Free-boundary nearly rigid displacements are considered in a plasma confined by a magnetic field consisting of one part generated by the plasma current density and one part due to steady currents in fixed external conductors. No conducting wall is assumed to surround the plasma. An induced surface current effect and a related force on the plasma arise when the externally applied field is inhomogeneous in the direction of displacement. This additional force has not been taken into account in conventional magnetohydrodynamic theory. In the particular case of tokamaks, this induced surface current effect has two impacts on vertical nearly rigid displacements. First, there arises an additional restoring force and a positive contribution to the change in potential energy when the externally applied field is inhomogeneous in the vertical direction. A special design of poloidal field coils can thus provide new means for vertical position control in tokamaks, which is also the case with strongly elongated cross sections. Second, an earlier simplified model, in which the plasma is represented by a line current interacting with the currents of the external coil system, has to be modified since the plasma is a highly conducting body of finite size.