= Resistive induction equation
{title2=$\partial_t\mathbf B=\nabla\times(\mathbf u\times\mathbf B)+\eta\nabla^2\mathbf B$}
For a <solenoidal vector field> $\mathbf B$ and constant <magnetic diffusivity> $\eta$, the magnetic induction equation is
$$
\partial_t\mathbf B=\nabla\times(\mathbf u\times\mathbf B)+\eta\nabla^2\mathbf B,\qquad \nabla\cdot\mathbf B=0.
$$
It combines <Faraday's law> with the moving-conductor relation for <electrical conductivity>, neglecting displacement current. With <incompressible flow>, its advective form is $\partial_t\mathbf B+(\mathbf u\cdot\nabla)\mathbf B=(\mathbf B\cdot\nabla)\mathbf u+\eta\nabla^2\mathbf B$.
Back to article page
