What effect does amplitude have on EM waves? There have been previous questions like this regarding light/photons specifically. My question is what the effect of the amplitude is on, say, a gamma wave? It would change its energy, as $E = h f$. Maybe amplitude doesn't even exist with an EM wave.
 A: The electric and magnetic field strength ($\bf{E}$ and $\bf{B}$)
is a concept of classical electrodynamics.
It emerges from quantum electrodynamics
as a collective effect only when there are many photons.
When considering only a few photons or even a single photon
(like when you have gamma rays) it doesn't make sense
to talk about electric and magnetic field or their amplitude.
The energy of a single photon is $E=hf$ (where $f$
is the frequency of the EM wave).
Therefore, the amplitude of electric and magnetic field
has no effect on the energy of the single photons.
Instead, the amplitude effects the number of photons.
The energy density of the EM field (i.e. energy per volume) is
$$\rho=\frac{\epsilon_0}{2}\bf{E}^2 + \frac{1}{2\mu_0}\bf{B}^2.$$
Therefore in an EM wave the number $N$ of photons within a volume
can be calculated from the volume integral
$$\begin{align}
N hf &= \int_V \rho\ dV \\
 &= \int_V \left( \frac{\epsilon_0}{2}\bf{E}^2 + \frac{1}{2\mu_0}\bf{B}^2 \right)dV \\
 &= \frac{1}{2} \left( \frac{\epsilon_0}{2}\hat{\bf{E}}^2 + \frac{1}{2\mu_0}\hat{\bf{B}}^2 \right) V
\end{align}$$
where $\hat{\bf{E}}$ and $\hat{\bf{B}}$ are the amplitudes of the EM wave.
