The heat conduction in a hollow cylindrical electric resistor which carries an alternating current is analysed. The hole within the cylinder is either empty or filled with a dielectric solid. The non-uniform power generated per unit volume in the resistor by the Joule effect is evaluated, and the steady periodic Fourier equation is written in a dimensionless form both in the domain occupied by the resistor and in that occupied by the dielectric, if present. A boundary condition of the third kind is assigned at the external surface of the cylinder. The dimensionless temperature field is determined analytically as a function of position, time and a proper set of dimensionless parameters.