ELECTRIC FIELDS IN MATTER — Edward Purcell & David Morin
Physics — E&M · maxwells-equations-em-waves
Problem
In part (b) of Problem 6.24 we dealt with approximate expressions for the electric and magnetic fields due to a slowly moving charge, valid in the limit v≪c. In this problem we will use the exact forms. The exact E, for any value of v, is given in Eq. (5.13) or Eq. (5.15). The Lorentz transformation then gives the exact B as B=(1/c2)v×E; see Problem 6.24(a). Verify that these exact expressions for E and B satisfy Maxwell's equations in vacuum. That is:
(a) Show that ∇⋅B=0. (We already showed in Problem 5.4 that ∇⋅E=0.) The vector identity for ∇⋅(A×B) in Appendix K will come in handy.
(b) Show that ∇×E=−∂B/∂t. (The ∇×B=∂E/∂t calculation is nearly the same, so you can skip that.) Note: Although the calculation is doable if you use the spherical-coordinate expression for E in Eq. (5.15) (don't forget that both r and θ vary with time), it's a bit easier if you use the Cartesian-coordinate expression in Eq. (5.13).