Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/419

$r_{1},\,r_{2},$ &c. the specific resistance referred to unit of volume.

$K_{1},\,K_{2},$ &c. the specific inductive capacity.

$k_{1},\,k_{2},$ &c. the reciprocal of the specific inductive capacity.

$E$ the electromotive force due to a voltaic battery, placed in the part of the circuit leading from the last stratum towards the first, which we shall suppose good conductors.

$Q$ the total quantity of electricity which has passed through this part of the circuit up to the time $t$.

$R$ the resistance of the battery with its connecting wires.

$\sigma _{12}$ the surface-density of electricity on the surface which separates the first and second strata.

Then in the first stratum we have, by Ohm's Law,

$X_{1}=r_{1}p_{1}$

(1)

By the theory of electrical displacement,

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

(2)

By the definition of the total current,

$u_{1}=p_{1}+{\frac {df_{1}}{dt}},$

(3)

with similar equations for the other strata, in each of which the quantities have the suffix belonging to that stratum.

To determine the surface-density on any stratum, we have an equation of the form

$\sigma _{12}=f_{2}-f_{1},$

(4)

and to determine its variation we have

${\frac {\sigma _{12}}{dt}}=p_{1}-p_{2}.$

(5)

By differentiating (4) with respect to $t,$ and equating the result to (5), we obtain

$p_{1}+{\frac {df_{1}}{dt}}=p_{2}+{\frac {df_{2}}{dt}}=u,\quad {\mbox{say,}}$

(6)

or, by taking account of (3),

$u_{1}=u_{2}=\mathrm {\&c.} =u.$

(7)

That is, the total current u is the same in all the strata, and is equal to the current through the wire and battery.

We have also, in virtue of equations (1) and (2),

$u={\frac {1}{r_{1}}}X_{1}+{\frac {1}{4\pi k_{1}}}{\frac {dX_{1}}{dt}},$

(8)

from which we may find $X_{l}$ by the inverse operation on $u,$

$X_{1}=\left({\frac {1}{r_{1}}}+{\frac {1}{4\pi k}}{\frac {d}{dt}}\right)^{-1}u.$

(9)