What produces a standing electromagnetic wave?

What boundary condition does a perfect conducting wall impose on the tangential electric field?

For a cavity of length \(L\) with electric-field nodes at both ends, write the allowed wavelengths.

A vacuum cavity has length \(0.50\,\mathrm{m}\). Find the first three resonant frequencies.

Adjacent electric-field nodes in a standing EM wave are separated by \(12.5\,\mathrm{cm}\). Find the frequency in vacuum.

Add \(E_1=E_0\sin(kx-\omega t)\) and \(E_2=E_0\sin(kx+\omega t)\).

For \(E=2E_0\sin(kx)\cos(\omega t)\), find the positions of electric-field nodes.

For \(E=2E_0\sin(kx)\cos(\omega t)\), find the positions of electric-field antinodes.

A cavity of length \(1.20\,\mathrm{m}\) resonates at \(f=500\,\mathrm{MHz}\). Determine the mode number in vacuum.

A cavity is filled with dielectric of refractive index \(1.50\). How do the resonant frequencies change compared with vacuum?

A vacuum cavity has length \(0.80\,\mathrm{m}\). It is filled with material of \(n_r=2.0\). Find the fundamental frequency.

Explain why the time-averaged net energy flow in an ideal standing wave is zero even though energy is present.

In a simple standing EM wave, electric-field nodes coincide with magnetic-field antinodes. Explain why this does not contradict energy conservation.

A cavity has one end enforcing an electric node and the other enforcing an electric antinode. Derive the allowed wavelengths.

For the node-antinode cavity in the previous question, find the allowed frequencies in vacuum.

A conducting cavity length expands thermally by fractional amount \(\alpha\Delta T\). Derive the fractional change in each resonant frequency.

A cavity supports modes up to \(1.20\,\mathrm{GHz}\). If \(L=0.75\,\mathrm{m}\), how many node-node vacuum modes are at or below this frequency?

A measured standing wave has electric nodes at \(x=0\), \(0.20\,\mathrm{m}\), \(0.40\,\mathrm{m}\), and \(0.60\,\mathrm{m}\). If the cavity ends are at \(0\) and \(0.60\,\mathrm{m}\), identify the mode number and frequency in vacuum.

A student claims that because a standing wave has zero average energy flow, its Poynting vector is zero at every instant. Correct the claim.

Prove that a node-node cavity has integer-spaced resonant frequencies, then contrast with a node-antinode cavity.