  We consider the Unipolar Inductor Model (Goldreich & Lynden-Bell 1969)
applied to Double Degenerate Binaries (DDBs) with ultrashort periods (Wu et al.
2002). In this model a magnetized primary white dwarf has a slight asynchronism
between its spin and orbital motion, so that the (non-magnetic) secondary
experiences a motional electric field when moving through the primary field
lines. This induces a current flow between the two stars and provides an
electric spin-orbit coupling mechanism for the primary. We study the combined
effect of Gravitational Wave emission and electric spin-orbit coupling on the
evolution of the primary degree of asynchronism and the associated rate of
electric current dissipation in such systems, assuming that the primary's spin
is not affected by any other mechanisms. In particular, we show that in
ultrashort period binaries the emission of GW pumps energy in the electric
circuit as to keep it steadily active. This happens despite the fact that
spin-orbit coupling can rapidly synchronize the primary, because GW represent a
slow desynchronizing mechanism steadily substracting orbital angular momentum
to the system. A slightly asynchronous steady-state is thus achieved,
determined by the balance between these two competing effects. This can be
shown to correspond to a condition where the total available electric energy is
conserved, because of GW emission, while dissipation, synchronization and
orbital shrinking continue.
