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 very long straight line of charge, treated as infinite and held at rest in vacuum, carries a uniform line charge density λ=2.5×10−8Cm−1. Its field at perpendicular distance s from the line is E=λ/2πε0s, directed radially away from the line. Take ε0=8.854×10−12Fm−1 and k=8.99×109Nm2C−2. A closed path lies in a plane perpendicular to the line and is traversed in this order: outward along a straight radial leg from s=2.0cm to s=8.0cm; then a quarter circle of radius 8.0cm centred on the line; then inward along a straight radial leg back to s=2.0cm; then a quarter circle of radius 2.0cm back to the start.
((a)) Derive an expression for ∫E⋅dl along a radial leg from s1 to s2 and evaluate it for the outward leg, in volts.
((b)) Hence obtain ∮E⋅dl round the whole closed path.
((c)) Two concentric uniformly charged spherical shells are held at rest: the inner, of radius a=2.0cm, carries +3.0nC, and the outer, of radius b=6.0cm, carries −3.0nC. The field is zero for r<a and for r>b, and equals Q/4πε0r2 radially outward for a<r<b, with Q=3.0nC. Using u=21ε0E2 and layers 4πr2dr, derive the total field energy of this pair of shells and evaluate it.