A Treatise on Electricity and Magnetism/Part IV/Chapter XX

by James Clerk Maxwell

CHAPTER XX.

ELECTROMAGNETIC THEORY OF LIGHT.

781.] In several parts of this treatise an attempt has been made to explain electromagnetic phenomena by means of mechanical action transmitted from one body to another by means of a medium occupying the space between them. The undulatory theory of light also assumes the existence of a medium. We have now to shew that the properties of the electromagnetic medium are identical with those of the luminiferous medium.

To fill all space with a new medium whenever any new phenomenon is to be explained is by no means philosophical, but if the study of two different branches of science has independently suggested the idea of a medium, and if the properties which must be attributed to the medium in order to account for electromagnetic phenomena are of the same kind as those which we attribute to the luminiferous medium in order to account for the phenomena of light, the evidence for the physical existence of the medium will be considerably strengthened.

But the properties of bodies are capable of quantitative measurement. We therefore obtain the numerical value of some property of the medium, such as the velocity with which a disturbance is propagated through it, which can be calculated from electromagnetic experiments, and also observed directly in the case of light. If it should be found that the velocity of propagation of electromagnetic disturbances is the same as the velocity of light, and this not only in air, but in other transparent media, we shall have strong reasons for believing that light is an electromagnetic phenomenon, and the combination of the optical with the electrical evidence will produce a conviction of the reality of the medium similar to that which we obtain, in the case of other kinds of matter, from the combined evidence of the senses. 782.] When light is emitted, a certain amount of energy is expended by the luminous body, and if the light is absorbed by another body, this body becomes heated, shewing that it has received energy from without. During the interval of time after the light left the first body and before it reached the second, it must have existed as energy in the intervening space.

According to the theory of emission, the transmission of energy is effected by the actual transference of light-corpuscules from the luminous to the illuminated body, carrying with them their kinetic energy, together with any other kind of energy of which they may be the receptacles.

According to the theory of undulation, there is a material medium which fills the space between the two bodies, and it is by the action of contiguous parts of this medium that the energy is passed on, from one portion to the next, till it reaches the illuminated body.

The luminiferous medium is therefore, during the passage of light through it, a receptacle of energy. In the undulatory theory, as developed by Huygens, Fresnel, Young, Green, &c., this energy is supposed to be partly potential and partly kinetic. The potential energy is supposed to be due to the distortion of the elementary portions of the medium. We must therefore regard the medium as elastic. The kinetic energy is supposed to be due to the vibratory motion of the medium. We must therefore regard the medium as having a finite density.

In the theory of electricity and magnetism adopted in this treatise, two forms of energy are recognised, the electrostatic and the electrokinetic (see Arts. 630 and 636), and these are supposed to have their seat, not merely in the electrified or magnetized bodies, but in every part of the surrounding space, where electric or magnetic force is observed to act. Hence our theory agrees with the undulatory theory in assuming the existence of a medium which is capable of becoming a receptacle of two forms of energy ^[1].

783.] Let us next determine the conditions of the propagation of an electromagnetic disturbance through a uniform medium, which we shall suppose to be at rest, that is, to have no motion except that which may be involved in electromagnetic disturbances. Let $C$ be the specific conductivity of the medium, $K$ its specific capacity for electrostatic induction, and $\mu$ its magnetic permeability.

To obtain the general equations of electromagnetic disturbance, we shall express the true current ${\mathfrak {C}}$ in terms of the vector potential ${\mathfrak {A}}$ and the electric potential $\Psi$.

The true current ${\mathfrak {C}}$ is made up of the conduction current ${\mathfrak {i}}$ and the variation of the electric displacement ${\dot {\mathfrak {D}}}$, and since both of these depend on the electromotive force ${\mathfrak {E}}$, we find, as in Art. 611,

${\mathfrak {C}}=\left(C+{\frac {1}{4\pi }}K{\frac {d}{dt}}\right){\mathfrak {E}}$

(1)

But since there is no motion of the medium, we may express the electromotive force, as in Art. 599,

${\mathfrak {E}}=-{\dot {\mathfrak {A}}}-\nabla \Psi .$

(2)

Hence

${\mathfrak {C}}=-\left(C+{\frac {1}{4\pi }}K{\frac {d}{dt}}\right)\left({\frac {d{\mathfrak {A}}}{dt}}+\nabla \Psi \right).$

(3)

But we may determine a relation between ${\mathfrak {C}}$ and ${\mathfrak {A}}$ in a different way, as is shewn in Art. 616, the equations (4) of which may be written

$4\pi \mu {\mathfrak {C}}=\nabla ^{2}{\mathfrak {A}}+\nabla J,$

(4)

where

$J={\frac {dF}{dx}}+{\frac {dG}{dy}}+{\frac {dH}{dz}}$

(5)

Combining equations (3) and (4), we obtain

$\mu \left(4\pi C+K{\frac {d}{dt}}\right)\left({\frac {d{\mathfrak {A}}}{dt}}+\nabla \Psi \right)+\nabla ^{2}{\mathfrak {A}}+\nabla J=0,$

(6)

which we may express in the form of three equations as follows―

$\left.{\begin{aligned}\mu \left(4\pi C+K{\frac {d}{dt}}\right)\left({\frac {dF}{dt}}+{\frac {d\Psi }{dx}}\right)+\nabla ^{2}F+{\frac {dJ}{dx}}&=0,\\\mu \left(4\pi C+K{\frac {d}{dt}}\right)\left({\frac {dG}{dt}}+{\frac {d\Psi }{dy}}\right)+\nabla ^{2}G+{\frac {dJ}{dy}}&=0,\\\mu \left(4\pi C+K{\frac {d}{dt}}\right)\left({\frac {dH}{dt}}+{\frac {d\Psi }{dz}}\right)+\nabla ^{2}H+{\frac {dJ}{dz}}&=0.\end{aligned}}\right\}$

(7)

These are the general equations of electromagnetic disturbances.

If we differentiate these equations with respect to $x$, $y$ and $z$ respectively, and add, we obtain

$\mu \left(4\pi C+K{\frac {d}{dt}}\right)\left({\frac {dJ}{dt}}-\nabla ^{2}\Psi \right)=0.$

(8)

If the medium is a non-conductor, $C=0$, and $\nabla ^{2}\Psi$, which is proportional to the volume-density of free electricity, is independent of $t$. Hence $J$ must be a linear function of $t$, or a constant, or zero, and we may therefore leave $J$ and $\Psi$ out of account in considering periodic disturbances.

Propagation of Undulations in a Non-conducting Medium.

784.] In this case $C=0$, and the equations become

$\left.{\begin{aligned}K\mu {\frac {d^{2}F}{dt^{2}}}+\nabla ^{2}F&=0,\\K\mu {\frac {d^{2}G}{dt^{2}}}+\nabla ^{2}G&=0,\\K\mu {\frac {d^{2}H}{dt^{2}}}+\nabla ^{2}H&=0.\end{aligned}}\right\}$

(9)

The equations in this form are similar to those of the motion of an elastic solid, and when the initial conditions are given, the solution can be expressed in a form given by Poisson ^[2], and applied by Stokes to the Theory of Diffraction^[3].

Let us write

$V={\frac {1}{\sqrt {K\mu }}}.$

(10)

If the values of $F$, $G$, $H$, and of ${\frac {dF}{dt}}$, ${\frac {dG}{dt}}$, ${\frac {dH}{dt}}$ are given at every point of space at the epoch ($t=0$), then we can determine their values at any subsequent time, $t$, as follows.

Let $O$ be the point for which we wish to determine the value of $F$ at the time $t$. With $O$ as centre, and with radius $Vt$, describe a sphere. Find the initial value of $F$ at every point of the spherical surface, and take the mean, ${\overline {F}}$, of all these values. Find also the initial values of ${\frac {dF}{dt}}$ at every point of the spherical surface, and let the mean of these values be ${\frac {\overline {dF}}{dt}}$.

Then the value of $F$ at the point $O$, at the time $t$, is

$\left.{\begin{aligned}F&={\frac {d}{dt}}({\overline {F}}t)+t{\frac {\overline {dF}}{dt}},\\{\text{Similarly}}\;\;G&={\frac {d}{dt}}({\overline {G}}t)+t{\frac {\overline {dG}}{dt}},\\H&={\frac {d}{dt}}({\overline {H}}t)+t{\frac {\overline {dH}}{dt}}.\end{aligned}}\right\}$

(11)

785.] It appears, therefore, that the condition of things at the point $O$ at any instant depends on the condition of things at a distance $Vt$ and at an interval of time $t$ previously, so that any disturbance is propagated through the medium with the velocity $V$.

Let us suppose that when $t$ is zero the quantities ${\mathfrak {A}}$ and ${\dot {\mathfrak {A}}}$ are zero except within a certain space $S$. Then their values at at the time $t$ will be zero, unless the spherical surface described about $O$ as centre with radius $Vt$ lies in whole or in part within the space $S$. If $O$ is outside the space $S$ there will be no disturbance at $O$ until $Vt$ becomes equal to the shortest distance from $O$ to the space $S$. The disturbance at $O$ will then begin, and will go on till $Vt$ is equal to the greatest distance from $O$ to any part of $S$. The disturbance at $O$ will then cease for ever.

786.] The quantity $V$, in Art. 793, which expresses the velocity of propagation of electromagnetic disturbances in a non-conducting medium is, by equation (9), equal to ${\frac {1}{\sqrt {K\mu }}}$.

If the medium is air, and if we adopt the electrostatic system of measurement, $K=1$ and $\mu ={\frac {1}{v^{2}}}$, so that $V=v$, or the velocity of propagation is numerically equal to the number of electrostatic units of electricity in one electromagnetic unit. If we adopt the electromagnetic system, $K={\frac {1}{v^{2}}}$ and $\mu =1$, so that the equation $V=v$ is still true.

On the theory that light is an electromagnetic disturbance, propagated in the same medium through which other electromagnetic actions are transmitted, $V$ must be the velocity of light, a quantity the value of which has been estimated by several methods. On the other hand, $v$ is the number of electrostatic units of electricity in one electromagnetic unit, and the methods of determining this quantity have been described in the last chapter. They are quite independent of the methods of finding the velocity of light. Hence the agreement or disagreement of the values of $V$ and of $v$ furnishes a test of the electromagnetic theory of light.

787.] In the following table, the principal results of direct observation of the velocity of light, either through the air or through the planetary spaces, are compared with the principal results of the comparison of the electric units:―

Velocity of Light (mètres per second).			Ratio of Electric Units.
Fizeau		314000000	Weber	310740000
Aberration, &c., and Sun's Parallax	$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$	308000000	Maxwell	288000000
Foucault		298360000	Thomson	282000000

It is manifest that the velocity of light and the ratio of the units are quantities of the same order of magnitude. Neither of them can be said to be determined as yet with such a degree of accuracy as to enable us to assert that the one is greater or less than the other. It is to be hoped that, by further experiment, the relation between the magnitudes of the two quantities may be more accurately determined.

In the meantime our theory, which asserts that these two quantities are equal, and assigns a physical reason for this equality, is certainly not contradicted by the comparison of these results such as they are.

788.] In other media than air, the velocity $V$ is inversely proportional to the square root of the product of the dielectric and the magnetic inductive capacities. According to the undulatory theory, the velocity of light in different media is inversely proportional to their indices of refraction.

There are no transparent media for which the magnetic capacity differs from that of air more than by a very small fraction. Hence the principal part of the difference between these media must depend on their dielectric capacity. According to our theory, therefore, the dielectric capacity of a transparent medium should be equal to the square of its index of refraction.

But the value of the index of refraction is different for light of different kinds, being greater for light of more rapid vibrations. We must therefore select the index of refraction which corresponds to waves of the longest periods, because these are the only waves whose motion can be compared with the slow processes by which we determine the capacity of the dielectric.

789.] The only dielectric of which the capacity has been hitherto determined with sufficient accuracy is paraffin, for which in the solid form M. M. Gibson and Barclay found^[4]

$K=1.975.$

(12)

Dr. Gladstone has found the following values of the index of refraction of melted paraffin, ${\text{sp. g. 0.779}}$, for the lines A, D and H:―

Temperature	A	D	H
54℃	1.4306	1.4357	1.4499
57℃	1.4294	1.4343	1.4493

from which I find that the index of refraction for waves of infinite length would be about ${\text{1.422.}}$

The square root of $K$ is ${\text{1.405.}}$

The difference between these numbers is greater than can be ac counted for by errors of observation, and shews that our theories of the structure of bodies must be much improved before we can deduce their optical from their electrical properties. At the same time, I think that the agreement of the numbers is such that if no greater discrepancy were found between the numbers derived from the optical and the electrical properties of a considerable number of substances, we should be warranted in concluding that the square root of $K$, though it may not be the complete expression for the index of refraction, is at least the most important term in it.

Plane Waves.

790.] Let us now confine our attention to plane waves, the front of which we shall suppose normal to the axis of $z$. All the quantities, the variation of which constitutes such waves, are functions of $z$ and $t$ only, and are independent of $x$ and $y$. Hence the equations of magnetic induction, (A), Art. 591, are reduced to

$a=-{\frac {dG}{dz}},\;b={\frac {dF}{dz}},\;c=0,$

(13)

or the magnetic disturbance is in the plane of the wave. This agrees with what we know of that disturbance which constitutes light.

Putting $\mu \alpha$, $\mu \beta$ and $\mu \gamma$ for $a$, $b$ and $c$ respectively, the equations of electric currents, Art. 607, become

$\left.{\begin{aligned}4\pi \mu u&=-{\frac {db}{dz}}&=&-{\frac {d^{2}F}{dz^{2}}},\\4\pi \mu v&=\;\;{\frac {da}{dz}}&=&-{\frac {d^{2}G}{dz^{2}}},\\4\pi \mu w&=0.\end{aligned}}\right\}$

(14)

Hence the electric disturbance is also in the plane of the wave, and if the magnetic disturbance is confined to one direction, say that of $x$, the electric disturbance is confined to the perpendicular direction, or that of $y$.

But we may calculate the electric disturbance in another way, for ii $f,g,h$ are the components of electric displacement in a non conducting medium

$u={\frac {df}{dt}},\;v={\frac {dg}{dt}},\;v={\frac {dh}{dt}}.$

(15)

If $P,Q,R$ are the components of the electromotive force

$f={\frac {K}{4\pi }}P,\;g={\frac {K}{4\pi }}Q,\;h={\frac {K}{4\pi }}R;$

(16)

and since there is no motion of the medium, equations (B), Art. 598, become

$P=-{\frac {dF}{dt}}$,⁠$Q=-{\frac {dG}{dt}}$,⁠$R=-{\frac {dH}{dt}}$.

(17)

Hence

$u=-{\frac {K}{4\pi }}{\frac {d^{2}F}{dt^{2}}}$,⁠$v=-{\frac {K}{4\pi }}{\frac {d^{2}G}{dt^{2}}}$,⁠$w=-{\frac {K}{4\pi }}{\frac {d^{2}F}{dt^{2}}}$.

(18)

Comparing these values with those given in equation (14), we find

$\left.{\begin{aligned}{\frac {d^{2}F}{dz^{2}}}&=K\mu {\frac {d^{2}F}{dt^{2}}}{\mbox{,}}\\{\frac {d^{2}G}{dz^{2}}}&=K\mu {\frac {d^{2}G}{dt^{2}}}{\mbox{,}}\\0&=K\mu {\frac {d^{2}H}{dt^{2}}}\end{aligned}}\right\}$

(19)

The first and second of these equations are the equations of propagation of a plane wave, and their solution is of the well-known form

$\left.{\begin{aligned}F&=f_{1}(z-Vt)+f_{2}(z+Vt){\mbox{,}}\\G&=f_{3}(z-Vt)+f_{4}(z+Vt){\mbox{,}}\end{aligned}}\right\}$

(20)

The solution of the third equation is

$K\mu H=A+Bt$,

(21)

where $A$ and $B$ are functions of $z$. $H$ is therefore either constant or varies directly with the time. In neither case can it take part in the propagation of waves.

Fig. 66. 791.] It appears from this that the directions, both of the magnetic and the electric disturbances, lie in the plane of the wave. The mathematical form of the disturbance therefore, agrees with that of the disturbance which constitutes light, in being transverse to the direction of propagation.

If we suppose $G=0$, the disturbance will correspond to a plane-polarized ray of light.

The magnetic force is in this case parallel to the axis of $y$ and equal to ${\frac {1}{\mu }}{\frac {dF}{dz}}$, and the electromotive force is parallel to the axis of $x$ and equal to $-{\frac {dF}{dt}}$. The magnetic force is therefore in a plane perpendicular to that which contains the electric force.

The values of the magnetic force and of the electromotive force at a given instant at different points of the ray are represented in Fig. 66, for the case of a simple harmonic disturbance in one plane. This corresponds to a ray of plane-polarized light, but whether the plane of polarization corresponds to the plane of the magnetic disturbance, or to the plane of the electric disturbance, remains to be seen. See Art. 797.

 

Energy and Stress of Radiation.

792.] The electrostatic energy per unit of volume at any point of the wave in a non-conducting medium is

${\frac {1}{2}}fP={\frac {K}{8\pi }}P^{2}=\left.{\frac {K}{8\pi }}{\frac {\overline {dF}}{dt}}\right|^{2}$.

(22)

The electrokinetic energy at the same point is

${\frac {1}{8\pi }}b\beta ={\frac {1}{8\pi \mu }}b^{2}=\left.{\frac {1}{8\pi \mu }}{\frac {\overline {dF}}{dz}}\right|^{2}$

(23)

In virtue of equation (8) these two expressions are equal, so that at every point of the wave the intrinsic energy of the medium is half electrostatic and half electrokinetic.

Let $p$ be the value of either of these quantities, that is, either the electrostatic or the electrokinetic energy per unit of volume, then, in virtue of the electrostatic state of the medium, there is a tension whose magnitude is $p$, in a direction parallel to $x$, combined with a pressure, also equal to $p$, parallel to $y$ and $z$. See Art. 107.

In virtue of the electrokinetic state of the medium there is a tension equal to $p$ in a direction parallel to $y$, combined with a pressure equal to $p$ in directions parallel to $x$ and $z$. See Art. 643.

Hence the combined effect of the electrostatic and the electrokinetic stresses is a pressure equal to $2p$ in the direction of the propagation of the wave. Now $2p$ also expresses the whole energy in unit of volume.

Hence in a medium in which waves are propagated there is a pressure in the direction normal to the waves, and numerically equal to the energy in unit of volume.

793.] Thus, if in strong sunlight the energy of the light which falls on one square foot is 83.4 foot pounds per second, the mean energy in one cubic foot of sunlight is about 0.0000000882 of a foot pound, and the mean pressure on a square foot is 0.0000000882 of a pound weight. A flat body exposed to sunlight would experience this pressure on its illuminated side only, and would therefore be repelled from the side on which the light falls. It is probable that a much greater energy of radiation might be obtained by means of the concentrated rays of the electric lamp. Such rays falling on a thin metallic disk, delicately suspended in a vacuum, might perhaps produce an observable mechanical effect. When a disturbance of any kind consists of terms involving sines or cosines of angles which vary with the time, the maximum energy is double of the mean energy. Hence, if $P$ is the maximum electromotive force, and $\beta$ the maximum magnetic force which are called into play during the propagation of light,

${\frac {K}{8\pi }}P^{2}={\frac {\mu }{8\pi }}\beta ^{2}={}$ mean energy in unit of volume.

(24)

With Pouillet's data for the energy of sunlight, as quoted by Thomson, Trans. R.S.E., 1854, this gives in electromagnetic measure

$P={}$ 60000000, or about 600 Dainell's cells per metre;

$\beta ={}$ 0.193, or rather more than a tenth of the horizontal magnetic force in Britain.

 

Propagation of a Plane Wave in a Crystallized Medium.

794.] In calculating, from data furnished by ordinary electromagnetic experiments, the electrical phenomena which would result from periodic disturbances, millions of millions of which occur in a second, we have already put our theory to a very severe test, even when the medium is supposed to be air or vacuum. But if we attempt to extend our theory to the case of dense media, we become involved not only in all the ordinary difficulties of molecular theories, but in the deeper mystery of the relation of the molecules to the electromagnetic medium.

To evade these difficulties, we shall assume that in certain media the specific capacity for electrostatic induction is different in different directions, or in other words, the electric displacement, instead of being in the same direction as the electromotive force, and proportional to it, is related to it by a system of linear equations similar to those given in Art. 297. It may be shewn, as in Art. 436, that the system of coefficients must be symmetrical, so that, by a proper choice of axes, the equations become

$f={\frac {1}{4\pi }}K_{1}P$,⁠$g={\frac {1}{4\pi }}K_{2}Q$,⁠$h={\frac {1}{4\pi }}K_{3}R$,

(1)

where $K_{1}$, $K_{2}$, and $K_{3}$ are the principal inductive capacities of the medium. The equations of propagation of disturbances are therefore

$\left.{\begin{aligned}{\frac {d^{2}F}{dy^{2}}}+{\frac {d^{2}F}{dz^{2}}}-{\frac {d^{2}G}{dx\,dy}}-{\frac {d^{2}H}{dz\,dx}}&=K_{1}\mu \left({\frac {d^{2}F}{dt^{2}}}-{\frac {d^{2}\Psi }{dx\,td}}\right){\mbox{,}}\\{\frac {d^{2}G}{dz^{2}}}+{\frac {d^{2}G}{dx^{2}}}-{\frac {d^{2}H}{dy\,dz}}-{\frac {d^{2}F}{dx\,dy}}&=K_{2}\mu \left({\frac {d^{2}G}{dt^{2}}}-{\frac {d^{2}\Psi }{dy\,dt}}\right){\mbox{,}}\\{\frac {d^{2}H}{dx^{2}}}+{\frac {d^{2}H}{dy^{2}}}-{\frac {d^{2}F}{dz\,dx}}-{\frac {d^{2}G}{dy\,dz}}&=K_{3}\mu \left({\frac {d^{2}H}{dt^{2}}}-{\frac {d^{2}\Psi }{dz\,dt}}\right){\mbox{,}}\end{aligned}}\right\}$

(2)

795.] If $l$, $m$, $n$ are the direction-cosines of the normal to the wave-front, and $V$ the velocity of the wave, and if

$lx+my+nz-Vt=w$,

(3)

and if we write $F^{\prime \prime }$, $G^{\prime \prime }$, $H^{\prime \prime }$, $\Psi ^{\prime \prime }$ for the second differential coefficients of $F$, $G$, $H$, $\Psi$ respectively with respect to $w$, and put

$K_{1}\mu ={\frac {1}{a^{2}}}$,⁠$K_{2}\mu ={\frac {1}{b^{2}}}$,⁠$K_{3}\mu ={\frac {1}{c^{2}}}$,

(4)

where $a$, $b$, $c$ are the three principal velocities of propagation, the equations become

$\left.{\begin{aligned}\left(m^{2}+n^{2}-{\frac {V^{2}}{a^{2}}}\right)F^{\prime \prime }-lmG^{\prime \prime }-nlH^{\prime \prime }-V\Psi ^{\prime \prime }{\frac {l}{a^{2}}}&=0{\mbox{,}}\\-lmF^{\prime \prime }+\left(n^{2}+l^{2}-{\frac {V^{2}}{b^{2}}}\right)G^{\prime \prime }-mnH^{\prime \prime }-V\Psi ^{\prime \prime }{\frac {m}{b^{2}}}&=0{\mbox{,}}\\-nlF^{\prime \prime }-mnG^{\prime \prime }+\left(l^{2}+m^{2}-{\frac {V^{2}}{c^{2}}}\right)H^{\prime \prime }-V\Psi ^{\prime \prime }{\frac {n}{b^{2}}}&=0{\mbox{.}}\end{aligned}}\right\}$

(5)

796.] If we write

${\frac {l^{2}}{V^{2}-a^{2}}}+{\frac {m^{2}}{V^{2}-b^{2}}}+{\frac {n^{2}}{V^{2}-c^{2}}}=U$,

(6)

we obtain from these equations

$\left.{\begin{aligned}&VU(VF^{\prime \prime }-l\Psi ^{\prime \prime })&=0{\mbox{,}}\\&VU(VG^{\prime \prime }-m\Psi ^{\prime \prime })&=0{\mbox{,}}\\&VU(VH^{\prime \prime }-n\Psi ^{\prime \prime })&=0{\mbox{.}}\end{aligned}}\right\}$

(7)

Hence, either $V=0$, in which case the wave is not propagated at all; or, $U=0$, which leads to the equation for $V$ given by Fresnel; or the quantities within brackets vanish, in which case the vector whose components are $F^{\prime \prime }$, $G^{\prime \prime }$, $H^{\prime \prime }$ is normal to the wave-front and proportional to the electric volume-density. Since the medium is a non-conductor, the electric density at any given point is constant, and therefore the disturbance indicated by these equations is not periodic, and cannot constitute a wave. We may therefore consider $\Psi ^{\prime \prime }=0$ in the investigation of the wave.

797.] The velocity of the propagation of the wave is therefore completely determined from the equation $U=0$, or

${\frac {l^{2}}{V^{2}-a^{2}}}+{\frac {m^{2}}{V^{2}-b^{2}}}+{\frac {n^{2}}{V^{2}-c^{2}}}=0$,

(8)

There are therefore two, and only two, values of $V^{2}$ Corresponding to a given direction of wave-front.

If $\lambda$, $\mu$, $\nu$ are the direction-cosines of the electric current whose components are $u$, $v$, $w$

$\lambda :\mu :\nu ::{\frac {1}{a^{2}}}F^{\prime \prime }:{\frac {1}{b^{2}}}G^{\prime \prime }:{\frac {1}{c^{2}}}H^{\prime \prime }$,

(9)

then

$l\lambda +m\mu +n\nu =0$;

(10)

or the current is in the plane of the wave-front, and its direction in the wave-front is determined by the equation

${\frac {l}{\lambda }}(b^{2}-c^{2})+{\frac {m}{\mu }}(c^{2}-a^{2})+{\frac {n}{\nu }}(a^{2}-b^{2})=0$.

(11)

These equations are identical with those given by Fresnel if we define the plane of polarization as a plane through the ray perpendicular to the plane of the electric disturbance.

According to this electromagnetic theory of double refraction the wave of normal disturbance, which constitutes one of the chief difficulties of the ordinary theory, does not exist, and no new assumption is required in order to account for the fact that a ray polarized in a principal plane of the crystal is refracted in the ordinary manner^[5].

 

Relation between Electric Conductivity and Opacity.

798.] If the medium, instead of being a perfect insulator, is a conductor whose conductivity per unit of volume is $C$, the disturbance will consist not only of electric displacements but of currents of conduction, in which electric energy is transformed into heat, so that the undulation is absorbed by the medium.

If the disturbance is expressed by a circular function, we may write

$F=e^{-pz}\cos(nt-qz)$,

(1)

for this will satisfy the equation

${\frac {d^{2}F}{dz^{2}}}=\mu K{\frac {d^{2}F}{dt^{2}}}+4\pi \mu C{\frac {dF}{dt}}$,

(2)

$q^{2}-p^{2}=\mu Kn^{2}$,

(3)

$2pq=4\pi \mu Cn$.

(4)

The velocity of propagation is

$V={\frac {n}{q}}$,

(5)

and the coefficient of absorption is

$p=2\pi \mu CV$.

(6)

Let $R$ be the resistance, in electromagnetic measure, of a plate whose length is $l$, breadth $b$, and thickness $z$,

$R={\frac {l}{bzC}}$.

(7)

The proportion of the incident light which will be transmitted by this plate will be

$e^{-2pz}=e^{-4\pi \mu {\frac {l}{b}}{\frac {V}{R}}}$.

(8)

799.] Most transparent solid bodies are good insulators, and all good conductors are very opaque. There are, however, many exceptions to the law that the opacity of a body is the greater, the greater its conductivity.

Electrolytes allow an electric current to pass, and yet many of them are transparent. We may suppose, however, that in the case of the rapidly alternating forces which come into play during the propagation of light, the electromotive force acts for so short a time in one direction that it is unable to effect a complete separation between the combined molecules. When, during the other half of the vibration, the electromotive force acts in the opposite direction it simply reverses what it did during the first half. There is thus no true conduction through the electrolyte, no loss of electric energy, and consequently no absorption of light.

800.] Gold, silver, and platinum are good conductors, and yet, when formed into very thin plates, they allow light to pass through them. From experiments which I have made on a piece of gold leaf, the resistance of which was determined by Mr. Hockin, it appears that its transparency is very much greater than is consistent with our theory, unless we suppose that there is less loss of energy when the electromotive forces are reversed for every semi-vibration of light than when they act for sensible times, as in our ordinary experiments.

801.] Let us next consider the case of a medium in which the conductivity is large in proportion to the inductive capacity.

In this case we may leave out the term involving $K$ in the equations of Art. 783. and they then become

$\left.{\begin{aligned}\nabla ^{2}F+4\pi \mu C{\frac {dF}{dt}}&=0{\mbox{,}}\\\nabla ^{2}G+4\pi \mu C{\frac {dG}{dt}}&=0{\mbox{,}}\\\nabla ^{2}H+4\pi \mu C{\frac {dH}{dt}}&=0{\mbox{.}}\end{aligned}}\right\}$

(1)

Each of these equations is of the same form as the equation of the diffusion of heat given in Fourier's Traité de Chaleur.

802.] Taking the first as an example, the component $F$ of the vector-potential will vary according to time and position in the same way as the temperature of a homogeneous solid varies according to time and position, the initial and the surface-conditions heing made to correspond in the two cases, and the quantity $4\pi \mu C$ being numerically equal to the reciprocal of the thermometric conductivity of the substance, that is to say, the number of units of volume of the substance which would be heated one degree by the heat which passes through a unit cube of the substance, two opposite faces of which differ by one degree of temperature, while the other faces are impermeable to heat^[6].

The different problems in thermal conduction, of which Fourier has given the solution, may be transformed into problems in the diffusion of electromagnetic quantities, remembering that $F$, $G$, $H$ are the components of a vector, whereas the temperature, in Fourier's problem, is a scalar quantity.

Let us take one of the cases of which Fourier has given a complete solution^[7], that of an infinite medium, the initial state of which is given.

The state of any point of the medium at the time $t$ is found by taking the average of the state of every part of the medium, the weight assigned to each part in taking the average being

$e^{-{\frac {\pi \mu Cr^{2}}{t}}}$,

where $r$ is the distance of that part from the point considered. This average, in the case of vector-quantities, is most conveniently taken by considering each component of the vector separately.

803.] We have to remark in the first place, that in this problem the thermal conductivity of Fourier's medium is to be taken inversely proportional to the electric conductivity of our medium, so that the time required in order to reach an assigned stage in the process of diffusion is greater the higher the electric conductivity. This statement will not appear paradoxical if we remember the result of Art. 655, that a medium of infinite conductivity forms a complete barrier to the process of diffusion of magnetic force.

In the next place, the time requisite for the production of an assigned stage in the process of diffusion is proportional to the square of the linear dimensions of the system.

There is no determinate velocity which can be defined as the velocity of diffusion. If we attempt to measure this velocity by ascertaining the time requisite for the production of a given amount of disturbance at a given distance from the origin of disturbance, we find that the smaller the selected value of the disturbance the greater the velocity will appear to be, for however great the distance, and however small the time, the value of the disturbance will differ mathematically from zero.

This peculiarity of diffusion distinguishes it from wave-propagation, which takes place with a definite velocity. No disturbance takes place at a given point till the wave reaches that point, and when the wave has passed, the disturbance ceases for ever.

804.] Let us now investigate the process which takes place when an electric current begins and continues to flow through a linear circuit, the medium surrounding the circuit being of finite electric conductivity. (Compare with Art. 660).

When the current begins, its first effect is to produce a current of induction in the parts of the medium close to the wire. The direction of this current is opposite to that of the original current, and in the first instant its total quantity is equal to that of the original current, so that the electromagnetic effect on more distant parts of the medium is initially zero, and only rises to its final value as the induction-current dies away on account of the electric resistance of the medium.

But as the induction-current close to the wire dies away, a new induction-current is generated in the medium beyond, so that the space occupied by the induction-current is continually becoming wider, while its intensity is continually diminishing.

This diffusion and decay of the induction-current is a phenomenon precisely analogous to the diffusion of heat from a part of the medium initially hotter or colder than the rest. We must remember, however, that since the current is a vector quantity, and since in a circuit the current is in opposite directions at opposite points of the circuit, we must, in calculating any given component of the induction-current, compare the problem with one in which equal quantities of heat and of cold are diffused from neighbouring places, in which case the effect on distant points will be of a smaller order of magnitude.

805.] If the current in the linear circuit is maintained constant, the induction currents, which depend on the initial change of state, will gradually be diffused and die away, leaving the medium in its permanent state, which is analogous to the permanent state of the flow of heat. In this state we have

$\nabla ^{2}F=\nabla ^{2}G=\nabla ^{2}H=0$

(2)

throughout the medium, except at the part occupied by the circuit, in which

$\left.{\begin{aligned}\nabla ^{2}F&=4\pi u,\\\nabla ^{2}G&=4\pi v,\\\nabla ^{2}H&=4\pi w.\end{aligned}}\right\}$

(14)

These equations are sufficient to determine the values of $F,G,H$ throughout the medium. They indicate that there are no currents except in the circuit, and that the magnetic forces are simply those due to the current in the circuit according to the ordinary theory. The rapidity with which this permanent state is established is so great that it could not be measured by our experimental methods, except perhaps in the case of a very large mass of a highly conducting medium such as copper.

NOTE. In a paper published in Poggendorff's Annalen, June 1867, M. Lorenz has deduced from Kirchhoff's equations of electric currents (Pogg. Ann. cii. 1856), by the addition of certain terms which do not affect any experimental result, a new set of equations, indi cating that the distribution of force in the electromagnetic field may be conceived as arising from the mutual action of contiguous elements, and that waves, consisting of transverse electric currents, may be propagated, with a velocity comparable to that of light, in non-conducting media. He therefore regards the disturbance which constitutes light as identical with these electric currents, and he shews that conducting media must be opaque to such radiations.

These conclusions are similar to those of this chapter, though obtained by an entirely different method. The theory given in this chapter was first published in the Phil. Trans, for 1865.

1. For my own part, considering the relation of a vacuum to the magnetic force, and the general character of magnetic phenomena external to the magnet, I am more inclined to the notion that in the transmission of the force there is such an action, external to the magnet, than that the effects are merely attraction and repulsion at a distance. Such an action may be a function of the aether; for it is not at all unlikely that, if there be an aether, it should have other uses than simply the conveyance of radiations. Faraday's Experimental Researches, 3075.
2. Mem. de l' Acad., tom, iii, p. 130.
3. Cambridge Transactions, vol. ix, p. 10 (1850).
4. Phil. Trans. 1871, p. 573.
5. See Stokes' 'Report on Double Refraction'; Brit. Assoc. Reports, 1862, p. 255.
6. See Maxwell's Theory of Heat, p. 235.
7. Traité de la Chaleur, Art. 384. The equation which determines the temperature, $v$, at a point $(x,\;y,\;z)$ after a time $t$, in terms of $f(\alpha ,\;\beta ,\;\gamma )$, the initial temperature at the point $(\alpha ,\;\beta ,\;\gamma )$, is

   $v=\iiint {\frac {d\alpha \,d\beta \,d\gamma }{2^{3}{\sqrt {k^{3}\pi ^{3}t^{3}}}}}e^{-\left({\frac {(\alpha -x)^{2}+(\beta -y)^{2}+(\gamma -z)^{2}}{4kt}}\right)}f(\alpha ,\;\beta ,\;\gamma )$,

   where $k$ is the thermometric conductivity.