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AlonsoMcLaren

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## 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...

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