A thin insulating spherical shell of radius carries a charge spread uniformly over…

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Electrostatic fields and potential · Electrostatic fields and potential

Electricity, Magnetism and Electromagnetism, Ch Electrostatic potential, potential energy and the energy of the field

Problem

A thin insulating spherical shell of radius

R = 6.0\ \mathrm{cm}

carries a charge

Q = +12\ \mathrm{nC}

spread uniformly over its surface. All the charge is at rest, and

V = 0

at infinity. Chapter 2's Gauss's law argument gives

E = 0

inside the shell and a radial field outside it. ((a)) The potential outside the shell is

V(r) = Q/4\pi\varepsilon_0 r

. Recover the radial field from it by differentiation, check that it agrees with the Gauss's law result, and evaluate the field at

r = 12\ \mathrm{cm}

. ((b)) Find the total field energy by integrating the energy density

u = \tfrac12\varepsilon_0E^2

over spherical layers of volume

4\pi r^2\,dr

. Say why the region inside the shell contributes nothing. ((c)) Evaluate the potential at the surface of the shell and hence the charge-account energy

\tfrac12 QV(R)

, and compare it with your answer to (b).

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