Unstable motion of a point charge near an electric dipole

Abstract We reconsider the dynamics of a point charge in the field of a point-like electric dipole. The force is noncentral, so the angular momentum is not conserved. Our discussion is largely based on the third conserved quantity β (in addition to the total energy and the axial component of angular momentum) that we derive in spherical coordinates. The equations of motion admit orbits characterised by β = 0 which are traced on a sphere centred at the dipole. Transitions between orbits on the same sphere demand highly specialised, restricted perturbations of the angular momentum components. If the restrictions are not satisfied, then r ( t ) eventually goes to zero or becomes unbounded. This is due to the lack of a confining, radial effective potential. Transitions between orbits lying on a sphere are exceedingly unlikely under realistic conditions. The stability properties of this classical system are virtually unobservable and inconsequential from a physical point of view. Our approach clarifies and extends previous discussions of this matter, and should be accessible to intermediate or advanced undergraduate physics students.