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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?

Qmechanic
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EE18
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1 Answers1

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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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