A thin spherical shell of radius carries a total charge spread uniformly over its surface

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 thin spherical shell of radius R=8.0 cmR = 8.0\,\mathrm{cm} carries a total charge Q=5.0 nCQ = 5.0\,\mathrm{nC} spread uniformly over its surface, with vacuum everywhere else. The potential is V(r)=Q4πε0R (constant),r<R;V(r)=Q4πε0r,r>R.V(r) = \frac{Q}{4\pi\varepsilon_0 R} \ (\text{constant}), \quad r<R; \qquad V(r) = \frac{Q}{4\pi\varepsilon_0 r}, \quad r>R. ((a)) Show that the inside form satisfies Laplace's equation. ((b)) Show that the outside form satisfies Laplace's equation. ((c)) Find the jump in ∂V/∂r\partial V/\partial r across r=Rr=R and use it to find the surface charge density σ\sigma on the shell. Check the result against σ=Q/4πR2\sigma=Q/4\pi R^2.

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