Electromagnetic induction by a rotating sphere

[AlonsoMcLaren]

Homework Statement

http://physics.columbia.edu/files/physics/content/Quals2010Sec2.pdf
Problem 1

Consider a rigid, ideally conducting sphere of radius R, with total charge zero. The sphere rotates with angular velocity Ω, ΩR<<c. Suppose a dipole magnetic field threads the sphere. the dipole is centered.on the center of the sphere. The dipole moment μ is given and aligned with Ω.

(a) What voltage is induced between the equator and the poles of the sphere?

Homework Equations

Transformation of EM fields in special relativity?

The Attempt at a Solution

There is a solution on page 7 of the same document.
However, in (2) of this solution, the EM field transformation rules in special relativity are being used.
But the frame rotating with the sphere is non-inertial, can we still use these transformation rules?

If indeed, the EM field transformation rules are applicable to rotating frames, then I have a paradox:
Consider the notoriously painstaking problem of a rotating uniformly charged sphere, in which you are asked to find the magnetic field on the North Pole
The conventional solution to it is very nontrivial, the answer is B=2QΩ/5cR z^ (in Gaussian units)
But what if I do the EM field transformation trick? In the rotating frame, the electric field at North Pole is E'=Q/R2 z^ and there is no magnetic field. Since the North Pole is not rotating, the transformation rules just give B=B'=0... Contradiction...

 

[TSny]

The EM field transformation trick that was used in the solution of the exam problem is valid.

To get the fields at some point P in the rotating frame, you are essentially just transforming the fields between two inertial frames. One frame is the nonrotating lab frame and the other frame is an inertial frame that is instantaneously co-moving with P of the rotating frame. This co-moving inertial frame is, of course, not rotating. The fields at P in the rotating frame are obtained from the fields at the same point in the lab frame by transforming between the two inertial frames.

In your example, you have a rotating charged sphere. In the frame rotating with the sphere, all the charge is at rest. But, it is not correct to conclude that there would be no magnetic field in the rotating frame. At point P in the rotating frame, the fields will be the same as the fields in an instantaneously co-moving inertial frame. But in this co-moving inertial frame the charge on the sphere is not at rest (except at the one point P). So, there will, in general, be a magnetic field in this co-moving frame at P.

If you want the field at the north pole in the rotating frame, then you need to choose an inertial frame that is co-moving with the north pole. So, how does the B field at the north pole in the rotating frame compare to the B field at the north pole in the lab frame?

 

[AlonsoMcLaren]

The EM field transformation trick that was used in the solution of the exam problem is valid.

To get the fields at some point P in the rotating frame, you are essentially just transforming the fields between two inertial frames. One frame is the nonrotating lab frame and the other frame is an inertial frame that is instantaneously co-moving with P of the rotating frame. This co-moving inertial frame is, of course, not rotating. The fields at P in the rotating frame are obtained from the fields at the same point in the lab frame by transforming between the two inertial frames.

In your example, you have a rotating charged sphere. In the frame rotating with the sphere, all the charge is at rest. But, it is not correct to conclude that there would be no magnetic field in the rotating frame. At point P in the rotating frame, the fields will be the same as the fields in an instantaneously co-moving inertial frame. But in this co-moving inertial frame the charge on the sphere is not at rest (except at the one point P). So, there will, in general, be a magnetic field in this co-moving frame at P.

If you want the field at the north pole in the rotating frame, then you need to choose an inertial frame that is co-moving with the north pole. So, how does the B field at the north pole in the rotating frame compare to the B field at the north pole in the lab frame?

So basically in a noninertial frame, Maxwell's equations do not hold (Otherwise there should be no magnetic field in the rotating frame because there is no current source)?

Also, for the exam problem, why E=0 inside the metal sphere at the origin of the co-moving inertial frame (call it P)? Surely the point P is stationary but other points in the sphere are not stationary. Therefore can we still use the electrostatic result that inside a conductor E=0?

 

[TSny]

So basically in a noninertial frame, Maxwell's equations do not hold (Otherwise there should be no magnetic field in the rotating frame because there is no current source)?

I don't believe Maxwell's equations hold in a noninertial frame. If you do a search on Maxwell's equations in rotating frames you'll find some relevant articles. It looks complicated, especially for material media and rotation speeds that are relativistic. See for example
http://arxiv.org/pdf/astro-ph/9801194v1.pdf
http://physics.princeton.edu/~mcdonald/examples/rotatingEM.pdf

(I've only glanced at these articles.)

Imagine a large, uniformly charged ring that is at rest relative to an inertial frame. Imagine that you stand at the center of the ring and that you are at rest relative to the ring. Then you would of course find E = B = 0 at your location. Now imagine that you stay in place but rotate your body continuously so that from your point of view the ring now appears to be rotating about you in the opposite direction. It would be odd if this rotation of your body somehow produced a nonzero E or B field at your location (according to you). But, using Maxwell's equations, you see the rotating ring constituting a circular current loop and therefore it should produce a nonzero B field at your location. So, Maxwell's equations aren't working from your perspective.

Also, for the exam problem, why E=0 inside the metal sphere at the origin of the co-moving inertial frame (call it P)?

In the co-moving frame, the immediate neighborhood of point P in the conductor is essentially at rest. So, the electric field, E', inside the conductor in this co-moving frame must be zero. We therefore say that the electric field is zero at P also in the rotating frame since the rotating frame and the co-moving frame coincide at P.

Surely the point P is stationary but other points in the sphere are not stationary. Therefore can we still use the electrostatic result that inside a conductor E=0?

Yes, other points of the sphere will not be stationary relative to the frame co-moving with P. But, that doesn't affect the result that E' = 0 at P in the frame co-moving with P. The laws of electrostatics must hold in the co-moving inertial frame and, in this frame, the conducting material is at rest at P. So, E' = 0 at P in this frame. But at some other point, Q, of the sphere, E' ≠ 0 for the frame co-moving with P. But of course E' = 0 at Q for the frame co-moving with Q.

Anyway, that's my understanding. I might be overlooking something.