# A History of the Theories of Aether and Electricity/Chapter 10

Chapter X.
The Followers of Maxwell.
The most notable imperfection in the electromagnetic theory of light, as presented in Maxwell's original memoirs, was the absence of any explanation of reflexion and refraction. Before the publication of Maxwell's Treatise, however, a method of supplying the omission was indicated by Helmholtz. The principles on which the explanation depends are that tho normal component of the electric displacement D, the tangential components of the electric force E, and the magnetic vector B or H, are to be continuous across the interface at which the reflexion takes place; the optical difference between the contiguous bodies being represented by a difference in their dielectric constants, and the electric vector being assumed to be at right angles to the plane of polarization. The analysis required is a mere transcription of MacCullagh's theory of reflexion, if the derivate of MacCullagh's displacement e with respect to the time be interpreted as the magnetic force; μ curl e as the electric force, and curl e as the electric displacement. The mathematical details of the solution were not given by Helmholtz himself, but were supplied a few years later in the inaugural dissertation of H. A. Lorentz.
In the years immediately following the publication of Maxwell's Treatise, a certain amount of evidence in favour of ​his theory was furnished by experiment. That an electric field is closely concerned with the propagation of light was demonstrated in 1875, when John Kerr showed that dielectrics subjected to powerful electrostatic force acquire the property of double refraction, their optical behaviour being similar to that of uniaxal crystals whose axes are directed along the lines of force.
Other researches undertaken at this time had a more direct bearing on the questions at issue between the hypothesis of Maxwell and the older potential theories. In 1875-6 Helmholtz and his pupil Schiller attempted to discriminate between the various doctrines and formulae relative to unclosed circuits by performing a crucial experiment.
It was agreed in all theories that a ring-shaped magnet, which returns into itself so as to have no poles, can exert no ponderomotive force on other magnets or on closed electric currents. Helmholtz had, however, shown in 1873 that according to the potential-theories such a magnet would exert a ponderomotive force on an unclosed current. The matter was tested by suspending a magnetized steel ring by a long fibre in a closed metallic case, near which was placed a terminal of a Holtz machine. No ponderomotive force could be observed when the machine was put in action so as to produce a brush discharge from the terminal: from which it was inferred that the potential-theories do not correctly represent the phenomena, at least when displacement-currents and convection-currents (such as that of the electricity carried by the electrically repelled air from the terminal) are not taken into account.
The researches of Helmholtz and Schiller brought into prominence the question as to the effects produced by the ​translatory motion of electric charges. That the convection of electricity is equivalent to a current had been suggested long before by Faraday. "If," he wrote in 1838, "a ball be electrified positively in the middle of a room and be then moved in any direction, effects will be produced as if a current in the same direction had existed." To decide the matter a new experiment inspired by Helmholtz was performed by H. A. Rowland in 1876. The electrified body in Rowland's disposition was a disk of ebonite, coated with gold leaf and capable of turning rapidly round a vertical axis between two fixed plates of glass, each gilt on one side. The gilt faces of the plates could be earthed, while the ebonite disk received electricity from a point placed near its edge; each coating of the disk thus formed a condenser with the plate nearest to it. An astatic needle was placed above the upper condenser-plate, nearly over the edge of the disk; and when the disk was rotated a magnetic field was found to be produced. This experiment, which has since been repeated under improved conditions by Rowland and Hutchinson, H. Penders, and Eichenwald, shows that the "convection-current" produced by the rotation of a charged disk, when the other ends of the lines of force are on an earthed stationary plate parallel to it, produces the same magnetic field as an ordinary conduction-current flowing in a circuit which coincides with the path of the convection-current. When two disks forming a condenser are rotated together, the magnetic action is the sum of the magnetic actions of cach of the disks separately. It appears, therefore, that electric charges cling to the matter of a conductor and move with it, so far as Rowland's phenomenon is concerned.
The first examination of the matter from the point of view of Maxwell's theory was undertaken by J. J. Thomson, in 1881, If an electrostatically charged body is in motion, the change in ​the location of the charge must produce a continuous alteration of the electric field at any point in the surrounding medium; or, in the language of Maxwell's theory, there must be displacement-currents in the medium. It was to these displacement-currents that Thomson, in his original investigation, attributed the magnetic effects of moving charges. The particular system which he considered was that formed by a charged spherical conductor, moving uniformly in a straight line. It was assumed that the distribution of electricity remains uniform over the surface during the motion, and that the electric field in any position of the sphere is the same as if the sphere were at rest; these assumptions are true so long as quantities of order (v/c)$^2$ are neglected, where v denotes the velocity of the sphere and c the velocity of light.
Thomson's method was to determine the displacement-currents in the space outside the sphere from the known values of the electric field, and then to calculate the vectorpotential due to these displacement-currents by means of the formula
$\mathbf {A} =\textstyle \iiint (\mathbf {S^{\prime }} /r)\ dx^{\prime }\ dy^{\prime }\ dz^{\prime }$.
where S′ denotes the displacement-current at (x′y′z′). The magnetic field was then determined by the equation
$\mathbf {H} =\mathrm {curl} \ \mathbf {A}$.
A defect in this investigation was pointed out by FitzGerald, who, in a short but most valuable note, published a few months afterwards, observed that the displacement-currents of Thomson do not satisfy the circuital condition. This is most simply seen by considering the case in which the system consists of two parallel plates forming a condenser; if one of the plates is fixed, and the other plate is moved towards it, the electric field is annihilated in the space over which the moving plate travels: this destruction of electric displacement constitutes a displacement-current, which, considered alone, is evidently not a closed ​current. The detect, as FitzGerald showed, may be immediately removed by assuming that a moving charge itself is to be counted as a current-element: the total current, thus composed of the displacement-currents and the convection-current, is circuital. Making this correction, FitzGerald found that the magnetic force due to a sphere of charge e moving with velocity v along the axis of z is curl(0, 0, ev/r)—a formula which shows that the displacement-currents have no resultant magnetic effect, since the term ev/r would be obtained from the convection-current alone.
The expressions obtained by Thomson and FitzGerald were correct only to the first order of the small quantity v/c. The effect of including terms of higher order was considered in 1889 by Oliver Heaviside, whose solution may be derived in the following manner:—
Suppose that a charged system is in motion with uniform velocity v parallel to the axis of z; the total current consists of the displacement-current Ė/4πc$^2$ where E denotes the electric force, and the convection-current ρv where ρ denotes the volume-density of electricity. So the equation which connects magnetic force with electric current may be written
$\mathbf {\dot {E}} /c^{2}={\text{curl }}\mathbf {H} -4\pi \rho \mathbf {v}$.
Eliminating E between this and the equation
${\text{curl }}\mathbf {E} =-\mathbf {\dot {H}}$,
and remembering that H is here circuital, we have
$\mathbf {\ddot {H}} /c^{2}-nabla^{2}\mathbf {H} =4\pi {\text{curl}}\rho \mathbf {v}$.
If, therefore, a vector-potential a be defined by the equation
$\mathbf {\ddot {a}} /c^{2}-\nabla ^{2}\mathbf {a} =4\pi \rho \mathbf {v}$,
the magnetic force will be the curl of a; and from the equation for a it is evident that the components a$_x$ and a$_y$ are zero, and that a$_z$, is to be determined from the equation
${\ddot {a}}_{z}/c^{2}-\nabla ^{2}a_{z}=4\pi \rho v$.
​Now, let (x, y, ζ) denote coordinates relative to axes which are parallel to the axes (x, y, z), and which move with the charged bodies; then (a$_z$, is a function of (x, y, ζ) only; so we have
${\frac {\partial }{\partial z}}={\frac {\partial }{\partial \zeta }},\qquad {\text{and}}\qquad {\frac {\partial }{\partial t}}=-v{\frac {\partial }{\partial \zeta }}$;
and the preceding equation is readily seen to be equivalent to
${\frac {\partial ^{2}a_{z}}{\partial x^{2}}}+{\frac {\partial ^{2}a_{z}}{\partial y^{2}}}+{\frac {\partial ^{2}a_{z}}{\partial \zeta _{1}^{2}}}=-4\pi \rho v$,
where ζ$_1$ denotes (1 - v$^2$/c$^2$)$^{-1/2}$ζ. But this is simply Poisson's equation, with ζ$_1$ substituted for z; so the solution may be transcribed from the known solution of Poisson's equation: it is
$a_{z}=\iiint {\frac {\rho ^{\prime }v^{\prime }\ dx^{\prime }\ dy^{\prime }\ d\zeta _{1}^{\prime }}{\{(\zeta _{1}-\zeta _{1}^{\prime })^{2}+(x-x^{\prime })^{2}+(y-y^{\prime })^{2}\}^{\frac {1}{2}}}}$,
the integrations being taken over all the space in which there are moving charges; or
$a_{z}=\iiint {\frac {\rho ^{\prime }v^{\prime }\ dx^{\prime }\ dy^{\prime }\ d\zeta ^{\prime }}{\{(\zeta -\zeta ^{\prime })^{2}+(1-v^{2}/c^{2})(x-x^{\prime })^{2}+(1-v^{2}/c^{2})(y-y^{\prime })^{2}\}^{\frac {1}{2}}}}$.
If the moving system consists of a single charge e at the point ζ = 0, this gives
$a_{z}={\frac {ev}{r(1-v^{2}\sin ^{2}\theta /c^{2})^{\frac {1}{2}}}}$.
where sin$^2$ θ = (x$^2$ + y$^2$)/r$^2$.
It is readily seen that the lines of magnetic force due to the moving point-charge are circles whose centres are on the line of motion, the magnitude of the magnetic force being
${\frac {ev(1-v^{2}/c^{2})\sin \theta }{r^{2}(1-v^{2}\sin ^{2}\theta /c^{2})^{\frac {3}{2}}}}$.
The electric force is radial, its magnitude being
${\frac {e^{2}c(1-v^{2}/c^{2})}{r^{2}(1-v^{2}\sin ^{2}\theta /c^{2})^{\frac {3}{2}}}}$.
The fact that the electric vector due to a moving point-charge is everywhere radial led Heaviside to conclude that the same solution is applicable when the charge is distributed over ​a perfectly conducting sphere whose centre is at the point, the only change being that E and H would now vanish inside the sphere. This inference was subsequently found to be incorrect: a distribution of electric charge on a moving sphere could in fact not be in equilibrium if the electric force were radial, since there would then be nothing to balance the mechanical force exerted ou the moving charge (which is equivalent to a current) by the magnetic field. The moving system which gives rise to the same field as a moving point-charge is not a sphere, but an oblate spheroid whose polar axis (which is in the direction of motion) bears to its equatorial axis the ratio {{Wikimath|(1 - v$^2$/c$^2$)$^{1/2}$:1.
The energy of the field surrounding a charged sphere is greater when the sphere is in motion than when it is at rest. To determine the additional energy quantitatively (retaining only the lowest significant powers of v/c), we have only to integrate, throughout the space outside the sphere, the expression H$^2$/8π, which represents the electrokinetic energy per unit volume: the result is e$^2$v$^2$/3a, where e denotes the charge, v the velocity, and a the radius of the sphere.
It is evident from this result that the work required to be done in order to communicate a given velocity to the sphere is greater when the sphere is charged than when it is uncharged; that is to say, the virtual mass of the sphere is increased by an amount 2e$^2$/3a, owing to the presence of the charge. This may be regarded as arising from the self-induction of the convection-current which is formed when the charge is set in motion. It was suggested by J. Larmor and by W. Wien that the inertia of ordinary ponderable matter may ultimately prove to be of this nature, the atoms being constituted of systems of electrons. ​It may, however, be remarked that this view of the origin of mass is not altogether consistent with the principle that the electron is an indivisible entity. For the so-called self-induction of the spherical electron is really the mutual induction of the convection-currents produced by the elements of electric charge which are distributed over its surface; and the calculation of this quantity presupposes the divisibility of the total charge into elements capable of acting severally in all respects as ordinary electric charges; a property which appears scarcely consistent with the supposed fundamental nature of the electron.
After the first attempt of J. J. Thomson to determine the field produced by a moving electrified sphere, the mathematical development of Maxwell's theory proceeded rapidly. The problems which admit of solution in terms of known functions are naturally those in which the conducting surfaces involved have simple geometrical forms—planes, spheres, and cylinders.
A result which was obtained by Horace Lamb, when investigating electrical motions in a spherical conductor, led to interesting consequences. Lamb found that if a spherical conductor is placed in a rapidly alternating held, the induced currents are almost entirely confined to a superficial layer; and his result was shortly afterwards generalized by Oliver Heaviside, who showed that whatever be the form of a conductor rapidly alternating currents do not penetrate far into its substance. The reason for this may be readily understood: it is virtually an application of the principle that a perfect conductor is impenetrable to magnetic lines of force. No perfect conductor is known to exist; but if the alternations of magnetic force to which a good conductor such as copper is exposed are very ​rapid, the conductor bas not time (so to speak) to display the imperfection of its conductivity, and the magnetic field is therefore unable to extend far below the surface.
The same conclusion may be reached by different reasoning. When the alternations of the current are very rapid, the ohmic resistance ceases to play a dominant part, and the ordinary equations connecting electromotive force, induction, and current are equivalent to the conditions that the currents shall be so distributed as to make the electrokinetic or magnetic energy a minimum. Consider now the case of a single straight wire of circular cross-section. The magnetic energy in the space outside the wire is the same whatever be the distribution of current in the cross-section (so long as it is symmetrical about the centre), since it is the same as if the current were flowing along the central axis, so the condition is that the magnetic energy in the wire shall be a minimum; and this is obviously satisfied when the current is concentrated in the superficial layer, since then the magnetic force is zero in the substance of the wire.
In spite of the advances which were effected by Maxwell and his earliest followers in the theory of electric oscillations, the gulf between the classical electrodynamics and the theory of light was not yet completely bridged. For in all the cases considered in the former science, energy is merely exchanged between one body and another, remaining within the limits of a given system; while in optics the energy travels freely through space, unattached to any material body. The first discovery of a more complete connexion between the two theories was made by FitzGerald, who argued that if the unification which had been indicated by Maxwell is valid, it ought to be possible to generate radiant energy by purely electrical means; and in 1883 he described methods by which this could be done.
FitzGerald's system is what has since become known as the magnetic oscillator: it consists of a small circuit, in which ​the strength of the current is varied according to the simple periodic law. The circuit will be supposed to be a circle of small area S, whose centre is the origin and whose plane is the plane of xy; and the surrounding medium will be supposed to be free aether. The current may be taken to be of strength A cos (2πt/T), so that the moment of the equivalent magnet is SA cos (2πt/T). Now in the older electrodynamics, the vector-potential due to a magnetic molecule of (vector) moment M at the origin is (1/4π) curl (M/r), where r denotes distance from the origin. The vector-potential due to FitzGerald's magnetic oscillator would therefore be (1/4π) curl K, where Kdenotes a vector parallel to the axis of z, and of magnitude (1/r) SA cos (2πt/T). The change which is involved in replacing the assumptions of the older electrodynamics by those of Maxwell's theory is in the present case equivalent to retarding the potential; so that the vector-potential a due to the oscillator is (1/4π) curl K where K is still directed parallel to the axis of z, and is of magnitude
$K={\frac {SA}{r}}\cos {\frac {2\pi }{T}}\left(t-{\frac {r}{c}}\right)$.
The electric force E at any point of space is $-\mathbf {\dot {a}}$, and the magnetic force H is curl a: so that these quantities may be calculated without difficulty. The electric energy per unit volume is E$^2$/8πc$^2$: performing the calculations, it is found that the value of this quantity averaged over a period of the oscillation and also averaged over the surface of a sphere of radius r is
${\frac {\pi A^{2}S^{2}}{6c^{2}r^{4}T^{2}}}\left(1+{\frac {4\pi ^{2}r^{2}}{c^{2}T}}\right)$.
The part of this which is radiated is evidently that which is proportional to the inverse square of the distance, so the ​average value of the radiant energy of electric type at distance r from the oscillator is $2\pi ^{3}A^{2}S^{2}/3c^{4}r^{2}T^{4}$ per unit volume. The radiant energy of magnetic type may be calculated in a similar way, and is found to have the same value; so the total radiant energy at distance r is $4\pi ^{3}A^{2}S^{2}/3c^{4}r^{2}T^{4}$ per unit volume; and therefore the energy radiated in unit time is $16\pi ^{4}A^{2}S^{2}/3c^{3}T^{4}$. This is small, unless the frequency is very high; so that ordinary alternating currents would give no appreciable radiation. FitzGerald, however, in the same year indicated a method by which the difficulty of obtaining currents of sufficiently high frequency might be overcome: this was, to employ the alternating currents which are produced when a condenser is discharged.
The FitzGerald radiator constructed on this principle is closely akin to the radiator afterwards developed with such success by Hertz: the only difference is that in FitzGerald's arrangement the condenser is used merely as the store of energy (its plates being so close together that the electrostatic field due to the charges is practically confined to the space between them), and the actual source of radiation is the alternating magnetic field due to the circular loop of wire: while in Hertz's arrangement the loop of wire is abolished, the condenser plates are at some distance apart, and the source of radiation is the alternating electrostatic field due to their charges.
In the study of electrical radiation, valuable help is afforded by a general theorem on the transfer of energy in the electromagnetic field, which was discovered in 1884 by John Henry Poynting. We have seen that the older writers on electric currents recognized that an electric current is associated with the transport of energy from one place (e.g. the voltaic cell which maintains the current) to another (e.g. an electromotor which is worked by the current); but they supposed the energy to be conveyed by the current itself within the wire, in much ​or the same way as dynamical energy is carried by water flowing in a pipe; whereas in Maxwell's theory, the storehouse and vehicle of energy is the dielectric medium surrounding the wire. What Poynting achieved was to show that the flux of energy at any place might be expressed by a simple formula in terms of the electric and magnetic forces at the place.
${\tfrac {1}{2}}\mathbf {ED} +(1/8\pi )\mathbf {BH}$;
$\mathbf {E{\dot {D}}} +(1/4\pi )\mathbf {H{\dot {B}}}$
or $\mathbf {E} (\mathbf {S} -{\boldsymbol {\iota }})+(1/4\pi )\mathbf {H{\dot {B}}}$, where S denotes the total current, and ι the current of conduction; or (in virtue of the fundamental electromagnetic equations)
$-(\mathbf {E.} {\boldsymbol {\iota }})+(1/4\pi )(\mathbf {E} .{\text{curl }}\mathbf {H} )-(1/4\pi )(\mathbf {H} .{\text{curl }}\mathbf {E} )$,
or $-(\mathbf {E.} {\boldsymbol {\iota }})-(1/4\pi ){\text{div }}[\mathbf {E.H} ]$. Now (E. ι) is the amount of electric energy transformed into heat per unit volume per second; and therefore the quantity -(1/4π) div [E.H] must represent the deposit of energy in unit volume per second due to the streaming of energy; which shows that the flux of energy is represented by the vector -(1/4π) div [E.H]. This is Poynting's theorem: that the flux of energy at any place is represented by the vector-product of the electric and magnetic forces, divided by 4π.
In the special case of the field which surrounds a straight wire carrying a continuous current, the lines of magnetic force are circles round the axis of the wire, while the lines of electric force are directed along the wire; hence energy must be flowing in the medium in a direction at right angles to the axis of the wire. A current in any conductor may therefore be regarded as consisting essentially of a convergence of electric and magnetic energy from the medium upon the conductor, and its transformation there into other forms.
This association of a current with motions at right angles to the wire in which it flows doubtless suggested to Poynting the conceptions of a memoir which he published in the following year. When an electric current flowing in a straight wire is gradually increased in strength from zero, the surrounding space becomes filled with lines of magnetic force, which have the form of circles round the axis of the wire. Poynting, adopting Faraday's idea of the physical reality of lines of force, assumed that these lines of force arrive at their places by moving outwards from the wire; so that the magnetic field grows by a continual emission from the wire of lines of force, which enlarge and spread out like the circular ripples from the place where a stone is dropped into a pond. The electromotive force which is associated with a changing magnetic field was now attributed directly to the motion of the lines of force, so that wherever electromotive force is produced by change in the magnetic field, or by motion of matter through the field, the electric intensity is equal to the number of tubes of magnetic force intersected by unit length in unit time.
A similar conception was introduced in regard to lines of electric force. It was assumed that any change in the total ​electric induction through a curve is caused by the passage of tubes of force in or out across the boundary; so that whenever magnetomotive force is produced by change in the electric field, or by motion of matter through the field, the magnetomotive force is proportional to the number of tubes of electric force intersected by unit length in unit time.
Poynting, moreover, assumed that when a steady current C flows in a straight wire, C tubes of electric force close in upon the wire in unit time, and are there dissolved, their energy appearing as heat. If E denote the magnitude of the electric force, the energy of each tube per unit length is 1/2E, so the amount of energy brought to the wire is 1/2CE per unit length per unit time. This is, however, only half the energy actually transformed into heat in the wire: so Poynting further assumed that E tubes of magnetic force also move in per unit length per unit time, and finally disappear by contraction to infinitely small rings. This motion accounts for the existence of the electric field; and since each tube (which is a closed ring) contains energy of amount 1/2C, the disappearance of the tubes accounts for the remaining 1/2CE units of energy dissipated in the wire.
The theory of moving tubes of force has been extensively developed by Sir Joseph Thomson. Of the two kinds of tubes—magnetic and electric—which had been introduced by Faraday and used by Poynting, Thomson resolved to discard the former and employ only the latter. This was a distinct departure from Faraday's conceptions, in which, as we have seen, great significance was attached to the physical reality of the magnetic lines; but Thomson justified his choice by inferences drawn from the phenomena of electric conduction in liquids and gases. As will appear subsequently, these phenomena indicate that molecular structure is closely connected with tubes of electrostatic force—perhaps much more closely than with tubes of magnetic force; and Thomson therefore decided to regard ​magnetism as the secondary effect, and to ascribe magnetic fields, not to the presence of magnetic tubes, but to the motion of electric tubes. In order to account for the fact that magnetic fields may occur without any manifestation of electric force, he assumed that tubes exist in great numbers everywhere in space, either in the form of closed circuits or else terminating on atoms, and that electric force is only perceived when the tubes have a greater tendency to lie in one direction than in another. In a steady magnetic field the positive and negative tubes might be conceived to be moving in opposite directions with equal velocities.
A beam of light might, from this point of view, be regarded simply as a group of tubes of force which are moving with the velocity of light at right angles to their own length. Such a conception almost amounts to a return to the corpuscular theory; but since the tubes have definite directions perpendicular to the direction of propagation, there would now be no difficulty in explaining polarization.
The energy accompanying all electric and magnetic phenomena was supposed by Thomson to be ultimately kinetic energy of the aether; the electric part of it being represented by rotation of the aether inside and about the tubes, and the magnetic part being the energy of the additional disturbance set up in the aether by the movement of the tubes. The inertia of this latter motion he regarded as the cause of induced electromotive force.
There was, however, one phenomenon of the electromagnetic field as yet unexplained in terms of these conceptions—namely, the ponderomotive force which is exerted by the field on a conductor carrying an electric current. Now any ponderomotive force consists in a transfer of mechanical momentum from the agent which exerts the force to the body which experiences it; and it occurred to Thomson that the ponderomotive forces of the electromagnetic field might be explained if the moving tubes of force, which enter a conductor carrying a clurent and are there dissolved, were supposed to possess ​mechanical momentum, which could be yielded up to the conductor. It is readily seen that such momentum must be directed at right angles to the tube and to the magnetic induction—a result which suggests that the momentum stored in unit volume of the aether may be proportional to the vector-product of the electric and magnetic vectors.
For this conjecture reasons of a more definite kind may be given. We have already seen that the ponderomotive forces on material bodies in the electromagnetic field may be accounted for by Maxwell's supposition that across any plane in the aether whose unit normal is N, there is a stress represented by
$\mathbf {P_{N}} =(\mathbf {D.N} )\mathbf {E} -{\tfrac {1}{2}}(\mathbf {D.E} )\mathbf {N} +{\tfrac {1}{4\pi }}(\mathbf {B.N} )\mathbf {H} -{\tfrac {1}{8\pi }}(\mathbf {B.H} )\mathbf {N}$.
So long as the field is steady (i.e. electrostatic or magnetostatic) the resultant of the stresses acting on any element of volume of the aether is zero, so that the element is in equilibrium. But when the field is variable, this is no longer the case. The resultant stress on the aether contained within a surface S is
$\textstyle \iint \mathbf {P_{N}} .dS$
integrated over the surface: transforming this into a volume- integral, the term (D.N)E gives a term div D.E + (D.∇)E, where ∇ denotes the vector operator (∂/∂x, ∂/∂y, ∂/∂z); and the first of these terms vanishes, since D is a circuital vector; the term - 1/2(D.E)N gives in the volume-integral a term 1/2 grad (D.E); and the magnetic terms give similar results. So the resultant force on unit-volume of the aether is
$(\mathbf {D.} \nabla )\mathbf {E} +{\tfrac {1}{2}}{\text{ grad}}(\mathbf {D.E} )+(1/r\pi )(\mathbf {B.} \nabla )\mathbf {E} +(1/8\pi ){\text{ grad}}(\mathbf {B.H} )$,
which may be written
$[{\text{curl }}\mathbf {E.D} ]+(1/4\pi )[{\text{curl }}\mathbf {H.B} ]$;
$[-\mathbf {{\dot {B}}.D} ]+[\mathbf {{\dot {D}}.B} ]$, or $(\partial /\partial t)[\mathbf {D.B} ]$.
This result compels us to adopt one of three alternatives: either to modify the theory so as to reduce to zero the resultant force on an element of free aether; this expedient has not met with general favour; or to assume that the force in question sets the aether in motion: this alternative was chosen by Helmholtz, but is inconsistent with the theory of the aether which was generally received in the closing years of the century; or lastly, with Thomson, to accept the principle that the aether is itself the vehicle of mechanical momentum, of amount [D.B] per unit volume.
Maxwell's theory was now being developed in ways which could scarcely have been anticipated by its author. But although every year added something to the superstructure, the foundations remained much as Maxwell had laid them, the doubtful argument by which he had sought to justify the introduction of displacement-currents was still all that was offered in their defence. In 1884, however, the theory was established on a different basis by a pupil of Helmholtz', Heinrich Hertz (b. 1857, d. 1894).
The train of Hertz' ideas resembles that by which Ampère, on hearing of Oersted's discovery of the magnetic field produced by electric currents, inferred that electric currents should exert ponderomotive forces on each other. Ampère argued that a current, being competent to originate a magnetic field, must be equivalent to a magnet in other respects; and therefore that currents, like magnets, should exhibit forces of mutual attraction and repulsion.
​Ampère's reasoning rests on the assumption that the magnetic field produced by a current is in all respects of the same nature as that produced by a magnet; in other words, that only one kind of magnetic force exists. This principle of the "unity of magnetic force" Hertz now proposed to supplement by asserting that the electric force generated by a changing magnetic field is identical in nature with the electric force due to electrostatic charges; this second principle he called the "unity of electric force." Suppose, then, that a system of electric currents ι exists in otherwise empty space. According to the older theory, these currents give rise to a vector-potential a$_1$, equal to Pot ι; and the magnetic force H$_1$, is the curl of a$_1$: while the electric force E$_1$, at any point in the field, produced by the variation of the currents, is $-\mathbf {\dot {a}} _{1}$.
It is now assumed that the electric force so produced is indistinguishable from the electric force which would be set up by electrostatic charges, and therefore that the system of varying currents exerts ponderomotive forces on electrostatic charges; the principle of action and reaction then requires that electrostatic charges should exert ponderomotive forces on a system of varying currents, and consequently (again appealing to the principle of the unity of electric force) that two systems of varying currents should exert on each other ponderomotive forces due to the variations.
But just as Helmholtz, by aid of the principle of conservation of energy, deduced the existence of an electromotive force of induction from the existence of the ponderomotive forces between electric currents (i.e. variable electric systems), so from the existence of ponderomotive forces between variable systems of currents (i.e. variable magnetic systems) we may infer that variations in the rate of change of a variable magnetic system give rise to induced magnetic forces in the surrounding space. The analytical formulae which determine these forces ​will be of the same kind as in the electric case; so that the induced magnetic force H′ is given by an equation of the form
$\mathbf {H} ^{\prime }=(1/c^{2})\mathbf {b} _{1}$,
where e denotes some constant, and b$_1$, which is analogous to the vector-potential in the electric case, is a circuital vector whose curl is the electric force E$_1$, of the variable magnetic system. The value of b$_1$, is therefore $(1/4\pi ){\text{ curl Pot }}\mathbf {E} _{2}$: so we have
$\mathbf {H} ^{\prime }=-{\frac {1}{4\pi c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}{\text{ curl Pot }}\mathbf {a} _{1}$.
This must be added to H$_1$. Writing H$_2$, for the sum, H + H′, we see that H$_2$ is the curl of a$_2$, where
$\mathbf {a} _{2}=\mathbf {a} _{1}-{\frac {1}{4\pi c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}{\text{Pot }}\mathbf {a} _{1}$;
and the electric force E$_2$, will then be $-\mathbf {\dot {a}} _{2}$. This system is not, however, final; for we must now perform the process again with these improved values of the electric and magnetic forces and the vector-potential; and so we obtain for the magnetic force the value curl a$_3$, and for the electric force the value $-\mathbf {\dot {a}} _{3}$, where
${\begin{matrix}\mathbf {a} _{3}&=&\mathbf {a} _{1}-{\frac {1}{4\pi c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}{\text{Pot }}\mathbf {a} _{2}&\ \\\ &=&\mathbf {a} _{1}-{\frac {1}{4\pi c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}{\text{Pot }}\mathbf {a} _{1}&+{\frac {1}{(4\pi c^{2})^{2}}}{\frac {\partial ^{4}}{\partial t^{4}}}{\text{Pot Pot }}\mathbf {a} _{1}\end{matrix}}$
This process must again be repeated indefinitely; so finally we obtain for the magnetic force H the value curl a, and for the electric force E the value $-\mathbf {\dot {a}}$, where
* $\mathbf {a} =\mathbf {a} _{1}-{\frac {1}{4\pi c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}{\text{Pot }}\mathbf {a} _{1}+{\frac {1}{(4\pi c^{2})^{2}}}{\frac {\partial ^{4}}{\partial t^{4}}}{\text{Pot Pot }}\mathbf {a} _{1}$
* $-\ {\frac {1}{(4\pi c^{2})^{3}}}{\frac {\partial ^{6}}{\partial t^{6}}}{\text{Pot Pot Pot }}\mathbf {a} _{1}+$…
$\nabla ^{2}\mathbf {a} =\nabla ^{2}\mathbf {a} _{1}+{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {a}$,
or $\nabla ^{2}\mathbf {a} -{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {a} =-4\pi {\boldsymbol {\iota }}$. This equation may be written
${\text{curl }}\mathbf {H} =(1/c^{2})\mathbf {\dot {E}} +4\pi {\boldsymbol {\iota }}$,
while the equations H = curl a, $\mathbf {E} =-\mathbf {\dot {a}}$ give
${\text{curl }}\mathbf {E} =-\mathbf {\dot {H}}$.
These are, however, the fundamental equations of Maxwell's theory in the form given in his memoir of 1868.
That Hertz's deduction is ingenious and interesting will readily be admitted. That it is conclusive may scarcely be claimed: for the argument of Helmholtz regarding the induction of currents is not altogether satisfactory; and Hertz, in following his master, is on no surer ground.
In the course of a discussion on the validity of Hertz's assumptions, which followed the publication of his paper, E. Aulinger brought to light a contradiction between the principles of the unity of electric and of magnetic force and the electrodynamics of Weber. Consider an electrostatically charged hollow sphere, in the interior of which is a wire carrying a variable current. According to Weber's theory, the sphere would exert a turning couple on the wire; but according to Hertz's principles, no action would be exerted, since charging the sphere makes no difference to either the electric or the magnetic force in its interior. The experiment thus suggested would be a crucial test of the correctness of Weber's theory; it has the advantage of requiring nothing but closed currents and electrostatic charges at rest; but the quantities to be observed would be on the limits of observational accuracy.
​After his attempt to justify the Maxwellian equations on theoretical grounds, Hertz turned his attention to the possibility of verifying them by direct experiment. His interest in the matter had first been aroused some years previously, when the Berlin Academy proposed as a prize subject "To establish experimentally a relation between electromagnetic actions and the polarization of dielectrics." Helmholtz suggested to Hertz that he should attempt the solution; but at the time he saw no way of bringing phenomena of this kind within the limits of observation. From this time forward, however, the idea of electric oscillations was continually present to his mind; and in the spring of 1886 he noticed an effect which formed the starting-point of his later researches. When an open circuit was formed of a piece of copper wire, bent into the form of a rectangle, so that the ends of the wire were separated only by a short air-gap, and when this open circuit was connected by a wire with any point of a circuit through which the spark-discharge of an induction-coil was taking place, it was found that a spark passed in the air-gap of the open circuit. This was explained by supposing that the change of potential, which is propagated along the connecting wire from the induction-coil, reaches one end of the open circuit before it reaches the other, so that a spark passes between them; and the phenomenon therefore was regarded as indicating a finite velocity of propagation of electric potential along wires.
​Continuing his experiments, Hertz found that a spark could be induced in the open or secondary circuit oven when it was not in metallic connexion with the primary circuit in which the electric oscillations were generated; and he rightly interpreted the phenomenon by showing that the secondary circuit was of such dimensions as to make the free period of electric oscillations in it nearly equal to the period of the oscillations in the primary circuit; the disturbance which passed from one circuit to the other by induction would consequently be greatly intensified in the secondary circuit by resonance.
The discovery that sparks may be produced in the air-gap of a secondary circuit, provided it has the dimensions proper for resonance, was of great importance: for it supplied a method of detecting electrical effects in air at a distance from the primary disturbance; a suitable detector was in fact all that was needed in order to observe the propagation of electric waves in free space, and thereby decisively test the Maxwellian theory. To this work Hertz now addressed himself.
The radiator or primary source of the disturbances studied by Hertz may be constructed of two sheets of metal in the same plane, each sheet carrying a stiff wire which projects towards the other sheet and terminates in a knob; the sheets are to be excited by connecting them to the terminals of an induction coil. The sheets may be regarded as the two coatings of a modified Leyden jar, with air as the dielectric between thein; the electric field is extended throughout the air, instead of being confined to the narrow space between the coatings, as in the ordinary Leyden jar. Such a disposition ensures that the system shall lose a large part of its energy by radiation at each oscillation.
​As in the jar discharge, the electricity surges from one sheet to the other, with a period proportional to (CL)$^{1/2}$, where C denotes the electrostatic capacity of the system formed by the two sheets, and L denotes the self-induction of the connexion. The capacity and induction should be made as small as possible in order to make the period small. The detector used by Hertz was that already described, namely, a wire bent into an incompletely closed curve, and of such dimensions that its free period of oscillation was the same as that of the primary oscillation, so that resonance might take place.
Towards the end of the year 1887, when studying the sparks induced in the resonating circuit by the primary disturbance, Hertz noticed that the phenomena were distinctly modified when a large mass of an insulating substance was brought into the neighbourhood of the apparatus; thus confirming the principle that the changing electric polarization which is produced when an alternating electric force acts on a dielectric is capable of displaying electromagnetic effects.
Early in the following year (1888) Hertz determined to verify Maxwell's theory directly by showing that electromagnetic actions are propagated in air with a finite velocity. For this purpose he transmitted the disturbance from the primary oscillator by two different paths, viz., through the air and along a wire; and having exposed the detector to the joint influence of the two partial disturbances, he observed interference between them. In this way he found the ratio of the velocity of electric waves in air to their velocity when conducted by wires; and the latter velocity he determined by observing the distance between the nodes of stationary waves in the wire, and calculating the period of the primary oscillation. The velocity of propagation of electric disturbances in air was in ​this way shown to be finite and of the same order as the velocity of light.
Later in 1888 Hertz showed that electric waves in air are reflected at the surface of a wall; stationary waves may thus be produced, and interference may be obtained between direct and reflected beams travelling in the same direction.
The theoretical analysis of the disturbance emitted by a Hertzian radiator according to Maxwell's theory was given by Hertz in the following year.
The effects of the radiator are chiefly determined by the free electric charges which, alternately appearing at the two sides, generate an electric field by their presence and a etic field by their motion. In each oscillation, as the charges on the poles of the radiator increase from zero, lines of electric force, having their ends on these poles, move outwards into the surrounding space. When the charges on the poles attain their greatest values, the lines cease to issue outwards, and the existing lines begin to retreat inwards towards the poles; but the outer lines of force contract in such a way that their upper and lower parts touch each other at some distance from the radiator, and the remoter portion of each of these lines thus takes the form of a loop; and when the rest of the line of force retreats inwards towards the radiator, this loop becomes detached and is propagated outwards as radiation. In this way the radiator emits a series of whirl-rings, which as they move grow thinner and wider; at a distance, the disturbance ​is approximately a plane wave, the opposite sides of the ring representing the two phases of the wave. When one of these rings has become detached from the radiator, the energy contained may subsequently be regarded as travelling outwards with it.
To discuss the problem analytically we take the axis of the radiator as axis of z, and the centre of the spark-gap as origin. The field may be regarded as due to an electric doublet formed of a positive and an equal negative charge, displaced from each other along the axis of the vibrator, and of moment
$Ae^{-p_{1}t}\sin(2\pi ct/\lambda )$,
the factor $e^{-p_{1}t}$ being inserted to represent the damping.
The simplest method of proceeding, which was suggested by FitzGerald, is to form the retarded potentials φ and a of L. Lorenz. These are determined in terms of the charges and their velocities by the equations
$\phi =\textstyle \sum {\frac {(e)_{t-r/c}}{r}},\qquad a_{z}=\textstyle \sum {\frac {(e{\dot {z}})_{t-r/c}}{r}}$,
whence it is readily shown that in the present case
$\phi =-\partial F/\partial z,\qquad \mathbf {a} =(0,0,\partial F/\partial t)$,
where
$F={\frac {Ae^{-p_{1}(t-r/c)}}{r}}\sin {\frac {2\pi }{\lambda }}(ct-r)$.
The electric and magnetic forces are then determined by the equations
$\mathbf {E} =c^{2}{\text{ curl }}\phi -\mathbf {\dot {a}} ,\qquad \mathbf {H} ={\text{curl }}\mathbf {a}$.
It is found that the electric force may be regarded as compounded of a force φ$_2$, parallel to the axis of the vibrator and depending at any instant only on the distance from the vibrator, together with a force φ$_1$ sin θ acting in the meridian plane ​perpendicular to the radius from the centre, where di depends at any instant only on the distance from the vibrator, and 0 denotes the angle which the radius makes with the axis of the oscillator. At points on the axis, and in the equatorial plane, the electric force is parallel to the axis. At a great distance from the oscillator, φ$_2$ is small compared with φ$_1$, so the wave is purely transverse. The magnetic force is directed along circles whose centres are on the axis of the radiator; and its magnitude may be represented in the form φ$_3$ sin θ, where φ$_3$ depends only on r and t; at great distances from the radiator, cφ$_3$ is approximately equal to φ$_1$.
If the activity of the oscillator be supposed to be continually maintained, so that there is no damping, we may replace p$_1$, by zero, and may proceed as in the case of the magnetic oscillator to determine the amount of energy radiated. The mean outward flow of energy per unit time is found to be 1/3c$^3$A$^2$(2π/λ)$^4$; from which it is seen that the rate of loss of energy by radiation increases greatly as the wave-length decreases.
The action of an electrical vibrator may be studied by the aid of mechanical models. In one of these, devised by Larmor, the aether is represented by an incompressible elastic solid, in which are two cavities, corresponding to the conductors of the vibrator, filled with incompressible fluid of negligible inertia, The electric force is represented by the displacement of the solid. For such rapid alternations as are here considered, the metallic poles behave as perfect conductors; and the tangential components of electric force at their surfaces are zero, This condition may be satisfied in the model by supposing the lining of each cavity to be of flexible sheet-metal, so as to be incapable of tangential displacement; the normal displacement of the lining then corresponds to the surface-density of electric charge on the conductor.
In order to obtain oscillations in the solid resembling those of an electric vibrator, we may suppose that the two cavities ​have the form of semicircular tubes forming the two halves of a complete circle. Each tube is enlarged at each of its ends, so as to present a front of considerable area to the corresponding front at the end of the other tube. Thus at each end of one diameter of the circle there is a pair of opposing fronts, which are separated from each other by a thin sheet of the elastic solid.
The disturbance may be originated by forcing an excess of liquid into one of the enlarged ends of one of the cavities. This involves displacing the thin sheet of clastic solid, which separates it from the opposing front of the other cavity, and thus causing a corresponding deficiency of liquid in the enlarged end behind this front. The liquid will then surge backwards and forwards in each cavity between its enlarged ends; and, the motion being communicated to the elastic solid, vibrations will be generated resembling those which are produced in the aether by a Hertzian oscillator.
In the latter part of the year 1888 the researches of Hertz yielded more complete evidence of the similarity of electric waves to light. It was shown that the part of the radiation from an oscillator which was transmitted through an opening in a screen was propagated in a straight line, with diffraction effects. Of the other properties of light, polarization existed in the original radiation, as was evident from the manner in which it was produced; and polarization in other directions was obtained by passing the waves through a grating of parallel metallic wires; the component of the electric force parallel to the wires was absorbed, so that in the transmitted beam the electric vibration was at right angles to the wires. This effect obviously resembled the polarization of ordinary light by a plate of tourmaline. Refraction was obtained by passing the radiation through prisms of hard pitch.
​The old question as to whether the light-vector is in, or at right angles to, the plane of polarization now presented itself in a new aspect. The wave-front of an electric wave contains two vectors, the electric and magnetic, which are at right angles to each other. Which of these is in the plane of polarization? The answer was furnished by FitzGerald and Trouton, who found on reflecting Hertzian waves from a wall of masonry that no reflexion was obtained at the polarizing angle when the vibrator was in the plane of reflexion. The inference from this is that the magnetic vector is in the plane of polarization of the electric wave, and the electric vector is at right angles to the plane of polarization. An interesting development followed in 1890, when O. Wiener succeeded in photographing stationary waves of light. The stationary waves were obtained by the composition of a beam incident on a mirror with the reflected beam, and were photographed on a thin film of transparent collodion, placed close to the mirror and slightly inclined to it. If the beam used in such an experiment is plane-polarized, and is incident at an angle of 45°, the stationary vector is evidently that perpendicular to the plane of incidence; but Wiener found that under these conditions the effect was obtained only when the light was polarized in the plane of incidence; so that the chemical activity must be associated with the vector perpendicular to the plane of polarization—i.e., the electric vector.
In 1890 and the years immediately following appeared several memoirs relating to the fundamental equations of electro-magnetic theory. Hertz, after presenting the general ​content of Maxwell's theory for bodies at rest, proceeded to extend the equations to the case in which material bodies are in motion in the field.
In a really comprehensive and correct theory, as Hertz remarked, a distinction should be drawn between the quantities which specify the state of the aether at every point, and those which specify the state of the ponderable matter entangled with it. This anticipation has been fulfilled by later investigators; but Hertz considered that the time was not ripe for such a complete theory, and preferred, like Maxwell, to assume that the state of the compound system—matter plus aether—can be specified in the same way when the matter moves as when it is at rest; or, as Hertz himself expressed it, that "the aether contained within ponderable bodies moves with them."
Maxwell's own hypothesis with regard to moving systems amounted merely to a modification in the equation
$\mathbf {\dot {B}} =-{\text{curl }}\mathbf {E}$,
which represents the law that the electromotive force in a closed circuit is measured by the rate of decrease in the number of lines of magnetic induction which pass through the circuit. This law is true whether the circuit is at rest or in motion; but in the latter case, the E in the equation must be taken to be the electromotive force in a stationary circuit whose position momentarily coincides with that of the moving circuit; and since an electromotive force [w. B] is generated in matter by its motion with velocity w in a magnetic field B, we see that E is connected with the electromotive force E′ in the moving ponderable body by the equation
$\mathbf {E} ^{\prime }=\mathbf {E} +[\mathbf {w.B} ]$
so that the equation of electromagnetic induction in the moving body is
$\mathbf {\dot {B}} =-{\text{curl }}\mathbf {E} ^{\prime }+{\text{curl }}[\mathbf {w.B} ]$
$\mathbf {D} =\epsilon \mathbf {E} ^{\prime }/4\pi c^{2},\qquad {\text{div }}\mathbf {D} =0,\qquad 4\pi ({\boldsymbol {\iota }}+\mathbf {\dot {D}} )={\text{curl }}\mathbf {H}$,
Hertz, however, impressed by the duality of electric and magnetic phenomena, modified the last of these equations by assuming that a magnetic force 4π [D.w] is generated in a dielectric which moves with velocity w in an electric field; such a force would be the magnetic analogue of the electromotive force of induction. A term involving curl [D.w] is then introduced into the last equation.
The theory of Hertz resembles in many respects that of Heaviside, who likewise insisted much on the duplex nature of the electromagnetic field, and was in consequence disposed to accept the term involving curl [D.w] in the equations of moving media. Heaviside recognized more clearly than his predecessors the distinction between the force E′, which determines the flux D, and the force E, whose curl represents the electric current; and, in conformity with his principle of duality, he made a similar distinction between the magnetic force H′, which determines the flux B, and the force H, whose curl represents the "magnetic current" This distinction, as Heaviside showed, is of importance when the system is acted on by "impressed forces," such as voltaic electromotive forces, or permanent magnetization; these latter must be included in E′ and K′, since they help to give rise to the fluxes D and B; but they must not be included in E and H since their curls are not electric or magnetic currents; so that in general We have
$\mathbf {E} ^{\prime }=\mathbf {E} +\mathbf {e} ,\qquad \mathbf {H} ^{\prime }=+\mathbf {h}$,
where e and h denote the impressed forces.
Developing the theory by the aid of these conceptions, Heaviside was led to make a further modification, An im​pressed force is best defined in terms of the energy which it communicates to the system; thus, if e be an impressed electric force, the energy communicated to unit volume of the electromagnetic system in unit time is e × the electric current. In order that this equation may be true, it is necessary to regard the electric current in a moving medium as composed of the conduction-current, displacement-current, convectioncurrent, and also of the term curl [D.w], whose presence in the equation we have already noticed. This may be called the current of dielectric convection. Thus the total current is
$\mathbf {S} =\mathbf {\dot {D}} ={\boldsymbol {\iota }}+\rho \mathbf {w} +{\text{curl}}[\mathbf {D.w} ]$,
where ρw denotes the conduction-current; and the equation connecting current with magnetic force is
${\text{curl }}(\mathbf {H^{\prime }} -\mathbf {h} _{0})=4\pi \mathbf {S}$,
where h$_0$, denotes the impressed magnetic forces other than that induced by motion of the medium.
We must now consider the advances which were effected during the period following the publication of Maxwell's Treatise in some of the special problems of electricity and optics.
We have seen that Maxwell accounted for the rotation of the plane of polarization of light in a medium subjected to a magnetic field K by adding to the kinetic energy of the aether, which is represented by 1/2ρė$^2$, a term 1/2σ(ė. curl ∂e/∂θ, where σ is a magneto-optic constant characteristic of the substance through which the light is transmitted, and ∂/∂θ stands for K$_x$∂/∂x + K$_y$∂/∂y + K$_z$∂/∂z. This theory was developed further in 1879 by FitzGerald, who brought it into closer connexion with the electromagnetic theory of light by identifying the curl of the displacement e of the aethereal particles with the electric displacement; the derivate of e with respect to the time then corresponds to the magnetic force. Being thus in possession of a definitely electromagnetic theory of the magnetic rotation of ​light, FitzGerald proceeded to extend it so as to take account of a closely related phenomenon. In 1876 J. Kerr had shown experimentally that when plane-polarized light is regularly reflected from either pole of an iron electromagnet, the reflected ray has a component polarized in a plane at right angles to the ordinary reflected ray. Shortly after this discovery had been made known, FitzGerald had proposed to explain it by means of the same term in the equations which accounts for the magnetic rotation of light in transparent bodies. His argument was that if the incident plane-polarized ray be resolved into two rays circularly polarized in opposite senses, the refractive index will have different values for these two rays, and hence the intensities after reflexion will be different; so that on recompounding them, two plane-polarized rays will be obtained—one polarized in the plane of incidence, and the other polarized at right angles to it.
The analytical discussion of Kerr's phenomenon, which was given by FitzGerald in his memoir of 1879, was based on these ideas; the most essential features of the phenomenon were explained, but the investigation was in some respects imperfect.
A new and fruitful conception was introduced in 1879–1880, when H. A. Rowland suggested a connexion between the magnetic rotation of light and the phenomenon which had been discovered by his pupil Hall. Hall's effect may be regarded ​as a rotation of conduction-currents under the influence of a magnetic field; and if it be assumed that displacement-currents in dielectrics are rotated in the same way, the Faraday effect may evidently be explained. Considering the matter from the analytical point of view, the Hall effect may be represented by the addition of a term k [K.S] to the electromotive force, where K denotes the impressed magnetic force, and S denotes the current: so Rowland assumed that in dielectrics there is an additional term in the electric force, proportional to [K.Ḋ], i.e. proportional to the rate of increase of [K.D]. Now it is universally true that the total electric force round a circuit is proportional to the rate of decrease of the total magnetic induction through the circuit: so the total magnetic induction through the circuit must contain a term proportional to the integral of [K.D] taken round the circuit: and therefore the magnetic induction at any point must contain a term proportional to [curl K.D]. We may therefore write
$\mathbf {B} =\mathbf {H} +\sigma {\text{ curl }}[\mathbf {K.D} ]$,
where σ denotes a constant. But if this be combined with the customary electromagnetic equations
${\text{curl }}\mathbf {H} =4\pi \mathbf {\dot {D}} ,\qquad {\text{curl }}\mathbf {E} =-\mathbf {\dot {B}} ,\qquad \mathbf {D} =\epsilon \mathbf {E} /4\pi c^{2}$,
and all the vectors except B be eliminated (K being treated as a constant), we obtain the equation
$\mathbf {\ddot {B}} =(c^{2}/\epsilon )\nabla ^{2}\mathbf {B} +(\sigma /4\pi ){\text{ curl }}(\partial ^{2}\mathbf {B} /\partial t\partial \theta )$,
where ∂/∂θ stands for K$_x$∂/∂x + K$_y$∂/∂y + K$_z$∂/∂z (K20/0c + Ky0/0y + K20/02); and this is identical with the equation which Maxwell had given for the motion of the aether in magnetized media. It follows that the assumptions of Maxwell and of Rowland, different though they are physically, lead to the same analytical equations-at any rate so far as concerns propagation through a homogeneous medium.
The connexions of Hall's phenomenon with the magnetic rotation of light, and with the reflexion of light from magnetized ​metals, were extensively studied
in the years following the publication of Rowland's memoir: but it was not until the modern theory of electrons had been developed that a satisfactory representation of the molecular processes involved in magnetooptic phenomena was attained.
The allied phenomenon of rotary polarization in naturally active bodies was investigated in 1892 by Goldhammer. It ​will be remembered that in the clastic-solid theory of Boussinesq, the rotation of the plane of polarization of saccharine solutions had been represented by substituting the equation
$\mathbf {e} ^{\prime }=\mathbf {Ae} +\mathbf {B} {\text{curl }}\mathbf {e}$
in place of the usual equation
$\mathbf {e} ^{\prime }=\mathbf {Ae}$.
Goldhammer now proposed to represent rotatory power in the electromagnetic theory by substituting the equation
$\mathbf {E} =(4\pi c^{2}/\epsilon )\mathbf {D} +k{\text{ curl }}\mathbf {D}$,
in place of the customary equation
$\mathbf {E} =(4\pi c^{2}/\epsilon )\mathbf {D}$:
the constant k being a measure of the natural rotatory power of the substance concerned. The remaining equations are as usual,
${\text{curl }}\mathbf {H} =4\pi \mathbf {\dot {D}} ,\qquad -{\text{curl }}\mathbf {E} =\mathbf {H}$.
Eliminating H and E, we have
$\mathbf {\ddot {D}} =(c^{2}/\epsilon )\nabla ^{2}\mathbf {D} +(k/4\pi )\nabla ^{2}{\text{curl }}\mathbf {D}$.
For a plane wave which is propagated parallel to the axis of x, this equation reduces to
${\begin{cases}{\frac {\partial ^{2}D_{y}}{\partial t^{2}}}={\frac {c^{2}}{\epsilon }}{\frac {\partial ^{2}D_{y}}{\partial x^{2}}}-{\frac {k}{4\pi }}{\frac {\partial ^{3}D_{z}}{\partial x^{3}}}\\{\frac {\partial ^{2}D_{z}}{\partial t^{2}}}={\frac {c^{2}}{\epsilon }}{\frac {\partial ^{2}D_{z}}{\partial x^{2}}}-{\frac {k}{4\pi }}{\frac {\partial ^{3}D_{y}}{\partial x^{3}}}\end{cases}}$
and, as MacCullagh had shown in 1836, these equations are competent to represent the rotation of the plane of polarization.
In the closing years of the nineteenth century, the general theory of aether and electricity assumed a new form. But before discussing the memoirs in which the new conception was unfolded, we shall consider the progress which had been made since the middle of the century in the study of conduction in liquid and gaseous media.
* ↑ Journal für Math. lxxii (1870), p. 68, note.
* ↑ Helmholtz (loc. cit.) pointed out that if the optical difference between the media were assumed to be due to a difference in their magnetic permeabilities, it would be necessary to suppose the magnetic vector at right angles to the plane of polarization in order to obtain Fresnel's sine and tangent formulae of reflexion.
* ↑ Cf. pp. 148, 149, 154-156.
* ↑ Zeitschrift für Muth. 1. Phys. xxii (1877), pp. 1, 205: Over de theorie der terugkaatsing en breking van het licht, Årnhem, 1875. Lorentz's work was based on Helmholtz's equations, but remains substantiully unchangod when Maxwell's formulae are substituted.
* ↑ Phil. Mag. (4) 1 (1875), pp. 337, 446; (5) viii (1879), pp. 85, 229; xiii (1882), pp. 153, 248.
* ↑ Monatsberichte d. Acad. d. Berlin, 1875, p. 400. Ann. d. Phys., clviii (1876), p. 87.
* ↑ Ann. d. Phys. clix (1876), pp. 456, 537; clx (1877), p. 333.
* ↑ The valuable memoirs by Helmholtz in Journal für Math. lxxii (1870), p. 57; lxxv (1873), p. 35; lxxviii (1874), p. 273, to which reference has already been made, contain a full discussion of the various possibilities of the potential-theories.
* ↑ Exper. Res., 1644.
* ↑ Monatsberichte d. Akad.d. Berlin, 1876, p. 211: Ann, d. Phys, clviii (1876), p. 487: Annales de Chit. et de Phys. xii (1877) p. 119.
* ↑ Phil. Mag. xxvii (1889), p. 446.
* ↑ Ibid. ii (1901), 7. 179: v (1903), p. 34.
* ↑ Ann. d. Phys. xi (1901), p. 1.
* ↑ Phil. Mag, xi (1881), p. 229.
* ↑ Proc. Roy. Dublin Soc., November, 1881; FitzGerald's Scientific Writings, p. 102.
* ↑ Phil. Mag. xxvii (1889), p. 324.
* ↑ By G. F. C. Searle.
* ↑ Cf. Searle, Phil, Trans. clxxxvii (1896), p. 676, and Phil. Mag. xliv (1897), p. 329. On the theory of the moving electrified sphere, cf. also J. J. Thomson, Recent Researches in Elect, and Mag., p. 16; O. Heaviside, Electrical Papers, ii, p. 514; Electromag. Theory, i, p. 269; W. B. Morton, Phil. Mag. xli (1896), P. 488; A. Schuster, Phil. Mag. xliii (1897), p. 1.
* ↑ Phil. Trans. clxxxvi (1895), p. 697.
* ↑ Arch. Néerl (3) v (1900), p. 96.
* ↑ Experimental evidence that the inertia of elections is purely electromagnetic was afterwards furnished by W. Kaufmann, Gött. Nach, 1901, p. 143; 1902, p 291.
* ↑ Cf., e.g., c. Niven, Phil. Trans. clxxii (1881), p. 307; H. Lamb. Phil. Trans. clxxiv (1883), p. 519; J.J. Thomson, Proc. Lond. Math. Soc. xv (1884), p. 197: H. A. Rowland, Phil. Mag. xvii (1884), p. 413; J. Thomson, Proc. Lond. Math. Soc. xvii (1886), p. 310; xix (1888), p. 520; and many investigations of Oliver Heaviside, collected in his Electrical Papers.
* ↑ Loc. cit.
* ↑ Electrician, Jan. 1885.
* ↑ The mathematical theory was given by Lord Rayleigh, Phil. Mag. xxi. (1886), p. 381. Cf. Maxwell's Treatise, § 689.
* ↑ Cf. p. 313.
* ↑ As was first remarked by Lord Rayleigh, Phil. Mag. xiii (1882), p. 314.
* ↑ Cf. J. Stefan, Wiener, Sitzungsber. xcix (1890), p. 319; Ann. d. Phys. xli (1890), p. 400.
* ↑ Trane, Roy. Dublin Soc. iii (1883); FitzGerald's Scient. Writings, p. 122.
* ↑ Cf. pp. 298, 299.
* ↑ The other term, which is neglected, is very small compared to the term retained, at great distances from the origin; it is what would be obtained if the effects of induction of the displacement-currents were neglected: i.e. it is the energy of the forced displacement-currents which are produced directly by the variation of the primary current, and which originate the radiating displacement-currents.
* ↑ Brit. Assoc. Rep., 1883; FitzGerald's Scientific Writings, p. 129.
* ↑ Phil. Trans. clxxv (1884), p. 343.
* ↑ Cf. pp. 248, 250, 282.
* ↑ Of course any circuital vector may be added. II. M. Macdonald, Electric Waves, p. 72, propounded a form which differs from Poynting's by a non-circuital vector.
* ↑ The analogue of Poynting's theorem in the theory of the vibrations of an isotropic elastic solid may be easily obtained; for from the equation of motion of an elastic solid, $\rho {\ddot {\mathbf {e} }}=(k+4n/3){\text{ grad div }}\mathbf {e} -n{\text{ curl curl }}\mathbf {e}$, it follows that ${\frac {\partial }{\partial t}}\{{\tfrac {1}{2}}\rho \mathbf {\dot {e}} ^{2}+{\tfrac {1}{2}}(k+{\tfrac {4}{3}}n)({\text{div }}\mathbf {e} )^{2}+{\tfrac {1}{2}}n({\text{curl }}\mathbf {e} )^{2}\}=-{\text{div }}\mathbf {W}$, where W denotes the vector $-(k+4n/3){\text{div }}\mathbf {e.e} +n[{\text{curl }}\mathbf {e.{\dot {e}}} ]$; and since the expression which is differentiated with respect to t represents the sum of the kinetic and potential energies per unit volume of the solid (save for terms which give only surface-integrals), it is seen that W is the analogue of the Poynting vector. Cf. L. Donati, Bologna Mem. (5) vii (1899), p. 633.
* ↑ Phil. Trans, clxxvi (1885), p. 277.
* ↑ Phil. Mag. xxxi (1891), p. 149; Thomson's Recent Researches in Elect, and Mag. (1893), chapter i.
* ↑ The hypothesis that the aether is a store house of mechanical momentum, which was first advanced by J. J. Thomson (Recent Researches in Elect, and Mug. (1893), p. 13), was afterwards developed by H. Poincaré, Archives Neérl. (2) v (1900), p. 252, and by M. Abraham, Gött, Nach., 1902, p. 20.
* ↑ Cf. p. 802.
* ↑ It was, however, adopted by G. T. Walker, Aberration and the Electromagnetic Field, Camb., 1900.
* ↑ Berlin Sitzungsberichte, 1893, p. 649; Am. d. Phys. lii (1894), p. 135. Helmholtz supposed the aether to behave as a frictionless incompressible fluid.
* ↑ Loc. cit.
* ↑ Ant. d. Phys. xxiii (1884), p. 84: English version in Hertz's Miscellaneous Pipers, translated by D. E. Jones and G. A. Schott, p. 273.
* ↑ $\alpha ={\text{Pot }}\beta$ is used to denote the solution of the equation $\nabla ^{2}\alpha +4\pi \beta =0$.
* ↑ Cf. p. 248.
* ↑ Cf. p. 287.
* ↑ Lorberg, Ann. d. Phys. xxvii (1886), p. 666; xxxi (1887), p. 131. Boltzmann, ibid. xxix (1886), p. 598.
* ↑ Ann. d. Phys. xxvii (1886), p. 119.
* ↑ Ann. d. Phys. xxxi (1887), p. 421. Hertz's Electric Waves, translated by D. E. Jones, p. 29.
* ↑ Unknown to Hertz, the transmission of electric waves along wires had been observed in 1870 by Wilhelm von Bezold, München Sitzungsberichte, i (1870), p. 113; Phil. Mag. xl (1870), p. 42. "If," he wrote at the conclusion of a series of experiments, "electrical waves be sent into a wire insulated at the end, they will be reflected at that end. The phenomena which accompany this process in alternating discharges appear to owe their origin to the interference of the advancing and reflected waves," and, "an electric discharge travels with the same rapidity in wires of equal length, without reference to the materials of which these wires are made." The subject was investigated by O. J. Lodge and A. P. Chattock at almost the same time as Hertz's experiments were being curried out: mention was made of their researches at the meeting of the British Association in 1888.
* ↑ Loc. cit.
* ↑ Sir Oliver Lodge was about this time independently studying electric oscillations in air in connexion with the theory of lightning-conductors: cf. Lodge, Phil. Mag. xxvi (1888), p. 217. So long before as 1842, Joseph Henry, of Washington, had noticed that the inductive effects of the Leyden jar discharge could be observed at considerable distances, and had even suggested a comparison with "a spark from flint and steel in the case of light."
* ↑ Cf. p. 253.
* ↑ Ann. d. Phys. xxxiv, p. 373. Electric Waves (English edition), p. 95.
* ↑ Ann. d. Phys. xxxiv (1888), p. 551. Electric Waves (English edition) p. 107.
* ↑ Hertz's experiments gave the value 45/28 for the ratio of the velocity of electric waves in air to the velocity of electric waves conducted by the wires, und 2 x 10$^{10}$ ems. per sec. for the latter velocity. These numbers were afterwards found to be open to objection: Poincaré (Comptes Rendus, cxi (1890). p. 322) showed that the period calculated by Hertz was ${\sqrt {2}}\times$ the true period, which would make the velocity of propagation in air equal to that of light $\times {\sqrt {2}}$ . Ernst Lecher (Wiener Berichte, May 8, 1890; Phil. Mag. xxx (1890), p. 128), experimenting on the velocity of propagation of electric vibrations in wires, found instead of Hertz's 2 x 10$^{10}$ ems. per sec., a value within two per cent. of the velocity of light. E. Sarasin and L. De La Rive at Geneva (Archives des Sc. Phys. xxix (1893)) finally proved that the velocities of propagation in air and along wires are equal.
* ↑ Ann. d. Phys. xxxix (1898), p. 610. Electric Waves (English edition), p. 124.
* ↑ Ibid., xxxvi (1889), p. 1. Electric Waves (English edition), p. 137.
* ↑ Cf. Karl Pearson and A. Lee, Phil, Trans, cxciii (1899), p. 165.
* ↑ Brit. Assoc. Rep., Leeds (1890), p. 755.
* ↑ Cf. p. 298. The use of retarded potentials was also recommended in the following year by Poincaré, Comptes Rendus, cxiii (1891), p. 515.
* ↑ Cf. p. 346.
* ↑ Proc. Camb. Phil. Soc, vii (1891), p. 165.
* ↑ Ann. d. Phys. xxxvi (1889), p. 769; Electric Waves (English ed.), p. 172.
* ↑ O. J. Lodge and J. L. Howard in the same year showed that electric radiation might be refracted and concentrated by means of large lenses. Cf. Phil. Mag. xxvii (1889), p. 48.
* ↑ Cf. pp. 168 et sqq.
* ↑ Nature, xxxix (1889), p. 391.
* ↑ Ann. d. Phys. xl (1890), p. 203. Cf. a controversy regarding the results; Comptes Rendus, cxii (1891), pp. 186, 325, 329, 365, 383, 456; and Ann. d. Phys. xli (1890), p. 154; xliii (1891), p. 177; xlviii (1893), p. 119.
* ↑ Gott. Nach. 1890, p. 106; Ann, d. Phys. xl (1890), p. 577; Electric Waves (English ed.), p. 195. In this memoir Hertz advocated the form of the equations which Maxwell had used in his paper of 1868 (cf. supra, p. 287) in preference to the earlier farm, which involved the scalar and vector potentials.
* ↑ Ann. d. Phys. xli (1890), p. 369; Electric Waves (English ed.), p. 241. The propagation of light through a moving dielectric bad been discussed previously, on the basis of Maxwell's equations for moving bodies, by J.J. Thomason, Phil. Mag. ix (1880), p. 284; Proc. Camb. Phil. Soc. v (1885), p. 250.
* ↑ Cf. p. 288.
* ↑ Heaviside's general theory was published in a series of papers in the Electrician, from 1885 onwards. His earlier work was republished in his Electrical Papers (2 vols., 1892), and his Electromagnetic Theory (2 vols., 1894). Mention may be specially made of a memoir in Phil. Trans. clxxxiii (1892), P. 423.
* ↑ Cf. p. 308.
* ↑ Phil. Trans., 1879, p. 691. FitzGerald's Scient. Writings, p. 45.
* ↑ Phil. M (5) iii (1877), p. 321.
* ↑ Proc. R. S. xxv (1877), p. 447; FitzGerald's Scient. Writings, p. 9.
* ↑ Cf. Larmor's remarks in his Report on the Action of Magnetism on Light, Brit. Assoc. Rep., 1893; and his editorial comments in FitzGerald's Scientific Writings. Larmor traced to its source an inconsistency in the equations by which FitzGerald had represented the boundary-conditions at an interface between the media. FitzGerald had indeed made the mistake, similar to that which was so often made by the earlier writers on the elastic-solid theory of light, of forgetting that when a medium is assumed to be incompressible, the condition of incompressibility must be introduced into the variational equation of motion (as was done supra, p. 172). Larmor showed that when this correction was made, new terms (resembling the terms in p, supra, p. 172) made their appearance; and the inconsistency in the equations was thus removed.
* ↑ Amer. Jour. Math. ii, p. 354, iii, p. 89; Phil. Mag. xi (1881), p. 254.
* ↑ Cf. p. 327.
* ↑ Cf. p. 308.
* ↑ The theory of Basset (Phil. Trans. clxxxii (1891), p. 371) was, like Rowland's, based on the idea of extending Hall's phenomenon to dielectric media. An objection to this theory was that the tangential component of the electromotive force was not continuous a cross the interface between a magnetized and an unmagnetized medium; but Basset subsequently overcame this difficulty (Nature, lii (1895), p. 618; liii (1895), p. 130; Amer. Jour. Math. xix (1897), p. 60)—the effect analogous to Hall's being introduced into the equation connecting electric displacement with electric force, so that the equation took the form $\mathbf {E} =(4\pi c^{2}/\epsilon )\mathbf {D} +\sigma [\mathbf {K.{\dot {D}}} ]$. Basset, in 1893 (Proc. Camb. Phil. Soc. viii, p. 68), derived analytical expressions which represent Kerr's magneto-optic phenomenon by substituting a complex quantity for the refractive index in the formulae applicable to transparent magnetized substances. The magnetic rotation of light and Kerr's phenomenon have been investigated also by R. T. Glazebrook, Phil. Mag, xi (1881), p. 397; by J. J. Thomson, Recent Researches, p. 482: by D. A. Goldhammer, Ann. d. Phys. xlvi (1892), p. 71; xlvii (1892), p. 345; xlviii (1893), p.740; 1 (1893), p. 772: by P. Drude, Ann. d. Phys. xlvi (1892), p. 353; xlviii (1893), p. 122; xlix (1893), p. 690; lii (1894)) p. 496: by C. H. Wind, Verslagen Kon, Akad, Amsterdam, 29th Sept., 1894: by Reiff, Ann. d. Phys. lvii (1896), p. 281: by J. G. Leathem, Phil. Trans. exe (1897), p. 89; Trans. Camb. Phil. Soc. xvii (1898), p. 16: and by W. Voigt in many memoirs, und in his treatise, Magneto- und Elektro-optik. Larmor's report presented to the British Association in 1893 has been already mentioned. In most of the later theories the equations of propagation of light in magnetized metals are derived from the two fundamental electromagnetic equations ${\text{curl }}\mathbf {H} =4\pi \mathbf {S} ,\qquad {\text{curl }}\mathbf {E} =\mathbf {\dot {H}}$; the total current S being assumed to consist of a part (the displacement-current) proportional to Ė, a part (the conduction-current) proportional to E, and a part proportional to the vector-product of E and the magnetization. Various mechanical models of media in which magneto-optic phenomena take place have been devised at different times. W. Thomson (Proc. Lond. Math. Soc. vi (1875)) investigated the propagation of waves of displacement along a stretched chain whose links contain rotating fly-wheels: cf. also Larmor, Proc. Lond. Math. Soc. xxi. (1890), p.423; xxiii (1891), p. 127; F. Hasenöhrl, Wien Sitzungsherichte cvii, 2a (1898), p. 1015; W. Thomson (Kelvin), Phil. Mag. xlviii (1899), p. 236, and Baltimore Lectures; and FitzGerald, Electrician, Aug. 4, 1899, FitzGerald's Scientific Writings, p. 481.
* ↑ Journal de Physique (3) i, pp. 205, 345.
* ↑ Cf. p. 186.
* ↑ Cf. p. 175.