= Solution
<Electromagnetic induction> is the production of an <electromotive force> by changing <magnetic flux> or by moving a conductor through a <magnetic field>. With the loop and surface orientations related by the <right-hand rule>, <Faraday's law of induction> is
$$
\boxed{\mathcal E=\oint(\mathbf E+\mathbf v\times\mathbf B)\cdot d\boldsymbol\ell=-\frac{d}{dt}\int\mathbf B\cdot d\mathbf S.}
$$
Here $\mathbf v$ is the local wire velocity. For a stationary loop, the induced <electric field> can have a nonzero closed-path integral; its differential form is $\nabla\times\mathbf E=-\partial\mathbf B/\partial t$. For a straight rod moving perpendicularly to itself and to a uniform <magnetic field>, the motional <electromotive force> has magnitude $B\ell v$.
The minus sign expresses <Lenz's law>: the induced <electric current> produces a <magnetic field> that opposes the change in linked <magnetic flux>. Pulling a conducting loop through a <magnetic field> therefore requires work against an electromagnetic resisting <force>; that work supplies the resulting heat or stored <energy>.
For an <inductor> with constant self-<inductance>, $\mathcal E=-L\,dI/dt$ and its stored <magnetic energy> is $LI^2/2$. A changing <electric current> in one coil can induce an <electromotive force> in another through <mutual inductance>, the operating principle of a <electrical transformer>. These examples connect the flux law, the direction of the response and <conservation of energy>.
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