My super basic question is, the (magnetic) force between two steady current loops obeys Newton's third but the (magnetic) force between two charges doesn't. This is surprising given that the former is built out of the latter, so is there any significance to this fact?
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2Why do you think that the 3rd law does not hold for charges? It holds for the full force between them. It does not hold for infinitesimal current elements but it does not have to for those are non-physical, it holds for the full current loop. – hyportnex Sep 06 '23 at 21:38
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1@hypnortex OP is correct here - the Lorentz force breaks the 3rd law. See Apparent Violation of Newton's 3rd Law and the Conservation of Momentum (and Angular Momentum) For a Pair of Charged Particles for details. – Emilio Pisanty Sep 06 '23 at 21:42
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@hyportnex mechanical momentum is not conserved in EM, and thus newtons third law is wrong. – jensen paull Sep 06 '23 at 21:49
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1@EmilioPisanty not if you take into account the field momentum see https://iopscience.iop.org/article/10.1088/1361-6404/aca5d1/meta – hyportnex Sep 06 '23 at 21:52
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1@jensenpaull total momentum is conserved – hyportnex Sep 06 '23 at 21:53
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@hyportnex hence why I said mechanical, which is OP's question – jensen paull Sep 06 '23 at 22:27
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@hyportnex et al: my apologies, I should have been clear. I am here concerned with a violation of the principle of conservation of mechanical momentum. Outside of this narrow view, I am aware that the principle is recovered by ascribing momentum to the fields. At any rate, I’m wondering why it holds for current loops but not for particles proper. – EE18 Sep 06 '23 at 22:31
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The total force is $\mathbf F_1(2) + \mathbf F_2(1) + \frac{1}{c^2}\frac{d}{dt} \int dV \mathbf E \times \mathbf H$, see McDonald and its 90 references... For stationary currents you have only Biot-Savart of the direct forces without $d/dt$ of the field momentum; individual moving charges do not form a stationary current. – hyportnex Sep 06 '23 at 22:45
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Thank you @hyportnex. I will take a look at that link as it seems it may well answer my question! – EE18 Sep 06 '23 at 23:19
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In general, a moving set of charges will create time-dependent electric and magnetic fields, and so the Poynting vector will in general be time-dependent. But the Poynting vector is (proportional to) the field momentum density; so generically we should expect the field momentum to be changing with time.
But Newton's Third Law expresses the idea that the momentum of an isolated system is conserved; and while the total momentum will be constant, the mechanical momentum will have to change to compensate for the change in field momentum. This means that charges will get a net change in mechanical momentum even though there are no forces being applied to them, which will manifest as unequal forces between the charges and an apparent violation of Newton's Third Law.
Any stationary charge or current situation will, of course, have a constant amount of field momentum and so the mechanical momentum will be separately conserved. In effect, the time-dependence in the fields that we see in a situation with a few point charges is cancelled out by the symmetry of the configuration for a current loop, leaving a time-independent field momentum and no violations of Newton's Third Law.
Michael Seifert
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Thank you for this very nice answer. If it's OK, I have two quick questions: (1) you write "Any stationary charge or current situation will, of course, have a constant amount of field momentum" -- my electrodynamics is rusty, but to your point earlier this is because the electric and magnetic fields themselves are constant (so their cross product, which I believe is proportional to the field momentum) is too? – EE18 Sep 07 '23 at 01:34
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(2) you write "In effect, the time-dependence in the fields that we see in a situation with a few point charges is cancelled out by the symmetry of the configuration for a current loop", but I am a bit confused. From your quoted statement in (1), is it symmetry that's relevant here, or simply that the sources of the fields (the charges and/or currents) are constant which is relevant? – EE18 Sep 07 '23 at 01:35
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@EE18: (1) Yes, that's correct. You could conceive of a situations where the fields depend on time but $\vec{S}$ doesn't (or the integral of $\vec{S}$ over all space doesn't); but it's at least a necessary condition to get time-dependence of the field momentum. (2) You know, I'm not entirely sure. The key point is that all of the time-dependent fields from the individual charges sum up to make a time-independent field. The fact that the individual charges form a steady current in the aggregate is key to this. But I'm not exactly sure how to think of this "symmetry" in mathematical terms. – Michael Seifert Sep 07 '23 at 12:45
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Thank you very much for all of your help, these were lovely answers! With respect to (2), my sense is that it's the constancy of the sources which matters (e.g. force due to two straight wires carrying constant current also obeys Newton 3) but I'm likewise unsure. – EE18 Sep 07 '23 at 12:47