= Solution
For a fixed circuit, <Faraday's law of induction> reads
$$
\oint_C\mathbf E\cdot d\mathbf l=-\frac{d}{dt}\int_S\mathbf B\cdot d\mathbf A,
\qquad \nabla\times\mathbf E=-\frac{\partial\mathbf B}{\partial t}.
$$
Surface orientation and contour direction obey the right-hand rule. The negative sign is <Lenz's law>: a passive induced current acts against the flux change, consistently with <energy> conservation. The induced <electric field> generally has nonzero <curl> and cannot be represented by one single-valued <electrostatic potential>.
For a material circuit moving with local <velocity> $\mathbf u$, the measured <electromotive force> includes the magnetic <Lorentz force> on its carriers:
$$
\mathcal E=\oint_{C(t)}(\mathbf E+\mathbf u\times\mathbf B)\cdot d\mathbf l
=-\frac{d\Phi_B[C(t)]}{dt}.
$$
This distinguishes transformer induction in a stationary coil from motional induction in a changing circuit. A straight sliding rod of <length> $l$ moving perpendicular to itself and to uniform $B$ has $|\mathcal E|=Blv$. With circuit resistance $R$, the electrical power $(Blv)^2/R$ equals the external mechanical power needed to oppose magnetic drag. An ideal transformer gives $V_2/V_1=N_2/N_1$ in magnitude from common <flux linkage>; real leakage, resistance and core losses limit that simple result. <pickup coils> measure flux change rather than an absolute static <magnetic field>, so integration and a reference state are needed to recover a field history.
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