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Imagine one charge rotating in a circular orbit about a point, and another charge rotating about another point slightly below the first, tracing out two parallel circles on the outside of a cylinder.

Assume you're a stationary observer, relative to the charges rotating around the cylinder.

Because their relative velocities are zero, Biot-Savart seems to imply no magnetic force will be produced between the two charges. But the law assumes a rectilinear velocity, so the question is, does it also hold for rotational velocities, and if not, what is the correct equation?

https://en.wikipedia.org/wiki/Biot%E2%80%93Savart_law

aditya_stack
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Feynmanfan85
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  • What does relative velocities being zero has to do with the magnetic field? Maybe if you were an observer going with the charges on the same path and then you won't see any moving charges. But in this case you're not an inertial frame of reference (always rotating) – Ofek Gillon Dec 29 '19 at 22:11
  • Assume an observer stationary relative to the two charges. If the two charges have equal rectilinear velocities, they do not produce a magnetic field. I'm asking whether this is also the case for two charges that have equal rotating velocities. – Feynmanfan85 Dec 29 '19 at 22:12
  • Well, that's true for inertial reference frames. The observer's frame isn't inertial. It's like saying that zero net force results in zero acceleration and then you take an accelerating observer which disagrees. – Ofek Gillon Dec 29 '19 at 22:13
  • Newton's and Maxwell's laws are true for all inertial frames, not inertial ones – Ofek Gillon Dec 29 '19 at 22:14
  • OK let's drop the jargon - imagine standing in a lab, and in the lab, there's a cylinder. It has two charges rotating at equal velocities around the outside. Field or no? – Feynmanfan85 Dec 29 '19 at 22:14
  • field of course, why not? – Ofek Gillon Dec 29 '19 at 22:15
  • Because in the case of two rectilinear charges with equal velocities, you do not have a magnetic field. So you're saying that in the case of two charges moving with equal rotational velocities, you do have a field? – Feynmanfan85 Dec 29 '19 at 22:16
  • "Because in the case of two rectilinear charges with equal velocities, you do not have a magnetic field". What? I think I don't understand what you mean by saying that, because 2 charges on the same wire going in the same direction will produce a magnetic field (magnetic field from an infinite wire is one of the first exercises encountered in magnetostatics) – Ofek Gillon Dec 29 '19 at 22:18
  • Two charges moving with equal rectilinear velocities will not exert a magnetic force on each other. The question is, will two charges moving with equal rotational velocities exert a magnetic force on each other? – Feynmanfan85 Dec 29 '19 at 22:19
  • OHHH, now I understand you, you mean that they don't apply a field on eachother, that was an important clarification. I suggest you edit your question :) – Ofek Gillon Dec 29 '19 at 22:21
  • I did, thanks, that's a fair point, it was unclear. – Feynmanfan85 Dec 29 '19 at 22:21
  • I don't understand any of this discussion. Two equal charges moving parallel to each other certainly exert a force on each other, whatever their relative velocities. Part of this Lorentz force will be the magnetic field generated by a moving charge. – ProfRob Dec 29 '19 at 23:24
  • Two charges moving with equal velocities relative to some third observer are not moving at all with respect to one other. Therefore, there magnetic force between them is zero, and any force between them is the result of electrostatic charge. – Feynmanfan85 Dec 29 '19 at 23:44
  • Related : Magnetic Field due to a single moving charge. – Frobenius Dec 30 '19 at 00:34
  • Also related: https://physics.stackexchange.com/questions/244230/magnetic-force-between-moving-charges] – Tofi Dec 30 '19 at 10:16

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They will apply a force on each other. Actually, that's what produces the magnetic pressure inside a solenoid (That's why it's hard to get steady, strong magnetic fields).

Moreover, this is similar to 2 charges moving in 2 straight parallel lines: that's exactly how 2 wires attract each other.

Ofek Gillon
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  • Thanks, do you have an equation for the quantity of the force exerted? – Feynmanfan85 Dec 29 '19 at 22:26
  • I don't have a unique quantity because I don't know the formula of the magnetic field from a point charge moving in space (I think using retarded potentials can help) but if I knew it was simply the Lorentz force. Also, please note I edited my answer – Ofek Gillon Dec 29 '19 at 22:32
  • Thanks, noted, I'll have another look at the literature. – Feynmanfan85 Dec 29 '19 at 22:56