I can technically follow the derivation of the EM energy density expression from Maxwell's equations, as here for example. But it seems really strange that, for waves, it does not depend on frequency. So I'm wondering how we could illustrate that more concretely.
The problem is, what kind of thought experiment can you do to show how much energy is in the field, or at least compare it between two different fields? It seems to me you would have to find a situation where the entire field is converted into motion of charges, and then the field energy would just be the resulting kinetic energy of the charges.
How can we show that, for a fixed amplitude and volume, a high-frequency EM wave has the same potential to generate motion as a low-frequency wave?
I am motivated by trying to understand Rayleigh scattering of sunlight in the atmosphere. Here, different frequencies produce the same amplitude of motion in electrons, so higher frequencies produce higher acceleration, hence greater energy. So that's why the scattered light turns more blue -- but that only makes sense if we can understand how the different colors had the same energy to begin with.