= Solution
<Mutual inductance> expresses how a changing <electric current> in one circuit induces an <electromotive force> in another. Choose winding orientations and define their linked <magnetic fluxes> by
$$
\Lambda_1=L_1I_1+MI_2,\qquad \Lambda_2=MI_1+L_2I_2.
$$
For a reciprocal linear magnetic system, the same $M$ appears in both expressions. <Faraday's law of induction> gives $\mathcal E_2=-M\,dI_1/dt$ when the secondary <electric current> does not itself change. The sign depends on winding orientation and is consistent with <Lenz's law>.
The stored <magnetic energy> is
$$
U=\frac12L_1I_1^2+MI_1I_2+\frac12L_2I_2^2.
$$
Its nonnegativity for arbitrary <electric currents> requires $M^2\leq L_1L_2$. Strong coupling makes $|M|$ approach $\sqrt{L_1L_2}$; flux leakage makes it smaller.
An ideal <electrical transformer> has two windings linked by the same time-varying core <magnetic flux>. The induced winding <voltages> are proportional to their turn counts. For appropriately chosen terminal polarities,
$$
\boxed{\frac{V_2}{V_1}=\frac{N_2}{N_1},\qquad V_1I_1=V_2I_2,\qquad \frac{I_2}{I_1}=\frac{N_1}{N_2}.}
$$
The <electric current> convention in the <power> relation takes primary power into the device and secondary power out to the load. Consequently a load <electrical impedance> $Z_L$ appears at the primary as $Z_{\rm in}=(N_1/N_2)^2Z_L$. A larger secondary <voltage> therefore comes with a smaller available secondary <electric current>, rather than extra <energy>.
A steady direct <electric current> produces no sustained transformer action after its switching transient. Practical departures from the ideal include winding <electrical resistance>, leakage <magnetic flux>, finite magnetizing <electric current>, core saturation, <eddy currents> and <magnetic hysteresis> losses.
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