electromagnetism - How do the electric or magnetic fields contain momentum?

I have recently come to know that the electric and magnetic field contain both linear and angular momenta, which are known functions of the electric and magnetic fields at any given point in space and time.

I don't understand how this is the case; could you explain how this works? Is it related to photons being emitted by the accelerating charges, or with the Abraham-Lorentz force?

Answer

In terms of a photon picture, this is not really mysterious at all. The electromagnetic force is mediated, in its quantum mechanical description, by the exchange of photons. These can be real - i.e. represent real light beams - or virtual, which means that the energy for the photon's existence has been 'borrowed' for a small amount of time as allowed by the Heisenberg uncertainty principle. Electrostatic and magnetostatic fields consist, in the quantum picture, of a huge number of virtual photons flying back and forth.

Now, each of these photons carries a certain amount of momentum. They must, because they will impart a force on the charged particles that absorb or emit them. Since each photon carries momentum, it is no surprise that the field as a whole can contain some net amount of momentum! Sometimes this will be zero - the contributions from the different photons will cancel out either locally at each point or globally once all points are considered - but this need not be the case. Thus, the electromagnetic field can carry momentum.

Now, this is a nice and intuitive picture, but it draws on a very exotic concept, so I'd understand if it weirds you out a little. More than that, since the existence of electromagnetic field momentum is required within classical electrodynamics, one would also want an answer which does not require quantum mechanics to explain it. (Think about this last bit carefully - it's not a trivial argument.)

In the end, whether the field "has" momentum or not is a matter of the definition of the word "have", which is a human construct. Strictly speaking, what is true is that

• it is possible to arrange situations where charged particles interact in a way in which their total mechanical momentum is not conserved, but once all the particles are separated again then their final total momentum equals the initial one.

This is augmented by the fact that

• there exists a quantity, with units of momentum, and which can be calculated from the electric and magnetic fields at each point, which will give a conserved quantity if it is added to the particles' total mechanical momentum.

It is important to note that the conservation of momentum is not a given; it is a property of physical theories which any particular theory may or may not have. (As it happens, all physical theories which we observe in the real world do observe it in some form, but that is not guaranteed a priori.)

One example of this is newtonian mechanics with forces which obey Newton's third law. In this case, it is a theorem of the theory that the total mechanical moementum is conserved.

Another example is Noether's theorem, which guarantees a momentum conservation law in dynamical systems of a certain class, whose laws are translationally invariant. For certain systems this invariance exists and hence momentum is conserved; for others it is not and momentum is not conserved.

For charged mechanical particles interacting electromagnetically, Newton's third law does not hold, so our old theorem is not applicable (and in fact its conclusion is false, as the mechanical momentum is not conserved). However, this does not mean that we cannot find a smarter, more sophisticated theorem which does imply a conservation law.

One therefore needs to sit for a bit and jiggle at the maths, but the theorem is indeed provable. In essence, what you do is

• write down the total force on the mechanical particles,


• express it in terms of electromagnetic fields, charges and currents,


• use Maxwell's equations to transform the charges and currents into electric and magnetic fields, and thus


• derive an expression for the total mechanical force on the system in terms of the integral of a certain function of the electric and magnetic fields at each point.


• One then needs to transform this quantity into the total time derivative of a simpler expression, which will be interpreted as the electromagnetic field momentum. This is possible but it leaves a remainder which depends on what volume you're integrating over.


• One can then prove that, for localized systems, this remainder vanishes. When it does, the total dynamical momentum - mechanical plus electromagnetic - is conserved.

In general, I would discourage you from attempting this calculation until you have taken solid courses in electromagnetism and vector calculus at university level, or you will just bruise yourself up against it. Focus, instead, on the physics, on a qualitative level.

If you have more specific questions I'm happy to try and reply, but if you want details on the mathematics you do need to specify what your background is so that we can give answers you will understand.