Boundary-Value Problems in Electrostatics: II — John David Jackson
Physics — E&M · wedge-region-potential-fields
Problem
The two-dimensional region, ρ≥a, 0≤ϕ≤β, is bounded by conducting surfaces at ϕ=0, ρ=a, and ϕ=β held at zero potential, as indicated in the sketch. At large ρ the potential is determined by some configuration of charges and/or conductors at fixed potentials.
(a) Write down a solution for the potential Φ(ρ,ϕ) that satisfies the boundary conditions for finite ρ.
(b) Keeping only the lowest nonvanishing terms, calculate the electric field components Eρ and Eϕ and also the surface-charge densities σ(ρ,0), σ(ρ,β), and σ(a,ϕ) on the three boundary surfaces.
(c) Consider β=π (a plane conductor with a half-cylinder of radius a on it). Show that far from the half-cylinder the lowest order terms of part b give a uniform electric field normal to the plane. Sketch the charge density on and in the neighborhood of the half-cylinder. For fixed electric field strength far from the plane, show that the total charge on the half-cylinder (actually charge per unit length in the z direction) is twice as large as would reside on a strip of width 2a in its absence. Show that the extra portion is drawn from regions of the plane nearby, so that the total charge on a strip of width large compared to a is the same whether the half-cylinder is there or not.