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The Lagrangian for electromagnetic waves shows that its energy has a kinetic part as well as a potential part (just like everything else). The potential part only exists in a medium whereas in the vacuum all energy in electromagnetic waves are kinetic.

Similar formulation exists for mechanical (e.g. elastic) waves. In this case, I understand easily that the kinetic energy is simply the intensity of shaking/amplitude of waves (e.g. in an earthquake) whereas the potential energy is the so-called strain energy which is basically just the potential energy associated for a spring.

Returning back to electromagnetic waves, I can understand that the kinetic energy is, again, the amplitude of the waves that is manifested by e.g. light brightness or heat (which originates from the motion/collision of particles). However, I am having a hard time thinking of an intuitive interpretation for potential energy for electromagnetic waves. I only know that this only exists in non-vacuum space. So, what is the potential energy for all the electromagnetic waves we create here on earth, which are traveling in the air that is non-vacuum. Are there examples of familiar electromagnetic waves that does not have (or have very little) kinetic energy but is purely all potential energy?

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  • $\begingroup$ I suppose what is meant here is the polarization and the magnetization of the medium, i.e., electric and magnetic response of the media - often characterized by permittivity and permeability constants in classical electrodynamics. en.wikipedia.org/wiki/… $\endgroup$
    – Roger V.
    Commented Jun 27, 2022 at 8:42
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    $\begingroup$ There are two terms in the energy density of an electromagnetic wave, but they do not correspond in any intelligible way to a "kinetic" part and a "potential" part. $\endgroup$
    – Buzz ♦
    Commented Jun 27, 2022 at 21:43
  • $\begingroup$ > "The potential part only exists in a medium whereas in the vacuum all energy in electromagnetic waves are kinetic." -- Where did you learn this terminology? It is not standard to call part of EM wave energy as kinetic and other part as potential. $\endgroup$
    – Ján Lalinský
    Commented Jun 28, 2022 at 1:26
  • $\begingroup$ @JánLalinský. I read it here, from the answer of Viktor T. Toth. I am very sure that for all classical physics, the Lagrangian is kinetic - potential energy, and the link I provided shows the Lagrangian for classical electromagnetism. Another reason I am so eager to understand this is because I see this separation for mechanical waves, also from its Lagrangian (For these waves, Lagrangian density is the right word), so this should have a parallel for EM. I guess the key here is to understand the 4-vector j_mu. $\endgroup$
    – Axel Wang
    Commented Jun 28, 2022 at 4:26
  • $\begingroup$ @AxelWang that is misguided a leads to confusion. The term -FF looks like kinetic energy density and the term jA looks like potential energy density in non-relativistic Lagrangian form $T-V$, but that formal analogy is all there is to it. The -FF term has value proportional to $E^2/c^2 - B^2$ which is zero for plane EM wave, and Mr. Toth would be forced to say that a plane wave has zero kinetic energy. But in macroscopic EM theory, plane wave carries non-zero EM energy. $\endgroup$
    – Ján Lalinský
    Commented Jun 28, 2022 at 15:48

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Potential energy is not an intrinsic quantity of any physical object, and therefore it is not meaningful to ask of the potential energy of an electromagnetic wave. In some sense, you can view the EM wave as a collection of photons, all of which can interact with any system (or not, i.e., for the sake of this question, we can say that photons do not interact with other photons), and thus may have potential energy with respect to that system. For example, photons interact with gravity, and will gain kinetic energy (via E = hf, for E is the energy, h is Planck's constant, an f is the frequency of the light) by changing frequency as they propagate through the gravitational field.

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