# What is the Lenz reaction due to Radial EMF generated by a rotating cylindrical magnet with a steady magnetic field?

Per the top answer to the question "What is the electric field generated by a spinning magnet?", a rotating magnet can generate an electric field even if it does so on its axis of cylindrical-symmetry. We know from the Maxwell-Faraday equation that if the magnetic field is non-changing, the electric field does not possess a curl. So therefore, it follows that a magnet rotating on its axis of cylindrical-symmetry produces an irrotational electric field.

Suppose I had a circular ring of electric charge oriented co-axially with this rotating magnetic cylinder such that the static electric field it produces is cylindrically-symmetric around the same axis as the electric field produced by the rotating magnetic cylinder. It is evident that if we then allowed this circular charged ring to expand or contract at some rate, then positive or negative work would be done on the charged ring. Also, if the magnitude of velocity of the expansion or contraction of the charged ring was much less than some "effective" rotational speed of the magnet, then we could consider the magnetic Lorentz force on the charged ring to be negligible in comparison to the electric force exerted onto it. Thus the acquired motion of linear elements of this charged ring can be largely constrained to the radial and axial directions.

If furthermore, we decided to position the charged loop such that the electric force acting on each linear element of the charged loop is perpendicular to the axis, then the acquired motion of the loop elements would be strictly perpendicular to the axis of symmetry with no longitudinal components. This constraint still permits that electric work is done on the charged loop as its linear elements are displaced while being subjected to the electric field of the rotating magnet.

After taking into account the additional constraint that our charged loop is cylindrically-symmetric around the same axis of symmetry as the rotating cylindrical magnet, it follows that the magnetic field produced by the expansion or contraction of the charged ring lacks radial and axial components. Because of the cylindrical-symmetry of our magnet, the effective current density $$\nabla \times \mathbf{M}$$ (i.e. the magnetiztion current density) of the magnet also lacks radial and axial components. Therefore the magnetic force on the magnet due to the magnetic field of the expanding or contracting charged ring lacks an azimuthal component and cannot contibute to the change of the rotational speed of the magnet. Thus, we are able to produce an electric current in the form of an expanding or contracting charged ring without a magnetic Lorentz force that could slow down the magnet responsible for the radial emf acting on the charged ring.

The expanding or contracting charged ring produces a changing electric field. I think that from the point of view of hypothetical charge elements in the magnet, this transforms to a changing magnetic field of $$v/c$$ dependence plus a correction to the electric field of $$v^2/c^2$$ dependence, where $$v$$ refers to some "effective" speed of these hypothetical charges relative to the frame $$S$$ in which we initially constructed the problem. The changing radial electric fields of the expanding charged loop acting on the magnet would, after transformation to the "rotating" frame $$S'$$ of charge elements in the magnet, be accompanied by changing magnetic fields with axial components. So it would seem that a transformer EMF would be perceived by charge elements in the magnet, such that the work done on the expanding or contracting ring does so at the expense of one or more kinds of energy associated with the "magnetization" currents of the cylindrical magnet. If so, how would we characterize such transformer EMF in frame $$S$$, the non-rotating frame where the expanding or contracting charged ring produces a magnetic field that is purely azimuthal?