Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/417

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.