A test electrode of radius sits in a region with no applied field

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 test electrode of radius a=0.15 ma = 0.15\,\mathrm{m} sits in a region with no applied field. Its surface is made of many narrow metal bands running round the zz axis, insulated from one another, and a control circuit holds each band at its own potential, so that the surface potential is V(a,θ)=V0cos⁡θV(a,\theta) = V_0\cos\theta, with V0=300 VV_0 = 300\,\mathrm{V} and θ\theta the polar angle. Treat the bands as narrow enough that this is a smooth function of θ\theta. There is no charge anywhere outside the sphere, and V→0V \to 0 as r→∞r \to \infty. ((a)) Write the general azimuthally symmetric Legendre series allowed in the region r≥ar \ge a, state which family of radial powers is discarded and why, and hence find 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=60∘\theta = 60^\circ.

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