# What is the potential energy of electromagnetic waves traveling in air?

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?

• 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/… Jun 27, 2022 at 8:42
• 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.
– Buzz
Jun 27, 2022 at 21:43
• > "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. Jun 28, 2022 at 1:26
• @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. Jun 28, 2022 at 4:26
• @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. Jun 28, 2022 at 15:48