  It is expected on both evolutionary and empirical grounds that many merging
neutron star (NS) binaries are composed of a highly magnetized NS in orbit with
a relatively low magnetic field NS. I study the magnetic interactions of these
binaries using the framework of a unipolar inductor model. The e.m.f. generated
across the non-magnetic NS as it moves through the magnetosphere sets up a
circuit connecting the two stars. The exact features of this circuit depend on
the uncertain resistance in the space between the stars R_space. Nevertheless,
I show that there are interesting observational and/or dynamical effects
irrespective of its exact value. When R_space is large, electric dissipation as
great as ~10^{46} erg/s (for magnetar-strength fields) occurs in the
magnetosphere, which would exhibit itself as a hard X-ray precursor in the
seconds leading up to merger. With less certainty, there may also be an
associated radio transient, but this would be observed well past merger (~hrs)
because of interstellar dispersion. When R_space is small, electric dissipation
largely occurs in the surface layers of the magnetic NS. This can reach
~10^{49} erg/s during the final ~1 sec before merger, similar to the energetics
and timescales of short gamma-ray bursts. In addition, for dipole fields
greater than ~10^{12} G and a small R_space, magnetic torques spin up the
magnetized NS. This drains angular momentum from the binary and accelerates the
inspiral. A faster coalescence results in less orbits occurring before merger,
which would impact matched-filtering gravitational-wave searches by
ground-based laser interferometers and could create difficulties for studying
alternative theories of gravity with compact inspirals.
