Consider a cylinder of permanently magnetized material, with uniform magnetization pointing along the cylindrical symmetry axis (the $z$-direction). The magnet is rotating about its cylindrical symmetry axis with angular velocity $\omega$. What electric field does the rotating magnet generate?
Backstory: Moving permanent magnets generally generate an electric field, even in cases where $d\vec{M}/d t = 0$. In the case of uniform motion, this electric field is straightforward to determine using a Lorentz boost. I'm interested in cases where the simple Lorentz boost does not work.
EDIT:
As perceived by some of the answers, I am not specifically interested in a cylinder. If your solution is for a ring, a sphere, or pretty much any nontrivial cylindrically symmetric object rotating about its cylindrical symmetry axis, I'm interested, as long as $d\vec{M}/d t = 0$.
Landau and Lifshitz describe a similar, interesting case where the rotating magnet is also a conductor. I'm interested in the case where the rotating object is not a conductor.
Unipolar induction is very interesting, but again, involves a rotating conductor, which I am not asking about.
Answer
The electric field is nonzero. For a cylinder of finite length, it's nonvanishing everywhere. In the limiting case of an infinitely long cylinder, the field is only nonvanishing inside the cylinder.
The easiest way to solve this is to use the fact that the electric and magnetic polarizations $(-\textbf{P},\textbf{M})$ transform in exactly the same way as the fields $(\textbf{E},\textbf{B})$ (Hnizdo 2011). Taking the low-velocity limit for convenience, we have $\textbf{P}=\textbf{v}\times\textbf{M}$. This produces a radial polarization with magnitude $P=\omega r M$, corresponding to a constant interior charge density plus a surface charge of the opposite sign. (This agrees with Kostya's answer.) The interior field is clearly nonvanishing. Applying Gauss's law in the limit of an infinitely long cylinder, the exterior field is found to vanish.
Hnizdo and McDonald, "Fields and Moments of a Moving Electric Dipole," 2011, http://www.physics.princeton.edu/~mcdonald/examples/movingdipole.pdf
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