A rapidly rotating perfectly conducting sphere and the electrodynamics of a neutron star

 . We discuss the electrodynamics of a rapidly rotating perfectly conducting magnetized sphere rotating with constant angular velocity about an axis which is inclined to the magnetic dipole field. An exact special relativistic solution is obtained for the electromagnetic field in the interior. The exterior solution is also determined and matched to the interior by using the appropriate boundary conditions. A formula for the relativistic energy emission from the sphere is derived, and compared with the known non-relativistic formula. It is shown that in the slow rotation limit, all the relativistic solutions and formulae obtained reduce to the known non-relativistic expressions in the literature. Some of the main objectives of this work are to obtain the expressions for the electromagnetic field vectors in the interior and exterior and to see how large the special relativistic corrections are in the energy emission formula. The latter, but also other expressions and formulae, may be of relevance to rapidly rotating neutron stars, because it will be found that the classical non-relativistic formula for the energy emission, is inaccurate for such stars.