= Magnetic flux transport for a moving circuit
{title2=$\frac{d\Phi}{dt}=\int_S\partial_t\mathbf B\cdot d\mathbf S-\oint_C(\mathbf v\times\mathbf B)\cdot d\mathbf r$}
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.
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