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Exercises

Consider an electromagnetic wave propagating through a non-dielectric, non-magnetic medium containing free charge density $\rho$ and free current density ${\bf j}$ . Demonstrate from Maxwell's equations that the associated wave equations take the form

$\displaystyle \nabla^{\,2}{\bf E} - \frac{1}{c^{\,2}}\,\frac{\partial^{\,2} {\bf E}}{\partial t^{\,2}}$ $\displaystyle = \frac{\nabla \rho}{\epsilon_0} + \mu_0\,\frac{\partial {\bf j}}{\partial t},$

$\displaystyle \nabla^{\,2}{\bf B} - \frac{1}{c^{\,2}}\,\frac{\partial^{\,2} {\bf B}}{\partial t^{\,2}}$ $\displaystyle = - \mu_0\,\nabla\times{\bf j}.$
A spherically symmetric charge distribution undergoes purely radial oscillations. Show that no electromagnetic waves are emitted. [Hint: Show that there is no magnetic field.]

Richard Fitzpatrick 2014-06-27

$\displaystyle \nabla^{\,2}{\bf E} - \frac{1}{c^{\,2}}\,\frac{\partial^{\,2} {\bf E}}{\partial t^{\,2}}$	$\displaystyle = \frac{\nabla \rho}{\epsilon_0} + \mu_0\,\frac{\partial {\bf j}}{\partial t},$
$\displaystyle \nabla^{\,2}{\bf B} - \frac{1}{c^{\,2}}\,\frac{\partial^{\,2} {\bf B}}{\partial t^{\,2}}$	$\displaystyle = - \mu_0\,\nabla\times{\bf j}.$