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When I was an undergraduate student, I was comfortable with the concepts of momentum and angular momentum of a particle or of a system of particles. However, the concept of electromagnetic fields carrying momentum and angular momentum occurred to me as somewhat odd to digest. I don't know if everyone feels the same. What could be a compelling way of introducing the concept of momentum and angular momentum of fields, both at undergraduate and graduate-level courses on classical electrodynamics, so that it becomes natural? Of course, without introducing the particle picture of light i.e. photons.

Solidification
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    https://en.m.wikipedia.org/wiki/Maxwell_stress_tensor under the motivation section, here we have momentum conservation for the em field from which we can interpret the results similarly to poyntings theorem, noting that f is the rate of change of momentum density in the final expression, the stress tensor as the momentumesque poynting vector and the actual momentum density of the em field. If the em field takes time to exert forces, where is this momentum stored whilst the wave hasn't reached the charge yet – jensen paull Apr 18 '23 at 13:00
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    An interesting motivation that I found was the idea that if you analyse a problem of two moving charges that move in orthogonal directions, Newton's third law may not apply (specially if you neglect that fields can carry some momentum). – Mauricio Apr 18 '23 at 13:13
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    The only motivation I needed is to accept that nothing could move at infinite speed. If you, too, accept that then you do not need any more motivation because between closing a switch to turn the current on and turning the motor shaft there is a time interval during which all "motions" are residing in the ether. – hyportnex Apr 18 '23 at 14:13
  • A classical EM field is a region where a test charge feels an EM force. The force is ultimately caused by another charge somewhere else. But the universe isn't classical. In QM, a photon is something like a classical particle and something like a classical wave. But it isn't really either one. It is a thing that has a frequency and wavelength, and also carries energy and momentum. See How can a red light photon be different from a blue light photon? – mmesser314 Apr 18 '23 at 14:36
  • On a general note... what irks me in hindsight about my physics education is that it made me "so comfortable" about concepts like momentum without asking deeper questions (Noether's theorem was discussed in my first year). Another thing that goes sideways in high school, in my opinion, is that not enough emphasis is being put on the fact that "particles" are approximations of the actual motion of extended objects, but that may not actually be so, but might only be my selective memory at work. There is a lot of fine print that students don't pay attention to. I won't exclude myself. – FlatterMann Apr 18 '23 at 18:33

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I found C. E. Mungan’s explanation of radiation pressure convincing (presented as an exercise in Griffiths’ Introduction to Electrodynamics). You can extend it by looking at the effect of a circularly polarized EM mode on a test charge. It shows that there is injected energy, momentum and angular momentum, which must come from the field.

To make things concrete, let $r$ be the position of the particle, $m$ its mass, $q$ its charge, $\gamma$ the damping factor, $\omega_0$ the proper frequency of oscillator, $E,B$ the EM fields. In the long wavelength limit (which justifies that the fields are not applied at $r$ and that you can neglect the magnetic force): $$ \ddot r+\gamma \dot r+\omega_0^2r=\frac{q}{m}E(x=0) $$ You can now solve the linear equation and in the stationary regime, you’ll have momentum, angular momentum and energy balance.

Let the angular frequency be $\omega$ and wave vector be $k$ so that: $$ B=\frac{1}{\omega}k\times E $$ and choose cartesian coordinates $x,y,z$ such that $k$ is along $z$ so $E$ and $r$ are in the $xy$ plane. Let $\pm \pi/2$ be the phase difference between $E_x,E_y$ so that RCP corresponds to $-\pi/2$ and LCP to $+\pi/2$. Let $E_0$ be the amplitude of the mode so that $\langle E_x^2+E_y^2\rangle=E_0^2$.

Taking the average over a period of the oscillation of the field of angular velocity $\omega$, you have the power (injected energy): $$ \begin{align} P &=\langle qE \cdot v\rangle \\ &= \omega\frac{q^2}{m}\frac{\gamma\omega}{(\omega^2-\omega_0^2)^2+(\gamma\omega)^2}E_0^2 \end{align} $$ For the force (injected momentum), the electric force vanishes, so only the magnetic one contributes: $$ \begin{align} F &=\langle qv \times B\rangle \\ &= k\frac{q^2}{m}\frac{\gamma\omega}{(\omega^2-\omega_0^2)^2+(\gamma\omega)^2}E_0^2 \end{align} $$ and the torque (injected angular momentum): $$ \begin{align} T &=\langle r \times qE\rangle \\ &=\pm e_z \frac{q^2}{m}\frac{\gamma\omega}{(\omega^2-\omega_0^2)^2+(\gamma\omega)^2}E_0^2 \end{align} $$ You can intepret the common factor: $$ \frac{q^2}{m}\frac{\gamma\omega}{(\omega^2-\omega_0^2)^2+(\gamma\omega)^2}E_0^2=\hbar \dot N $$ as the rate of absorption of photons. Note that a non vanishing $\gamma$ is crucial for it to be non zero. There is no quantum mechanics, the discussion has been purely classical, but you need to introduce the constant $\hbar$ to match the dimensions. You recover the expression of energy (sorry for the conflict), momentum and angular momentum of the mode: $$ \begin{align} E &= \omega \hbar N \\ P &= k \hbar N \\ L &= \pm e_z \hbar N \end{align} $$ which are the expected relations from field theory (note that this shows that the field has spin $1$).

The benefit of this approach is that the calculations are accessible at an undergraduate level.

Hope this helps.

LPZ
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My answer is rather throwaway.

Light can carry energy. You can invoke photoelectric effect, or just feel it with heat lamps (the Sun is one huge one).

By the Special Theory of Relativity, if light can carry energy, it must also be able to carry momentum, and thus angular momentum.

You can also invoke experiments like Compton scattering and Raman spectroscopy to directly motivate that they carry momentum, this time with the expectation that the wavelength has something to do with it.

naturallyInconsistent
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