A reformulation of a known conservation law is employed to study certain zero energy trajectories of a classical point test charge (e.g., a classical electron) moving under the influence of the electrostatic force due to a fixed electric dipole. It is found that the motion takes place along a “folded” ellipse with foci at the dipole charges and a fold along an axis perpendicular to the line joining these foci and lying in the plane of the motion. The motion is determined to be periodic with period ${T(d,y}_0)=\sqrt{m}\sqrt{4\pi {\epsilon }_0/qQ}(\sqrt{\pi /2\mathrm{}}\left[\Gamma \mathrm{}\right(1/4)/\Gamma \mathrm{}(3/4\left)\mathrm{}\right]\left)\right[\frac{1}{3}{d}^{3/2}+(2/\sqrt{d}{)y}_0^2],$ where d is the separation of the dipole charges that are placed symmetrically about the origin on the x axis, and $y_0$ is the initial y position of the test charge that starts from rest at ${(0,y}_0).\mathrm{}$

• Received 28 September 1998

DOI:https://doi.org/10.1103/PhysRevE.59.4614