(a) Show that Maxwell’s equations with magnetic charge (Eq. 7.44) are invariant under the duality transformation
\left. \begin{matrix} \text { (i) } \nabla \cdot E =\frac{1}{\epsilon_{0}} \rho_{e} , & \text { (iii) } \nabla \times E =-\mu_{0} J _{m}-\frac{\partial B }{\partial t} \text {, } \\ \text { (ii) } \nabla \cdot B =\mu_{0} \rho_{m} , &\text { (iv) } \nabla \times B =\mu_{0} J _{e}+\mu_{0} \epsilon_{0} \frac{\partial E }{\partial t}. \end{matrix} \right\} (7.44)
\left.\begin{array}{l} E ^{\prime}= E \cos \alpha+c B \sin \alpha ,\\c B ^{\prime}=c B \cos \alpha- E \sin \alpha ,\\c q_{e}^{\prime}=c q_{e} \cos \alpha+q_{m} \sin \alpha ,\\q_{m}^{\prime}=q_{m} \cos \alpha-c q_{e} \sin \alpha.\end{array}\right\} (7.68)
where c \equiv 1 / \sqrt{\epsilon_{0} \mu_{0}} and α is an arbitrary rotation angle in “E/B-space.” Charge and current densities transform in the same way as q_{e} \text { and } q_{n}. [This means, in particular, that if you know the fields produced by a configuration of electric charge, you can immediately (using \alpha=90^{\circ}) write down the fields produced by the corresponding arrangement of magnetic charge.]
(b) Show that the force law (Prob. 7.38)
F =q_{e}( E + v \times B )+q_{m}\left( B -\frac{1}{c^{2}} v \times E \right) (7.69)
is also invariant under the duality transformation.