Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/422

and the instantaneous discharge after any time $t$ is $EC.$ This is called the residual discharge.

If the ratio of $r$ to $k$ is the same for all the strata, the value of $E$ will be reduced to zero. If, however, this ratio is not the same, let the terms be arranged according to the values of this ratio in descending order of magnitude.

The sum of all the coefficients is evidently zero, so that when $t=0,\,E=0.$ The coefficients are also in descending order of magnitude, and so are the exponential terms when $t$ is positive. Hence, when $t$ is positive, $E$ will be positive, so that the residual discharge is always of the same sign as the primary discharge.

When $t$ is indefinitely great all the terms disappear unless any of the strata are perfect insulators, in which case $r_{1}$ is infinite for that stratum/, and $R$ is infinite for the whole system, and the final value of $E$ is not zero but

$E=E_{0}(1-4\pi a_{1}k_{1}C)$

(25)

Hence, when some, but not all, of the strata are perfect insulators, a residual discharge may be permanently preserved in the system.

330.] We shall next determine the total discharge through a wire of resistance $R_{0}$ kept permanently in connexion with the extreme strata of the system, supposing the system first charged by means of a long-continued application of the electromotive force $E.$

At any instant we have

$E=a_{1}r_{1}p_{1}+a_{2}r_{2}p_{2}+\mathrm {\&c.} +R_{0}u=0,$

(26)

and also, by (3),

$u=p_{1}+{\frac {df_{1}}{dt}}.$

(27)

Hence

$(R+R_{0})u=a_{1}r_{1}{\frac {df_{1}}{dt}}+a_{2}r_{2}{\frac {df_{2}}{dt}}+\mathrm {\&c.}$

(28)

Integrating with respect to $t$ in order to find $Q,$ we get

$(R+R_{0})Q=a_{1}r_{1}(f_{1}'-f_{1})+a_{2}r_{2}(f_{2}'-f_{2})+\mathrm {\&c.} ,$

(29)

where $f_{1}$ is the initial, and $f_{1}'$ the final value of $f_{1}.$

In this case $f_{1}'=0,$ and $f_{1}=E\left({\frac {r_{1}}{4\pi k_{1}R}}-C\right).$

Hence

$(R+R_{0})Q={\frac {E_{0}}{4\pi R}}\left({\frac {a_{1}r_{1}^{2}}{k_{1}}}+{\frac {a_{2}r_{2}^{2}}{k_{2}}}+\mathrm {\&c.} \right)-E_{0}CR,$

(30)

$=-{\frac {CE_{0}}{R}}\Sigma \Sigma \left[a_{1}a_{2}k_{1}k_{2}\left({\frac {r_{1}}{k_{1}}}-{\frac {r_{2}}{k_{2}}}\right)^{2}\right],$

(31)

where the summation is extended to all quantities of this form belonging to every pair of strata.