We obtain three stationary axially symmetric solutions of the Einstein–Maxwell field equations for rigidly rotating charged dust in a force–free electromagnetic field. The first solution, expressed in terms of Bessel functions of the first and second kind and hyperbolic functions, is not discussed beyond the derivation of one of the metric functions, because it already exists in the literature in a number of different situations. The other two solutions do not seem to exist in the literature. The second one is expressed in terms of modified Bessel functions of the first and second kind and circular functions. We obtain all the metric tensor components and electromagnetic 4–potential as well as the mass and charge densities for particular values of the arbitrary constants involved, but we do not discuss it beyond this point. The third solution is based on four arbitrary constants and different combinations of their values and of relations between them give rise to a number of different space–times, some of which are known solutions, but some are not. The cylindrically symmetric space–time of Som & Raychaudhuri is a particular case of this solution. A stationary axially symmetric space–time, which is a new solution for rotating neutral dust, is another and the interior van Stockum solution is a particular case of this. A number of interesting properties of the solution are pointed out.