Coherent electromagnetic fields arising from an azimuthally modulated beam are considered. The beam is completely enclosed in a toroidal vacuum tank of rectangular cross section and highly conducting walls. Expressions are given for the image currents arising from low harmonics of the beam circulation frequency. These expressions are then used to evaluate resistive losses in the walls of the chamber. Expressions are given for fields arising from harmonics of the revolution frequency high enough that the beam may be in resonance with a characteristic mode of the vacuum chamber. The results are generalized to provide a description of the electric field in the neighborhood of a resonance. Numerical examples of resistive losses are given, indicating that these effects will not be serious for circulating currents of the order of 1 amp. Some properties of high‐order Bessel functions, required for a description of the resonant chamber modes and the energy lost in their excitation, are developed in an appendix.