  Interstellar magnetic fields influence all stages of the process of star
formation, from the collapse of molecular cloud cores to the formation of
protostellar jets. This requires us to have a full understanding of the
physical properties of magnetized plasmas of different degrees of ionization
for a wide range of densities and temperatures. We derive general equations
governing the magneto-hydrodynamic evolution of a three-fluid medium of
arbitrary ionization, also including the possibility of charged dust grains as
the main charge carriers. In a companion paper (Pinto & Galli 2007), we
complement this analysis computing accurate expressions of the collisional
coupling coefficients. Over spatial and temporal scales larger than the
so-called large-scale plasma limit and the collision-dominated plasma limit,
and for non-relativistic fluid speeds, we obtain an advection-diffusion for the
magnetic field. We derive the general expressions for the resistivities, the
diffusion time scales and the heating rates in a three-fluid medium and we use
them to estimate the evolution of the magnetic field in molecular clouds and
protostellar jets. Collisions between charged particles significantly increase
the value of the Ohmic resistivity during the process of cloud collapse,
affecting in particular the decoupling of matter and magnetic field and
enhancing the rate of energy dissipation. The Hall resistivity can take larger
values than previously found when the negative charge is mostly carried by dust
grains. In weakly-or mildy-ionized protostellar jets, ambipolar diffusion is
found to occur on a time scale comparable to the dynamical time scale, limiting
the validity of steady-state and nondissipative models to study the jet's
structure.
