Electricity and Magnetism · Problem 9.1

ELECTRIC FIELDS IN MATTER — Edward Purcell & David Morin

Physics — E&M · maxwells-equations-displacement-current

Problem

Due to the contradiction between Eqs. (9.2) and (9.5), we know that there must be an extra term in the ∇×B\nabla \times \mathbf{B} relation, as we found in Eq. (9.10). Call this term W\mathbf{W}. In the text, we used the Lorentz transformations to motivate a guess for W\mathbf{W}. Find W\mathbf{W} here by taking the divergence of both sides of ∇×B=μ0J+W\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mathbf{W}. Assume that the only facts you are allowed to work with are (1) ∇⋅E=ρ/ϵ0\nabla \cdot \mathbf{E} = \rho/\epsilon_0, (2) ∇⋅B=0\nabla \cdot \mathbf{B} = 0, (3) ∇⋅J=−∂ρ/∂t\nabla \cdot \mathbf{J} = -\partial\rho/\partial t, and (4) ∇×B=μ0J\nabla \times \mathbf{B} = \mu_0 \mathbf{J} in the case of steady currents.

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