Stewart-Tolman Effect (Current caused by Acceleration of wires)

[phantomvommand]

Homework Statement

Rings are wrapped around a cylinder, like a solenoid. Number of rings per unit length = n, radius of ring = r. Rings are fixed. Cylinder is rotated with angular acceleration a. Find the magnetic field at the centre of the cylinder.

Relevant Equations

B = mu n I
I = V/R

Consider the inertial reference frame in which the positive ions forming the crystal lattice of some portion of a metallic ring are at rest.

In this frame, an inertial force of mra exists. Consider the electrons in this portion of the metallic ring. The inertial force of mra is exerted on the electrons as well. The electrons cannot accelerate forever, as they are bound by some electrostatic force of attraction due to the positive ions.

I am not sure about the following:
Apparently, a fictitious electric field exists. At some point, the electrostatic force must become strong enough to balance the inertial force of mra on the electrons. --> eE = mra, where E is the magnitude of the electric field.

I am very confused about the fictitious electric field. What exactly is causing it?

I get that there is no other candidate for the force which opposes mra. (normal contact force, friction etc are all unlikely) But that still doesn't convince me that it must be an electric field. What is the explanation for why the electric field exists?

Also, it seems intuitive that the electrons cannot forever accelerate. However, what exactly is the law that states this?

Thanks to all for any help.

 

[Steve4Physics]

Since no one else has replied yet, see if this helps...

It is ambiguous to say ‘the rings are fixed’ as this could imply they are stationary (‘fixed’) in the lab’ frame of reference. I believe the rings are meant to be attached to the cylinder and rotate with the cylinder. Or the question makes no sense.

“Consider the inertial reference frame in which the positive ions forming the crystal lattice of some portion of a metallic ring are at rest. “

This is not an inertial frame. A ring’s crystal lattice is not only rotating but has angular acceleration. So this frame is non-inertial.

We can ignore radial motion. To an observer in the rotating frame, electrons are subject to two forces – the tangential fictitious Euler force (look it up if needed) and a ‘drag’ force produced by collisions between the electrons and the lattice.

The electrons reach a terminal speed relative to the lattice when the Euler and ‘drag’ forces become equal. (In the same way that raindrops reach a terminal velocity when their weight and the drag force become equal.)

That means there is a current of electrons flowing relative to the lattice. The ‘fictitious electric field’ is poorly named. It should be called the ‘equivalent electric field’. It is simply the electric field which would be needed to produce the same current in an inertial (e.g. stationary) system.

Finally, as the angular speed continues to increase, relativistic effects will start to become significant and would need to accounted for. I expect the question simply requires you to consider the non-relativistic regime (ωr<<c) so you can ignore the classical possibility of the angular speed tending to infinity!

 

[phantomvommand]

Since no one else has replied yet, see if this helps...

It is ambiguous to say ‘the rings are fixed’ as this could imply they are stationary (‘fixed’) in the lab’ frame of reference. I believe the rings are meant to be attached to the cylinder and rotate with the cylinder. Or the question makes no sense.

“Consider the inertial reference frame in which the positive ions forming the crystal lattice of some portion of a metallic ring are at rest. “

This is not an inertial frame. A ring’s crystal lattice is not only rotating but has angular acceleration. So this frame is non-inertial.

We can ignore radial motion. To an observer in the rotating frame, electrons are subject to two forces – the tangential fictitious Euler force (look it up if needed) and a ‘drag’ force produced by collisions between the electrons and the lattice.

The electrons reach a terminal speed relative to the lattice when the Euler and ‘drag’ forces become equal. (In the same way that raindrops reach a terminal velocity when their weight and the drag force become equal.)

That means there is a current of electrons flowing relative to the lattice. The ‘fictitious electric field’ is poorly named. It should be called the ‘equivalent electric field’. It is simply the electric field which would be needed to produce the same current in an inertial (e.g. stationary) system.

Finally, as the angular speed continues to increase, relativistic effects will start to become significant and would need to accounted for. I expect the question simply requires you to consider the non-relativistic regime (ωr<<c) so you can ignore the classical possibility of the angular speed tending to infinity!

Thanks for the reply; it has been very helpful. Apologies for the confusion over inertial and non-inertial.

I suppose the equivalent electric field also implies an equivalent circular emf, which can be used to find current?

 

[Steve4Physics]

I suppose the equivalent electric field also implies an equivalent circular emf, which can be used to find current?

Yes. In this situation the emf is simply the magnitude of the electric field multiplied by the circuital distance.