I don't remember when I missed it, but someone suggested that a disk
magnet spinning on its axis of symmetry might have a radial electric
field. Application of Gauss's law should convince one that a purely
radial electric field cannot exist in a radially symmetric geometry
for which translational symmetry holds in the direction of the axis
unless there is a nontrivial radially symmetric distribution of
electric charge. While this is not the situation in the complicated
case under discussion here, reflection on gaussian surfaces can be
enlightening.