# Is the energy of the electromagnetic wave proportional to the amplitude or to the amplitude squared of the wave? [duplicate]

I've read that in mechanical waves that it's(energy) is proportional to the amplitude squared but in electromagnetic waves it's only proportional to the amplitude, is that really true?

• Where have you read that the energy of an electromagnetic wave is proportional to its amplitude (and not amplitude square)? Nov 17, 2017 at 13:46
• In my physics book. Nov 17, 2017 at 13:48
• Please show some research effort before asking here, for instance, search for something like "energy electromagnetic wave" with your favourite search engine. If there's something unclear to you about the results, then you can ask a more specific question about that here. Nov 17, 2017 at 13:58

Seems wrong then. The energy density of an electromagnetic wave is

$$u=\frac{\varepsilon_{0}|E|^2}{2}+\frac{|B|^2}{2\mu_{0}}$$

which is certainty proportional to its amplitude squared. For example, a plane wave of amplitude $\vec{E}_{0}$ also satisfies $|\vec{E}_{0}|=c|\vec{B}_{0}|$ such that

$$u=\frac{\varepsilon_{0}|\vec{E}_{0}|^2}{2}+\frac{|\vec{E}_{0}|^2}{2\mu_{0}c^2}=\varepsilon_{0}|\vec{E}_{0}|^2$$

since $c^2=\frac{1}{\varepsilon_{0}\mu_{0}}$.

• although I gave +1, and it is correct as far as energy density goes, BUT I think that there is a difference between energy density and the energy in the amplitude, see ccrma.stanford.edu/~jos/pasp/Acoustic_Energy_Density.html where "Thus, half of the acoustic intensity $I$ in a plane wave is kinetic, and the other half is potential". There is not potential as such in the em plane wave . So maybe there is no "wrong" in the book but a different deffinition Nov 17, 2017 at 14:21
• @annav I agree with this distinction. It is somewhat interesting to see that if one compares the Lagrangian of an acoustic string $\mathcal{L}=\frac{\rho}{2}\left(\frac{\partial y}{\partial t}\right)^2-\frac{T}{2}\left(\frac{\partial y}{\partial x}\right)^2$ to that of an electromagnetic field in vacuum $\mathcal{L}=\frac{\varepsilon_{0}|E|^2}{2}-\frac{|B|^2}{2\mu_{0}}$ a correspondence between potential energy and the magnetic field can be seen. Nov 17, 2017 at 14:31
• Isn’t $E_{0}$ the intensity of the electric field component rather then a amplitude? Nov 17, 2017 at 18:39
• @HolgerFiedler The intensity of the electromagnetic field is related to the Poynting vector $|\vec{S}|=|\frac{1}{\mu_0}\vec{E}\times\vec{B}|=\sqrt{\frac{\varepsilon_0}{\mu_0}}|\vec{E}_{0}|^2$ where the last equality follows assuming a plane wave. Nov 17, 2017 at 18:43
• @HolgerFiedler the question doesn't ask about photons. The amplitude of the wave can be readily confirmed by for example measuring the accelerating effect it has on electrons. I.e. Thomson scattering. Similarly, interference patterns result from the adding of amplitudes, not intensities. Your interjections on the subject of classical electromagnetism are increasingly bizarre. Nov 18, 2017 at 0:28