A theory of convergent, high‐efficiency electron guns for the production of dense hollow beams is presented. An approximate solution to Poisson's equation is obtained in a cylindrical coordinate system modified to account for the toroidal geometry. Experimental measurements on a series of toroidal guns tend to confirm the space‐charge theory. They also show that the annular slit in the anode very seriously affects the uniformity of cathode emission and diverges the beam more than can be accounted for by present theories. These effects of the slit appear to confine the usefulness of these guns to high‐power applications.