Motional electromotive force (${\mathcal{E}}_{\mathrm{mot}}={\oint }_C(\mathbf{v}\times \mathbf{B})\cdot d\mathbf{l}$)

Motional electromotive force is the contribution caused by a conductor moving through a magnetic field. The magnetic part of the Lorentz force separates charges within the conductor. For a bar of length $L$ moving at velocity $v$ perpendicular to a uniform field, its magnitude is $BL|v|$; the sign follows the chosen circuit orientation. Together with the electric-field contribution it enters Faraday's law for moving circuits.

For a smoothly moving circuit and a divergence-free magnetic field, the magnetic flux derivative is $d\Phi /dt={\int }_S{\partial }_t\mathbf{B}\cdot d\mathbf{S}-{\oint }_C(\mathbf{v}\times \mathbf{B})\cdot d\mathbf{r}$. The minus sign follows by orienting the swept side surface compatibly with the two spanning surfaces. Combining this transport identity with Faraday's law yields the moving-loop electromotive-force law.