Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/285

If $B$ is the distance between two of the planes of the series, $B=\pi b$, so that the additional distance is

$\alpha =B{\frac {\log _{e}2}{\pi }}$

(25)

198.] Let us next consider the space included between two of the equipotential surfaces, one of which consists of a series of parallel waves, while the other corresponds to a large value of $\phi$, and may be considered as approximately plane.

If $D$ is the depth of these undulations from the crest to the trough of each wave, then we find for the corresponding value of $\phi$,

$\phi ={\frac {1}{2}}\log {\frac {e^{\frac {D}{b}}+1}{e^{\frac {D}{b}}-1}}$

(26)

The value of $x'$ at the crest of the wave is

$b\log {\frac {1}{2}}\left(e^{\phi }+e^{-\phi }\right)$

(27)

Hence, if $A$ is the distance from the crests of the waves to the opposite plane, the capacity of the system composed of the plane surface and the undulated surface is the same as that of two planes at a distance $A+\alpha '$ where

$\alpha '={\frac {B}{\pi }}\log _{e}{\frac {2}{1+e^{-\pi {\frac {D}{B}}}}}$

199.] If a single groove of this form be made in a conductor having the rest of its surface plane, and if the other conductor is a plane surface at a distance $A$, the capacity of the one conductor with respect to the other will be diminished. The amount of this diminution will be less than the ${\tfrac {1}{n}}$th part of the diminution due to $n$ such grooves side by side, for in the latter case the average electrical force between the conductors will be less than in the former case, so that the induction on the surface of each groove will be diminished on account of the neighbouring grooves.

If $L$ is the length, $B$ the breadth, and $D$ the depth of the groove, the capacity of a portion of the opposite plane whose area is $S$ will be

${\frac {S}{4\pi A}}-{\frac {LB}{4\pi A}}{\frac {\alpha '}{A+\alpha '}}$

(29)

If $A$ is large compared with $B$ or $\alpha '$, the correction becomes

${\frac {L}{4\pi ^{2}}}{\frac {B^{2}}{A^{2}}}\log _{e}{\frac {2}{1+e^{-\pi {\frac {D}{B}}}}}$

(30)