A long straight conducting rod of circular cross-section

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 The field as a differential equation: Poisson, Laplace and the well-posed problem

Problem

A long straight conducting rod of circular cross-section, radius a=1.0 cma = 1.0\,\mathrm{cm}, is isolated in vacuum far from any other conductor. It carries a fixed free charge per unit length λ=3.0 nC m−1\lambda = 3.0\,\mathrm{nC\,m^{-1}}, and is idealised as infinitely long, so that nothing changes along its axis and the potential depends only on the cylindrical radius ss from that axis. ((a)) Write down the region Ω\Omega outside the rod, the equation holding in Ω\Omega, and the boundary data at s=as=a and as s→∞s\to\infty. ((b)) Classify the condition at s=as=a as Dirichlet, Neumann, or the third kind used for an isolated conductor of known charge, and explain why the usual far-away condition V→0V\to0 cannot be used here.

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