A patterned electrode of radius

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Conductors, capacitance and boundary-value electrostatics

The Boundary-Value Problem, Ch Separation of variables in spherical geometry

Problem

A patterned electrode of radius a=0.20 ma = 0.20\,\mathrm{m}, with no applied field, has a surface made of many narrow metal bands running round the zz axis, insulated from one another. A control circuit holds each band at its own potential, so that the surface potential is V(a,θ)=V0(3cos⁡2θ−1),V0=50 V.V(a,\theta) = V_0(3\cos^2\theta - 1), \qquad V_0 = 50\,\mathrm{V}. There is no charge outside the sphere and V→0V \to 0 as r→∞r \to \infty. ((a)) Rewrite V(a,θ)V(a,\theta) as a Legendre polynomial and find the exterior potential V(r,θ)V(r,\theta) for r≥ar \ge a. ((b)) Find the surface charge density σ(θ)\sigma(\theta) on the outer faces of the bands. ((c)) Evaluate the radial surface field Er(a,θ)E_r(a,\theta) and σ(θ)\sigma(\theta) at θ=30∘\theta = 30^\circ.

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