On differences between the B- and E-approaches and the implications for the Solar atmosphere

A simple collisional three-component plasma model consisting of electrons, ions, and neutrals with arbitrary collision frequencies and dynamic time scales is considered. It is shown that the usual MHD-approach dealing with magnetic field perturbations can give other results than the approach in which all perturbations are expressed via the perturbed electric field. For the partially ionized plasma with strong collisional coupling of neutrals with ions, magnetosonic (nondamping) and Alfv\'{e}n (weakly damping) waves modified by the presence of neutrals are obtained. It is shown that the magnetic diffusivity for Alfv\'{e}n waves appears only due to the longitudinal current connected with the field $E_{1z}$ at the angular propagation of perturbations relatively to the background magnetic field. The model can be applied to different parts of a solar atmosphere and prominences.