A conventional field-emission electron gun consists of a pointed cathode and an extraction electrode at a positive potential which produces a strong electric field at the cathode surface. Electrons are field-emitted from the cathode, pass through an axial hole in the extractor electrode : and must then be accelerated to full energy. The field emission current density at the cathode surface is very sensitive to changes in the electric field strength. The distribution of electric field on the end of the cathode can be changed by an additional nearby electrode if its size is not large compared to the cathode radius. A cylindrical shield electrode has been proposed which surrounds the cathode, and for convenience can be at the same potential as the cathode. This configuration has been examined theoretically using second-order finite element analysis. The results of this analysis show that the change in distribution of the electric field on the cathode will depend upon the relative dimensions of the cathode and the shield, and, for a significant change in emission distribution, the shield radius must not be larger than approximately five times the cathode radius. The addition of the shield leaves the virtual source size unchanged, so the brightness of the source is approximately proportional to the axial current density. It has been suggested that combined cathodes and shields can be fabricated by ion milling to form a pointed cathode in the end of a shield.