A point charge is held fixed at the origin in vacuum
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 point charge Q=4.0nC is held fixed at the origin in vacuum. A small test charge q0=2.0nC is carried slowly round a closed path that lies in a plane through Q, so that kinetic energy may be ignored and the only work is that of the electrostatic force. The path has four legs, traversed in this order. Leg 1 runs straight, radially outward from r=0.10m to r=0.30m along one ray from Q; leg 2 is a quarter circle of radius 0.30m centred on Q; leg 3 runs straight, radially inward from r=0.30m to r=0.10m; leg 4 is a quarter circle of radius 0.10m centred on Q, back to the start. Take k=1/4πε0=8.99×109Nm2C−2.
((a)) Starting from the line integral of the force in short straight steps, derive an expression for the work done by the electrostatic force on q0 along a radial leg running from r1 to r2, and evaluate it for leg 1.
((b)) Show that legs 2 and 4 do no work, and hence give the total work done by the field round the whole closed path.
((c)) A uniformly charged spherical shell of radius R=5.0cm carries charge Q=4.0nC spread uniformly over its surface and held at rest. Its field is zero inside and equal to Q/4πε0r2 radially outward at distance r≥R from the centre. Using the energy density u=21ε0E2 and spherical layers of volume 4πr2dr, derive the total field energy of the shell and evaluate it.