Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/286

and for a slit of infinite depth, putting $D=\infty$, the correction is

${\frac {L}{4\pi ^{2}}}{\frac {B^{2}}{A^{2}}}\log _{e}2$

(31)

To find the surface-density on the series of parallel plates we must find $\sigma ={\frac {1}{4\pi }}{\frac {d\psi }{dx'}}$ when $\phi =0$. We find

$\sigma ={\frac {1}{4\pi b}}{\frac {1}{\sqrt {e^{-2{\frac {x'}{b}}}-1}}}$

(32)

The average density on the plane plate at distance $A$ from the edges of the series of plates is ${\bar {\sigma }}={\frac {1}{4\pi b}}$. Hence, at a distance from the edge of one of the plates equal to $n\alpha$ the surface-density is ${\tfrac {1}{\sqrt {2^{2n}-1}}}$ of this average density.

200.] Let us next attempt to deduce from these results the distribution of electricity in the figure formed by rotating the plane of the figure about the axis $y'=-R$. In this case, Poisson s equation will assume the form

${\frac {d^{2}V}{dx'^{2}}}+{\frac {d^{2}V}{dy'^{2}}}+{\frac {1}{R+y'}}{\frac {dV}{dy'}}+4\pi \rho =0$

(33)

Let us assume $V=\phi$, the function given in Art. 193, and determine the value of $\rho$ from this equation. We know that the first two terms disappear, and therefore

$\rho =-{\frac {1}{4\pi }}{\frac {1}{R+y'}}{\frac {d\phi }{dy'}}$

(34)

If we suppose that, in addition to the surface-density already investigated, there is a distribution of electricity in space according to the law just stated, the distribution of potential will be represented by the curves in Fig. XI.

Now from this figure it is manifest that ${\frac {d\phi }{dy'}}$ is generally very small except near the boundaries of the plates, so that the new distribution may be approximately represented by what actually exists, namely a certain superficial distribution near the edges of the plates.

If therefore we integrate $\iint \rho \ dx'dy'$ between the limits $y'=0$ and $y'={\frac {\pi }{2}}b$, and from $x'=-\infty$ to $x=+\infty$, we shall find the whole additional charge on one side of the plates due to the curvature.