Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/408

the spheres cause the resistance of the compound medium to be different in different directions.

Application of the Principle of Images.

315.] Let us take as an example the case of two media separated by a plane surface, and let us suppose that there is a source $S$ of electricity at a distance a from the plane surface in the first medium, the quantity of electricity flowing from the source in unit of time being $S.$

If the first medium had been infinitely extended the current at any point $P$ would have been in the direction $SP,$ and the potential at $P$ would have been ${\frac {E}{r_{1}}}$ where $E={\frac {Sk_{1}}{4\pi }}$ and $r_{1}=SP.$

In the actual case the conditions may be satisfied by taking a point $I,$ the image of $S$ in the second medium, such that $IS$ is normal to the plane of separation and is bisected by it. Let $r_{2}$ be the distance of any point from $I,$ then at the surface of separation

$r_{1}=r_{2},$

(1)

${\frac {dr_{1}}{d\nu }}=-{\frac {dr_{2}}{d\nu }}.$

(2)

Let the potential $V_{1}$ at any point in the first medium be that due to a quantity of electricity $E$ placed at $S,$ together with an imaginary quantity $E_{2}$ at $I,$ and let the potential $V_{2}$ at any point of the second medium be that due to an imaginary quantity $E_{1}$ at $S,$ then if

$V_{1}={\frac {E}{r_{1}}}+{\frac {E_{2}}{r_{1}}}\quad \quad \mathrm {and} \quad \quad V_{2}={\frac {E_{1}}{r_{1}}},$

(3)

the superficial condition $V_{1}=V_{2}$ gives

$E+E_{2}=E_{1},$

(4)

and the condition

${\frac {1}{k_{1}}}{\frac {dV_{1}}{d\nu }}={\frac {1}{k_{2}}}{\frac {dV_{2}}{d\nu }}$

(5)

gives

${\frac {1}{k_{1}}}(E-E_{2})={\frac {1}{k_{2}}}E_{1},$

(6)

whence

$E_{1}={\frac {2k_{2}}{k_{1}+k_{2}}}E,\quad \quad E_{2}={\frac {k_{2}-k_{1}}{k_{1}+k_{2}}}E.$

(7)

The potential in the first medium is therefore the same as would be produced in air by a charge $E$ placed at $S,$ and a charge $E_{1}$ at $I$ on the electrostatic theory, and the potential in the second medium is the same as that which would be produced in air by a charge $E_{1}$ at $S.$