Self-force on moving electric and magnetic dipoles: Dipole radiation, Vavilov-Čerenkov radiation, friction with a conducting surface, and the Einstein-Hopf effect

The classical electromagnetic self-force on an arbitrary time-dependent electric or magnetic dipole moving with constant velocity in vacuum, and in a medium, is considered. Of course, in vacuum there is no net force on such a particle. Rather, because of loss of mass by the particle due to radiation, the self-force precisely cancels this inertial effect, and thus the spectral distribution of the energy radiated by dipole radiation is deduced without any consideration of radiation fields or of radiation reaction, in both the nonrelativistic and relativistic regimes. If the particle is moving in a homogeneous medium faster than the speed of light in the medium, Vavilov-Čerenkov radiation results. This is derived for the different polarization states, in agreement with the earlier results of Frank. The friction experienced by a point (time-independent) dipole moving parallel to an imperfectly conducting surface is examined. Finally, the relativistic quantum/thermal Einstein-Hopf effect is rederived. We obtain a closed form for the spectral distribution of the force and demonstrate that, even if the atom and the blackbody background have independent temperatures, the force is indeed a drag when the imaginary part of the polarizability is proportional to a power of the frequency. The unifying theme of these investigations is that friction on an atom requires a dissipative mechanism, be it through radiation or resistivity in the environment.

Figure 1

Quantum frictional force written in the form of the dimensionless integral (5.28) plotted as a function of the velocity $v$. The lower set of curves (black) is for $n=1$ and the upper set (red) is for $n=3$. In each case, three situations are envisaged: $\beta ={\beta }^{\prime }$ (solid), $\beta ={\beta }^{\prime }/2$ (dashed), and $\beta =2{\beta }^{\prime }$ (dotted).