This paper is a study of a possible mechanism of magnetic field amplification in the thin transition layer just outside the magnetopause of a quickly spinning planet. Here we present analysis of a new magnetospheric magnetic field source arising due to the rapid rotation. When the ionosphere and magnetopause conductivities of the rapidly rotating planet are sufficiently high (but less than the field-aligned conductivity) the circulation exists at the magnetopause in the open field line bundles. The toroidal magnetic field is generated there from the poloidal ambient magnetosheath magnetic field by differential rotation. The requirement of the &agr; effect existence is satisfied due to the deviation from axial symmetry about the planet's rotation axis of the initial magnetosheath velocity and magnetic fields. The increasing poloidal magnetic field can be regenerated from the toroidal one by the &agr; effect. This mechanism may act as a hydromagnetic dynamo. The generated magnetic field penetrates into the magnetosphere and adds to the magnetospheric magnetic field. ¿ American Geophysical Union 1996