Experiments are under way at the University of New England to measure the optical absorption of excited gas particles in a pre-breakdown discharge. Such measurements can be used to deduce the number density of electrons in the discharge. By comparing this experimental density map with the predictions of theory, electron transport parameters may be determined. In this paper, new theoretical expressions are derived for the number density distributions of electrons in a uniform electric field. These are found by solving the electron diffusion equation in a plane parallel electrode geometry with a radially symmetric cathodic current source. The contribution of ion-induced secondary current is included, and problems posed by non-equilibrium conditions near the electrodes are addressed. Techniques of data reduction are discussed with a particular emphasis on the avoidance of these problems.