Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/420

The total electromotive force $E$ is

$E=a_{1}X_{1}+a_{2}X_{2}+\mathrm {\&c.} ,$

(10)

or

$E=\left\{a_{1}\left({\frac {1}{r_{1}}}+{\frac {1}{4\pi k_{1}}}{\frac {d}{dt}}\right)^{-1}+a_{2}\left({\frac {1}{r_{2}}}+{\frac {1}{4\pi k_{2}}}{\frac {d}{dt}}\right)^{-1}+\mathrm {\&c.} \right\}u,$

(11)

an equation between $E,$ the external electromotive force, and $u,$ the external current.

If the ratio of $r$ to $k$ is the same in all the strata, the equation reduces itself to

$E+{\frac {r}{4\pi k}}{\frac {dE}{dt}}=(a_{1}r_{1}+a_{2}r_{2}+\mathrm {\&c.} )u,$

(12)

which is the case we have already examined, and in which, as we found, no phenomenon of residual charge can take place.

If there are $n$ substances having different ratios of $r$ to $k,$ the general equation (11), when cleared of inverse operations, will be a linear differential equation, of the nth order with respect to $E$ and of the $(n-1)$th order with respect to $u,\,t$ being the independent variable.

From the form of the equation it is evident that the order of the different strata is indifferent, so that if there are several strata of the same substance we may suppose them united into one without altering the phenomena.

329.] Let us now suppose that at first $f_{1},\,f_{2},$ &c. are all zero, and that an electromotive force $E$ is suddenly made to act, and let us find its instantaneous effect.

Integrating (8) with respect to $t,$ we find

$Q=\int u\,dt={\frac {1}{r_{1}}}\int X_{1}\,dt+{\frac {1}{4\pi k_{1}}}X_{1}+\mathrm {const.}$

(13)

Now, since $X_{1}$ is always in this case finite, $\int X_{1}\,dt,$ must be insensible when $t$ is insensible, and therefore, since $X_{1}$ is originally zero, the instantaneous effect will be

$X_{1}=4\pi k_{1}Q.$

(14)

Hence, by equation (10),

$E+4\pi (k_{1}a_{1}+k_{2}a_{2}+\mathrm {\&c.} )Q,$

(15)

and if $C$ be the electric capacity of the system as measured in this instantaneous way,

$C={\frac {Q}{E}}={\frac {1}{4\pi (k_{1}a_{1}+k_{2}a_{2}+\mathrm {\&c.} )}}.$

(16)