  We investigate the linear growth and vertical structure of the MRI in
protoplanetary discs when dust grains are well mixed with the gas over the
entire disc thickness. All the grains have the same radius (a = 0.1, 1 or 3
micron) and constitute 1 % of the total mass of the gas. Solutions are obtained
at R = 5 and 10 AU for a minimum-mass solar nebula model and different choices
of the initially vertical magnetic field strength (B), configuration of the
diffusivity tensor and grain sizes. We find that when no grains are present, or
they are > 1 micron, the midplane remains magnetically coupled for B up to a
few gauss at both radii. In contrast, when a population of small grains (a =
0.1 micron) is present, the disc is magnetically inactive for z/H < 2 and only
B < 50 mG couple to the fluid. At 5 AU, Ohmic diffusion dominates for z/H < 1
when B < a few mG, irrespective of the properties of the grain population.
Conversely, at 10 AU this diffusion term is unimportant in all the scenarios
studied here. For z/H > 5, ambipolar diffusion is severe and prevents the field
from coupling to the gas for all B. Hall diffusion is dominant for a wide range
of field strengths at both radii when dust grains are present. The growth rate,
wavenumber and range of magnetic field strengths for which MRI-unstable modes
exist are all drastically diminished when dust grains are present, particularly
when they are small (a ~ 0.1 micron). We conclude that in protoplanetary discs,
the magnetic field is able to couple to the gas and shear over a wide range of
fluid conditions even when small dust grains are well mixed with the gas.
Despite the low magnetic coupling, MRI modes grow for an extended range of
magnetic field strengths and Hall diffusion largely determines the properties
of the perturbations in the inner regions of the disc (abridged).
