Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/292

The surface-density $\sigma _{1}$ on the first plane is got from the equation (6)

$4\pi \sigma _{1}={\frac {dV_{1}}{db_{1}}}=-{\frac {4\pi }{a}}(\lambda +\lambda ')$

(9)

That on the second plane $\sigma _{2}$ from the equation (7)

$4\pi \sigma _{2}=-{\frac {dV_{2}}{db_{2}}}=-{\frac {4\pi }{a}}(\lambda -\lambda ')$

(10)

If we now write

$\alpha =-{\frac {a}{2\pi }}\log _{e}\left(2\sin {\frac {\pi c}{a}}\right)$

(11)

and eliminate $\lambda$ and $\lambda '$ from the equations (6), (7), (8), (9), (10), we find

$4\pi \sigma _{1}\left(b_{1}+b_{2}+{\frac {2b_{1}b_{2}}{\alpha }}\right)=V_{1}\left(1+2{\frac {b_{2}}{\alpha }}\right)-V_{2}-V{\frac {2b_{2}}{\alpha }}$

(12)

$4\pi \sigma _{2}\left(b_{1}+b_{2}+{\frac {2b_{1}b_{2}}{\alpha }}\right)=-V_{1}+V_{2}\left(1+2{\frac {b_{1}}{\alpha }}\right)-V{\frac {2b_{1}}{\alpha }}$

(13)

When the wires are infinitely thin, $\alpha$ becomes infinite, and the terms in which it is the denominator disappear, so that the case is reduced to that of two parallel planes without a grating interposed.

If the grating is in metallic communication with one of the planes, say the first, $V=V_{1}$, and the right-hand side of the equation for $\sigma _{1}$ becomes $V_{1}-V_{2}$. Hence the density $\sigma _{1}$ induced on the first plane when the grating is interposed is to that which would have been induced on it if the grating were removed, the second plane being maintained at the same potential, as 1 to $1+{\tfrac {2b_{1}b_{2}}{\alpha \left(b_{1}+b_{2}\right)}}$.

We should have found the same value for the effect of the grating in diminishing the electrical influence of the first surface on the second, if we had supposed the grating connected with the second surface. This is evident since $b_{1}$ and $b_{2}$ enter into the expression in the same way. It is also a direct result of the theorem of Art. 88.

The induction of the one electrified plane on the other through the grating is the same as if the grating were removed, and the distance between the planes increased from $b_{1}+b_{2}$ to

$b_{1}+b_{2}+2{\frac {b_{1}b_{2}}{\alpha }}$

If the two planes are kept at potential zero, and the grating electrified to a given potential, the quantity of electricity on the grating will be to that which would be induced on a plane of equal area placed in the same position as

$2b_{1}b_{2}$ is to $2b_{1}b_{2}+\alpha \left(b_{1}+b_{2}\right)$.