Page:A Dynamical Theory of the Electromagnetic Field.pdf/46

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