Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/295

The quantity of electricity induced on one of the planes protected by the grating, the other plane being at a given difference of potential, will be the same as if the plates had been at a distance

${\frac {(\alpha -\gamma )\left(b_{1}+b_{2}\right)+\beta _{1}\beta _{2}-4\alpha \gamma }{\alpha +\gamma }}$ instead of $b_{1}+b_{2}$.

The values of $\alpha$ and $\gamma$ are approximately as follows,

$\alpha ={\frac {a}{2\pi }}\left\{\log {\frac {a}{2\pi c}}-{\frac {5}{3}}\cdot {\frac {\pi ^{4}c^{4}}{15a^{4}+\pi ^{4}c^{4}}}+2e^{-4\pi {\frac {b_{1}+b_{2}}{a}}}\left(1+e^{-4\pi {\frac {b_{1}}{a}}}+e^{-4\pi {\frac {b_{2}}{a}}}+\mathrm {etc} .\right)+\mathrm {etc} .\right\}$

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$\gamma ={\frac {3\pi ac^{2}}{3a^{2}+\pi ^{2}c^{2}}}\left({\frac {e^{-4\pi {\frac {b_{1}}{a}}}}{1-e^{-4\pi {\frac {b_{1}}{a}}}}}-{\frac {e^{-4\pi {\frac {b_{2}}{a}}}}{1-e^{-4\pi {\frac {b_{2}}{a}}}}}\right)+\mathrm {etc} .$

(25)