In a recent paper (Inan et al., 2003) a method of remediating enhanced energetic electron fluxes in the radiation belt was proposed in which injection of VLF whistler mode waves from spacecraft within the radiation belts would dramatically increase the pitch angle scattering of the relativistic electrons and cause these particles to be rapidly lost from the belts, thereby mitigating the flux enhancement. The VLF wave transmitting system discussed by Inan et al. (2003) involves electric dipole antennas. One of the most important characteristics of such an antenna is the current distribution along the length of the dipole, since it is this current which ultimately determines the amount of VLF power which can be radiated from the antenna into the plasma. In past work it has been assumed without proof that the dipole current has a triangular distribution. In the present work we determine the dipole antenna current distribution from first principles, constructing an integral equation of the Hallén type relating the current distribution to the wave vector potential. In this development it is assumed that the length of the thin cylindrical dipole antenna is small compared to the wavelength of whistler mode waves which propagate parallel to the Earth's magnetic field Bo. In the case of the dipole antenna oriented parallel to Bo, it is found that the assumption of a triangular current distribution is reasonable for antenna lengths up to hundreds of meters. For the case of the antenna perpendicular to Bo, it is found that the current decays exponentially along the antenna from the feed points to the antenna ends. In this case we find the conditions under which a triangular current distribution is still a reasonable approximation. We also give the conditions under which the quasi-static model of Balmain (1964) reasonably describes the electric fields associated with the dipole antenna.