When a dielectric resonator is used for a filter or for frequency stabilization of an oscillator, often it is necessary to know its external Q. This paper studies theoretically the external Q values under the assumption that the dielectric resonator is installed in a cylindrical metal case and a loop is used for coupling between the resonator and the external circuit. In the analysis, instead of using the electromagnetic field in the dielectric resonator, the resonator is replaced with a polarization current from which the vector potential is derived. From the vector potential, the magnetic field distribution is calculated. The power is supplied to the external circuit by the loop coupling of this magnetic field. From the energy stored in the dielectric resonator and the power delivered to the circuit, the external Q is derived. To study the effect of the metal case on the external Q value, cases with various heights have been considered at 3.5 GHz and the effectiveness of the method is confirmed by comparison with experiments.