Wednesday 11 May 2016

Does a changing magnetic field impart a force on a stationary charged particle?


Does a collapsing and re-establishing magnetic field impart a force on a stationary charged particle? Does the charge particle get repelled and or attracted? Does it move or spin?



Answer



Yes, it will create a force. The force is directed solenoidally around the change in magnetic field.


To see this, look at maxwell's equation $\nabla \times \mathbf{E} = -\partial_t \mathbf{B}$. This is analagous to the equation from magnetostatics: $\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$. Thus a changing magnetic field sources an electric field the same way a current sources a magnetic field.


So for a concrete example, suppose you have a solenoid and you turn on a current so the magnetic field strength increases at a constant rate. Then $\partial_t \mathbf{B}$ is constant in the solenoid, and the electric field you get will be the same as the magnetic field you would get from a constant $\mathbf{J}$ in the region of the solenoid. That is, the electric field you get will look like the magnetic field from a wire. So outside the solenoid, you will get an electric field wrapping around the axis of the solenoid. This electric field will cause a force on the charge. Note, the force is directed in a circle, but it will not cause circular motion. Instead the charge will eventually spiral away from the solenoid.


Notice that in some sense you would say the force is caused directly by the electric field, and it is only indirectly caused by the magnetic field. However, I am still going to say that a "yes" answer is more appropriate in this case, and anyway I think this indirect effect is what you were trying to get at in the first place.


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