91Ó°ÊÓ

Show directly from Maxwell's equations that the charge density \(\rho\) and the electric current density \(j\) obey the conservation law $$ \frac{\partial \rho}{\partial t}+\nabla \cdot j=0 $$

Show from Maxwell's equations in a vacuum that the magnetic field \(B\) obeys the wave equation.

Using Maxwell's equations in a vacuum, obtain an equation in the form of a conservation law for the rate of change of the energy \(w=\) \(|\boldsymbol{B}|^{2} / 2+|\boldsymbol{E}|^{2} / 2 c^{2}\) of an electromagnetic wave.

Show that for an isotropic elastic solid in equilibrium, the deformation \(v\) must obey \((\lambda+\mu) \nabla \nabla \cdot v+\mu \nabla^{2} v=0\)

An isotropic elastic solid with Lamé constants \(\lambda\) and \(\mu\) is subjected to a deformation \(v_{1}=a x_{1} x_{2}, v_{2}=b\left(x_{1}^{2}-x_{2}^{2}\right), v_{3}=0\) (a) Find the strain tensor \(E_{i j}\). (b) Find the stress tensor \(P_{i j}\). (c) Determine whether it is possible for the material to be in equilibrium.

A compressible fluid with negligible viscosity is initially at rest with uniform density \(\rho_{0}\) and pressure \(p_{0}\), with no body forces. A small perturbation is then introduced so that there is a velocity \(u(r, t)\) and the density becomes \(\rho_{0}+\rho_{1}(r, t)\). (a) Assuming that products of the small quantities \(\boldsymbol{u}\) and \(\rho_{1}\) can be neglected, show that the equation for conservation of mass (5.9) becomes $$ \frac{\partial \rho_{1}}{\partial t}+\rho_{0} \nabla \cdot u=0 $$ (b) Assuming that the perturbation \(p_{1}\) to the pressure is related to \(\rho_{1}\) by \(p_{1}=a \rho_{1}\) where \(a\) is constant, show that the Navier- Stokes equation reduces to $$ \rho_{0} \frac{\partial u}{\partial t}=-a \nabla \rho_{1} $$ (c) Hence show that the density perturbation \(\rho_{1}\) obeys the wave equation and interpret this result physically.