Numerical and analytical treatment of the smoothed Tamm problem



We study the motion of an electric charge in a medium over a finite spatial distance. We consider at first the motion described by continuous functions of time, that is, its velocity and all its time derivatives are continuous. If the velocity v of the charge is greater than the velocity cn of light in the medium, then the intensity of the radiation is proportional to the frequency in some angular region and decreases exponentially outside of it. If the motion has no jumps in the velocity but jumps in the acceleration, the exponential decrease changes to a reciprocal one with 1/ω. For a purely decelerated motion, when both the initial and the final velocities are greater than cn, the intensity of the radiation contains a plateau. For practical application, the case with a vanishing final charge velocity is important: the intensity of the radiation is then maximal at the Cherenkov angle θc that corresponds to the initial charge velocity whereas it decreases sharply for θ>θc. For this case, provided the velocity of the charge exceeds cn, the total intensity (obtained by integrating the angular intensity of the radiation over the solid angle) is a linear function of the frequency, even though the Tamm condition is strongly violated.