In this paper, the effect of cathode geometry on the radial characteristics of a downstream plasma column is presented for a cold cathode type of hollow cathode (HC) system. It is observed that when an axial magnetic field is applied to a cylindrical cathode, the downstream plasma exhibits an off-centered peak in the plasma density. However, as the magnetic field increases, the discharge extinguishes rapidly above a critical value, due to the suppression in secondary electron emission from the cylindrical cathode surface. On the other hand, by replacing the hollow cylinder with a cone-shaped cathode an oblique sheath is formed with respect to the axial magnetic field. This configuration helps in sustaining the discharge at twice the magnetic field than in the cylindrical case. It is also found that the downstream plasma exhibits a peak plasma density at the center for the case of the conical cathode. The above experimental observations for each HC setup has been qualitatively explained with the help of a phenomenological model.