Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/282

Hence from the origin to $x'=a$ the amount is

$E={\frac {1}{4\pi }}\log \left(e^{\frac {a}{b}}+{\sqrt {e^{\frac {2a}{b}}-1}}\right)$

(11)

If $a$ is large compared with $b$, this becomes

${\begin{array}{ll}E&={\frac {1}{4\pi }}\log 2e^{\frac {a}{b}}\\\\&={\frac {a+b\log _{e}2}{4\pi b}}\end{array}}$

(12)

Hence the quantity of electricity on the plane hounded by the straight edge is greater than it would have been if the electricity had been uniformly distributed over it with the same density that it has at a distance from the boundary, and it is equal to the quantity of electricity having the same uniform surface-density, but extending to a breadth equal to $b\log _{e}2$ beyond the actual boundary of the plate.

This imaginary uniform distribution is indicated by the dotted straight lines in Fig. XI. The vertical lines represent lines of force, and the horizontal lines equipotential surfaces, on the hypo thesis that the density is uniform over both planes, produced to infinity in all directions.

196.] Electrical condensers are sometimes formed of a plate placed midway between two parallel plates extending considerably beyond the intermediate one on all sides. If the radius of curvature of the boundary of the intermediate plate is great compared with the distance between the plates, we may treat the boundary as approximately a straight line, and calculate the capacity of the condenser by supposing the intermediate plate to have its area extended by a strip of uniform breadth round its boundary, and assuming the surface-density on the extended plate the same as it is in the parts not near the boundary.

Thus, if $S$ be the actual area of the plate, $L$ its circumference, and $B$ the distance between the large plates, we have

$b={\frac {1}{\pi }}B$

(13)

and the breadth of the additional strip is

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

(14)

so that the extended area is

$S'=S+BL{\frac {1}{\pi }}\log _{e}2$

(15)