Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/277

If in this way, or in any other, we have determined the distribution of potential for the case of a given curve of section when the charge is placed at a given point taken as origin, we may pass to the case in which the charge is placed at any other point by an application of the general method of transformation.

Let the values of $\alpha$ and $\beta$ for the point at which the charge is placed be $\alpha _{1}$ and $\beta _{1}$, then substituting in equation (11) $\alpha -\alpha _{0}$ for $\rho$, and $\beta -\beta _{1}$ for $\theta$, we find for the potential at any point whose coordinates are $\alpha$ and $\beta$,

${\begin{array}{ll}\phi =&E\log \left(1-2e^{\alpha -\alpha _{1}}\cos \left(\beta -\beta _{1}\right)+e^{2\left(\alpha -\alpha _{1}\right)}\right)\\&-E\log \left(1-2e^{\alpha -\alpha _{1}-2\alpha _{0}}\cos \left(\beta -\beta _{1}\right)+e^{2\left(\alpha +\alpha _{1}-2\alpha _{0}\right)}\right)+2E\left(\alpha _{1}-\alpha _{0}\right)\end{array}}$

(14)

This expression for the potential becomes zero when $\alpha =\alpha _{0}$, and is finite and continuous within the curve $\alpha _{0}$ except at the point $\alpha _{1}\beta _{1}$, at which point the first term becomes infinite, and in its immediate neighbourhood is ultimately equal to $2E\log r'$, where $r'$ is the distance from that point.

We have therefore obtained the means of deducing the solution of Green's problem for a charge at any point within a closed curve when the solution for a charge at any other point is known.

The charge induced upon an element of the curve $\alpha _{0}$ between the points $\beta$ and $\beta +d\beta$ by a charge $E$ placed at the point $\alpha _{1}\beta _{1}$ is

${\frac {E}{2\pi }}{\frac {1-e^{2\left(\alpha _{1}-\alpha _{0}\right)}}{1-2e^{\left(\alpha _{1}-\alpha _{0}\right)}\cos \left(\beta -\beta _{1}\right)+e^{2\left(\alpha _{1}-\alpha _{0}\right)}}}d\beta$

(15)

From this expression we may find the potential at any point $\alpha _{1}\beta _{1}$ within the closed curve, when the value of the potential at every point of the closed curve is given as a function of $\beta$, and there is no electrification within the closed curve.

For, by Theorem II of Chap. Ill, the part of the potential at $\alpha _{1}\beta _{1}$, due to the maintenance of the portion $d\beta$ of the closed curve at the potential $V$, is $nV$, where $n$ is the charge induced on $d\beta$ by unit of electrification at $\alpha _{1}\beta _{1}$. Hence, if $V$ is the potential at a point on the closed curve defined as a function of $\beta$, and $\phi$ the potential at the point $\alpha ,\beta$ within the closed curve, there being no electrification within the curve,

$\phi ={\frac {1}{2\pi }}\int _{0}^{2\pi }{\frac {\left(1-e^{2\left(\alpha _{1}-\alpha _{0}\right)}\right)V\ d\beta }{1-2e^{\left(\alpha _{1}-\alpha _{0}\right)}\cos \left(\beta -\beta _{1}\right)+e^{2\left(\alpha _{1}-\alpha _{0}\right)}}}$

(16)