= Solution
For a moving closed circuit $C(t)$ bounding an oriented surface $S(t)$, <Faraday's law> is
$$
\boxed{\mathcal E_{\rm em}(t)=\oint_{C(t)}[\mathbf E+\mathbf v\times\mathbf B]\cdot d\mathbf l
=-\frac{d}{dt}\int_{S(t)}\mathbf B\cdot\mathbf n\,dS.}
$$
Here $\mathbf E$ is the <electric field>, $\mathbf B$ the <magnetic field>, $\mathbf v$ the local velocity of a circuit element, and $\mathcal E_{\rm em}$ the electromagnetic <electromotive force> per unit charge around the circuit. The line orientation and normal $\mathbf n$ obey the right-hand rule. The flux derivative is a total derivative and includes both changing field and changing surface. The term $\mathbf v\times\mathbf B$ is the <motional electromotive force>. A battery contributes an additional non-electromagnetic source electromotive force.
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