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We have a radial magnetic field, say $$B= \frac{k}{\sqrt{x^2+y^2}}(\cos{\theta}\cdot\hat{i}+\sin\theta\cdot\hat{j})$$ and we have a conducting ring, with its axis along the $z$ axis. What happens if we give this ring a velocity in the $z$ direction?

From Faraday's Law, there is no change in flux, and hence there should be no induced EMF. But on each element, there is a magnetic Lorentz Force acting on the electrons along the ring, so they should move, and there should be a current which implies that there is an EMF.

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From Faraday's Law, there is no change in flux

The magnetic field you propose violates Maxwell's equation

$$ \nabla \cdot \mathbf B = 0 $$

on the line $(x=0,y=0)$, which means flux cannot be uniquely assigned to the ring, only to some specific choice of surface attached to the ring. If that surface is plane disk, then flux is zero, but if that surface is a long cylinder hat, then flux is not zero and is increasing as the ring moves along $z$.

and hence there should be no induced EMF.

Correct, but for different reason: induced EMF in general is due to induced electric field, which is not present in this case at all (vanishes everywhere). What you meant to say is that due to zero change in flux, there should be no EMF at all . That is not true (because we can't say flux is not changing, because there isn't unique way to assign flux). In the hypothetical case the magnetic field was as you proposed (which would be a major discovery contradicting standard EM theory), there would be motional EMF $$ \oint_{ring} (\mathbf v \times \mathbf B) \cdot d\mathbf l \neq 0 $$ and Faraday's law wouldn't be obeyed (because there would be no unique flux).

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  • $\begingroup$ So Faraday's law is only applicable if flux can be uniquely associated with a loop? Since in a general case, there exist infinitely many surfaces, how would one determine if flux can be uniquely associated or not? $\endgroup$
    – Aspirant
    Jul 27, 2021 at 6:26
  • $\begingroup$ In real cases magnetic flux (as opposed to electric flux) can always be uniquely associated with the loop, because all known magnetic fields have always zero flux through any closed surface (which is equivalent to obeying the Maxwell equation $\nabla \cdot \mathbf B =0$ everywhere). So in real cases Faraday's law is sensible. Your hypothetical example of magnetic field does not obey this condition (which contradicts our knowledge of magnetic fields), and then magnetic flux can only be associated with definite surface, not with the loop. Faraday's law then makes no sense. $\endgroup$
    – Ján Lalinský
    Jul 27, 2021 at 8:25
  • $\begingroup$ Thanks for the help! $\endgroup$
    – Aspirant
    Jul 28, 2021 at 16:44
  • $\begingroup$ Just a last question; you say induced EMF vanishes everywhere. Is a time varying magnetic field the only thing that causes an induced EMF? For instance, is there an induced EMF in the coil of an AC generator, or is that also motional EMF? $\endgroup$
    – Aspirant
    Jul 28, 2021 at 16:45
  • $\begingroup$ Induced EMF (the standard meaning of the term) is due to induced electric field, which was not present in your example. Induced electric field at some point does not require varying magnetic field present at that same point of space, but it does require it to be present somewhere in space. Electric power generators convert mechanical energy to EM energy, using primarily motional EMF (movement of current-carrying conductor against ponderomotive forces due to magnetic field), but usually there is induced electric field and corresp. EMF present as well, because magnetic field changes in time. $\endgroup$
    – Ján Lalinský
    Jul 28, 2021 at 21:28

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