Introduction to Electrodynamics · Problem 7.64

Conservation Laws — David J. Griffiths

Physics · duality-transformation

Problem

(a) Show that Maxwell's equations with magnetic charge (Eq. 7.44) are invariant under the duality transformation

\begin{aligned} \mathbf{E}' &= \mathbf{E}\cos\alpha + c\mathbf{B}\sin\alpha,\\ c\mathbf{B}' &= c\mathbf{B}\cos\alpha - \mathbf{E}\sin\alpha,\\ cq_e' &= cq_e\cos\alpha + q_m\sin\alpha,\\ q_m' &= q_m\cos\alpha - cq_e\sin\alpha, \end{aligned} \tag{7.68}

where

c \equiv 1/\sqrt{\epsilon_0\mu_0}

and

\alpha

is an arbitrary rotation angle in "

\mathbf{E}/\mathbf{B}

-space." Charge and current densities transform in the same way as

q_e

and

q_m

. (b) Show that the force law (Prob. 7.38)

\mathbf{F} = q_e(\mathbf{E} + \mathbf{v} \times \mathbf{B}) + q_m\left(\mathbf{B} - \frac{1}{c^2}\mathbf{v} \times \mathbf{E}\right) \tag{7.69}

is also invariant under the duality transformation.