Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/421

This is the same result that we should have obtained if we had neglected the conductivity of the strata.

Let us next suppose that the electromotive force $E$ is continued uniform for an indefinitely long time, or till a uniform current of conduction equal top is established through the system.

We have then $X_{1}=r_{1}p,$ and therefore

$E=(r_{1}+a_{1}+r_{2}a_{2}+\mathrm {\&c.} )p.$

(17)

If $R$ be the total resistance of the system,

$R={\frac {E}{P}}=r_{1}a_{1}+r_{2}a_{2}+\mathrm {\&c.}$

(18)

In this state we have by (2),

$f_{1}={\frac {r_{1}}{4\pi k_{1}}}p,$

so that

$\sigma _{12}=\left({\frac {r_{2}}{4\pi k_{2}}}-{\frac {r_{1}}{4\pi k_{1}}}\right)p.$

(19)

If we now suddenly connect the extreme strata by means of a conductor of small resistance, $E$ will be suddenly changed from its original value $E_{0}$ to zero, and a quantity $Q$ of electricity will pass through the conductor.

To determine $Q$ we observe that if $X_{1}'$ be the new value of $X_{l},$ then by (13),

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

(20)

Hence, by (10), putting $E=0,$

$0=a_{1}X_{1}+\mathrm {\&c.} +4\pi (a_{1}k_{1}+a_{2}k_{2}+\mathrm {\&c.} )Q,$

(21)

or

$0=E_{0}+{\frac {1}{C}}Q.$

(22)

Hence $Q=-CE_{0}$ where $C$ is the capacity, as given by equation (16). The instantaneous discharge is therefore equal to the instantaneous charge.

Let us next suppose the connexion broken immediately after this discharge. We shall then have $u=0,$ so that by equation (8),

$X_{1}=X'e^{-{\frac {4\pi k_{1}}{r_{1}}}t},$

(23)

where $X'$ is the initial value after the discharge.

Hence, at any time $t,$

$X_{1}=E_{0}\left\{{\frac {r_{1}}{R}}-4\pi k_{1}C\right\}e^{-{\frac {4\pi k_{1}}{r_{1}}}}.$

The value of $E$ at any time is therefore

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

(24)