Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/289

then $x$ and $y$ will be conjugate with respect to $\phi$ and $\psi$, and $\phi$ and $\psi$ will be conjugate with respect to $x$ and $y$.

Now let $x$ and $y$ be rectangular coordinates, and let $k\psi$ be the potential, then $k\phi$ will be conjugate to $k\psi$, $k$ being any constant.

Let us put $\psi =\pi$, then $y=A\pi$, $x=A\left(\phi -e^{\phi }\right)$.

If $\phi$ varies from $-\infty$ to 0, and then from 0 to $+\infty$, $x$ varies from $-\infty$ to $-A$ and from $-A$ to $-\infty$. Hence the equipotential surface for which $b=\pi A$ is a plane parallel to $x$ at a distance $b=\pi A$ from the origin, and extending from $-\infty$ to $x=-A$.

Let us consider a portion of this plane, extending from

$x=-(A+a)$ to $x=-A$ and from $z=0$ to $z=c$,

let us suppose its distance from the plane of $xz$ to be $y=B=A\pi$, and its potential to be $V=k\psi =k\pi$.

The charge of electricity on any portion of this part of the plane is found by ascertaining the values of $\phi$ at its extremities.

If these are $\phi _{1}$ and $\phi _{2}$, the quantity of electricity is

${\frac {1}{4\pi }}ck\left(\phi _{2}-\phi _{1}\right)$

We have therefore to determine $\phi$ from the equation

$x=-(A+a)=A\left(\phi -e^{\phi }\right)$

$\phi$ will have a negative value $\phi _{1}$ and a positive value $\phi _{2}$ at the edge of the plane, where $x=-A,\ \phi =0$.

Hence the charge on the negative side is $-ck\phi _{1}$, and that on the positive side is $ck\phi _{2}$.

If we suppose that $a$ is large compared with $A$,

${\begin{array}{l}\phi _{1}=-{\frac {a}{A}}-1+e^{-{\frac {a}{A}}-1+e^{-{\frac {a}{A}}-+\mathrm {etc.} }}\\\\\phi _{e}=\log \left\{{\frac {a}{A}}+1+\log \left({\frac {a}{A}}+1+\mathrm {etc.} \right)\right\}\end{array}}$

If we neglect the exponential terms in $\phi _{1}$ we shall find that the charge on the negative surface exceeds that which it would have if the superficial density had been uniform and equal to that at a distance from the boundary, by a quantity equal to the charge on a strip of breadth $A={\frac {b}{\pi }}$ with the uniform superficial density.

The total capacity of the part of the plane considered is

$C={\frac {c}{4\pi ^{2}}}\left(\phi _{2}-\phi _{1}\right)$