Nonexistence of the self-accelerating dipole and related questions

We calculate the self-force of a constantly accelerating electric dipole, showing, in particular, that classical electromagnetism does not predict that an electric dipole could self-accelerate, nor could it levitate in a gravitational field. We also resolve a paradox concerning the inertial mass of a longitudinally accelerating dipole, showing that the combined system of dipole plus field can be assigned a well-defined energy-momentum four-vector, so that the principle of relativity is satisfied. We then present some general features of electromagnetic phenomena in a reference frame described by the Rindler metric, showing in particular that an observer fixed in a gravitational field described everywhere by the Rindler metric will find any charged object supported in the gravitational field to possess an electromagnetic self-force equal to that observed by an inertial observer relative to which the body undergoes rigid hyperbolic motion. It follows that the principle of equivalence is satisfied by these systems.

Figure 1

Field lines of the electric field in the instantaneous rest frame of a positively charged small object undergoing constant proper acceleration in the $+x$ direction. A negative charge situated anywhere on the dashed line will experience a force with a component in the $+x$ direction and will itself produce an electric field similarly tending to accelerate the first charge.

Figure 2

A pair of charged spherical shells separated by a rigid rod.

Figure 3

The contribution $f_{\text{dip}}$ to the self-force of a pair of oppositely charged spherical shells of radius $R$ with centers separated by $d$ and undergoing rigid hyperbolic motion in the transverse direction. The force is shown in units of $e^2/L^2$, for nine equispaced values of $R$ between $0.1L$ and $0.9L$. The total self-force of the pair of spheres is $f=2(f_{\text{dip}}+f_{\text{shell}})$ where $f_{\text{shell}}$ is given by Eq. (25). As $d\to 0$ one finds that $f_{\text{dip}}\to -f_{\text{shell}}$ (see text) so $f\to 0$. The spheres are just touching when $d/R=2$. At $d\gg R$, $f_{\text{dip}}$ is independent of $R$ and is given by the $d$-dependent term in Eq. (6).

Figure 4

The $x$ component of the field ${\overline{\mathbf{E}}}_{\text{shell}}$, which is the field of an accelerating sphere with a radially symmetric contribution removed, so as to leave a continuous function whose integral can be used to calculate the self-force for the case of a transversely oriented dipole. The figure shows the case $L=1$, $R=1/2$ for illustration. Note that $|{\overline{\mathbf{E}}}_{\text{shell},x}|$ falls monotonically with $y$ for points outside the shell.