ELEMENTS OF THE MATHEMATICAL THEOKY OF ELECTRICITY AND MAGNETISM BY SIR J. J. THOMSON, M.A., D.Sc., LL.D., Pn.D, F.R.S., FELLOW OF TRINITY COLLEGE, CAMBRIDGE; CAVENDISH PROFESSOR OF EXPERIMENTAL PHYSICS IN THE UNIVERSITY OF CAMBRIDGE; PROFESSOR OF NATURAL PHILOSOPHY IN THE RQYAL INSTITUTION, LONDON FOURTH EDITION CAMBRIDGE : AT THE UNIVERSITY PRESS 1909 First Edition 1895. Second Edition 1897. Third Edition 1904. Fourth Edition 1909. PREFACE TO FIRST EDITION IN the following work I have endeavoured to give an account of the fundamental principles of the Mathematical theory of Electricity and Magnetism and their more important applications, using only simple mathematics. With the exception of a few paragraphs no more advanced mathematical knowledge is required from the reader than an acquaintance with the Elementary principles of the Differential Calculus. It is not at all necessary to make use of advanced analysis to establish the existence of some of the most important electromagnetic phenomena. There are always some cases which will yield to very simple mathematical treatment and yet which establish and illustrate the physical phenomena as well as the solution by the most elaborate analysis of the most general cases which could be given. The study of these simple cases would, I think, often be of advantage even to students whose mathematical attainments are sufficient to enable them to follow the solution of the more general cases. For in these simple cases the absence of analytical difficulties allows attention to be more easily concentrated on the physical aspects of the question, and thus gives the student a more vivid 236 VI PREFACE idea and a more manageable grasp of the subject than he would be likely to attain if he merely regarded electrical phenomena through a cloud of analytical symbols. I have received many valuable suggestions and much help in the preparation of this book from my friends Mr H. F. Newall of Trinity College and Mr G. F. C. Searle of Peterhouse who have been kind enough to read the proofs. I have also to thank Mr W. Hayles of the Cavendish Laboratory who has prepared many of the illustrations. J. J. THOMSON. CAVENDISH LABORATORY, CAMBRIDGE. September 3, 1895. PREFACE TO THE SECOND EDITION IN this Edition I have through the kindness of several correspondents been able to correct a considerable number of misprints. I have also made a few verbal alterations in the hope of making the argument clearer in places where experience has shown that students found unusual difficulties. J. J. THOMSON. CAVENDISH LABORATORY, CAMBRIDGE. November, 1897. PREFACE TO THE THIRD EDITION THE most important of the alterations made in this Edition is a new chapter on the properties of moving electrified bodies ; many of these properties may be proved in a simple way, and the important part played by moving charges in Modern Physics seems to warrant a discussion of their properties in even an Elementary Treatise. I have much pleasure in thanking Mr G. F. C. Searle of Peterhouse for many valuable suggestions, and for his kindness in reading the proof sheets of the first five chapters; to Mr P. V. Bevan of Trinity College I am indebted for similar assistance with the subsequent chapters. J. J. THOMSON. CAVENDISH LABORATORY, CAMBRIDGE. October 4, 1904. PREFACE TO THE FOURTH EDITION IN this Edition a few additions and corrections have been made. J. J. THOMSON. CAVENDISH LABORATORY, CAMBRIDGE. April 26, 1909. TABLE OF CONTENTS CHAP. PAGES I. General Principles of Electrostatics . . . 1 59 II. Lines of Force 60 83 III. Capacity of Conductors. Condensers . . 84 119 IV. Specific Inductive Capacity 120 144 V. Electrical Images and Inversion . . . 145 190 VI. Magnetism 191231 VII. Terrestrial Magnetism 232245 VIII. Magnetic Induction 246282 IX. Electric Currents 283-328 X. Magnetic Force due to Currents . . . 329386 XL Electromagnetic Induction 387 456 XII. Electrical Units : Dimensions of Electrical Quantities 457479 XIII. Dielectric Currents and the Electromagnetic Theory of Light 480505 XIV. Thermoelectric Currents 506-518 XV. The Properties of Moving Electric Charges . 519546 INDEX 547550 ELEMENTS OF THE MATHEMATICAL THEOEY OF ELECTEICITY AND MAGNETISM CHAPTER I GENERAL PRINCIPLES OF ELECTROSTATICS 1. Example of Electric Phenomena. Electri fication. Electric Field. A stick of sealing-wax after being rubbed with a well dried piece of flannel attracts light bodies such as small pieces of paper or pith balls covered with gold leaf. If such a ball be suspended by a silk thread, it will be attracted towards the sealing-wax, and, if the silk thread is long enough, the ball will move towards the wax until it strikes against it. When it has done this, however, it immediately flies away from the wax ; and the pith ball is now repelled from the wax instead of being attracted towards it as it was before the two had been in contact. The piece of flannel used to rub the sealing-wax also exhibits similar attractions for the pith balls, and these attractions are also changed into repulsions after the balls have been in contact with the flannel. The effects we have described are called electric phenomena, a title which as we shall see includes an T. E. 1 2 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I enormous number of effects of the most varied kinds. The example we have selected, where electrical effects are pro duced by rubbing two dissimilar bodies against each other, is the oldest electrical experiment known to science. The sealing-wax and the flannel are said to be electri fied, or to be in a state of electrification, or to be charged with electricity ; and the region in which the attractions and repulsions are observed is called the electric field. 2. Positive and Negative Electrification. If we take two pith balls A and B, coated with gold leaf and suspended by silk threads, and let them strike against the stick of sealing-wax which has been rubbed with a piece of flannel, they will be found to be repelled, not merely from the sealing-wax but also from each other. To observe this most conveniently remove the pith balls to such a distance from the sealing-wax and the flannel that the effects due to these are inappreciable. Now take another pair of similar balls, G and D, and let them strike against the flannel; G and D will be found to be repelled from each other when they are placed close together. Now take the ball A and place it near C; A and G will be found to be attracted towards each other. Thus, a ball which has touched the sealing-wax is repelled from another ball which has been similarly treated, but is attracted towards a ball which has been in contact with the flannel. The electricity on the balls A and E is thus of a kind different from that on the balls G and D, for while the ball A is repelled from B it is attracted towards D, while the ball C is attracted towards B and repelled from D ; thus when the ball A is attracted the ball G is repelled and vice versd. 2] GENERAL PRINCIPLES OF ELECTROSTATICS 3 The state of the ball which has touched the flannel is said to be one of positive electrification, or the ball is said to be positively electrified ; the state of the ball which has touched the sealing-wax is said to be one of negative electrification, or the ball is said to be negatively electri fied. We may for the present regard l positive and nega tive as conventional terms, which when applied to electric phenomena denote nothing more than the two states of electrification described above. As we proceed in the subject, however, we shall see that the choice of these terms is justified, since the properties of positive and negative electrification are, over a wide range of pheno mena, contrasted like the properties of the signs plus and minus in Algebra. The two balls A and B must be in similar states of electrification since they have been similarly treated; the two balls C and D will also for the same reason be in similar states of electrification. Now A and B are repelled from each other, as are also C and D ; hence we see that two bodies in similar states of electrification are repelled from each other : while, since one of the pair A, B is attracted towards either of the pair C, D, we see that two bodies, one in a positive state of electrification, the other in a negative state, are attracted towards each other. In whatever way a state of electrification is produced on a body, it is found to be one or other of the preceding kinds ; i.e. the ball A is either repelled from the electrified body or attracted towards it. In the former case the electrification is negative, in the latter positive. A method, which is sometimes convenient, of detecting whether the electrification of a body is positive or negative 12 4 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I is to dust it with a mixture of powdered red lead and yellow sulphur which has been well shaken ; the friction of the one powder against the other electrifies both powders, the sulphur becoming negatively, the red lead positively electrified. If now we dust a negatively electrified surface with this mixture, the positively electrified red lead will stick to the surface, while the negatively electrified sulphur will be easily detached, so that if we blow on the powdered surface the sulphur will come off while the red lead will remain, and thus the surface will be coloured red : if a posi tively electrified surface is treated in this way it will be come yellow in consequence of the sulphur sticking to it. 3. Electrification by Induction. If the negatively electrified stick of sealing-wax used in the preceding ex periments is held near to, but not touching, one end of an elongated piece of metal supported entirely on glass or ebonite stems, and if the metal is dusted over with the mixture of red lead and sulphur, it will be found, after blowing off the loose powder, that the end of the metal nearest to the sealing-wax is covered with the yellow sulphur, while the end furthest away is covered with red lead, showing that the end of the metal nearest the negatively electrified stick of sealing-wax is positively, the end remote from it negatively, electrified. In this experiment the metal, which has neither been rubbed nor been in contact with an electrified body, is said to be electrified by induction; the electrification on the metal is said to be induced by the electrification on the stick of sealing-wax. The electrification on the part of the metal nearest the wax is of the kind opposite to that on the wax, while the electrification on the more remote [...]... inside the normal induction over the surface is surface equal to sum of the areas cut off the unit sphere by cones with their bases on the meshes and their vertices e times the at 0, and since the meshes completely surface the sum of the areas cut off fill up the the cones will be the area of the sphere, which T E closed the unit sphere by is equal 2 GENERAL PRINCIPLES OF ELECTROSTATICS 18 [CH Thus the. .. N the com ponent of the electric intensity resolved along the normal drawn from the outside of the surface at a point in a mesh, and w meshes is the area of the mesh sum of the products Nco drawn on the surface that the With the notation is : the symbol S denotes to be taken for all the of the integral calculus I-JffdS, where dS is extending all 10 an element of the surface, the integration over the. .. PRINCIPLES OF ELECTROSTATICS 24 intensity and is [CH I everywhere parallel to the axis of the cylinder, no normal component over the curved has, therefore, Q Fig 7 surface of the cylinder, the total normal induction over R the surface thus arises entirely from the flat ends Let be the magnitude of the electric intensity at any point in the field, the area of either of the CD cylindrical surface Then the induction... the electroscope, the wire and the ground, If, when the electrified the elongated piece of metal in the described in Art 3 Thus, supposing the body experiment to be negatively electrified, the positive electrification will will correspond to while the negative will go to the most remote part of the system consisting of the metal of the the wire and the ground, i.e the negative be on the disc, electroscope,... direction of the electric intensity where 6 is The normal to the surface is at right angles to PQRS, at right angles to the area Ruvw, and hence is the angle between the normal to the surface and and Ruvw to the angles between the planes equal OR and is OR PQRS Hence area PQRS x cos 6 the area of the projection of the area PQRS on the plane Ruvw = area Ruvw (1) Ru is Consider the figures Ruwv and rspq... over the surface through Q P is Rw PQRS The part due will also be equal to this flat and to will ends of the normal due to the flat end the flat end through be of the same sign, part of the total the intensity at Q is along the outward drawn normal Thus since the normal intensity vanishes over since the curved surface of PQRS the total normal induction If a is the quantity of unit area of the plane the. .. The normal induction over the surface total R x (area But its area is therefore of the curved surface of the cylinder) since the length of the curved surface is unity is equal to 2-Trr, where r is the distance of P E from the axis of the cylinder If is the charge per unit length on the electrified cylinder, then by Gauss s theorem the total normal induction over the surface is The total normal induction... portion of the surface into meshes and taking the sum of the contributions of each By mesh, we see that the total normal induction over the is equal to e times the area cut off a sphere of unit radius with its centre at by a cone having the surface boundary of the surface as base and its vertex at 0 Let us now apply the results we have obtained to the case of a closed surface First The total take the case... glass rod and, holding the rod by the other end, place the wire no so that it is in contact with A and B simultaneously : ; alteration in the divergence of the gold leaves will be pro duced by this process, showing that the sum of the charges Take away the wire and remove is unaltered and on A B GENERAL PRINCIPLES OF ELECTROSTATICS 12 B from the vessel, of the gold leaves be the same as ; it [CH I and now... point in the direction of the GENERAL PRINCIPLES OF ELECTROSTATICS 14 [CH I normal drawn from the outside of the surface at that point, and multiply this normal component by the area Fig 3 of the mesh the ; on the surface is sum of these products for all the meshes denned induction over the surface by the to be the total This is normal electric algebraically expressed relation where / is the total . Edition 1895. Second Edition 1897. Third Edition 1904. Fourth Edition 1909. PREFACE TO FIRST EDITION IN the following work I have endeavoured to give an account of the fundamental principles of the Mathematical theory of Electricity and Magnetism and their more important applications, using. Art. 3. Thus, supposing the body to be negatively electrified, the positive electrification will be on the disc, while the negative will go to the most remote part of the system consisting of the metal of the electroscope, the wire and the ground, i.e. the negative electrification will go to the ground and. a distance r apart, r being very large compared with the greatest linear dimension of either of the bodies, the repulsion between them is proportional to the product of their charges and inversely proportional to the square of the distance