Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/411

For the potential in the third medium we find

$V=(1+\rho ')(1-\rho )E\left\{{\frac {1}{PS}}+{\frac {\rho \rho '}{PJ_{1}}}+\,\mathrm {\&c.} +{\frac {(\rho \rho ')^{n}}{PJ_{n}}}\right\}.$

(16)

If the first medium is the same as the third, then $k_{1}=k_{3}$ and $\rho =\rho ',$ and the potential on the other side of the plate will be

$V=(1-\rho ^{2})E\left\{{\frac {1}{PS}}+{\frac {\rho ^{2}}{PJ_{1}}}+\,\mathrm {\&c.} +{\frac {\rho ^{2n}}{PJ_{n}}}\right\}.$

(17)

If the plate is a very much better conductor than the rest of the medium, $\rho$ is very nearly equal to 1 . If the plate is a nearly perfect insulator, $\rho$ is nearly equal to -1, and if the plate differs little in conducting power from the rest of the medium, $\rho$ is a small quantity positive or negative.

The theory of this case was first stated by Green in his 'Theory of Magnetic Induction (Essay, p. 65). His result, however, is correct only when $\rho$ is nearly equal to 1^[1]. The quantity $g$ which he uses is connected with $\rho$ by the equations

$g={\frac {2\rho }{3-\rho }}={\frac {k_{1}-k_{2}}{k_{1}+k_{2}}},\quad \quad \rho ={\frac {3g}{2+g}}={\frac {k_{1}-k_{2}}{k_{1}+k_{2}}}.$

If we put $\rho ={\frac {2\pi \kappa }{1+2\pi \kappa }},$ we shall have a solution of the problem of the magnetic induction excited by a magnetic pole in an infinite plate whose coefficient of magnetization is $\kappa$.

On Stratified Conductors.

319.] Let a conductor be composed of alternate strata of thickness $c$ and $c'$ of two substances whose coefficients of conductivity are different. Required the coefficients of resistance and conductivity of the compound conductor.

Let the plane of the strata be normal to $Z.$ Let every symbol relating to the strata of the second kind be accented, and let every symbol relating to the compound conductor be marked with a bar thus, ${\overline {X}}$. Then

${\begin{array}{rl}{\overline {X}}=X=X',&(c+c'){\overline {u}}=cu+c'u',\\{\overline {Y}}=Y=Y',&(c+c'){\overline {v}}=cv+c'v';\\(c+c'){\overline {Z}}=cZ+c'Z',&\quad {\overline {w}}=w=w'.\end{array}}$

We must first determine $u,\,u',\,v,\,v',\,Z$ and $Z'$ in terms of ${\overline {X}},\,{\overline {Y}}$ and ${\overline {w}}$ from the equations of resistance, Art. 297, or those

1. See Sir W. Thomson's 'Note on Induced Magnetism in a Plate,' Camb. and Dub. Math. Journ., Nov. 1845, or Reprint, art. ix. § 156.