[en] Recent experiments have shown severe surface damage and a reduction of the maximum accelerating gradient for an rf cavity that is operating under external magnetic fields. This implies that serious problems may occur in lattices where rf cavities and external magnetic fields coexist, such as those of the proposed neutrino factory and muon collider. Although existing data suggest that this magnetic field dependent breakdown is associated with the emission of electrons from locally enhanced field regions on the cavity surface, the mechanism that drives this effect is not yet well understood. Here, we show that such field emitted electrons are accelerated by the cavity and focused by the magnetic field to the other side of the cavity where they heat its surface. We show that if the magnetic field is strong, significant surface deformation can occur that eventually could limit the accelerating gradient of the cavity. Results of our model are compared to the existing experimental data from an 805 MHz cavity. The geometry of the pillbox cavity in our case is more complicated, and the analysis depends on the electron energies, focused dimensions, and angle of impact, but damage may reasonably be expected with similar cyclical heating above 40 C. However, it is not yet known what the mechanism is for such surface damage to cause a cavity to breakdown. One possibility is that if electrons are focused on a location with a high surface gradient, then the local damage will generate new asperities with higher FN enhancement factors, thus initiating breakdown. While our preliminary analysis offers some quantification on the effects of the magnetic fields on the cavity's operation, other theoretical issues were not addressed. For instance, emission from secondary electrons was disregarded, the asperity was placed on axis, the magnetic field was assumed as uniform, the thermal-diffusion calculation ignored the shape of the rise time, and adopted an approximate calculation. On the theoretical level, it will be interesting to pursue additional simulations exploring these effects in detail. Experimentally, there is a clear need for more well-designed experiments to study, systematically, the effect of external fields on the cavity's operation.