TEM Waves

Now we can start looking at waveforms in various cavities. Suppose we let $E_z = B_z = 0$ . Then the wave in the cavity is a pure transverse electromagnetic (TEM) wave just like a plane wave, except that it has to satisfy the boundary conditions of a perfect conductor at the cavity boundary!

Note from the equations above that: _&perp#perp;· _&perp#perp;& = & 0
_&perp#perp;×E_&perp#perp;& = & 0 from which we can immediately see that: &nabla#nabla;_&perp#perp;^2 _&perp#perp;= 0 and that _&perp#perp;= -&phis#phi; for some suitable potential that satisfies $\nabla_\perp^2\phi = 0$ . The solution looks like a propagating electrostatic wave. From the wave equation we see that: &mu#mu;&epsi#epsilon;&omega#omega;^2 = k^2 or k = ±&omega#omega;&mu#mu;&epsi#epsilon; which is just like a plane wave (which can propagate in either direction, recall).

Again referring to our list of mutilated Maxwell equations above, we see that: ik _&perp#perp;& = & -i&omega#omega;( × _&perp#perp;)
_&perp#perp;& = & -&omega#omega;&mu#mu;&epsi#epsilon;k ( × _&perp#perp;)
_&perp#perp;& = & ±&mu#mu;&epsi#epsilon; ( × _&perp#perp;) or working the other way, that: _&perp#perp;= ±&mu#mu;&epsi#epsilon; ( × _&perp#perp;) so we can easily find one from the other.

TEM waves cannot be sustained in a cylinder because the surrounding (perfect, recall) conductor is equipotential. Therefore $\Vec{E}_\perp$ is zero as is $\Vec{B}_\perp$ . However, they are the dominant way energy is transmitted down a coaxial cable, where a potential difference is maintained between the central conductor and the coaxial sheathe. In this case the fields are very simple, as the $\Vec{E}$ is purely radial and the $\Vec{B}$ field circles the conductor (so the energy goes which way?) with no $z$ components.

Finally, note that all frequencies are permitted for a TEM wave. It is not ``quantized'' by the appearance of eigenvalues due to a constraining boundary value problem.