The Electrostatic Energy of a Two-Dimensional System

LL. G. Chambers
1959 Edinburgh Mathematical Notes  
The use of the complex variable z( -x+iy) and the complex potential W(=U + iV) for two-dimensional electrostatic systems is well known and the actual system in the (x, y) plane has an image system in the (U, V) plane. It does not seem to have been noticed previously that the electrostatic energy per unit length of the actual system is simply related to the area of the image domain in the (U, V) plane. The electrostatic energy density is \ e E 2 (1) where E is the electric field and e is the
more » ... ld and e is the permittivity. The energy per unit length in a two-dimensional system, is therefore given by (1) where the integration is over the whole field (which will be bounded by conductors in general). Now if W = U + i V =/(z) =f(x + iy) Then (2) dW E\ = dz and the electrostatic energy per unit length is dW dxdy dz and this is equal to (3) where the integration is taken over the image domain in the (U, V) plane of the domain of the field in the (x, y) plane. Formula (2) does not seem to have been developed in this form previously, and I cannot find any mention of it in the standard treatises. The method is best illustrated by means of an example. Consider a capacitor composed of two coaxal cylindrical conductors of radii a, b(>a) whose potentials are respectively 0, F o . Then the field between the two conductors is given by use, available at https://doi.
doi:10.1017/s0950184300003189 fatcat:zud5fkctnjbspok25grn7u6rli