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ELECTRICITY AND MATTER
J.J. THOMSON

KF 14581

In whitthead

YALE UNIVERSITY MRS. HEPSA ELY SILLIMAN MEMORIAL LECTURES
ELECTRICITY AND MATTER

ELECTRICITY AND MATTER
BY J. J. THOMSON, D.Sc., LL.D. , PH.D. , F.R.S.
FFLLOW OF TRINITY COLLEGE, CAMBRIDGE ; CAVENDISH PROFESSOR OF EXPERIMENTAL PHYSICS, CAMBRIDGE
WITH DIAGRAMS
LONDON ARCHIBALD CONSTABLE & CO. , LTD.
1909

KF14581
HARVARD UNIVERSITY
LIBRARY
Copyright, 1904, by Yale University, for the United States of America
Printed by the Scribner Press New York, U. S. A.

THE SILLIMAN FOUNDATION.
In the year 1883 a legacy of eighty thousand dollars was left to the President and Fellows of Yale College in the city of New Haven , to be held in trust, as a gift from her children, in memory of their beloved and honored mother Mrs. Hepsa Ely Silliman.
On this foundation Yale College was requested and directed to establish an annual course of lectures designed to illustrate the presence and providence, the wisdom and goodness of God, as manifested in the natural and moral world. These were to be designated as the Mrs. Hepsa Ely Silliman Memorial Lectures. It was the belief of the testator that any orderly presentation of the facts of nature or history contributed to the end of this foundation more effectively than any attempt to emphasize the elements of doctrine or of creed ; and he therefore provided that lectures on dogmatic or polemical theology should be excluded from the scope of this foundation , and that the subjects should be selected rather from the domains of natural science
and history, giving special prominence to astronomy, chemistry, geology, and anatomy.
It was further directed that each annual course should
be made the basis of a volume to form part of a series constituting a memorial to Mrs. Silliman. The memorial fund came into the possession of the Corporation of Yale University in the year 1902 ; and the present volume constitutes the first of the series of memorial lectures.

PREFACE
In these Lectures given at Yale University in May, 1903, I have attempted to discuss the bearing of the recent advances made in Electrical Science on our views of the Constitution of Matter and the Nature of Electricity ; two questions which are probably so intimately connected , that the solution of the one would supply that of the other. A characteristic feature of recent Electrical Researches, such as the study and discovery of Cathode and Röntgen Rays and Radio-active Substances, has been the very especial degree in which they have involved the relation between Matter and Electricity.
In choosing a subject for the Silliman Lectures, it seemed to me that a consideration of the bearing of recent work on this relationship might be suitable, especially as such a discussion suggests multitudes of questions which would furnish admirable subjects for further investigation by some of my hearers.
Cambridge, Aug., 1903. J. J. THOMSON.

CONTENTS

CHAPTER I PAGE

REPRESENTATION OF THE ELECTRIC FIELD BY LINES

OF FORCE •

1

CHAPTER II

ELECTRICAL AND BOUND MASS

36

CHAPTER III

EFFECTS DUE TO THE ACCELERATION OF FARADAY

TUBES .

53

CHAPTER IV

THE ATOMIC STRUCTURE OF ELECTRICITY

71

CHAPTER V

THE CONSTITUTION OF THE ATOM .

90

CHAPTER VI RADIO-ACTIVITY AND RADIO -ACTIVE SUBSTANCES .

• 140

ELECTRICITY AND MATTER
CHAPTER I
REPRESENTATION OF THE ELECTRIC FIELD BY LINES OF FORCE
My object in these lectures is to put before you in as simple and untechnical a manner as I can some views as to the nature of electricity, of the processes going on in the electric field, and of the connection between electrical and ordinary matter which have been suggested by the results of recent investigations.
The progress of electrical science has been greatly promoted by speculations as to the nature of electricity. Indeed , it is hardly possible to overestimate the services rendered by two theories as old almost as the science itself ; I mean the theories known as the two- and the one-fluid theories of electricity.
The two-fluid theory explains the phenomena of electro-statics by supposing that in the universe there are two fluids, uncreatable and indestruc-

2

ELECTRICITY AND MATTER

tible, whose presence gives rise to electrical effects ; one of these fluids is called positive, the other negative electricity, and electrical phenomena are explained by ascribing to the fluids the following properties. The particles of the positive fluid repel each other with forces varying inversely as the square of the distance between them, as do also the particles of the negative fluid ; on the other hand, the particles of the positive fluid attract those of the negative fluid. The attraction between two charges, m and m' , of opposite signs are in one form of the theory supposed to be exactly equal to the repulsion between two charges, m and m ' of the same sign, placed in the same position as the previous charges. In another development of the theory the attraction is supposed to slightly exceed the repulsion , so as to afford a basis for the explanation of gravitation.
The fluids are supposed to be exceedingly mo bile and able to pass with great ease through con. ductors. The state of electrification of a body is determined by the difference between the quantities of the two electric fluids contained by it ; if it contains more positive fluid than negative it is positively electrified , if it contains equal quantities it is uncharged. Since the fluids are uncreatable

LINES OF FORCE

3

and indestructible, the appearance of the positive fluid in one place must be accompanied by the departure of the same quantity from some other place, so that the production of electrification of one sign must always be accompanied by the production of an equal amount of electrification of the opposite sign.
On this view, every body is supposed to consist of three things : ordinary matter, positive electricity, negative electricity. The two latter are supposed to exert forces on themselves and on each other, but in the earlier form of the theory no action was contemplated between ordinary matter and the electric fluids ; it was not until a comparatively recent date that Helmholtz introduced the idea of a specific attraction between ordinary matter and the electric fluids. He did this to explain what is known as contact electricity, ¿.e., the electrical separation produced when two metals, say zinc and copper, are put in contact with each other, the zinc becoming positively, the copper negatively electrified . Helmholtz sup-
posed that there are forces between ordinary matter and the electric fluids varying for different kinds of matter, the attraction of zinc for positive electricity being greater than that of copper, so

4

ELECTRICITY AND MATTER

that when these metals are put in contact the zinc robs the copper of some of its positive electricity.
There is an indefiniteness about the two-fluid theory which may be illustrated by the consideration of an unelectrified body. All that the twofluid theory tells us about such a body is that it contains equal quantities of the two fluids. It gives no information about the amount of either ; indeed, it implies that if equal quantities of the two are added to the body, the body will be unaltered, equal quantities of the two fluids exactly neutraliz ing each other. If we regard these fluids as being anything more substantial than the mathematical symbols + and - this leads us into difficulties ; if we regard them as physical fluids, for example, we have to suppose that the mixture of the two fluids. in equal proportions is something so devoid of physical properties that its existence has never been detected.
The other fluid theory-the one-fluid theory of Benjamin Franklin- is not open to this objection. On this view there is only one electric fluid, the positive ; the part of the other is taken by ordinary matter, the particles of which are supposed to repel each other and attract the positive fluid ,

LINES OF FORCE

5

just as the particles of the negative fluid do on the two-fluid theory. Matter when unelectrified is supposed to be associated with just so much of the electric fluid that the attraction of the matter on a portion of the electric fluid outside it is just sufficient to counteract the repulsion exerted on the same fluid by the electric fluid associated with the matter. On this view, if the quantity of matter in a body is known the quantity of electric fluid is at once determined .
The services which the fluid theories have ren-
dered to electricity are independent of the notion of a fluid with any physical properties ; the fluids were mathematical fictions, intended merely to give a local habitation to the attractions and repulsions existing between electrified bodies, and served as the means by which the splendid mathematical development of the theory of forces varying inversely as the square of the distance which was inspired by the discovery of gravitation could be brought to bear on electrical phenomena. As long as we confine ourself to questions which only involve the law of forces between electrified bodies, and the simultaneous production of equal quantities of + and electricity, both theories must give the same results and there can be nothing to

4

ELECTRICITY AND MATTER

that when these metals are put in contact the zinc robs the copper of some of its positive electricity.
There is an indefiniteness about the two-fluid theory which may be illustrated by the consideration of an unelectrified body. All that the twofluid theory tells us about such a body is that it contains equal quantities of the two fluids. It gives no information about the amount of either ; indeed, it implies that if equal quantities of the two are added to the body, the body will be unaltered, equal quantities of the two fluids exactly neutraliz ing each other. If we regard these fluids as being anything more substantial than the mathematical symbols + and this leads us into difficulties ; if
we regard them as physical fluids, for example, we have to suppose that the mixture of the two fluids in equal proportions is something so devoid of physical properties that its existence has never been detected .
The other fluid theory-the one-fluid theory of Benjamin Franklin- is not open to this objection. On this view there is only one electric fluid, the positive ; the part of the other is taken by ordinary matter, the particles of which are supposed to repel each other and attract the positive fluid ,

LINES OF FORCE

5

just as the particles of the negative fluid do on the two-fluid theory. Matter when unelectrified is supposed to be associated with just so much of the electric fluid that the attraction of the matter on a portion of the electric fluid outside it is just sufficient to counteract the repulsion exerted on the same fluid by the electric fluid associated with the matter. On this view, if the quantity of matter in a body is known the quantity of electric fluid is at once determined.
The services which the fluid theories have ren-
dered to electricity are independent of the notion of a fluid with any physical properties ; the fluids were mathematical fictions, intended merely to give a local habitation to the attractions and repulsions existing between electrified bodies, and served as the means by which the splendid mathematical development of the theory of forces varying inversely as the square of the distance which was inspired by the discovery of gravitation could be brought to bear on electrical phenomena. As long as we confine ourself to questions which only involve the law of forces between electrified bodies, and the simultaneous production of equal quantities
of + and electricity, both theories must give the same results and there can be nothing to

PREFACE
In these Lectures given at Yale University in May, 1903, I have attempted to discuss the bearing of the recent advances made in Electrical Science on our views of the Constitution of Matter and the Nature of Electricity ; two questions which are probably so intimately connected , that the solution of the one would supply that of the other. A characteristic feature of recent Electrical Researches, such as the study and discovery of Cathode and Röntgen Rays and Radio-active Substances, has been the very especial degree in which they have involved the relation between Matter and Electricity.
In choosing a subject for the Silliman Lectures, it seemed to me that a consideration of the bearing of recent work on this relationship might be suitable, especially as such a discussion suggests multitudes of questions which would furnish admirable subjects for further investigation by some of my hearers.
Cambridge, Aug., 1903 . J. J. THOMSON.

1

CONTENTS

CHAPTER I PAGE
REPRESENTATION OF THE ELECTRIC FIELD BY LINES

OF FORCE

1

CHAPTER II

ELECTRICAL AND BOUND MASS

36

CHAPTER III

EFFECTS DUE TO THE ACCELERATION OF FARADAY

TUBES .

53

CHAPTER IV

THE ATOMIC STRUCTURE OF ELECTRICITY

71

14

CHAPTER V

THE CONSTITUTION OF THE ATOM .

90

CHAPTER VI RADIO-ACTIVITY AND RADIO- ACTIVE SUBSTANCES .

• 140

ELECTRICITY AND MATTER
CHAPTER I
REPRESENTATION OF THE ELECTRIC FIELD BY LINES OF FORCE
My object in these lectures is to put before you in as simple and untechnical a manner as I can some views as to the nature of electricity, of the processes going on in the electric field, and of the connection between electrical and ordinary matter which have been suggested by the results of recent investigations.
The progress of electrical science has been greatly promoted by speculations as to the nature of electricity. Indeed, it is hardly possible to overestimate the services rendered by two theories as old almost as the science itself ; I mean the theories known as the two- and the one-fluid theories of electricity.
The two-fluid theory explains the phenomena of electro-statics by supposing that in the universe there are two fluids, uncreatable and indestruc-

2

ELECTRICITY AND MATTER

tible, whose presence gives rise to electrical effects ; one of these fluids is called positive, the other negative electricity, and electrical phenomena are explained by ascribing to the fluids the following properties. The particles of the positive fluid repel each other with forces varying inversely as the square of the distance between them, as do also the particles of the negative fluid ; on the other hand, the particles of the positive fluid attract those of the negative fluid. The attraction between two charges, m and m' , of opposite signs are in one form of the theory supposed to be exactly equal to the repulsion between two charges, m and m ' of the same sign, placed in 1 the same position as the previous charges. In another development of the theory the attraction is supposed to slightly exceed the repulsion, so as to afford a basis for the explanation of gravitation.
The fluids are supposed to be exceedingly mo bile and able to pass with great ease through con. ductors. The state of electrification of a body is determined by the difference between the quantities of the two electric fluids contained by it ; if it contains more positive fluid than negative it is positively electrified , if it contains equal quantities it is uncharged. Since the fluids are uncreatable

LINES OF FORCE

3

and indestructible, the appearance of the positive fluid in one place must be accompanied by the departure of the same quantity from some other place, so that the production of electrification of one sign must always be accompanied by the production of an equal amount of electrification of the opposite sign.
On this view, every body is supposed to con- [ sist of three things : ordinary matter, positive elec tricity, negative electricity. The two latter are supposed to exert forces on themselves and on each other, but in the earlier form of the theory no action was contemplated between ordinary matter and the electric fluids ; it was not until a comparatively recent date that Helmholtz introduced the idea of a specific attraction between ordinary matter and the electric fluids. He did this to explain what is known as contact electricity, i.e., the electrical separation produced when two metals, say zinc and copper, are put in contact with each other, the zinc becoming positively, the copper negatively electrified . Helmholtz supposed that there are forces between ordinary matter and the electric fluids varying for different kinds of matter, the attraction of zinc for positive electricity being greater than that of copper, so

4

ELECTRICITY AND MATTER

that when these metals are put in contact the zinc robs the copper of some of its positive electricity.
There is an indefiniteness about the two-fluid theory which may be illustrated by the consideration of an unelectrified body. All that the twofluid theory tells us about such a body is that it contains equal quantities of the two fluids. It gives no information about the amount of either ; indeed, it implies that if equal quantities of the two are added to the body, the body will be unaltered, equal quantities of the two fluids exactly neutraliz ing each other. If we regard these fluids as being anything more substantial than the mathematical symbols + and - this leads us into difficulties ; if we regard them as physical fluids, for example, we have to suppose that the mixture of the two fluids in equal proportions is something so devoid of physical properties that its existence has never been detected .
The other fluid theory-the one-fluid theory of Benjamin Franklin- is not open to this objection. On this view there is only one electric fluid, the positive ; the part of the other is taken by ordinary matter, the particles of which are supposed to repel each other and attract the positive fluid ,

LINES OF FORCE

5

just as the particles of the negative fluid do on the two-fluid theory. Matter when unelectrified is supposed to be associated with just so much of the electric fluid that the attraction of the matter

on a portion of the electric fluid outside it is just sufficient to counteract the repulsion exerted on the same fluid by the electric fluid associated with the matter. On this view, if the quantity of mat-

ter in a body is known the quantity of electric fluid is at once determined .
The services which the fluid theories have ren-

dered to electricity are independent of the notion of a fluid with any physical properties ; the fluids were mathematical fictions, intended merely to give a local habitation to the attractions and repulsions existing between electrified bodies, and served as the means by which the splendid mathematical development of the theory of forces varying inversely as the square of the distance which was

inspired by the discovery of gravitation could be brought to bear on electrical phenomena. As long as we confine ourself to questions which only involve the law of forces between electrified bodies ,

and the simultaneous production of equal quantities

of

and - electricity, both theories must give

the same results and there can be nothing to

6

ELECTRICITY AND MATTER

decide between them. The physicists and mathematicians who did most to develop the " Fluid Theories " confined themselves to questions of this kind, and refined and idealized the conception of these fluids until any reference to their physical properties was considered almost indelicate. It is not until we investigate phenomena which involve the physical properties of the fluid that we can hope to distinguish between the rival fluid theories. Let us take a case which has actually arisen. We have been able to measure the masses associated with given charges of electricity in gases at low pressures, and it has been found that
the mass associated with a positive charge is immensely greater than that associated with a nega tive one. This difference is what we should expect on Franklin's one-fluid theory, if that theory were modified by making the electric flui.d correspond to negative instead of positive electricity, while we have no reason to anticipate so great a difference on the two-fluid theory. We shall, I am sure, be struck by the similarity between some of the views which we are led to take by the results of the most recent researches with those enunciated by Franklin in the very infancy of the subject.

LINES OF FORCE

7

Faraday's Line of Force Theory
The fluid theories, from their very nature, imply the idea of action at a distance. This idea, although its convenience for mathematical analysis has made it acceptable to many mathematicians, is one which many of the greatest physicists have felt utterly unable to accept, and have devoted much thought and labor to replacing it by something involving mechanical continuity. Pre-eminent among them is Faraday. Faraday was deeply influenced by the axiom, or if you prefer it, dogma that matter cannot act where it is not. Faraday, who possessed, I believe, almost unrivalled mathematical insight, had had no training in analysis, so that the convenience of the idea of action at a distance for purposes of calculation had no chance of mitigating the repugnance he felt to the idea of forces acting far away from their base and with no physical connection with their origin. He therefore cast about for some way of picturing to himself the actions in the electric field which would get rid of the idea of action at a distance, and replace it by one which would bring into prominence some continuous connection between the bodies exerting the forces. He was able to

8

ELECTRICITY AND MATTER

do this by the conception of lines of force. As I shall have continually to make use of this method, and as I believe its powers and possibilities have never been adequately realized, I shall devote some time to the discussion and development of this conception of the electric field.

FIG. 1.
The method was suggested to Faraday by the consideration of the lines of force round a bar magnet. If iron filings are scattered on a smooth surface near a magnet they arrange themselves as in Fig. 1 ; well-marked lines can be traced run-

LINES OF FORCE

9

ning from one pole of the magnet to the other ; the direction of these lines at any point coincides with the direction of the magnetic force, while the intensity of the force is indicated by the concentration of the lines. Starting from any point in the field and travelling always in the direction of the magnetic force, we shall trace out a line which will not stop until we reach the negative pole of the magnet ; if such lines are drawn at all points in the field , the space through which the magnetic field extends will be filled with a system of lines, giving the space a fibrous structure like that possessed by a stack of hay or straw, the grain of the structure being along the lines of force. I have spoken so far only of lines of magnetic force ; the same considerations will apply to the electric field, and we may regard the electric field as full of lines of electric force, which start from positively and end on negatively electrified bodies. Up to this point the process has been entirely geometrical, and could have been employed by those who looked at the question from the point of view of action at a distance ; to Faraday, however, the lines of force were far more than mathematical abstractions - they were physical realities. Faraday materialized the lines of force and endowed

10

ELECTRICITY AND MATTER

them with physical properties so as to explain the phenomena of the electric field. Thus he supposed that they were in a state of tension, and that they repelled each other. Instead of an intangible action at a distance between two electrified bodies, Faraday regarded the whole space between the bodies as full of stretched mutually repellent springs. The charges of electricity to which alone an interpretation had been given on the fluid theories of electricity were on this view just the ends of these springs, and an electric charge, instead of being a portion of fluid confined to the electrified body, was an extensive arsenal of springs spreading out in all directions to all parts of the field.
To make our ideas clear on this point let us consider some simple cases from Faraday's point of view. Let us first take the case of two bodies with equal and opposite charges, whose lines of force are shown in Fig. 2. You notice that the lines of force are most dense along A B, the line joining the bodies, and that there are more lines of force on the side of A nearest to BВ than on the opposite side. Consider the effect of the lines of force on A ; the lines are in a state of tension and are pulling away at A ; as there

LINES OF FORCE

11

are more pulling at A on the side nearest to B than on the opposite side, the pulls on A toward B overpower those pulling A away from B, so that A will tend to move toward B ; it was in this way that Faraday pictured to himself the attraction between oppositely electrified bodies. Let us now consider the condition of one of the curved lines of force, such as PQ; it is in a state

FIG. 2.
of tension and will therefore tend to straighten itself, how is it prevented from doing this and maintained in equilibrium in a curved position ? We can see the reason for this if we remember that the lines of force repel each other and that the lines are more concentrated in the region between PQ and AB than on the other side of PQ; thus the repulsion of the lines inside PQ will be greater than the repulsion of those outside and the line PQ will be bent outwards.

12

ELECTRICITY AND MATTER

Let us now pass from the case of two oppositely electrified bodies to that of two similarly electrified ones, the lines of force for which are shown in Fig. 3. Let us suppose A and B are positively electrified ; since the lines of force start from positively and end on negatively electrified bodies, the lines starting from A and B will travel away to join some body or bodies possessing the

FIG. 3.
negative charges corresponding to the positive ones on A and B; let us suppose that these charges are a considerable distance away, so that the lines of force from A would, if B were not present, spread out, in the part of the field under consideration, uniformly in all directions. Consider now the effect of making the system of lines of force attached to A and B approach each other ;

LINES OF FORCE

13

since these lines repel each other the lines of force on the side of A nearest B will be pushed to the opposite side of A, so that the lines of force will now be densest on the far side of A ; thus the pulls exerted on A in the rear by the lines of force will be greater than those in the front and the result will be that A will be pulled away

from B. We notice that the mechanism producing this repulsion is of exactly the same type as that which produced the attraction in the previous case, and we may if we please regard the repulsion between A and B as due to the attractions on A and B of the

complementary negative charges which must exist in

other parts of the field. The results of the repul
sion of the lines of force

are clearly shown in the case represented in Fig. 4, that of two oppositely electrified plates ; you will notice that the lines of force between the plates are straight except

FIG. 4.

near the edges of the plates ; this is what we should expect as the downward pressure exerted

14

ELECTRICITY AND MATTER

by the lines of force above a line in this part of the field will be equal to the upward pressure exerted by those below it. For a line of force near the edge of the plate, however, the pressure of the lines of force below will exceed the pressure from those above, and the line of force will bulge out until its curvature and tension counteract the squeeze from inside ; this bulging is very plainly shown in Fig. 4.
So far our use of the lines of force has been descriptive rather than metrical ; it is, however, easy to develop the method so as to make it metrical. We can do this by introducing the idea of tubes of force. If through the boundary of any small closed curve in the electric field we draw the lines of force, these lines will form a tubular surface, and if we follow the lines back to the positively electrified surface from which they start and forward on to the negatively electrified surface on which they end, we can prove that the positive charge enclosed by the tube at its origin is equal to the negative charge enclosed by it at its end. By properly choosing the area of the small curve through which we draw the lines of force, we may arrange that the charge enclosed by the tube is equal to the unit charge. Let us call

LINES OF FORCE

15

such a tube a Faraday tube-then each unit of positive electricity in the field may be regarded as the origin and each unit of negative electricity as the termination of a Faraday tube. We regard these Faraday tubes as having direction, their direction being the same as that of the electric force, so that the positive direction is from the positive to the negative end of the tube. If we draw any closed surface then the difference between the number of Faraday tubes which pass out of the surface and those which pass in will be equal to the algebraic sum of the charges inside the surface ; this sum is what Maxwell called the electric displacement through the surface. What Maxwell called the electric displacement in any direction at a point is the number of Faraday tubes which pass through a unit area through the point drawn at right angles to that direction, the number being reckoned algebraically ; i.e. , the lines which pass through in one direction being taken as positive, while those which pass through in the opposite direction are taken as negative, and the number passing through the area is the difference between the number passing through positively and the number passing through negatively.
For my own part, I have found the conception

16

ELECTRICITY AND MATTER

of Faraday tubes to lend itself much more readily to the formation of a mental picture of the processes going on in the electric field than that of electric displacement, and have for many years abandoned the latter method .
Maxwell took up the question of the tensions and pressures in the lines of force in the electric field , and carried the problem one step further than Faraday. By calculating the amount of these tensions he showed that the mechanical effects in the electrostatic field could be explained by supposing that each Faraday tube force exerted a tension equal to R, R being the intensity of the electric force, and that, in addition to this tension, there was in the medium through which the tubes pass a hydrostatic pressure equal to NR, N being the density of the Faraday tubes ; i.e., the number passing through a unit area drawn at right angles to the electric force. If we consider the effect of these tensions and pressure on a unit volume of the medium in the electric field, we see that they are equivalent to a tension NR along the direction of the electric force and an equal pressure in all directions at right angles to that force.

LINES OF FORCE

17

Moving Faraday Tubes

Hitherto we have supposed the Faraday tubes

to be at rest, let us now proceed to the study of

the effects produced by the motion of those tubes . Let us begin with the consideration of a very

simple case that of two parallel

plates, A and B, charged, one with

positive the other with negative electricity, and suppose that after
AB being charged the plates are connected by a conducting wire, EFG. This wire will pass through some

of the outlying tubes ; these tubes, when in a conductor, contract to molecular dimensions and the repul

sion they previously exerted on neighboring tubes will therefore disappear. Consider the effect of this on a tube PQ between the

FIG. 5.

plates ; PQ was originally in equilibrium under its own tension, and the repulsion exerted by the neighboring tubes. The repulsions due to those

cut by E F G have now, however, disappeared so that PQ will no longer be in equilibrium, but will

be pushed towards EFG. Thus, more and more tubes will be pushed into EFG, and we shall

8

ELECTRICITY AND MATTER

do this by the conception of lines of force. As I shall have continually to make use of this method, and as I believe its powers and possibilities have never been adequately realized, I shall devote some time to the discussion and development of this conception of the electric field.

FIG. 1.
The method was suggested to Faraday by the consideration of the lines of force round a bar magnet. If iron filings are scattered on a smooth surface near a magnet they arrange themselves as in Fig. 1 ; well-marked lines can be traced run-

LINES OF FORCE

9

ning from one pole of the magnet to the other ; the direction of these lines at any point coincides with the direction of the magnetic force, while the intensity of the force is indicated by the concentration of the lines. Starting from any point in the field and travelling always in the direction of the magnetic force, we shall trace out a line which will not stop until we reach the negative pole of the magnet ; if such lines are drawn at all points in the field, the space through which the magnetic field extends will be filled with a system of lines, giving the space a fibrous structure like that possessed by a stack of hay or straw, the grain of the structure being along the lines of force. I have spoken so far only of lines of magnetic force ; the same considerations will apply to the electric field, and we may regard the electric field as full of lines of electric force, which start from positively and end on negatively electrified bodies. Up to this point the process has been entirely geometrical, and could have been employed by those who looked at the question from the point of view of action at a distance ; to Faraday, however, the lines of force were far more than mathematical abstractions - they were physical realities. Faraday materialized the lines of force and endowed

10

ELECTRICITY AND MATTER

them with physical properties so as to explain the phenomena of the electric field . Thus he sup posed that they were in a state of tension, and that they repelled each other. Instead of an intangible action at a distance between two electri fied bodies, Faraday regarded the whole space
· between the bodies as full of stretched mutually repellent springs. The charges of electricity to which alone an interpretation had been given on the fluid theories of electricity were on this view just the ends of these springs, and an electric charge, instead of being a portion of fluid confined to the electrified body, was an extensive arsenal of springs spreading out in all directions to all parts of the field.
To make our ideas clear on this point let us consider some simple cases from Faraday's point of view. Let us first take the case of two bodies with equal and opposite charges, whose lines of force are shown in Fig. 2. You notice that the lines of force are most dense along A B, the line joining the bodies, and that there are more lines. of force on the side of A nearest to B than on the opposite side. Consider the effect of the lines of force on A ; the lines are in a state of tension and are pulling away at A ; as there

LINES OF FORCE

11

are more pulling at A on the side nearest to B than on the opposite side, the pulls on A toward B overpower those pulling A away from B, so that A will tend to move toward B; it was in this way that Faraday pictured to himself the attraction between oppositely electrified bodies. Let us now consider the condition of one of the curved lines of force, such as PQ; it is in a state

FIG. 2.
of tension and will therefore tend to straighten itself, how is it prevented from doing this and maintained in equilibrium in a curved position ? We can see the reason for this if we remember that the lines of force repel each other and that the lines are more concentrated in the region between PQ and AB than on the other side of PQ; thus the repulsion of the lines inside PQ will be greater than the repulsion of those outside and the line PQ will be bent outwards.

12

ELECTRICITY AND MATTER

Let us now pass from the case of two oppositely electrified bodies to that of two similarly electrified ones, the lines of force for which are shown in Fig. 3. Let us suppose A and B are positively electrified ; since the lines of force start from positively and end on negatively electrified bodies, the lines starting from A and B will travel away to join some body or bodies possessing the

FIG. 3.
negative charges corresponding to the positive ones on A and B ; let us suppose that these charges are a considerable distance away, so that the lines of force from A would , if B were not present, spread out, in the part of the field under consideration, uniformly in all directions. Consider now the effect of making the system of lines of force attached to A and B approach each other ;

LINES OF FORCE

13

since these lines repel each other the lines of force on the side of A nearest B will be pushed to the opposite side of A, so that the lines of force will now be densest on the far side of A ; thus the

pulls exerted on A in the rear by the lines of force will be greater than those in the front and the result will be that A will be pulled away

from B. We notice that the mechanism producing this repulsion is of exactly the same type as that which produced the attraction in the previous case, and we may if we please regard the repulsion between A and B as due to the attrac tions on A and B of the

complementary negative charges which must exist in other parts of the field.
The results of the repulsion of the lines of force

are clearly shown in the case represented in Fig. 4, that

of two oppositely electrified plates ; you will notice that the lines of force between the plates are straight except

FIG. 4.

near the edges of the plates ; this is what we

should expect as the downward pressure exerted

14

ELECTRICITY AND MATTER

by the lines of force above a line in this part of the field will be equal to the upward pressure exerted by those below it. For a line of force near the edge of the plate, however, the pressure of the lines of force below will exceed the pressure from those above, and the line of force will bulge out until its curvature and tension counteract the squeeze from inside ; this bulging is very plainly shown in Fig. 4.
So far our use of the lines of force has been
descriptive rather than metrical ; it is, however, easy to develop the method so as to make it metrical. We can do this by introducing the idea of tubes of force. If through the boundary of any small closed curve in the electric field we draw the lines of force, these lines will form a tubular surface, and if we follow the lines back to the positively electrified surface from which they start and forward on to the negatively electrified surface on which they end, we can prove that the positive charge enclosed by the tube at its origin is equal to the negative charge enclosed by it at its end. By properly choosing the area of the small curve through which we draw the lines of force, we may arrange that the charge enclosed by the tube is equal to the unit charge. Let us call

LINES OF FORCE

15

such a tube a Faraday tube-then each unit of positive electricity in the field may be regarded as the origin and each unit of negative electricity as the termination of a Faraday tube. We regard these Faraday tubes as having direction, their di rection being the same as that of the electric force, so that the positive direction is from the positive to the negative end of the tube. If we draw any closed surface then the difference between the number of Faraday tubes which pass out of the surface and those which pass in will be equal to the algebraic sum of the charges inside the surface ; this sum is what Maxwell called the electric displacement through the surface. What Maxwell called the electric displacement in any direction at a point is the number of Faraday tubes which pass through a unit area through the point drawn at right angles to that direction, the number being reckoned algebraically ; i.e., the lines which pass through in one direction being taken as positive, while those which pass through in the opposite direction are taken as negative, and the number passing through the area is the difference between the number passing through positively and the number passing through negatively.
For my own part, I have found the conception

16

ELECTRICITY AND MATTER

of Faraday tubes to lend itself much more readily to the formation of a mental picture of the processes going on in the electric field than that of electric displacement, and have for many years abandoned the latter method.
Maxwell took up the question of the tensions and pressures in the lines of force in the electric field , and carried the problem one step further than Faraday. By calculating the amount of these tensions he showed that the mechanical effects in the electrostatic field could be explained by supposing that each Faraday tube force exerted a tension equal to R, R being the intensity of the electric force, and that, in addition to this tension, there was in the medium through which the tubes pass a hydrostatic pressure equal to NR, N being the density of the Faraday tubes ; i.e., the number passing through a unit area drawn at right angles to the electric force. If we consider the effect of these tensions and pressure on a unit volume of the medium in the electric field, we see that they are equivalent to a tension NR along the direction of the electric force and an equal pressure in all directions at right angles to that force.

LINES OF FORCE

17

Moving Faraday Tubes

Hitherto we have supposed the Faraday tubes

to be at rest, let us now proceed to the study of the effects produced by the motion of those tubes. Let us begin with the consideration of a very

simple case that of two parallel

plates, A and B, charged, one with

positive the other with negative

electricity, and suppose that after

being charged the plates are con-

B

nected by a conducting wire, EFG.

This wire will pass through some

of the outlying tubes ; these tubes, E
when in a conductor, contract to

molecular dimensions and the repul

sion they previously exerted on neighboring tubes will therefore

disappear. Consider the effect of this on a tube PQ between the

FIG. 5 .

plates ; PQ was originally in equilibrium under

its own tension, and the repulsion exerted by the neighboring tubes. The repulsions due to those cut by E F G have now, however, disappeared so

that PQ will no longer be in equilibrium, but will be pushed towards EFG. Thus, more and more tubes will be pushed into EFG, and we shall

18

ELECTRICITY AND MATTER

have a movement of the whole set of tubes between the plates toward EFG. Thus, while the discharge of the plates is going on, the tubes between the plates are moving at right angles to themselves. What physical effect accompanies this movement of the tubes ? The result of con-

necting the plates by EFG is to produce a current of electricity flowing from the positively 1 charged plate through EF G to the negatively charged plate ; this is, as we know, accompanied by a magnetic force between the plates. This magnetic force is at right angles to the plane of the paper and equal to 4 times the intensity of the current in the plate, or, if σ is the density of the charge of electricity on the plates and v the velocity with which the charge moves, the magnetic force is equal to 4πσv.
Here we have two phenomena which do not take place in the steady electrostatic field, one the movement of the Faraday tubes, the other the existence of a magnetic force ; this suggests that there is a connection between the two , and that motion of the Faraday tubes is accompanied by the production of magnetic force. I have followed up the consequences of this supposition and

have shown that, if the connection between the

LINES OF FORCE

19

magnetic force and the moving tubes is that given below, this view will account for Ampère's laws connecting current and magnetic force, and for Faraday's law of the induction of currents. Maxwell's great contribution to electrical theory, that variation in the electric displacement in a dielectric produces magnetic force, follows at once from this view. For, since the electric displacement is measured by the density of the Faraday tubes, if the electric displacement at any place changes, Faraday tubes must move up to or away from the place, and motion of Faraday tubes, by hypothesis, implies magnetic force.
The law connecting magnetic force with the motion of the Faraday tubes is as follows : À Faraday tube moving with velocity v at a point P, produces at P a magnetic force whose magnitude is 4 v sin 0, the direction of the magnetic force being at right angles to the Faraday tube, and also to its direction of motion ; is the angle between the Faraday tube and the direction in which it is moving. We see that it is only the motion of a tube at right angles to itself which produces magnetic force ; no such force is produced by the gliding of a tube along its length.

20

ELECTRICITY AND MATTER

Motion of a Charged Sphere

We shall apply these results to a very simple and important case the steady motion of a charged sphere. If the velocity of the sphere is small compared with that of light then the Faraday tubes will, as when the sphere is at rest, be uniformly distributed and radial in direction. They will be carried along with the sphere. If e is the charge on the sphere, O its centre, the
e density of the Faraday tubes at P is 4п Opx ; so that if v is the velocity of the sphere, the
P

FIG. 6 .

angle between OP and the direction of motion of

the sphere, then, according to the above rule, the

ev sin o

magnetic force at P will be

the direc-

p2

tion of the force will be at right angles to OP,

and at right angles to the direction of motion of

the sphere ; the lines of magnetic force will thus

LINES OF FORCE

21

be circles, having their centres on the path of the centre of the sphere and their planes at right angles to this path. Thus, a moving charge of electricity will be accompanied by a magnetic field. The existence of a magnetic field implies energy; we know that in a unit volume of the

field at a place where the magnetic force is H
there are µH² units of energy, where μ is the 8πT

magnetic permeability of the medium. In the

case of the moving sphere the energy per unit

μe²v² sin² 0

volume at P is

Taking the sum of

8пT OP

this energy for all parts of the field outside the

μ e²v²

sphere, we find that it amounts to

where a

За ,

is the radius of the sphere. If m is the mass of

the sphere, the kinetic energy in the sphere is

m v² ; in addition to that we have the energy

outside the sphere, which as we have seen is !

μ e²v²
; so that the whole kinetic energy of the 3a

2με

system is

m+

+(

3 α ) v², or the energy is

the same as if the mass of the sphere were

2μ ρ

m+

instead of m. Thus, in consequence of

3a

22

ELECTRICITY AND MATTER

the electric charge, the mass of the sphere is

2μ e²

measured by 3a

This is a very important re-

sult, since it shows that part of the mass of a charged sphere is due to its charge. I shall later on have to bring before you considerations which

show that it is not impossible that the whole mass of a body may arise in the way.
Before passing on to this point, however, I should like to illustrate the increase which takes

place in the mass of the sphere by some analogies drawn from other branches of physics. The first of these is the case of a sphere moving through a

frictionless liquid. When the sphere moves it sets the fluid around it moving with a velocity proportioned to its own, so that to move the sphere we

we have not merely to move the substance of the sphere itself, but also the liquid around it ; the consequence of this is, that the sphere behaves as

if its mass were increased by that of a certain volume of the liquid . This volume, as was shown by Green in 1833, is half the volume of the sphere. In the case of a cylinder moving at right angles to its length, its mass is increased by the mass of an equal volume of the liquid. In the case of an elongated body like a cylinder, the amount by

LINES OF FORCE

23

which the mass is increased depends upon the direction in which the body is moving, being much smaller when the body moves point foremost than when moving sideways. The mass of such a body depends on the direction in which it is moving.
Let us, however, return to the moving electrified sphere. We have seen that in consequence of

its charge its mass is increased by 2 ; thus, if it 3a
is moving with the velocity v, the momentum is
not me, but (m + 2 ) v The additional mo-

mentum 2με v is not in the sphere, but in the space 3a
surrounding the sphere. There is in this space ordinary mechanical momentum, whose resultant is 2μ e²
v and whose direction is parallel to the di3a
rection of motion of the sphere. It is important to bear in mind that this momentum is not in any
way different from ordinary mechanical momentum and can be given up to or taken from the momentum of moving bodies. I want to bring the existence of this momentum before you as vividly and forcibly as I can, because the recognition of it makes the behavior of the electric field

i

14

ELECTRICITY AND MATTER

by the lines of force above a line in this part of the field will be equal to the upward pressure exerted by those below it. For a line of force near the edge of the plate, however, the pressure of the lines of force below will exceed the pressure from those above, and the line of force will bulge out until its curvature and tension counteract the squeeze from inside ; this bulging is very plainly shown in Fig. 4.
So far our use of the lines of force has been
descriptive rather than metrical ; it is, however, easy to develop the method so as to make it metrical. We can do this by introducing the idea of tubes of force. If through the boundary of any small closed curve in the electric field we draw the lines of force, these lines will form a tubular surface, and if we follow the lines back to the positively electrified surface from which they start and forward on to the negatively electrified surface on which they end, we can prove that the positive charge enclosed by the tube at its origin is equal to the negative charge enclosed by it at its end. By properly choosing the area of the small curve through which we draw the lines of force, we may arrange that the charge enclosed by the tube is equal to the unit charge. Let us call

LINES OF FORCE

15

such a tube a Faraday tube- then each unit of positive electricity in the field may be regarded as the origin and each unit of negative electricity as the termination of a Faraday tube. We regard these Faraday tubes as having direction, their direction being the same as that of the electric force, so that the positive direction is from the positive to the negative end of the tube. If we draw any closed surface then the difference between the number of Faraday tubes which pass out of the surface and those which pass in will be equal to the algebraic sum of the charges inside the surface ; this sum is what Maxwell called the electric displacement through the surface. What Maxwell called the electric displacement in any direction at a point is the number of Faraday tubes which pass through a unit area through the point drawn at right angles to that direction, the number being reckoned algebraically ; i.e. , the lines which pass through in one direction being taken as positive, while those which pass through in the opposite direction are taken as negative, and the number passing through the area is the difference between the number passing through positively and the number passing through negatively.
For my own part, I have found the conception

16

ELECTRICITY AND MATTER

of Faraday tubes to lend itself much more readily to the formation of a mental picture of the processes going on in the electric field than that of electric displacement, and have for many years abandoned the latter method .
Maxwell took up the question of the tensions and pressures in the lines of force in the electric field , and carried the problem one step further than Faraday. By calculating the amount of these tensions he showed that the mechanical effects in the electrostatic field could be explained by supposing that each Faraday tube force exerted a tension equal to R, R being the intensity of the electric force, and that, in addition to this tension, there was in the medium through which the tubes pass a hydrostatic pressure equal to NR, N being the density of the Faraday tubes ; i.e., the number passing through a unit area drawn at right angles to the electric force. If we consider the effect of these tensions and pressure on a unit volume of the medium in the electric field, we see that they are equivalent to a tension NR along the direction of the electric force and an equal pressure in all directions at right angles to that force.

LINES OF FORCE

17

Moving Faraday Tubes

Hitherto we have supposed the Faraday tubes

to be at rest, let us now proceed to the study of the effects produced by the motion of those tubes. Let us begin with the consideration of a very simple case that of two parallel plates, A and B, charged, one with

positive the other with negative

electricity, and suppose that after

being charged the plates are con- A

B

+

nected by a conducting wire, EFG. PQ

This wire will pass through some

of the outlying tubes ; these tubes, when in a conductor, contract to molecular dimensions and the repul-

sion they previously exerted on

neighboring tubes will therefore

disappear. Consider the effect of this on a tube PQ between the

FIG. 5.

plates ; PQ was originally in equilibrium under

its own tension, and the repulsion exerted by the neighboring tubes. The repulsions due to those

cut by E F G have now, however, disappeared so that PQ will no longer be in equilibrium, but will be pushed towards EFG. Thus, more and more tubes will be pushed into EFG, and we shall

18

ELECTRICITY AND MATTER

have a movement of the whole set of tubes between the plates toward EFG. Thus, while the discharge of the plates is going on, the tubes between the plates are moving at right angles to themselves. What physical effect accompanies this movement of the tubes ? The result of con-

necting the plates by EFG is to produce a current of electricity flowing from the positively charged plate through EFG to the negatively charged plate ; this is, as we know, accompanied by a magnetic force between the plates . This magnetic force is at right angles to the plane of the paper and equal to 4 times the intensity of the current in the plate, or, if σ is the density of the charge of electricity on the plates and v the velocity with which the charge moves, the magnetic force is equal to 4πσv.
Here we have two phenomena which do not take place in the steady electrostatic field , one the movement of the Faraday tubes, the other the existence of a magnetic force ; this suggests that there is a connection between the two, and that motion of the Faraday tubes is accompanied by the production of magnetic force. I have followed up the consequences of this supposition and have shown that, if the connection between the

LINES OF FORCE

19

magnetic force and the moving tubes is that given below, this view will account for Ampère's laws connecting current and magnetic force, and for Faraday's law of the induction of currents. Maxwell's great contribution to electrical theory, that variation in the electric displacement in a dielectric produces magnetic force, follows at once from this view. For, since the electric displacement is measured by the density of the Faraday tubes, if the electric displacement at any place changes, Faraday tubes must move up to or away from the place, and motion of Faraday tubes, by hypoth esis, implies magnetic force.
The law connecting magnetic force with the motion of the Faraday tubes is as follows : A Faraday tube moving with velocity v at a point P, produces at P a magnetic force whose magnitude is 47 v sin 0, the direction of the magnetic force being at right angles to the Faraday tube, and also to its direction of motion ; 0 is the angle between the Faraday tube and the direction in which it is moving. We see that it is only the motion of a tube at right angles to itself which produces magnetic force ; no such force is produced by the gliding of a tube along its length.

20

ELECTRICITY AND MATTER

Motion of a Charged Sphere
We shall apply these results to a very simple and important case the steady motion of a charged sphere. If the velocity of the sphere is small compared with that of light then the Faraday tubes will , as when the sphere is at rest, be uniformly distributed and radial in direction. They will be carried along with the sphere. If e is the charge on the sphere, O its centre, the
e density of the Faraday tubes at P is 4п OPz; so that if v is the velocity of the sphere, ◊ the
P

o

FIG. 6.

angle between OP and the direction of motion of

the sphere, then, according to the above rule, the

ev sin

magnetic force at P will be

" the direc22

tion of the force will be at right angles to OP,

and at right angles to the direction of motion of the sphere ; the lines of magnetic force will thus

LINES OF FORCE

21

be circles, having their centres on the path of the centre of the sphere and their planes at right

angles to this path. Thus, a moving charge of

electricity will be accompanied by a magnetic field. The existence of a magnetic field implies energy ; we know that in a unit volume of the

field at a place where the magnetic force is H

there

are

µH2 units

of

energy,

where

μ is the

8πT

magnetic permeability of the medium. In the

case of the moving sphere the energy per unit

volume at P is μe²v³ sin² 0 Taking the sum of 8пT OP

this energy for all parts of the field outside the

μ e²v2

sphere, we find that it amounts to

where a

За

is the radius of the sphere. If m is the mass of

the sphere, the kinetic energy in the sphere is

mv² ; in addition to that we have the energy

outside the sphere, which as we have seen is

μ e²v² ; so that the whole kinetic energy of the
3a
e system is &
(m +2342α ) v , or the energy is

the same as if the mass of the sphere were

2μ e²

m+

instead of m. Thus, in consequence of

3a

22

ELECTRICITY AND MATTER

the electric charge, the mass of the sphere is

2μ e²

measured by

This is a very important re-

3a

sult, since it shows that part of the mass of a charged sphere is due to its charge. I shall later on have to bring before you considerations which show that it is not impossible that the whole mass of a body may arise in the way.
Before passing on to this point, however, I should like to illustrate the increase which takes

place in the mass of the sphere by some analogies drawn from other branches of physics. The first of these is the case of a sphere moving through a frictionless liquid . When the sphere moves it sets the fluid around it moving with a velocity proportioned to its own, so that to move the sphere we we have not merely to move the substance of the sphere itself, but also the liquid around it ; the consequence of this is, that the sphere behaves as if its mass were increased by that of a certain volume of the liquid . This volume, as was shown by Green in 1833, is half the volume of the sphere. In the case of a cylinder moving at right angles to its length, its mass is increased by the mass of an equal volume of the liquid . In the case of an elongated body like a cylinder, the amount by

LINES OF FORCE

23

which the mass is increased depends upon the direction in which the body is moving, being much smaller when the body moves point foremost than when moving sideways. The mass of such a body depends on the direction in which it is moving.
Let us, however, return to the moving electrified sphere. We have seen that in consequence of

its charge its mass is increased by 2 ; thus, if it За
is moving with the velocity v, the momentum is
not mv, but (m + 2 ) v. The additional mo-

2με

mentum

v is not in the sphere, but in the space

3a

surrounding the sphere. There is in this space ordinary mechanical momentum, whose resultant is

2μ e² v and whose direction is parallel to the di-
3a

rection of motion of the sphere. It is important to bear in mind that this momentum is not in any

way different from ordinary mechanical momen-

tum and can be given up to or taken from the momentum of moving bodies. I want to bring the existence of this momentum before you as

vividly and forcibly as I can, because the recognition of it makes the behavior of the electric field

24 .

ELECTRICITY AND MATTER

entirely analogous to that of a mechanical system. To take an example, according to Newton's Third Law of Motion, Action and Reaction are equal and opposite , so that the momentum in any direction of any self-contained system is invariable. Now, in the case of many electrical systems there are apparant violations of this principle ; thus, take the case of a charged body at rest struck by an electric pulse, the charged body when exposed to the electric force in the pulse acquires velocity and momentum, so that when the pulse has passed over it, its momentum is not what it was originally. Thus, if we confine our attention to the momentum in the charged body, i.e., if we suppose that momentum is necessarily confined to what we consider ordinary matter, there has been a violation of the Third Law of Motion, for the only momentum recognized on this restricted view has been changed. The phenomenon is, however, brought into accordance with this law if we recognize the existence of the momentum in the electric field ; for, on this view, before the pulse reached the charged body there was momentum in the pulse, but none in the body ; after the pulse passed over the body there was some momentum in the body and a smaller amount in the pulse,

LINES OF FORCE

25

the loss of momentum in the pulse being equal to the gain of momentum by the body.
We now proceed to consider this momentum more in detail. I have in my " Recent Researches on Electricity and Magnetism " calculated the amount of momentum at any point in the electric field , and have shown that if N is the number of Faraday tubes passing through a unit area drawn at right angles to their direction, B the magnetic induction, the angle between the induction and the Faraday tubes, then the momentum per unit volume is equal to NB sin 0, the direction of the momentum being at right angles to the magnetic induction and also to the Faraday tubes. Many of you will notice that the momentum is parallel to what is known as Poynting's vector-the vector whose direction gives the direction in which energy is flowing through the field.

Moment of Momentum Due to an Electrified Point and a Magnetic Pole
To familiarize ourselves with this distribution of momentum let us consider some simple cases in detail. Let us begin with the simplest, that of an electrified point and a magnetic pole; let A , Fig. 7, be the point, B the pole. Then, since the momen-

26

ELECTRICITY AND MATTER

tum at any point P is at right angles to A P, the direction of the Faraday tubes and also to BP, the magnetic induction , we see that the momentum will be perpendicular to the plane A B P ; thus, if we draw a series of lines such that their direc-

tion at any point coincides with the direction of

the momentum at that point, these lines will form a series of circles whose planes are perpendicular

to the line A B, and whose centres lie along that line. This distribution of momentum, as far as direction

goes, is that possessed by a top spinning around A B. Let us now find what this distribution of momentum

throughout the field is equivalent to. It is evident that the resultant

FIG. 7.

momentum in any direction is zero, but since the system is spinning

round A B, the direction of rotation being everywhere the same, there will be a finite moment of momentum round A B. Calculating the value of this from the expression for the momentum given above, we obtain the very simple expression em as the value of the moment of momentum about A B, e being the charge on the point and m the strength of the pole. By means of this

LINES OF FORCE

27

expression we can at once find the moment of momentum of any distribution of electrified points

and magnetic poles.

To return to the system of the point and pole, this conception of the momentum of the system leads directly to the evaluation of the force acting on a moving electric charge or a moving magnetic pole. For suppose that in the time & t the electrified point were to move from A to A', the moment of momentum is still em, but its axis is along A' B instead of A B. The moment of momentum of the field

has thus changed, but the whole moment of momentum of the system comprising

point, pole, and field must be constant, so

that the change in the moment of momen tum of the field must be accompanied

by an equal and opposite change in the

B

moment of momentum of the pole and FIG. 8 .

point. The momentum gained by the point must be equal and opposite to that gained by the pole, since the whole momentum is zero. If 0 is

the angle A B A', the change in the moment of
momentum is em sin 0, with an axis at right
angles to AB in the plane of the paper. Let -
SI be the change in the momentum of A, -

28

ELECTRICITY AND MATTER

SI that of B, then 8 I and - 8 I must be

equivalent to a couple whose axis is at right angles to AB in the plane of the paper, and whose moment is e m sin 0. Thus & I must be at right angles to the plane of the paper and

em A A' sin o SI.AB- em sin 0 =
AB

Where

is the angle BA A'. If v is the

velocity of A, A A' = vdt and we get SI= em v sin & St AB2

This change in the momentum may be sup

posed due to the action of a force F perpen-

dicular to the plane of the paper, F being the

81

•

rate of increase of the momentum, or

We thus

St

em v sin o

get F

A B2 ; or the point is acted on by a

force equal to e multiplied by the component of the magnetic force at right angles to the direction of motion. The direction of the force acting on

the point is at right angles to its velocity and also to the magnetic force. There is an equal

and opposite force acting on the magnetic pole. The value we have found for F is the ordinary

expression for the mechanical force acting on a

moving charged particle in a magnetic field ; it

LINES OF FORCE

29

may be written as ev Hsin o, where H's is the strength of the magnetic field. The force acting on unit charge is therefore v Hsin o. This mechanical force may be thus regarded as arising from an electric force v H sin o, and we may express the result by saying that when a charged body is moving in a magnetic field an electric force v Hsin is produced . This force is the well-known electromotive force of induction due to motion in a magnetic field.
The forces called into play are due to the relative motion of the pole and point ; if these are moving with the same velocity, the line joining them will not alter in direction, the moment of momentum of the system will remain unchanged and there will not be any forces acting either on the pole or the point.
The distribution of momentum in the system of pole and point is similar in some respects to that in a top spinning about the line A B. We can illustrate the forces acting on a moving elec trified body by the behavior of such a top. Thus, let Fig. 9 represent a balanced gyroscope spinning about the axis A B, let the ball at A represent the electrified point, that at B the magnetic pole. Suppose the instrument is spinning with AB

30

ELECTRICITY AND MATTER

horizontal, then if with a vertical rod I push against AB horizontally, the point A will not merely move horizontally forward in the direction in which it is pushed, but will also move vertically upward or downward, just as a charged

A B

FIG. 9 .
point would do if pushed forward in the same way, and if it were acted upon by a magnetic pole at B.
Maxwell's Vector Potential There is a very close connection between the momentum arising from an electrified point and a

LINES OF FORCE

31

magnetic system, and the Vector Potential of that

system, a quantity which plays a very large part in Maxwell's Theory of Electricity. From the expression we have given for the moment of momentum due to a charged point and a magnetic pole, we can at once find that due to a charge e of electricity at a point P, and a little magnet A B ; let the negative pole of this magnet be at A , the positive at B, and let m be the strength of either pole. A simple calculation shows that in this case the axis of the resultant moment of momen-

tum is in the plane P A B at right angles to P O,

O being the middle point of A B, and that the

magnitude of the moment of momentum is equal

sin o

to e.m. A B

where

OP2

is the angle AB

makes with OP. This moment of momentum is

equivalent in direction and magnitude to that due

sin o

to a momentum e. m. A B

at P directed

OP2

at right angles to the plane PA B, and another

momentum equal in magnitude and opposite in
sin direction at O. The vector m A B Op at P
OP2

at right angles to the plane PA B is the vector

called by Maxwell the Vector Potential at P due to the Magnet.

32

ELECTRICITY AND MATTER

Calling this Vector Potential I, we see that the momentum due to the charge and the magnet is equivalent to a momentum e I at P and a momentume I at the magnet.
We may evidently extend this to any complex system of magnets, so that if I is the Vector Potential at P of this system, the momentum in the field is equivalent to a momentum e I at P together with momenta at each of the magnets equal to
(Vector Potential at P due to that magnet).
If the magnetic field arises entirely from electric currents instead of from permanent magnets, the momentum of a system consisting of an electrified point and the currents will differ in some of its features from the momentum when the magnetic field is due to permanent magnets. In the latter case, as we have seen, there is a moment of momentum, but no resultant momentum. When, however, the magnetic field is entirely due to electric currents, it is easy to show that there is a resultant momentum, but that the moment of momentum about any line passing through the electrified particle vanishes. A simple calculation shows that the whole momentum in the field is equivalent to a momentum e I at the electrified

LINES OF FORCE 33
point I being the Vector Potential at P due to the currents .
Thus, whether the magnetic field is due to permanent magnets or to electric currents or partly to one and partly to the other, the momentum when an electrified point is placed in the field at P is equivalent to a momentum e I at P where I is the Vector Potential at P. If the magnetic field is entirely due to currents this is a complete representation of the momentum in the field ; if the magnetic field is partly due to magnets we have in addition to this momentum at P other momenta at these magnets ; the magnitude of the
momentum at any particular magnet is ―e times the Vector Potential at P due to that magnet.
The well-known expressions for the electromotive forces due to Electro-magnetic Induction follow at once from this result. For, from the Third Law of Motion, the momentum of any selfcontained system must be constant. Now the
momentum consists of ( 1 ) the momentum in the field ; ( 2 ) the momentum of the electrified point, and ( 3) the momenta of the magnets or circuits carrying the currents. Since ( 1 ) is equivalent to a momentum e I at the electrified particle, we see that changes in the momentum of the field must

34

ELECTRICITY AND MATTER

be accompanied by changes in the momentum of the particle. Let Mbe the mass of the electrified

particle, u, v, w the components parallel to the axes of x, y, z of its velocity, F, G, H, the components parallel to these axes of the Vector Po-

tential at P, then the momentum of the field is equivalent to momenta e F, e G, e H at P parallel to the axes of x, y, z ; and the momentum of the

charged point at P has for components Mu, Mo, Mw. As the momentum remains constant, Mu + e Fis constant, hence if Su and SF are simultaneous changes in u and F,

MSu + e & F = 0 ;

du or m
dt

dF e
dt

From this equation we see that the point with the

charge behaves as if it were acted upon by a me

chanical force parallel to the axis of x and equal to

dF

dF

e

i.e., by an electric force equal to

dt'

dt.

In a similar way we see that there are electric

dG dH

forces -

-

parallel to y and z respec-

dt ' at'

tively. These are the well-known expressions of the forces due to electro-magnetic induction, and we see that they are a direct consequence of the

LINES OF FORCE

35

principle that action and reaction are equal and opposite.
Readers of Faraday's Experimental Researches will remember that he is constantly referring to what he called the " Electrotonic State " ; thus he regarded a wire traversed by an electric current as being in the Electrotonic State when in a magnetic field. No effects due to this state can be detected as long as the field remains constant ; it is when it is changing that it is operative. This Electrotonic State of Faraday is just the momentum existing in the field.

CHAPTER II ELECTRICAL AND BOUND MASS .

I WISH in this chapter to consider the connec tion between the momentum in the electric field

and the Faraday tubes, by which, as I showed in

the last lecture, we can picture to ourselves the state of such a field. Let us begin by considering

the case of the moving charged sphere. The lines of electric force are radial ; those of magnetic force

are circles having for a common axis the line of mo-

tion of the centre of the sphere ; the momentum

P

at a point P is at right

angles to each of these

directions and so is at

right angles to OP in

FIG. 10.

the plane containing P and the line of motion of the centre of the

sphere. If the number of Faraday tubes passing through a unit area at P placed at right angles to O P is N, the magnetic induction at P is, if u is the magnetic permeability of the medium

ELECTRICAL MASS

37

surrounding the sphere, 4π μ Nv sin 0, v being the velocity of the sphere and the angle OP makes with the direction of motion of the sphere. By the rule given on page 25 the momentum in unit volume of the medium at P is N × 4π μµ Nv sin 0, or 4μ Nº v sin 0, and is in the direction of the component of the velocity of the Faraday tubes at right angles to their length. Now this is exactly the momentum which would be produced if the tubes were to carry with them, when they move at right angles to their length, a mass of the surrounding medium equal to 4π μ N2 per unit volume, the tubes pos-
sessing no mass themselves and not carrying any of the medium with them when they glide through it parallel to their own length. We suppose in fact the tubes to behave very much as long and narrow cylinders behave when moving through water ; these if moving endwise, i.e., parallel to their length, carry very little water along with them, while when they move sideways, i.e., at right angles to their axis, each unit length of the tube carries with it a finite mass of water. When the length of the cylinder is very great compared with its breadth, the mass of water carried by it when moving endwise may be neglected in com-

38

ELECTRICITY AND MATTER

parison with that carried by it when moving sideways ; if the tube had no mass beyond that which it possesses in virtue of the water it displaces, it would have mass for sideways but none for endwise motion.

We shall call the mass 4π μ N2 carried by the tubes in unit volume the mass of the bound ether.

It is a very suggestive fact that the electrostatic

energy E in unit volume is proportional to M the

mass of the bound ether in that volume. This can

2π N2

easily be proved as follows : E =

where

K'

K is the specific inductive capacity of the
medium ; while M = 4π μµ N², thus, M
E= μα

butx V where V is the velocity with which

light travels through the medium, hence

E = 1 MV² ;

thus E is equal to the kinetic energy possessed by the bound mass when moving with the velocity of light.
The mass of the bound ether in unit volume is

4π μ N² where N is the number of Faraday tubes ; thus, the amount of bound mass per unit length of

ELECTRICAL MASS

39

each Faraday tube is 4π μN. We have seen that this is proportional to the tension in each tube, so that we may regard the Faraday tubes as tightly stretched strings of variable mass and tension ; the
tension being, however, always proportional to the mass per unit length of the string.
Since the mass of ether imprisoned by a Faraday tube is proportional to Nthe number of Faraday tubes in unit volume, we see that the mass and momentum of a Faraday tube depend not merely upon the configuration and velocity of the tube under consideration, but also upon the number and velocity of the Faraday tubes in its neighborhood. We have many analogies to this in the case of dynamical systems ; thus, in the case of a number of cylinders with their axes parallel, moving about in an incompressible liquid, the momentum of any cylinder depends upon the positions and velocities of the cylinders in its neighborhood. The following hydro-dynamical system is one by which we may illustrate the fact that the bound mass is proportional to the square of the number of Faraday tubes per unit volume.
Suppose we have a cylindrical vortex column of strength m placed in a mass of liquid whose velocity, if not disturbed by the vortex column, would

40

ELECTRICITY AND MATTER

be constant both in magnitude and direction, and at right angles to the axis of the vortex column. The lines of flow in such a case are represented in Fig. 11 , where A is the section of the vortex

FIG. 11 .
column whose axis is supposed to be at right angles to the plane of the paper. We see that some of these lines in the neighborhood of the column are closed curves. Since the liquid does not cross the lines of flow, the liquid inside a closed curve will always remain in the neighborhood of the column and will move with it. Thus, the column will imprison a mass of liquid equal to that en closed by the largest of the closed lines of flow. If m is the strength of the vortex column and a the velocity of the undisturbed flow of the liquid, we can easily show that the mass of liquid imprisoned

ELECTRICAL MASS
m² by the column is proportional to

41 Thus, if

we take m as proportional to the number of Faraday tubes in unit area, the system illustrates the connection between the bound mass and the strength of the electric field.

Effective of Velocity on the Bound Mass
I will now consider another consequence of the idea that the mass of a charged particle arises from the mass of ether bound by the Faraday tube associated with the charge. These tubes, when they move at right angles to their length, carry with them an appreciable portion of the ether through which they move, while when they move parallel to their length, they glide through the fluid without setting it in motion. Let us consider how a long, narrow cylinder, shaped like a Faraday tube, would behave when moving through a liquid.
Such a body, if free to twist in any direction, will not, as you might expect at first sight, move point foremost, but will, on the contrary, set itself broadside to the direction of motion, setting itself so as to carry with it as much of the fluid through which it is moving as possible. Many illustra tions of this principle could be given, one very

42

ELECTRICITY AND MATTER

familiar one is that falling leaves do not fall edge first, but flutter down with their planes more or less horizontal.
If we apply this principle to the charged sphere, we see that the Faraday tubes attached to the sphere will tend to set themselves at right angles to the direction of motion of the sphere, so that if this principle were the only thing to be considered all the Faraday tubes would be forced up into the equatorial plane, i.e. , the plane at right angles to the direction of motion of the sphere, for in this position they would all be moving at right angles to their lengths. We must remember, however, that the Faraday tubes repel each other, so that if they were crowded into the equatorial region the pressure there would be greater than that near the pole. This would tend to thrust the Faraday tubes back into the position in which they are equally distributed all over the sphere. The actual distribution of the Faraday tubes is a compromise between these extremes. They are not all crowded into the equatorial plane, neither are they equally distributed, for they are more in the equatorial regions than in the others ; the excess of the density of the tubes in these regions increasing with the speed with which the charge is moving.

ELECTRICAL MASS

43

When a Faraday tube is in the equatorial region it imprisons more of the ether than when it is near the poles, so that the displacement of the Faraday tubes from the pole to the equator will increase the amount of ether imprisoned by the tubes, and therefore the mass of the body.
It has been shown (see Heaviside, Phil. Mag., April, 1889, " Recent Researches," p. 19) that the effect of the motion of the sphere is to displace each Faraday tube toward the equatorial plane, i.e., the plane through the centre of the sphere at right angles to its direction of motion, in such a way that the projection of the tube on this plane remains the same as for the uniform distribution of tubes, but that the distance of every point in the tube from the equatorial plane is reduced in the proportion of ✓ V - v² to V, where Vis the velocity of light through the medium and v the velocity of the charged body.
From this result we see that it is only when the velocity of the charged body is comparable with the velocity of light that the change in distribution of the Faraday tubes due to the motion of the body becomes appreciable.
In " Recent Researches on Electricity and Magnetism," p. 21 , I calculated the momentum I, in the

44

ELECTRICITY AND MATTER

space surrounding a sphere of radius a, having its

centre at the moving charged body, and showed that the value of I is given by the following expression :

e²

V2

V

I = 2a ( Vº — v²) ¹

{ 0 (1-4 v2) + sin 20

V2 ( 1 + + v2 cos 2200)) } ;

• (1)

where as before v and V are respectively the ve-
locities of the particle and the velocity of light,
and 0 is given by the equation
v sin =
V

The mass of the sphere is increased in conse-
I quence of the charge by , and thus we see from

equation ( 1 ) that for velocities of the charged body comparable with that of light the mass of the body will increase with the velocity. It is evident from equation (1 ) that to detect the influence of velocity on mass we must use exceedingly small particles moving with very high velocities. Now, particles having masses far smaller than the mass of any known atom or molecule are shot out from radium with velocities approaching in some cases to that of light, and the ratio of the electric charge to the mass for parti

ELECTRICAL MASS

45

cles of this kind has lately been made the subject of a very interesting investigation by Kaufmann , with the results shown in the following table ; the first column contains the values of the velocities

of the particle expressed in centimetres per second, the second column the value of the fraction e where e is the charge and m the mass of the m

particle : VX 10-10 2.83 2.72 2.59 2.48 2.36

e X 10-7
m .62
.77
.975
1.17
1.31

We see from these values that the value of dim
minishes as the velocity increases, indicating, if we suppose the charge to remain constant, that the mass increases with the velocity. Kaufmann's results give us the means of comparing the part of the mass due to the electric charge with the part independent of the electrification ; the second part of the mass is independent of the velocity. If then we find that the mass varies appreciably
with the velocity, we infer that the part of the

46

ELECTRICITY AND MATTER

mass due to the charge must be appreciable in comparison with that independent of it. To calculate the effect of velocity on the mass of an electrified system we must make some assumption as to the nature of the system, for the effect on a charged sphere for example is not quite the same as that on a charged ellipsoid ; but having made the assumption and calculated the theoretical effect of the velocity on the mass, it is easy to deduce the ratio of the part of the mass independent of the charge to that part which at any velocity depends upon the charge. Suppose that the part of the mass due to electrification is at a velocity v equal to m.f(v) where f(v) is a known function of v, then if M, M, are the observed masses at the velocities v and v¹ respectively and M the part of the mass independent of charge, then
M₁ = M + m。f(v), M¸¹ = M + m。f(v¹), two equations from which M and m, can be determined . Kaufmann, on the assumption that the charged body behaved like a metal sphere, the distribution of the lines of force of which
when moving has been determined by G. F. C. Searle, came to the conclusion that when the parti-
99 cle was moving slowly the " electrical mass was

ELECTRICAL MASS

47

about one-fourth of the whole mass. He was care. ful to point out that this fraction depends upon the assumption we make as to the nature of the moving body, as, for example, whether it is spheri cal or ellipsoidal, insulating or conducting ; and that with other assumptions his experiments might show that the whole mass was electrical, which he evidently regarded as the most probable result.
In the present state of our knowledge of the constitution of matter, I do not think anything is gained by attributing to the small negatively charged bodies shot out by radium and other

bodies the property of metallic conductivity, and I prefer the simpler assumption that the distribution of the lines of force round these particles is the same as that of the lines due to a charged point, provided we confine our attention to the field outside a small sphere of radius a having its centre at the charged point ; on this supposition the part of the mass due to the charge is the value
I of in equation (1 ) on page 44. I have calcu
v
lated from this expression the ratio of the masses of the rapidly moving particles given out by radium to the mass of the same particles when at rest, or moving slowly, on the assumption that the

48

ELECTRICITY AND MATTER

whole ofthe mass is due to the charge and have compared these results with the values of the same ratio as determined by Kaufmann's experiments. These results are given in Table ( II) , the first column of which contains the values of v, the veloci ties of the particles ; the second p, the number of times the mass of a particle moving with this velocity exceeds the mass of the same particle when at rest, determined by equation ( 1 ) ; the third column p¹ , the value of this quantity found by Kaufmann in his experiments.

TABLE II.

ст v X 10-10
sec
2.85
2.72
2.59
2.48
2.36

ρ 3.1 2.42 2.0 1.66 1.5

3.09 2.43 2.04 1.83 1.65

These results support the view that the whole mass of these electrified particles arises from their charge.
We have seen that if we regard the Faraday tubes associated with these moving particles as being those due to a moving point charge, and

ELECTRICAL MASS

49

confine our attention to the part of the field which is outside a sphere of radius a concentric with the charge, the mass m due to the charge e on the particle is, when the particle is moving slowly,
2 με given by the equation m = 3 a

In a subsequent lecture I will explain how the values of me and e have been determined ; the result of these determinations is that m
= 10 and e = 1.2 × 10-20 in C. G. S. elece
trostatic units. Substituting these values in the expression for m we find that a is about 5 X 10-14 cm, a length very small in comparison with the value 10-8 cm , which is usually taken as a good approximation to the dimensions of a molecule.
We have regarded the mass in this case as due to the mass of ether carried along by the Faraday tubes associated with the charge. As these tubes stretch out to an infinite distance, the mass of the particle is as it were diffused through space, and has no definite limit. In consequence, however, of the very small size of the particle and the fact that the mass of ether carried by the tubes (being proportional to the square of the density of the Faraday tubes) varies inversely as the fourth

50

ELECTRICITY AND MATTER

power of the distance from the particle, we find by a simple calculation that all but the most insig nificant fraction of mass is confined to a distance from the particle which is very small indeed compared with the dimensions ordinarily ascribed to atoms.
In any system containing electrified bodies a part of the mass of the system will consist of the mass of the ether carried along by the Faraday tubes associated with the electrification. Now one view of the constitution of matter—a view, I hope to discuss in a later lecture- is that the atoms of the various elements are collections of
positive and negative charges held together mainly by their electric attractions, and, moreover, that the negatively electrified particles in the atom (corpuscles I have termed them) are identical with those small negatively electrified particles whose properties we have been discussing. On this view of the constitution of matter, part of the mass of any body would be the mass of the ether dragged along by the Faraday tubes stretching across the atom between the positively and negatively electrified constituents . The view I wish to put before you is that it is not merely a part of the mass of a body which arises in this

ELECTRICAL MASS

51

way, but that the whole mass of any body is just the mass of ether surrounding the body which is carried along by the Faraday tubes associated with the atoms of the body. In fact, that all mass is mass of the ether, all momentum, momentum of the ether, and all kinetic energy, kinetic energy of the ether. This view, it should be said, requires the density of the ether to be immensely greater than that of any known substance.
It might be objected that since the mass has to be carried along by the Faraday tubes and since the disposition of these depends upon the relative position of the electrified bodies, the mass of a collection of a number of positively and negatively electrified bodies would be constantly changing with the positions of these bodies, and thus that mass instead of being, as observation and experiment have shown, constant to a very high degree of approximation , should vary with changes in the physical or chemical state of the body.
These objections do not, however, apply to such a case as that contemplated in the preceding theory, where the dimensions of one set of the electrified bodies the negative ones are excessively small in comparison with the distances separating the various members of the system of electrified bodies.