= Magnetic Galilean limit of the pre-Maxwell equations
The magnetic nonrelativistic limit neglects displacement current and uses $\partial_t\mathbf B=-\nabla\times\mathbf E$, $\nabla\times\mathbf B=\mu_0\mathbf J$, $\nabla\cdot\mathbf B=0$. Under a <Galilean transformation> $\mathbf x'=\mathbf x-\mathbf vt$, $t'=t$, its consistent field transformations are $\mathbf B'=\mathbf B$, $\mathbf J'=\mathbf J$ and $\mathbf E'=\mathbf E+\mathbf v\times\mathbf B$. Indeed $\partial_{t'}=\partial_t+\mathbf v\cdot\nabla$ and $\nabla\times(\mathbf v\times\mathbf B)=-(\mathbf v\cdot\nabla)\mathbf B$, giving covariance of the induction equation. The full charge-current transformation has $\mathbf J'=\mathbf J-\rho_e\mathbf v$; neglecting its convective correction belongs to the magnetic quasineutral approximation, not exact relativistic electrodynamics. With $\mathbf u'=\mathbf u-\mathbf v$, the combination $\mathbf E+\mathbf u\times\mathbf B$ is invariant. Vanishing electric field in the fluid rest frame therefore gives the <ideal magnetohydrodynamic induction equation>.
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