An instrumented spherical balloon of radius floats in the open atmosphere with no…

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

An instrumented spherical balloon of radius a=0.50 ma = 0.50\,\mathrm{m} floats in the open atmosphere with no external field. Its skin carries many narrow metal bands running round the vertical axis, insulated from one another, and an on-board circuit holds each band at its own potential, so that the surface potential is V(a,θ)=V0cos⁡θ,V0=−800 V.V(a,\theta) = V_0\cos\theta, \qquad V_0 = -800\,\mathrm{V}. There is no other charge nearby and V→0V \to 0 as r→∞r\to\infty. ((a)) Find the exterior potential V(r,θ)V(r,\theta), r≥ar\ge a. ((b)) Find Er(a,θ)E_r(a,\theta) and the charge density σ(θ)\sigma(\theta) on the outer faces of the bands at the pole (θ=0\theta=0) and at the equator (θ=90∘\theta=90^\circ).

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