Maxwell's method. Rimington's method, too, is apt to be tedious. The methods which have been found to be most satisfactory in the laboratory are Rayleigh's, Anderson's, and Owen's. These methods are therefore used generally when a choice of methods is available. Fig. 354 (1) Rayleigh's Method In this method the inductance, L, is placed in series with a small variable resistance, r, and arranged in the arm, AB, of the Wheatstone net, as shown in fig. 354. The remaining arms are of the same order of magnitude as the resistance, AB. G is a ballistic galvanometer preferably of the same order of resistance ; K x and K a are keys. The resistance, r, is reduced to zero and' the network is balanced for steady currents by closing the key, K 2 , before making the galvanometer circuit through K x . MEASUREMENT OF CAPACITY AND INDUCTANCE 563 To obtain an accurate balance, it may be necessary, when the arms are equal, to introduce a smaller resistance than is available in the Post Office Box which provides the three resistances, r 8 , r z and r A . This may be effected by having a length of platinoid wire in series with L, and adjusting the length until an accurate balance is obtained. If now a small E.M.F., E, is introduced in one arm, a current which depends on E will pass through each of the other resistances in the network. Thus, if the battery key is closed, K a being meanwhile closed, dc an E.M.F. of magnitude L-y is established in the arm, AB, resulting in a current in each of the arms in the net. Let the current in the galvanometer branch, BD, be k • (Lt), where k is- a constant which depends on the value of the resistances. Under such circumstances the total quantity of electricity which passes through the galvanometer due to this cause is : /* dc kL -j t dt = kLc , „ (9) where c is the maximum steady current flowing through AB. This quantity, Q, may be calculated from the observed throw in the ballistic galvanometer, and thus L is obtained in terms of the constants of the galvanometer, k and c . Assuming that a moving-needle galvanometer is employed we have, page 477, Q -v i • «i (*+!)- «*■ ™ (I0) where x is the logarithmic decrement. To eliminate k and c , a measurable small potential charge is introduced into the arm AB. This is brought about by adding a small resistance, r, to AB. (r should be not greater than g£o r 1 , usually j^ ohm does very well.) Assuming that the current, c , will not be materially affected by this small charge, the potential introduced in the arm AB is c& : this causes a current, kc^/m. the galvanometer, producing a steady deflection, r Then G being, as above, the galvanometer constant, i.e. the field strength at the centre of the coil due to unit current, we have : kc rG = H tan Q lt , H tan 0« . or kc = ~x • * • G r 564 ADVANCED PRACTICAL PHYSICS FOR STUDENTS Substituting this value in equation (10) above, we have : 0/ X^ rp sin -j L = r T sin M I+ ^ n tan0j If r is expressed in ohms and T in seconds, L is in henrys (io 9 C.G.S. E.M. units). If a moving coil type of ballistic galvanometer is used and r ? and r 4 or r x and r 3 are small, the galvanometer may give but a small deflection as it is shunted by these resistances. This is the case when the resistance of L is small. Measurable and reliable results can be obtained if the galvanometer circuit is broken the moment the discharge has passed through it. This condition is most conveniently brought about by using a single key as shown in fig. 298 on page 481, for both battery and galvanometer circuits. The three brass strips, A, B, and C, are insulated and connected to separate terminals. When C is depressed contact is made between C and B, then A and S, but not between B and A which are separated by an ebonite stop. S and A replace K 2 ; C and B replace K 2 , when C is pressed down and a steady current flows in the circuit, the galvanometer which is in the circuit shows no deflection for a balanced bridge. On releasing C, the bottom contact is first broken (i.e. the battery circuit), and a very short interval of time afterwards C and B are separated. This interval is sufficient to allow the impulse to be given to the galvanometer, coil which, due to the separation of B and C, swings without excessive damping due to induced currents. If the moving-coil instrument is used in this way, the end result is slightly modified, for in this case (see page 479) where is the corrected value of the first throw for no damping, i.e. is the observed throw X f J. Now for a steady current equal to kc^ the couple on the coil is GkCff, so that Gkc r = T0J, t being, as above, the restoring couple in the suspension, per unit angular displacement. Combining the last two equations we see that n 2 * 0i " MEASUREMENT OF CAPACITY AND INDUCTANCE 565 (2) Comparison with a Standard Inductance To effect a direct comparison of an unknown inductance, L^ with a standard inductance, L 2 , the arrangement of fig. 355 may be employed. / In series with L x is a non-inductive resistance of a variable Fig- 355 magnitude, say a length of manganin wire, which is adjusted so that the total resistance included between A and B is r x ohms. By means of a key, S^ the galvanometer, G, is included in the circuit to which a direct current from a 2 volt cell is supplied via key, S 2 , and a balance is obtained for steady currents, i.e. the balance is obtained by closing the battery circuit first, and when the current is established, the equality of potential at B and D is tested by closing the galvanometer circuit. Having balanced the bridge for steady currents the balance is now tested for variable current. The first steady balance will not be one which also is in adjustment for variable currents. The resistance in series with L x is given another value and the steady current balance obtained once more for the net, which is once more tested with variable current. This process, which is often a lengthy one, is repeated until the resistances, r v r 2 , r t , / 4 are of such magnitude that the points, B and D, are at the same potential both with steady and varying currents. It will be shown that under such circumstances Li = ri = U,, The variable current may be obtained by breaking and making the battery circuit; or by switches, S x and S 8 , a telephone ear-piece 566 ADVANCED PRACTICAL PHYSICS FOR STUDENTS may be made to replace the galvanometer, and an alternating current from a buzzer supplied instead of the direct current used in the steady balance experiment. Assuming an alternating E.M.F. be applied to AC and the network adjusted so that the potential at B is the same as that at D, at all times we have, using in the Wheatstone relation the ' quasi-resistances ' of the arms : Ljp + r x = Tj L/p + r z r 4 Equating the real parts we have : »V« = r 2 r 9 , or £ =~ 3 , (n) a condition which was fulfilled by first balancing for steady currents. Equating imaginary parts of the equation, we find T y *» or _-L=— 9 — _J i, from (n). (3) Maxwell's Method The arrangement of resistances and the inductance to be measured is shown in fig. 356. The method involves a comparison of L, with a capacity of known value. In the present method the resistances r lt r t , and r 8 , together with L, are made into a balanced Wheatstone net, using a cell and galvanometer in the usual way. In parallel with r x is placed a condenser whose capacity, K, is known. During the process of balancing the resistances for steady currents, care is, of course, taken to ensure that the current is established before the galvanometer key is depressed to test the balance. The experiment may now be completed either with the aid of the galvanometer across BD and the cell connected through a switch to AC, or by means of a telephone ear-piece as detector and an alternating E.M.F. applied from a hummer.* If the galvanometer is retained, the current is made and broken whilst the galvanometer is permanently in the circuit, and r % and r 8 are adjusted until no deflection is obtained in the galvanometer. If the alternating E.M.F. from a hummer is used, the resistances are adjusted until no noise is heard in the telephone ear-piece. In either case the steady current balance will, in general, be upset. This must be again obtained and then the test repeated, until finally the balance is equally good for steady and varying currents. * It should be noted that the galvanometer and steady E.M.F. may be dis- pensed with in experiments (2) (3) and (5). Alternating current may be used alone. The first adjustment is to reduce the noise in the phones to a minimum, MEASUREMENT OF CAPACITY AND INDUCTANCE 567 When the network is balanced for steady and alternating currents the relation between the four ' quasi-resistances ' may be applied. K B Fig. 356 Since the arm, AB, consists of — -r=~ and r x in parallel, the effective ' quasi-resistance ' is 1 -f- ipKr x ' whence : "\ r,(i + ipKrJ r 4 + Lip ' where r 4 is the total ohmic resistance in the arm, DC. Equating real quantities in the above equation : r*r« == y,r^ or -^ =— *, 2'3 (12) a condition which was experimentally established in the steady current balance. Equating imaginary terms : r % r z pKr x = r t Lp, or L = r 2 r 8 K, or from (12) above = r jftK, from which L may be calculated. (4) Rimington's Method A method due to Rimington enables a determination of L to be made in terms of a known capacity, making use of a modifica-" tion of the Maxwell arrangement. The self-inductance is placed and the second to reduce this minimum to zero. N.B. — The value of the resistance for intermittent current is only the same as that for alternating current when the frequency of the latter is small. 568 ADVANCED PRACTICAL PHYSICS FOR STUDENTS in the arm, DC (fig 357). The capacity, K, is placed in parallel with a variable portion, r, of the resistance r x (AB). The network is balanced for steady currents, and then the point, E, is chosen such that when the battery circuit is made and broken no deflection is produced in the galvanometer. This method has the advantage that in the second balance the adjustment of the resistances, r x , r s , r s and r 4 , is not disturbed. When the above conditions are fulfilled we have the following relation between the physical quantities involved. L = K — - henrys. Fig- 357 shows the arrangement of apparatus. For this experiment it is not possible to use a telephone and alternating current, as will be seen by working out the case by the general method given ; the conditions required for balance with such currents reduces the method to Maxwell's, i.e. r x — r = o. To establish the formula given above for the case of a balance with intermittent direct current, let c x be the steady current in AB (resistance, r x ) when the key is closed ; c 4 the steady current in DC. Then when the circuit is broken, the quantity of elec- Lc . /*° hdc tricity which passes through the inductance is -~- i.e./ ^-, where R is the resistance between the points D and C (r 4 ) together with the resistance of the rest of the net, i.e. R = * 4 + r 2 + G(r 3 + r x ) where G is the resistance of the galvanometer. MEASUREMENT OF CAPACITY AND INDUCTANCE 569 The fraction of this quantity which passes through the galvano- y i +r 3 of LC4 r x + r 3 + G R i.e. is which reduces to LC4K + r z ) G(r 3 + r t ) G + r 3 + r x , Lc 4 r x ('i + r z + G) (13) 'i('i + '3) + Gfo + r 2 ) Also when the circuit is broken, the condenser discharges a quantity of electricity through the galvanometer, in the opposite direction to that due to the inductance. Now the quantity of electricity on the condenser due to the steady current, c v in AB is Kfrcj). The part of this quantity passing through the resistances other than r is : Krc 1 . _ . - ri + f » + LG+r, + r; Of this amount, that passing through the galvanometer is : Kr 2 ci x Jr 2 + rj r + r -I G ( y 2 + y 4) (G + r. + ftf- 1 8 G+r 2 +y 4 This reduces to Kr 2 Cjr 2 (14) ' 2 ( r t + r 3 ) +G(r! + r 2 ) ' when substitutions are made using the fact that r#i — r 2 r z . When the quantities represented by (13) and (14) are equal and opposite, the galvanometer will be unaffected by making or breaking the circuit by means of the key shown in the figure, i.e. K> 2 CjT 2 = Lc^j, or We may further express — in terms of the resistance for when the currents have acquired the steady values, c x and c 4 ,we have : £1 = *z + U . c 4 ?i+r 2 ' '1 r 2 r x + r 2 c A r 2 Hence L = ^-^ 57© ADVANCED PRACTICAL PHYSICS FOR STUDENTS Experimental Details The chief disadvantage to this experiment is that much time may very easily be spent in a fruitless effort to determine the value of an inductance, unless the order of suitable resistances, etc., to balance the network, is known. If, however, the approxi- mate magnitude of L is known we may, as in the following example, obtain this information. Example An inductance (of 98*15 ohms resistance) is to be evaluated, using a capacity, one micro-farad. Now from above we have : L = K -^, and the inductance is known to be of the order of one henry, i.e. y 4 is of the order 10 2 ohms., K is of the order 10- 6 farad ; 10- 6 r 1 = — so x io 2 r 2 = io _4 r a « i.e. r is a fraction of r v or r = ar lf where a is not greater than 1, Zj__- -rn-4 2* 10* a 2 r, The arm, AB, of the figure could be therefore two Post Office boxes in series. To obtain a subdivision as at E, however, it will be convenient in this case to use three Post Office boxes, or boxes having a similar range of resistances, for this arm of the net, as in fig. 358. The condenser, K, is placed in parallel with boxes 1 and 2 as shown. MEASUREMENT OF CAPACITY AND INDUCTANCE 571 The balance for steady currents having been obtained, the value of r is varied by taking out plugs from 1 and 2, and inserting the same value in 3, thus keeping r x constant and of the value required fpr a steady current balance, until r is finally obtained, such that the potential drop along AE is such as to cause a current through the galvanometer equal and opposite to that due to the inductance. Values of r too small will be shown by a movement of the galvanometer to the one side due to the induct- ance effect predominating, whereas when c is too large the capacity effect will be large and cause a deflection in the opposite direction. This process was employed to find the value of L for the coil mentioned above, and the following values obtained, r x = 20000 r z = 1963, r = 14500 r % = 1000, r t = 98*15, K = 1 micro-farad, K-lxiHS'xio-xgiis _ x . 03 henrys- whence L = 20000 The self -inductance was a nominal 1 henry. (5) Anderson's Method The scheme of connexions shown in fig. 359 illustrates a very convenient method of finding the value of an inductance in terms of capacity and resistance. Fig- 359 The Wheatstone net, r t , r 2 , r 3 , r 4 (r 4 being the total resistance of the arm, DC, including the self-inductance) is arranged as shown. The condenser, K, is connected from A to a variable point, E, in a resistance r (BE). 572 ADVANCED PRACTICAL PHYSICS FOR STUDENTS The network is first balanced for steady currents using a battery between AC and a galvanometer, G, between ED. For such steady currents K and r do not have any effect. When a balance is obtained, by first depressing the battery key and then the galvanometer key, we have : ^i = - 3 . The network must now be balanced for varying currents. This may be done in one or two ways : either by using the galvanometer and an intermittent direct current, or an alternating current and telephone receiver. If the galvanometer is used as a detector, a tapping key is inserted in the battery circuit and r is so adjusted for the particular value of K, that on making and breaking the circuit no deflection of the galvanometer is produced. When this condition holds, it may be shown, by the application of Kirchoff's Laws to the circuits, ABED A, ABCDA, that L = K{fy, + r(r,+r«).} The more convenient way of proceeding with the experiment, however, is to replace the galvanometer by a telephone and the battery by a source of alternating current. In the general case, when these changes have been carried out to conform with fig. 359 a note will be heard in the ear-piece when AE is closed. By adjusting r (the resistance BE) this note may be reduced in intensity to zero. We may readily find the relation which exists between the various resistances, L and K, under such circumstances. The condition satisfied is that the potential drop between A and E is the same at all times as the potential drop between A and D. Let the currents in the arm, AB be c- l ; in AE, c 2 ; and in AC, c. Then, when no current passes in either direction along ED, the magnitude of the current along BC is c x + c 2 . Consider the circuit ADC, and let the applied potential be taken as the real part of E e*', The potential difference between A and D is r z c, where c is the solution of dc E e*' = r z c + Lj t +r 9 c. Putting c = he*** in the above equation we have : E U) + E e^ A = E o ; whence c = -. ; — » , T . . » and potential difference between A and D is therefore E e** r a +r] + Lip' r * ••' ( I5 > MEASUREMENT OF CAPACITY AND INDUCTANCE 573 Now let Q be the charge on the condenser, K, at any instant. From a consideration of the circuit, ABC, we have : E^*' = r x c x + r 2 (c x + c 2 ) (16) But the potential drop along AB is the same as the drop along AEB, or Q / * ClTl = K + c *' " * ' ^ Substituting for c x (17) in (16) and putting c 2 = -J^, we have : whence as before : Q *£L from which the potential difference between A and E ( ^ J is p Eof^ (18) ■( I+ &)+K*(, i+ a + r ) The condition specified above for no current through the telephone is that (15) and (18) are the same for all values of t. Equating these expressions we have : ('. + U + Lip) E«** = f ,|(i + £j) + ^(fif .+Wi+frJ JE^'; whence equating the real parts we have : r. + ,.-,.(*±r,). or — = — , a condition which is assumed by the initial balance for steady currents. Equating the imaginary quantities we have : Lp = ^Kfor, + rr x + rr 2 ), or L = K{r z r s + r(r s + u)} . (19) This result could have been obtained using the quasi-resistance method already outlined. For we may consider the circuit taken 574 ADVANCED PRACTICAL PHYSICS FOR STUDENTS above and write down the complete ' Ohm's Law Equation ' and find c,c x and c 2 and hence the potential drops, AE and AD, and equate. Thus, for circuit ADC : E/»< '* + y 4 + L # ial drop, AD = Circuit ABC and Potential drop, AD = r * B ° e *\ (20) r z + r * + ^P E e** = e 1 r 1 + (c s + cjr s (21) By Kirchoff' s Law since the E.M.F. in circuit, AEBA, is zero, Substituting in (21) above, whence and since ^f=*c Xi we have, integrating, Q = ^ and the potential between AE, which is g., is XV. EI**' .(22) Equate to the value given in equation (20), we have, equating the real quantities : '* u and equating the imaginaries, I* = K{r s r t + r(r t + r 4 )}. (6) Owen's Method The following method of finding the value of the self -inductance of a coil was described by Dr. D. Owen in the " Proceedings of the London Physical Society," 1914-15, vol. xxvii. The coil whose inductance, L, is to be found is placed in one arm of a Wheatstone network, in series with a variable non- inductive resistance, so that the total resistance is r 2 ohms. The scheme of connexions is shown in the diagram (fig. 360). K x is a standard capacity, K a a second capacity, and R an adjustable non-inductive resistance. MEASUREMENT OF CAPACITY AND INDUCTANCE 575 It will be noticed that in this method there is only alternating current supplied to the bridge. In the first case the resistance R is made equal to zero, and the resistance r 2 is adjusted so that a minimum intensity is heard in the telephone. R is then gradually increased until this minimum is reduced to silence. Under these circumstances the relation L = K^R^K^R is found to hold. Fig. 360 Now it was shown on page 561, as an example of the ' quasi- resistance ' method of solving such problems, that when such a balance is obtained L = K^R = K> 2 R. As a contrast with the method given on that page, we may show that /the above expression gives the relation between the capacity and resistances used, from a fuller consideration. For let the applied E.M.F. be E^*'; and, since no current flows through the galvanometer in the balanced state, let c x be the current in ABC. 576 ADVANCED PRACTICAL PHYSICS FOR STUDENTS For the circuit ABC we have : E^ = Cl (r 1+ r 2 )+L^; hence, putting o x = Ae*' and substituting to find A, we have : Ep^ '* ^ + r 2 + Lip' The potential difference between A and B is r x c x or /jE^ r% + U + L# Now, if c 2 is the current in ADC, we have for that circuit : E^ = Rc 2 + Q(^+^-), dQ it (23) or, since -jr= c lt E ^= R f+Gt + r> gives the following Hence putting Q = Be* 1 and -^ = B#d?*' we find B, which Q ** The potential difference, at any time, t, between A and D, is therefore ~- _x_ E g»* The condition that there is no sound in the telephone is that at all times, t, the potential at B = potential at D, i.e. for all values of t : i.e. r&^Rip + (± + i) | = (r x + r t ) + Up. Equating real quantities : ' iK ' it + r.) -'• + '• MEASUREMENT OF CAPACITY AND INDUCTANCE 577 Ki + K 2 _ r^jfyg K 2 r x * K 2 tr or rj&i = rz& 2 Equating imaginary quantities : or L = /iKjR or r 2 K 2 R MUTUAL INDUCTANCE Direct Measurement with a Ballistic Galvanometer The coefficient of mutual induction of two coils may be denned as the number of lines of magnetic force which pass through one coil when unit current circulates through the other. Thus, if a current of maximum strength, c , passes through one of the coils, whose mutual induction is M, Mc lines of mag- netic force thread the second ; and whilst the current grows in the primary, the number of lines of magnetic force threading the secondary is changing. Therefore an induced E.M.F. is set up in the secondary during the time of growth of the primary current. This E.M.F. is numerically equal to the rate of change of the number of lines of magnetic force in the secondary, i.e. = — (Mc)* where c is the instantaneous current in the primary at during the growth of that current. If L is the coefficient of self-inductance of the secondary coil, and c 1 is the current in the secondary corresponding to c in the primary, we have a further E.M.F. in the secondary due to the hdc 1 self-inductance numerically equal to -tj—> i- e - if R is the total resistance of the secondary coil circuit, neglecting signs, Rcl = L Tt + M Tf Now Q, the quantity of electricity passing through the second- ary, is / cHt where the integration is carried out over the whole time during which c rises to the steady value c . Q =jcHi -=f~dc^ +f~dc. * N.B. — This only applies to coils with non-magnetic cores. For if there is an iron core the value of the flux is not proportional to the current. 37 578 ADVANCED PRACTICAL PHYSICS FOR STUDENTS A = and rfc-"* The value of c 1 at the commencement and the end of this integration is zero, hence : !*•■ If, therefore, the second coil is connected to a ballistic galvano- meter and the throw is 0, due to the passage of this quantity of electricity, M may be calculated in terms of R, c , the constants of the galvanometer, and 0. In this case it will be well to calibrate the galvanometer in the circuit since R includes the resistance of the galvanometer and the second coil. G <2> Fig 361. Fig. 361 shows a convenient disposition of apparatus to carry out a direct measurement of M on these lines. The current in the primary coil may be regulated to a suitable value by means of the resistance R x . In series with R x is a small resistance r(A»). C is a four-segment commutator, or may be a switch consisting of four mercury filled holes in a block of paraffin wax. K 1 and 2 are connected together the ballistic galvanometer is in direct circuit with the secondary coil, and when K is depressed a deflection, 0, is obtained for the establishing of the steady current, c , in the primary. If a moving-needle ballistic galva- nometer is used, we have T H . 0„ Mc G 2 Tf- (25) where d is the value of the first deflection corrected for damping, i.e. O = f i + - 1 where x is the logarithmic decrement and is the observed throw.* * If a moving-coil instrument is used, the form of double key, described on page 481, should be used ; K is replaced by the lower pair of contacts in such a key and the upper pair act as a key in the galvanometer circuit. MEASUREMENT OF CAPACITY AND INDUCTANCE 579 If now C is arranged so that connexion is made between 1 and 3, 2 and 4 only, and a steady current, c , is passed in the primary circuit, the potential drop established at the ends of r is C&, i.e. the current through the galvanometer is -^ » smce r is very small compared with the resistance of the galvanometer. If this causes a steady deflection, lt we have : G. c ^=Htane 1 , (26) as shown on page 481. Combining (25) and (26), we obtain the following value for M : it T Smi If a moving-coil galvanometer must be used, since Q =-• ^ • — the value for M becomes : M = J . 1 . 1°. w 2 2 It will be noted that the value of R is eliminated : r may well be a standard ^ or jf ohm. T is obtained in the usual way by timing 20 or 30 swings, under the conditions of damping which o.btain during the observations above ; O = 6 1 1 + - j The value of x may be obtained by one of the methods given on page 480. Comparisons of Mutual and Self-Inductance. (Maxwell's Method) The one coil of self-inductance, L, is arranged in one arm of a Wheatstone network ; the other coil of the mutual inductance is connected, as shown in fig. 362, in series with the battery. For this method the resistance, r (AC), is not present. The coils are arranged by trial so that the self and mutual induction effects in the arm AB are opposed. The resistances, r lt r t , r z , r 4 , are balanced for steady currents. The battery circuit is then made and broken, the galvanometer circuit being closed. r z , r % , r 4 are adjusted until the minimum effect is produced in the galvanometer for such current changes. The steady balance is again tested. By repetition the values r v r z , r 4 , may be finally arranged such that for steady and varying currents the galvanometer remains unaffected. 58o ADVANCED PRACTICAL PHYSICS FOR STUDENTS The steady current balance gives in the usual way the relations between the resistances, i.e. During the fall of the current at break, the self-inductance dc effect is an E.M.F. L^- 1 , where c t is the current in the arm ABC. B s In the opposite direction an E.M.F., M ^-, is set up in the same arm due to the mutual inductance. If these two E.M.F.'s are equal and opposite, L-^ = M jt (numerically), where c = c x + Cj, Lc x = M(c x + c 2 ), L c x +c a M c x ' -* + '? '4 It is apparent from the above equation that the experiment may only be performed if L is larger than M. MEASUREMENT OF CAPACITY AND INDUCTANCE 581 This method is open to the same criticism as the Maxwell and comparison methods of finding L, and is therefore seldom used. A modification of this method is obtained by the insertion of the arm, AC, of a variable resistance, r, as shown in fig. 362 With K 2 open a steady current balance is first obtained. This having been accomplished, the resistances, r v r 2) r z , r 4 , are fixed ; K 2 is now closed and r adjusted until the galvanometer is unaffected by intermittent current in the network, e.g. when the current is made and broken or reversed. As before, no resultant E.M.F. is set up in AB during such changes of current, otherwise a current proportional to such E.M.F. would pass through the galvanometer. Thus we have again : I^i =M^ (numerically), at ctt where c is the current passing through the coil M, i.e. is {c x + c 2 + c 3 ). Integrating as before, we have : LCi = Mc, L _£_ 1,e# M c/ Now if R is the resistance of the whole net between A and C, we have : cR = c 1 {r 1 + r 2 ) L r x + r 9 M~ R ' = r(r t + r t ) + r{r 3 + u) + {r x + r t ) (r 3 + r 4 ) r{r 9 + rj or and ^3 + (rx + rj M r L = r{r 1 + r 9 ) + (^i + r 2 ) r s M rr z This arrangement of resistance eliminates the trouble- some double balancing of the last method. The steady balance is obtained with a cell and galvanometer in the usual way. The battery circuit is then made and broken or reversed ; r is 582 ADVANCED PRACTICAL PHYSICS FOR STUDENTS adjusted until the galvanometer shows no deflection during that process. As before, care must be taken that the coils are so connected that the effect of the self and mutual induction oppose each other. Alternatively a hummer may be used as a source of alternating E.M.F., instead of making and breaking the battery circuit, and a telephone used as detector. If the self-inductance is less than the mutual inductance a modification of the method above may be used. In this case the battery and the resistance, r, are interchanged, i.e. M is in series with r. For balance, the effect of the induced E.M.F. in the arm, AB, due to L and M, must neutralize each other. As before, equating the numerical values of such E.M.F.'s we have, if c 8 is the current through r and M, T dc i _m^ 3 Lc t = Mc 3 , and c 1 {r l + r 2 ) = c 3 (r -J- x), M c, r + x i.e. — - = - 1 = — ^— ~, where x is the resistance of the coil, M. This method, however, cannot be performed using alternating current and telephone. Determination of Mutual Inductance by the Fluxmeter The mutual inductance of two coils may be very readily obtained, using a fluxmeter (which is described on page 482). AJ } R — Mttl'l ' E Fig. 363 One coil, M, is connected directly to the fluxmeter, F, fig. 363. The other coil, L, is connected through a commutator, C, to a circuit consisting of an accumulator, E, an ammeter, A, and a 1 variable resistance, R. The current in L is adjusted to some convenient amount, c amperes, and then reversed, causing a deflection of x divisions in the fluxmeter. MEASUREMENT OF CAPACITY AND INDUCTANCE 583 If each division of the fluxmeter (as usual) corresponds to 10000 maxwells, the change in the number of maxwells in the secondary, M, due to a reversal of c amperes in L is ioooo* , i.e. for — E.M. units of current in L there are 5000* lines of magnetic force threading M Hence, since the coefficient of mutual inductance is denned as the number of lines in the one coil for unit current in the other, it has a value, 50000* . c for the coils used. The experiment is repeated using various values of c, and the mean value of - is obtained, and hence the coemcient of mutual c inductance in E.M. units may be calculated. The mutual inductance in henrys is io~ 9 times the above value, for two coils are said to have a coemcient of mutual inductance of 1 henry when a current change of 1 ampere per second in the primary causes an E.M.F. of 1 volt in the secondary, i.e. a flux of io 8 lines for a current of 10- 1 E.M.U., i.e. the henry is io 9 E.M. units. CHAPTER XXIII THE QUADRANT ELECTROMETER A Modern development of the Kelvin Quadrant Electrometer is seen in fig. 364, and is due to Dr. F. Dolezalek. The four quadrants, QQ, etc. are supported on ambroid pillars, AA. As shown in the figure, two quadrants are mounted on a pivot, and may be swung aside to allow of the introduction of the ' needle.' Alternate quadrants are joined together to terminals Fig. 364 under the base plate of the instrument, care being taken that such terminals are very well insulated from the case. The needle may be either a light paper frame (fig. 365), coated with a metal to make it conducting, or as in fig. 366, a thin 584 THE QUADRANT ELECTROMETER 585 mica sheet spluttered with silver. In some of the more recent instruments a very thin aluminium needle of the shape of the second form is used. The needle is attached to a light rod which carries a small mirror, M, and is supported by means of a thin quartz or phosphor-bronze strip from the torsion head, R. Fig. 365 Fig. 366 The method of observing the deflection produced by a difference of potential on the quadrant consists of the use of the usual lamp and scale, as with galvanometers. A beam of light from a lamp