# A Dynamical Theory of the Electromagnetic Field/Part VI

PART VI. — ELECTROMAGNETIC THEORY OF LIGHT.
(91) At the commencement of this paper we made use of the optical hypothesis of an elastic medium through which the vibrations of light are propagated, in order to show that we have warrantable grounds for seeking, in the same medium, the cause of other phenomena as well as those of light. We then examined electromagnetic phenomena, seeking for their explanation in the properties of the field which surrounds the electrified or magnetic bodies. In this way we arrived at certain equations expressing certain properties of the electromagnetic field. We now proceed to investigate whether these properties of that which constitutes the electromagnetic field, deduced from electromagnetic phenomena alone, are sufficient to explain the propagation of light through the same substance.
(92) Let us suppose that a plane wave whose direction cosines are $l,m,n$ is propagated through the field with a velocity $V$. Then all the electromagnetic functions will be functions of
$w=lx+my+nz-Vt$
The equations of Magnetic Force (B), p. 482, will become
${\begin{array}{l}\mu \alpha =m{\frac {dH}{dw}}-n{\frac {dG}{dw}},\\\\\mu \beta =n{\frac {dF}{dw}}-l{\frac {dH}{dw}},\\\\\mu \gamma =l{\frac {dG}{dw}}-m{\frac {dF}{dw}}.\end{array}}$
If we multiply these equations respectively by $l,m,n$, and add, we find
| $l\mu \alpha +m\mu \beta +n\mu \gamma =0\,$ | (62) |
which shows that the direction of the magnetization must be in the plane of the wave.
(93) If we combine the equations of Magnetic Force (B) with those of Electric Currents (C), and put for brevity
| ${\frac {dF}{dy}}+{\frac {dG}{dy}}+{\frac {dH}{dz}}=J,\ \mathrm {and} \ {\frac {d^{2}}{dx^{2}}}+{\frac {d^{2}}{dy^{2}}}+{\frac {d^{2}}{dz^{2}}}=\nabla ^{2}$ | (63) |
| $\left.{\begin{array}{l}4\pi \mu p'={\frac {dJ}{dx}}-\nabla ^{2}F,\\\\4\pi \mu q'={\frac {dJ}{dy}}-\nabla ^{2}G,\\\\4\pi \mu r'={\frac {dJ}{dz}}-\nabla ^{2}H,\end{array}}\right\$ | (64) |
$p',q',r'$
| $p'={\frac {df}{dt}},\ q'={\frac {dg}{dt}},\ r'={\frac {dh}{dt}}$ | (65) |
But these electric displacements are caused by electromotive forces, and by the equations of Electric Elasticity (E),
| $P=kf,\ Q=kg,\ R=kh$ | (66) |
These electromotive forces are due to the variations either of the electromagnetic or the electrostatic functions, as there is no motion of conductors in the field; so that the equations of electromotive force (D) are
| $\left.{\begin{array}{l}P=-{\frac {dF}{dt}}-{\frac {d\Psi }{dx}},\\\\Q=-{\frac {dG}{dt}}-{\frac {d\Psi }{dy}},\\\\R=-{\frac {dH}{dt}}-{\frac {d\Psi }{dz}}.\end{array}}\right\$ | (67) |
(94) Combining these equations, we obtain the following: –
| $\left.{\begin{array}{l}k\left({\frac {dJ}{dx}}-\nabla ^{2}F\right)+4\pi \mu \left({\frac {d^{2}F}{dt^{2}}}+{\frac {d^{2}\Psi }{dxdt}}\right)=0,\\\\k\left({\frac {dJ}{dy}}-\nabla ^{2}G\right)+4\pi \mu \left({\frac {d^{2}G}{dt^{2}}}+{\frac {d^{2}\Psi }{dydt}}\right)=0,\\\\k\left({\frac {dJ}{dy}}-\nabla ^{2}H\right)+4\pi \mu \left({\frac {d^{2}H}{dt^{2}}}+{\frac {d^{2}\Psi }{dzdt}}\right)=0.\end{array}}\right\$ | (68) |
If we differentiate the third of these equations with respect to $y$, and the second with respect to $z$, and subtract, J and $\Psi$ disappear, and by remembering the equations (B) of magnetic force, the results may be written
| $\left.{\begin{array}{l}k\nabla ^{2}\mu \alpha =4\pi \mu {\frac {d^{2}}{dt^{2}}}\mu \alpha ,\\\\k\nabla ^{2}\mu \beta =4\pi \mu {\frac {d^{2}}{dt^{2}}}\mu \beta ,\\\\k\nabla ^{2}\mu \gamma =4\pi \mu {\frac {d^{2}}{d^{2}t}}\mu \gamma .\end{array}}\right\$ | (69) |
(95) If we assume that $\alpha ,\beta ,\gamma$ are functions of $lx+my+nz-Vt=w$, the first equation become
| $k\mu {\frac {d^{2}\alpha }{dw^{2}}}=4\pi \mu ^{2}V^{2}{\frac {d^{2}\alpha }{dw^{2}}}$ | (70) |
or
| $V=\pm {\sqrt {\frac {k}{4\pi \mu }}}$ | (71) |
The other equations give the same value for V, so that the wave is propagated in either direction with a velocity V. ​This wave consists entirely of magnetic disturbances, the direction of magnetization being in the plane of the wave. No magnetic disturbance whose direction of magnetization is not in the plane of the wave can be propagated as a plane wave at all.
Hence magnetic disturbances propagated through the electromagnetic field agree with light in this, that the disturbance at any point is transverse to the direction of propagation, and such waves may have all the properties of polarized light.
(96) The only medium in which experiments have been made to determine the value of $k$ is air, in which $\mu =1$, and therefore, by equation (46),
| $V=v\,$ | (72) |
By the electromagnetic experiments of MM. Weber and Kohlrausch,
$v$ = 310,740,000 metres per second
is the number of electrostatic units in one electromagnetic unit of electricity, and this, according to our result, should be equal to the velocity of light in air or vacuum.
The velocity of light in air, by M. Fizeau's experiments, is
V = 314,858,000;
according to the more accurate experiments of M. Foucault
V = 298,000,000.
The velocity of light in the space surrounding the earth, deduced from the coefficient of aberration and the received value of the radius of the earth's orbit, is
V = 308,000,000.
(97) Hence the velocity of light deduced from experiment agrees sufficiently well with the value of $v$ deduced from the only set of experiments we as yet possess. The value of $v$ was determined by measuring the electromotive force with which a condenser of known capacity was charged, and then discharging the condenser through a galvanometer, so as to measure the quantity of electricity in it in electromagnetic measure. The only use made of light in the experiment was to see the instruments. The value of V found by M. Foucault was obtained by determining the angle through which a revolving mirror turned, while the light reflected from it went and returned along a measured course. No use whatever was made of electricity or magnetism.
The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.
(98) Let us now go back upon the equations in (94), in which the quantities J and $\Psi$ occur, to see whether any other kind of disturbance can be propagated through the medium depending on these quantities which disappeared from the final equations.
​If we determine $\chi$ from the equation
| $\nabla ^{2}\chi ={\frac {d^{2}\chi }{dx^{2}}}+{\frac {d^{2}\chi }{dy^{2}}}+{\frac {d^{2}\chi }{dz^{2}}}=J,$ | (73) |
and F', G', H' from the equations
| $F'=F-{\frac {d\chi }{dx}},\ G'=G-{\frac {d\chi }{dy}},\ H'=H-{\frac {d\chi }{dz}},$ | (74) |
then
| ${\frac {dF'}{dx}}+{\frac {dG'}{dy}}+{\frac {dH'}{dz}}=0$ | (75) |
and the equations in (94) become of the form
| $k\nabla ^{2}F'=4\pi \mu \left({\frac {a^{2}F'}{dt^{2}}}+{\frac {d}{dxdt}}\left(\Psi +{\frac {d\chi }{dt}}\right)\right)$ | (76) |
Differentiating the three equations with respect to $x,y$, and $z$, and adding, we find that
| $\Psi =-{\frac {d\chi }{dt}}+\varphi (x,y,z)$ | (77) |
and that
| $\left.{\begin{array}{l}k\nabla ^{2}F'=4\pi \mu {\frac {d^{2}F}{dt^{2}}},\\\\k\nabla ^{2}G'=4\pi \mu {\frac {d^{2}G'}{dt^{2}}},\\\\k\nabla ^{2}H'=4\pi \mu {\frac {d^{2}H'}{dt^{2}}}.\end{array}}\right\$ | (78) |
Hence the disturbances indicated by F', G', H' are propagated with the velocity $V={\sqrt {\tfrac {k}{4\pi \mu }}}$ through the field: and since
${\frac {dF'}{dx}}+{\frac {dG'}{dy}}+{\frac {dH'}{dz}}=0$
the resultant of these disturbances is in the plane of the wave.
(99) The remaining part of the total disturbances F, G, H being the part depending on $\chi$ is subject to no condition except that expressed in the equation
* ${\frac {d\Psi }{dt}}+{\frac {d^{2}\chi }{dt^{2}}}=0$
If we perform the operation $\nabla ^{2}$ on this equation, it becomes
| $ke={\frac {dJ}{dt}}-k\nabla ^{2}\varphi (x,y,z)$ | (79) |
Since the medium is a perfect insulator, $e$, the free electricity, is immoveable, and therefore ${\tfrac {dJ}{dt}}$ is a function of $x,y,z$, and the value of J is either constant or zero, or uniformly increasing or diminishing with the time; so that no disturbance depending on J can be propagated as a wave.
(100) The equations of the electromagnetic field, deduced from purely experimental evidence, show that transversal vibrations only can be propagated. If we were to go beyond our experimental knowledge and to assign a definite density to a substance which ​we should call the electric fluid, and select either vitreous or resinous electricity as the representative of that fluid, then we might have normal vibrations propagated with a velocity depending on this density. We have, however, no evidence as to the density of electricity, as we do not even know whether to consider vitreous electricity as a substance or as the absence of a substance.
Hence electromagnetic science leads to exactly the same conclusions as optical science with respect to the direction of the disturbances which can be propagated through the field; both affirm the propagation of transverse vibrations, and both give the same velocity of propagation. On the other hand, both sciences are at a loss when called on to affirm or deny the existence of normal vibrations.
Relation between the Index of Refraction and the Electromagnetic Character of the substance.
(101) The velocity of light in a medium, according to the Undulatory Theory, is
${\frac {1}{i}}V_{0}$
where $i$ is the index of refraction and $V_{0}$ is the velocity in vacuum. The velocity, according to the Electromagnetic Theory, is
${\sqrt {\frac {k}{4\pi \mu }}}$
where, by equations (49) and (71), $k={\tfrac {1}{D}}k_{0}$, and $k_{0}=4\pi V_{0}^{2}$.
Hence
| $D={\frac {i^{2}}{\mu }}$ | (80) |
or the Specific Inductive Capacity is equal to the square of the index of refraction divided by the coefficient of magnetic induction.
Propagation of Electromagnetic Disturbances in a Crystallized Medium.
(102) Let us now calculate the conditions of propagation of a plane wave in a medium for which the values of $k$ and $\mu$ are different in different directions. As we do not propose to give a complete investigation of the question in the present imperfect state of the theory as extended to disturbances of short period, we shall assume that the axes of magnetic induction coincide in direction with those of electric elasticity.
(103) Let the values of the magnetic coefficient for the three axes be $\lambda ,\mu ,\nu$, then the equations of magnetic force (B) become
| $\left.{\begin{array}{l}\lambda \alpha ={\frac {dH}{dy}}-{\frac {dG}{dz}},\\\\\mu \beta ={\frac {dF}{dz}}-{\frac {dH}{dx}},\\\\\nu \gamma ={\frac {dG}{dx}}-{\frac {dF}{dy}}.\end{array}}\right\$ | (81) |
The equations of electric elasticity (E) will be
| $\left.{\begin{array}{l}P=4\pi a^{2}f,\\Q=4\pi b^{2}g,\\R=4\pi c^{2}h,\end{array}}\right\$ | (82) |
where $4\pi a^{2}$, $4\pi b^{2}$, and $4\pi c^{2}$ are the values of $k$ for the axes of $x,y,z$.
Combining these equations with (A) and (D), we get equations of the form
| ${\frac {1}{\mu \nu }}\left(\lambda {\frac {d^{2}F}{dx^{2}}}+\mu {\frac {d^{2}F}{dy^{2}}}+\nu {\frac {d^{2}F}{dz^{2}}}\right)-{\frac {1}{\mu \nu }}{\frac {d}{dx}}\left(\lambda {\frac {dF}{dx}}+\mu {\frac {dG}{dy}}+\nu {\frac {dH}{dz}}\right)={\frac {1}{a^{2}}}\left({\frac {d^{2}F}{dt^{2}}}+{\frac {d^{2}\Psi }{dxdt}}\right)$ | (83) |
(104) If $l,m,n$ are the directions-cosines of the wave, and V its velocity, and if
| $lx+my+nz-Vt=w\,$ | (84) |
then F, G, H, and $\Psi$ will be functions of w, and if we put F', G', H', $\Psi '$ for the second differentials of these quantities with respect to $w$, the equations will be
| $\left.{\begin{array}{l}\left(V^{2}-a^{2}\left({\frac {m^{2}}{\nu }}+{\frac {n^{2}}{\mu }}\right)\right)F'+{\frac {a^{2}lm}{\nu }}G'+{\frac {a^{2}ln}{\mu }}H'-lV\Psi '=0,\\\\\left(V^{2}-b^{2}\left({\frac {n^{2}}{\lambda }}+{\frac {l^{2}}{\nu }}\right)\right)G'+{\frac {b^{2}mn}{\lambda }}H'+{\frac {b^{2}ml}{\nu }}F'-mV\Psi '=0,\\\\\left(V^{2}-c^{2}\left({\frac {l^{2}}{\mu }}+{\frac {m^{2}}{\lambda }}\right)\right)H'+{\frac {c^{2}nl}{\mu }}F'+{\frac {c^{2}nm}{\lambda }}G'-nV\Psi '=0.\end{array}}\right\$ | (85) |
If we now put
| $\left.{\begin{array}{r}V^{4}-V^{2}{\frac {1}{\lambda \mu \nu }}\left\{l^{2}\lambda \left(b^{2}\mu +c^{2}\nu \right)+m^{2}\mu \left(c^{2}\nu +a^{2}\lambda \right)+n^{2}\nu \left(a^{2}\lambda +b^{2}\mu \right)\right\}\\\\+{\frac {a^{2}b^{2}c^{2}}{\lambda \mu \nu }}\left({\frac {l^{2}}{a^{2}}}+{\frac {m^{2}}{b^{2}}}+{\frac {n^{2}}{c^{2}}}\right)\left(l^{2}\lambda +m^{2}\mu +n^{2}\nu \right)=U,\end{array}}\right\$ | (86) |
and shall find
| $F'V^{2}U-l\Psi 'VU=0\,$ | (87) |
with two similar equations for G' and H'. Hence either
| $V=0\,$ | (88) |
| $U=0\,$ | (89) |
or
| $VF'=l\Psi ',\ VG'=m\Psi '\ \mathrm {and} \ VH'=n\Psi '$ | (90) |
The third supposition indicates that the resultant of F', G', H' is in the direction normal to the plane of the wave; but the equations do not indicate that such a disturbance, if possible, could be propagated, as we have no other relation between $\Psi '$ and F, G', H'.
The solution $V=0$ refers to a case in which there is no propagation.
The solution $U=0$ gives two values for $V^{2}$ corresponding to values of F, G', H', which ​are given by the equations
| ${\frac {l}{a^{2}}}F'+{\frac {m}{b^{2}}}G'+{\frac {n}{c^{2}}}H'=0$ | (91) |
| ${\frac {a^{2}l\lambda }{F'}}\left(b^{2}\mu -c^{2}\nu \right)+{\frac {b^{2}m\mu }{G'}}\left(c^{2}\nu -a^{2}\lambda \right)+{\frac {c^{2}n\nu }{H'}}\left(a^{2}\lambda -b^{2}\mu \right)=0,$ | (92) |
(105) The velocities along the axes are as follows: –
${\begin{array}{lccccc}\mathrm {Direction\ of\ propagation} &&&x&y&z\\&&x&&{\frac {a^{2}}{\nu }}&{\frac {a^{2}}{\nu }}\\Direction\ of\ the\ electric\ displacements&&y&{\frac {b^{2}}{\nu }}&&{\frac {b^{2}}{\lambda }}\\&&z&{\frac {c^{2}}{\nu }}&{\frac {c^{2}}{\lambda }}\end{array}}$
Now we know that in each principal plane of a crystal the ray polarized in that plane obeys the ordinary law of refraction, and therefore its velocity is the same in whatever direction in that plane it is propagated.
If polarized light consists of electromagnetic disturbances in which the electric displacement is in the plane of polarization, then
| $a^{2}+b^{2}=c^{2}\,$ | (93) |
If, on the contrary, the electric displacements are perpendicular to the plane of polarization,
| $\lambda =\mu =\nu \,$ | (94) |
We know, from the magnetic experiments of Faraday, Plucker, &c, that in many crystals $\lambda ,\mu ,\nu$ are unequal.
The experiments of KnoblauchMagazine on electric induction through crystals seem to show that $a,b$ and $c$, may be different.
The inequality, however, of $\lambda ,\mu ,\nu$ is so small that great magnetic forces are required to indicate their difference, and the differences do not seem of sufficient magnitude to account for the double refraction of the crystals.
On the other hand, experiments on electric induction are liable to error on account of minute flaws, or portions of conducting matter in the crystal.
Further experiments on the magnetic and dielectric properties of crystals are required before we can decide whether the relation of these bodies to magnetic and electric forces is the same, when these forces are permanent as when they are alternating with the rapidity of the vibrations of light.
Relation between Electric Resistance and Transparency.
(106) If the medium, instead of being a perfect insulator, is a conductor whose resistance per unit of volume is $\rho$, then there will be not only electric displacements, but true currents of conduction in which electrical energy is transformed into heat, and the undulation is thereby weakened. To determine the coefficient of absorption, let us investigate the propagation along the axis of $x$ of the transverse disturbance G.
By the former equations
| ${\begin{array}{ll}{\frac {d^{2}G}{dx^{2}}}&=-4\pi \mu (q')\\\\&=-4\pi \mu ({\frac {df}{dt}}+q)\ \mathrm {by} \ (A)\\\\{\frac {d^{2}G}{dx^{2}}}&=+4\pi \mu \left({\frac {1}{k}}{\frac {d^{2}G}{dt^{2}}}-{\frac {1}{\rho }}{\frac {dG}{dt}}\right)\ \mathrm {by} \ (E)\ \mathrm {and} \ (F)\end{array}}$ | (95) |
If G is of the form
| $G=e^{-px}\cos(qx+nt)\,$ | (96) |
we find that
| $p={\frac {4\pi \mu }{\rho }}{\frac {n}{q}}={\frac {2\pi \mu }{\rho }}{\frac {V}{i}}$ | (97) |
where V is the velocity of light in air, and $i$ is the index of refraction. The proportion of incident light transmitted through the thickness $x$ is
| $e^{-2px}$ | (98) |
Let R be the resistance in electromagnetic measure of a plate of the substance whose thickness is $x$, breadth $b$, and length $l$, then
| ${\begin{array}{rl}R=&{\frac {l\rho }{bx}},\\\\2px=&4\pi \mu {\frac {V}{i}}{\frac {l}{bR}}\end{array}}$ | (99) |
(107) Most transparent solid bodies are good insulators, whereas all good conductors are very opaque.
Electrolytes allow a current to pass easily and yet are often very transparent. We may suppose, however, that in the rapidly alternating vibrations of light, the electromotive forces act for so short a time that they are unable to effect a complete separation between the particles in combination, so that when the force is reversed the particles oscillate into their former position without loss of energy.
Gold, silver, and platinum are good conductors, and yet when reduced to sufficiently thin plates they allow light to pass through them. If the resistance of gold is the same for electromotive forces of short period as for those with which we make experiments, the amount of light which passes through a piece of gold-leaf, of which the resistance was determined by Mr. C. Hockin, would be only $10^{-50}$ of the incident light, a totally imperceptible quantity. I find that between ${\tfrac {1}{500}}$ and ${\tfrac {1}{1000}}$ green light gets through ​such gold-leaf. Much of this is transmitted through holes and cracks; there is enough, however, transmitted through the gold itself to give a strong green hue to the transmitted light. This result cannot be reconciled with the electromagnetic theory of light, unless we suppose that there is less loss of energy when the electromotive forces are reversed with the rapidity of the vibrations of light than when they act for sensible times, as in our experiments.
Absolute Values of the Electromotive and Magnetic Forces called into play in the Propagation of Light.
(108) If the equation of propagation of light is
$F=A\cos {\frac {2\pi }{\lambda }}(z-Vt)$
the electromotive force will be
$P=-A{\frac {2\pi }{\lambda }}V\sin {\frac {2\pi }{\lambda }}(z-Vt);$
and the energy per unit of volume will be
${\frac {P^{2}}{8\pi \mu V^{2}}}$
where P represents the greatest value of the electromotive force. Half of this consists of magnetic and half of electric energy.
The energy passing through a unit of area is
$W={\frac {P^{2}}{8\pi \mu V}}$
so that
$P={\sqrt {8\pi \mu VW}}$
where V is the velocity of light, and W is the energy communicated to unit of area by the light in a second.
According to Pouillet's data, as calculated by Professor W. Thomson, the mechanical value of direct sunlight at the Earth is
83.4 foot-pounds per second per square foot.
This gives the maximum value of P in direct sunlight at the Earth's distance from the Sun,
P=60,000,000,
or about 600 Daniell's cells per metre.
At the Sun's surface the value of P would be about
13,000 Daniell's cells per metre.
At the Earth the maximum magnetic force would be .193.
At the Sun it would be 4.13.
These electromotive and magnetic forces must be conceived to be reversed twice in every vibration of light; that is, more than a thousand million million times in a second.
* ↑ Leipzig Transactions, vol. v. (1857), p. 260, or Poggendorff's 'Annalen,' Aug. 1856, p. 10.
* ↑ Comptes Rendus, vol. xxix. (1849), p. 90.
* ↑ Ibid. vol. lv. (1862), pp. 501, 792.
* ↑ Philosophical Magazine, 1852.
* ↑ Transactions of the Royal Society of Edinburgh, 1854 ("Mechanical Energies of the Solar System"),
* ↑ The horizontal magnetic force at Kew is about 1.76 in metrical units.