A small uncharged conducting sphere of radius (used as a collector electrode in an…

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

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

Problem

A small uncharged conducting sphere of radius a=5.0 cma = 5.0\,\mathrm{cm} (used as a collector electrode in an electrostatic precipitator) is placed in a uniform field E0=2.0×105 V m−1E_0 = 2.0\times10^5\,\mathrm{V\,m^{-1}} pointing along the polar axis, far from the sphere. The sphere carries no net charge. ((a)) Starting from the general exterior series with the regularity condition applied at infinity, derive the boundary conditions and hence the potential V(r,θ)V(r,\theta) for r≥ar\ge a. ((b)) Identify the 1/r21/r^2 term as a dipole potential and derive the induced dipole moment pp and the polarisability α≡p/E0\alpha \equiv p/E_0. ((c)) Evaluate α\alpha and pp numerically for the stated sphere and field.

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