= Solution
<Faraday's law of induction> relates an induced <electromotive force> to the rate of change of <magnetic flux>. Choose a positive orientation around a loop and its associated surface normal by the <right-hand rule>. For a stationary conducting loop,
$$
\Phi_B=\int_S\mathbf B\cdot d\mathbf S,\qquad
\boxed{\mathcal E=\oint\mathbf E\cdot d\mathbf l=-\frac{d\Phi_B}{dt}.}
$$
Unlike an electrostatic field, the induced <electric field> generally has nonzero <curl>; the differential form is $\nabla\times\mathbf E=-\partial\mathbf B/\partial t$. For a moving circuit, <magnetic flux transport for a moving circuit> includes the magnetic <Lorentz force> on the charges:
$$
\mathcal E=\oint(\mathbf E+\mathbf u\times\mathbf B)\cdot d\mathbf l
=-\frac{d\Phi_B}{dt},
$$
where $\mathbf u$ is the local wire <velocity>. Thus changing the field or moving a conductor can produce induction.
The minus sign is <Lenz's law>: an induced <electric current>, when a closed conducting path permits it, produces a <magnetic field> opposing the change of linked flux. It need not oppose the original field itself. For example, increasing flux through a loop produces a current tending to reduce that increase; decreasing the same flux reverses the current. The opposition is consistent with <conservation of energy>: external mechanical or electrical <work> is needed to maintain a change against its inductive reaction.
For a coil of $N$ turns, define the <magnetic flux linkage> $\Lambda=\sum_{j=1}^N\Phi_j$, reducing to $N\Phi_B$ when every turn links the same flux. For fixed geometry and a linear magnetic response, <self-inductance> $L$ is defined by $\Lambda=LI$, and the coil's induced <electromotive force> is
$$
\boxed{\mathcal E_{\rm self}=-L\frac{dI}{dt},\qquad [L]=\mathrm H=\mathrm{Wb\,A^{-1}}.}
$$
The corresponding terminal voltage drop, with the <passive sign convention>, is $V_L=L\,dI/dt$. It opposes changes in <electric current>; it does not require the current itself to vanish.
Building the current from zero requires power $P=IV_L=LI\,dI/dt$. Integrating gives the energy stored by an ideal <inductor>:
$$
\boxed{U_B=\int_0^I LI'\,dI'=\frac12LI^2.}
$$
This energy is stored in the <magnetic field> and can be returned when the current falls. In a series <RL circuit> driven by a constant voltage $V_0$, <Kirchhoff's voltage law> gives $L\dot I+RI=V_0$. Starting with zero current, $I=(V_0/R)(1-e^{-tR/L})$, with <time constant> $L/R$. This makes the resistance to rapid current changes directly observable. Two coils also have <mutual inductance>: changing primary current induces secondary <electromotive force> $-M\,dI_1/dt$ with a polarity determined by the chosen winding orientations, the principle behind a transformer.
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