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Electrical resistivity is the inverse of scalar electrical conductivity in an isotropic Ohmic medium. In its rest frame, $\mathbf{E}={\rho }_{\mathrm{el}}\mathbf{j}$, so the electric field does work per unit volume $\mathbf{j}\cdot \mathbf{E}={\rho }_{\mathrm{el}}|\mathbf{j}{|}^2$. In Gaussian units, finite conductivity gives magnetic diffusivity $\eta =c^2/(4\pi \sigma )$; neglecting this diffusion is a distinct approximation from neglecting electron inertia in Hall magnetohydrodynamics.

For field amplitudes of order the guide field, the ion Lorentz-force response over time $\tau$ is $\Delta u/v_A=O(v_A\tau /l)\ll 1$. The electron-current velocity is of order $v_A{d}_i/l$, larger than a bulk velocity of order $v_A$. Thus the leading Hall magnetohydrodynamics induction equation becomes the closed electron magnetohydrodynamics equation, while ions remain fixed to leading order on this fast scale. The first nonzero ion response need not have zero acceleration. This limit retains the fast branch of incompressible Hall-MHD wave dispersion, not the slow branch; very oblique modes must separately satisfy the fast-time ordering.

Hall magnetohydrodynamics retains bulk ion inertia but freezes magnetic flux into the Electron velocity ${\mathbf{u}}_e=\mathbf{u}-\mathbf{j}/(en)$. With constant density, negligible electrical resistivity and Electron inertia, Gaussian units give $\mathbf{j}=c\nabla \times \mathbf{B}/(4\pi )$ and ${\partial }_t\mathbf{B}=\nabla \times [(\mathbf{u}-c\nabla \times \mathbf{B}/(4\pi en))\times \mathbf{B}]$. The ion velocity still satisfies the magnetohydrodynamic momentum equation. Neglecting that velocity gives electron magnetohydrodynamics, a further approximation rather than a synonym for the complete Hall system.