CBSE Class 12 · 2009 · Electromagnetic InductionLong Answer

(a) Lenz's law: an induced current always flows in a direction that opposes the change in flux producing it. Example: pushing the north pole of a magnet towards a coil induces a current whose near face becomes a north pole and repels the magnet. Work must be done against this repulsion, and that work supplies the electrical energy of the induced current; if the current instead aided the motion, energy would appear without any work, which is impossible. Hence Lenz's law follows from conservation of energy.
(b) Take two long coaxial solenoids of equal length $\ell$: inner $S_1$ (radius $r_1$, $n_1$ turns/length) and outer $S_2$ (radius $r_2$, $n_2$ turns/length). Pass a current $I_2$ through $S_2$; its interior field is $B_2={\mu }_0{n}_2{I}_2$.
This field links the inner solenoid over its cross-section $\pi r_1^2$, with total turns $n_1\ell$:

Since $N_1{\varphi }_1=M{I}_2$,

The same value is obtained on reversing the roles, confirming $M_{12}=M_{21}$.