= Solution
\b[A current in one circuit links <magnetic flux> through another.] It produces <magnetic flux linkage> through another. With fixed geometry and a linear reciprocal medium, choose consistent winding orientations and write
$$
\Lambda_1=L_1I_1+MI_2,\qquad \Lambda_2=MI_1+L_2I_2.
$$
The shared coefficient is \b[<mutual inductance>]; reversal of one winding reverses its sign. <Faraday's law of induction> gives the mutual induced emf $\mathcal E_2=-M\dot I_1$ for fixed $M$. The minus sign expresses <Lenz's law>.
The <magnetic energy> is
$$
W=\tfrac12L_1I_1^2+MI_1I_2+\tfrac12L_2I_2^2.
$$
Nonnegative <energy> for every current pair requires $|M|\leq\sqrt{L_1L_2}$, motivating the coupling coefficient $M/\sqrt{L_1L_2}$. Transformers use strong common flux to exchange <energy> between circuits; leakage flux reduces coupling. A shared core increases linkage, but saturation, <hysteresis> and eddy-current losses limit the linear description. A moving geometry also contributes $-I_1\dot M$ to the mutual emf.
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