7.1.3 TEM, TM and TE waves

For a TEM wave $\nabla \times E$ has the $z$ component $\partial E_y/\partial x-\partial E_x/\partial y=-\mathrm{j}\omega \mu H_z=0$, and $\nabla \cdot E=\partial E_x/\partial x+\partial E_y/\partial y=0$ since $E_z=0$. A transverse field that is both curl-free and divergence-free in the cross-section is the gradient of a potential, $E_T=-{\nabla }_T\Phi$, with ${\nabla }_T^2\Phi =0$. On a perfect conductor the tangential $E$ vanishes, so each conductor is an equipotential. If there is only one conductor, enclosing the whole cross-section, $\Phi$ is constant on the boundary of the region, the solution of Laplace's equation is that constant everywhere, and $E_T=0$. With two separate conductors $\Phi$ can take two different values, and a nonzero TEM field exists: it is the electrostatic field of the two-conductor capacitor, travelling along $z$ with ${\mathrm{e}}^{-\mathrm{j}kz}$.