Do electromagnetic fields have momentum?

[cianfa72]

TL;DR Summary

About the Feynman's argument that electromagnetic fields have/hide momentum

Hi, reading Feynman's lectures on section 10-5 I came up with a question.

He claims that electromagnetic fields have momentum. He gives an example of two charges A and B in a electromagnetic field. When the charge A is suddenly moved (as I understand by an external force w.r.t. the "A + B" system) it will feel a reaction force (from the field I believe) meanwhile B has felt nothing.

He says that A will pick up some momentum (from the field reaction) while B will not since it has felt nothing and therefore it has not yet changed its momentum. So if we consider only the system "A + B" the conservation of momentum will not check out in that tiny time interval.

However my point is the following: the law of conservation of momentum only holds when the net external force is null. Therefore maybe here the point is that Feynman considers the small time interval from when the external force is actually removed from A, so that the system "A + B" will be acted by internal forces from the EM field alone.

Then if we assign momentum to the field the conservation of momentum "magically" checks out even in that tiny time interval (the system being now "A + B + EM field").

Note that in the above Feynman's argument charges A and B are actually the sources of the EM field considered.

What do you think about? Thanks.

 

[Frabjous]

Check out V2 Ch 27
https://www.feynmanlectures.caltech.edu/II_27.html

 

[Orodruin]

There is no guarantee that the momentum will "check out" in the A+B system even after the "tiny interval" has passed. The electromagnetic field will generally carry away some momentum as well.

The point is that generally A+B cannot be considered a closed system for which momentum conservation holds for any sort of time interval and if you add the EM field momentum into the mix, momentum conservation holds for all times - it is a local conservation law described by a source free continuity equation.

 

[Dale]

The laws of electromagnetism have spatial translation symmetry. So by Noether’s theorem we know that electromagnetism has a conserved momentum.

 

[cianfa72]

There is no guarantee that the momentum will "check out" in the A+B system even after the "tiny interval" has passed. The electromagnetic field will generally carry away some momentum as well.

The point is that generally A+B cannot be considered a closed system for which momentum conservation holds for any sort of time interval and if you add the EM field momentum into the mix, momentum conservation holds for all times

You mean A+B system cannot be considered closed in the sense that not only internal Newton 3rd law's force pairs exchanged between them are involved for any sort of time interval.

However, if one adds the EM field generated from them (the system being now "A + B + EM field"), in a sense the EM field is able to "store" some momentum and this way one can "restore" the (total) momentum conservation law valid for all the time.

- it is a local conservation law described by a source free continuity equation.

Ah ok, it is alike a continuity equation without source/sink terms (i.e. in the integral form there is no momentum "source" term inside the closed surface considered).

 

[cianfa72]

The laws of electromagnetism have spatial translation symmetry. So by Noether’s theorem we know that electromagnetism has a conserved momentum.

I believe here the point is different: in that lecture Feynman was considering the composite system "charges + charge generated's EM field" and not just the EM field itself.

 

[Dale]

I believe here the point is different: in that lecture Feynman was considering the composite system "charges + charge generated's EM field" and not just the EM field itself.

OK, but that is unnecessary. The EM field has momentum regardless. Therefore ...

Then if we assign momentum to the field the conservation of momentum "magically" checks out even in that tiny time interval (the system being now "A + B + EM field").

This is a fairly ridiculous statement. We are not assigning momentum to the field to magically balance the books of a system. It already has momentum on its own. And that intrinsic momentum naturally leads to conservation of total momentum.

 

[cianfa72]

The EM field has momentum regardless. Therefore ...We are not assigning momentum to the field to magically balance the books of a system. It already has momentum on its own. And that intrinsic momentum naturally leads to conservation of total momentum.

Sorry, I was just joking. I was taking, let's say, Feynman's point of view in his lecture 4 - conservation of energy.

Basically Dennis's mum has to be very clever to come up with new formulas to account for the different forms Dennis's blocks (i.e. energy) are hidden in. Since she believes in the law of conservation of blocks (energy), she invents new formulas for new situations in order to "check it out" the conservation of blocks (energy) in all the situations.

In this thread specific case, we already have a formula for the intrinsic EM field's momentum and it turns out that it leads to the conservation of total momentum (including the momentum of the charges sources of the EM field).

 

[Dale]

I disagree with Feynman on this approach. IMO it was out of date by several decades when he proposed it and it is out of date by more than a century now. Noether’s theorem is essential.

 

[cianfa72]

I disagree with Feynman on this approach. IMO it was out of date by several decades when he proposed it and it is out of date by more than a century now. Noether’s theorem is essential.

Ok, so by Noether’s theorem can one formulate the law of conservation of total momentum for the system "charge A + charge B + their generated EM field" ?

 

[Dale]

Ok, so by Noether’s theorem can one formulate the law of conservation of total momentum for the system "charge A + charge B + their generated EM field" ?

Yes. Without any of the Dennis' mum stuff. It can be found directly from the Lagrangian. This has been known since 1918.

 

[cianfa72]

It can be found directly from the Lagrangian.

Can you kindly point me to a formulation based on Lagrangian for "charge + EM generated field" ?

 

[Dale]

Sure, here you go:

https://phys.libretexts.org/Bookshe...ld/6.01:_Charged_Particle_in_a_Magnetic_Field

 

[cianfa72]

I read your link and sections 17 & 65 from "The Classic Theory of Fields" by Landau-Lifshitz.

In particular, they argue that for a system of interacting particles/charges - since the interaction propagation velocity isn't infinite - to describe it one must consider the composite system consisting of these charges + the field generated from them.

That means that, in order to rigorously describe such a system of interacting charges, the "overall" Lagrangian will depend on coordinates and velocities of the particles as well as some "degrees of freedom" of the EM field.

As you pointed out before, such a Lagrangian has got some symmetries. According Noether’s theorem, one of them corresponds to the (total) conservation of momentum.

Correct ?

 

[Dale]

As you pointed out before, such a Lagrangian has got some symmetries. According Noether’s theorem, one of them corresponds to the (total) conservation of momentum.

Yes, that is correct. Specifically, momentum is the conserved quantity that corresponds to the symmetry under spatial translation.

 

[TSny]

Here is a resource letter on electromagnetic momentum. It was written by David Griffiths, author of the widely used EM textbook.