= Solution
A parabolic isotropic band has <group velocity> $\mathbf v=\hbar\mathbf k/m^*$ and semiclassical force $\hbar\dot{\mathbf k}=-e(\mathbf E+\mathbf v\times\mathbf B)$ for electrons with charge $-e$, where $e>0$. In the <relaxation-time approximation> the nonequilibrium drift velocity loses momentum at rate $1/\tau$. Combining these gives
$$
\boxed{m^*(\dot{\mathbf v}+\mathbf v/\tau)=-e\mathbf E-e\mathbf v\times\mathbf B.}
$$
The relaxation term represents the average effect of scattering rather than a force on every carrier between collisions. Linear response, an approximately energy-independent $m^*$ and $\tau$ near the <Fermi surface>, and wavelengths long enough for a local bulk response are implicit. The equilibrium random velocities average to zero and do not themselves contribute a drift current.
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