Fields and Charge in Two-Dimensional Region Bounded by Conducting Surfaces

Question:
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 E ρ and E φ and also the surface-charge densities σ(ρ, 0), σ(ρ, β), and σ(ρ, φ) on the three boundary surfaces.

Solution:
(a) Inside the region bounded by ρ ≥ a, 0 ≤ φ ≤ β surfaces, where there isn’t any electric charge and current, potential is given by Laplace equation, i.e.,

2Φ = 0

In two dimension, in terms of the polar coordinates (ρ, φ), this equation can be written as,

…………………………………………………………………………………………………………………………………………………………………………………………………………………

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