The method of matched asymptotic expansions is employed for investigating the effect of a uniform current on the velocity field of a viscous, incompressible, conducting fluid streaming past a stationary conducting ellipsoid of revolution, assuming that the Reynolds number is small. It is also assumed that at infinity the velocity and uniform current are parallel to the axis of the ellipsoid. It is found that the presence of the current increases or decreases the drag coefficient, depending on whether the fluid conductivity is larger than that of the ellipsoid or vice versa. It is suggested that this effect of the current on the drag coefficient holds for all axisymmetric bodies that are also symmetric about a plane perpendicular to their axis. The case of a circular disk broadside on the undisturbed current, obtained as a special case of a planetary ellipsoid, is slightly different; when the conductivity of the disk is non-zero the electromagnetically induced flow field vanishes.