# Torque on a charged ring

I have been studying electromagnetic induction. As I understand, a magnetic field will exert a force on a moving charge. What I would like to know is whether a changing magnetic field will exert a force on a stationary charge?

For example, consider a charged ring that is placed in a magnetic field. I know that if a current was flowing through this ring, the torque exerted by a magnetic field would be given by $I(\hat{A} \times \hat B)$. Where I is the current flowing in the circuit, $A$ is the area of the coil and $B$ is the magnetic field.

As I understand, a current is just a flow of charge, or in other words, changing charge. Thus, a changing magnetic field should exert a torque on a stationary charge. Is this right?

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• That's how an induction motor works, isn't it? – CuriousOne May 12 '16 at 7:57
• @CuriousOne Isn't the torque in an induction motor developed due to a changing CURRENT rather than a changing magnetic field? – Gummy bears May 12 '16 at 8:34
• You can't get a current without a changing magnetic field, unless you have a superconductor. – CuriousOne May 12 '16 at 9:18

## 1 Answer

Yes, it is right :)
A changing magnetic field is equivalent to an electrical field. That's what Maxwells equations say.
So a static charge will feel a force, and a collection of charges might feel a torque.

But actually, as pointed out by Feynman (this some decades old, but I have not heard, that it's deprecated), there are two conceptually different mechanisms, the connection of which we do not understand yet - it seems a coincidence.
I mean, the force $F_L$ of a magnetic field on a moving charge, and the force from a changing magnetic field on a static charge. Both lead to the rule, that the emf is equal to the derivative of the magnetic flux. But it seems, there are two distinct reasons for this rule, not obviously derivable one from the other.

• It's surely not coincidence, but one can make the argument that classical field theory can't make causative statements about matter. One will probably have to look at quantum field theory (or even one level deeper) for that. – CuriousOne May 12 '16 at 9:27
• So is there a formula for force/torque on a collection of charges due to a changing magnetic field? – Gummy bears May 12 '16 at 11:53
• well, not one formula, you need to know the properties of the field $B(t)$ and your charges (the positions). Then, yes, en.wikipedia.org/wiki/Maxwell's_equations ;) - just calculate the resulting E-field, and the force is straightforward $E\cdot q$ – Ilja May 12 '16 at 12:06