The energy of photons in an electromagnetic wave is indeed directly proportional to the frequency of the wave. This relationship is described by Planck's equation, which states that the energy (E) of a photon is given by the equation E = hf, where h is Planck's constant (a fundamental constant of nature) and f is the frequency of the wave.
On the other hand, the energy density of an electromagnetic wave refers to the amount of energy carried by the wave per unit volume. It is a measure of the energy distributed throughout the space occupied by the wave. The energy density is not dependent on the frequency of the wave but rather on the square of the amplitude of the electric and magnetic fields in the wave.
In an electromagnetic wave, the energy is distributed across a large number of photons, and the total energy density of the wave is determined by the combined contributions of these photons. Each individual photon carries energy according to its frequency, but the overall energy density of the wave is determined by the amplitude of the fields rather than the frequency.
The energy density of an electromagnetic wave is given by the equation:
Energy Density = (1/2) * ε₀ * E² + (1/2) * B² / μ₀
where ε₀ is the permittivity of free space, E is the electric field amplitude, B is the magnetic field amplitude, and μ₀ is the permeability of free space.
As you can see, the energy density depends on the square of the electric and magnetic field amplitudes, not on the frequency of the wave. This means that waves of different frequencies can have the same energy density if their field amplitudes are appropriately adjusted.
In summary, the energy of photons in an electromagnetic wave is directly proportional to the frequency of the wave, but the energy density of the wave itself is determined by the square of the electric and magnetic field amplitudes, rather than the frequency.