Classical Electrodynamics · Problem 2.13

Boundary-Value Problems in Electrostatics: II — John David Jackson

Physics — E&M · hollow-cylinder-two-potentials

Problem

(a) Two halves of a long hollow conducting cylinder of inner radius

b

are separated by small lengthwise gaps on each side, and are kept at different potentials

V_1

and

V_2

. Show that the potential inside is given by

\Phi(\rho, \phi) = \frac{V_1 + V_2}{2} + \frac{V_1 - V_2}{\pi}\tan^{-1}\!\left(\frac{2b\rho}{b^2 - \rho^2}\cos\phi\right)

where

\phi

is measured from a plane perpendicular to the plane through the gap. (b) Calculate the surface-charge density on each half of the cylinder.