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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

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  • $\begingroup$ 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) $\endgroup$
    – Ofek Gillon
    Commented Dec 29, 2019 at 22:11
  • $\begingroup$ 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. $\endgroup$
    – Feynmanfan85
    Commented Dec 29, 2019 at 22:12
  • $\begingroup$ 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. $\endgroup$
    – Ofek Gillon
    Commented Dec 29, 2019 at 22:13
  • $\begingroup$ Newton's and Maxwell's laws are true for all inertial frames, not inertial ones $\endgroup$
    – Ofek Gillon
    Commented Dec 29, 2019 at 22:14
  • $\begingroup$ 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? $\endgroup$
    – Feynmanfan85
    Commented Dec 29, 2019 at 22:14

1 Answer 1

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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.

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  • $\begingroup$ Thanks, do you have an equation for the quantity of the force exerted? $\endgroup$
    – Feynmanfan85
    Commented Dec 29, 2019 at 22:26
  • $\begingroup$ 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 $\endgroup$
    – Ofek Gillon
    Commented Dec 29, 2019 at 22:32
  • $\begingroup$ Thanks, noted, I'll have another look at the literature. $\endgroup$
    – Feynmanfan85
    Commented Dec 29, 2019 at 22:56

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