Mutual inductance $M$ quantifies the coupling between two coils. If current $I_1$ in coil 1 produces flux $\Phi_{21}$ through coil 2: $N_2\Phi_{21} = MI_1$. EMF induced in coil 2: $\varepsilon_2 = -M\frac{dI_1}{dt}$. Reciprocity: $M_{12} = M_{21} = M$. For two coaxial solenoids: $M = \mu_0 n_1 n_2 A_{\text{inner}} l$ (the relevant area is that of the inner solenoid). General relation: $M = k\sqrt{L_1 L_2}$ where $k$ is the coupling coefficient (0 for no coupling, 1 for perfect coupling). In a transformer: $M \approx \sqrt{L_1 L_2}$ (tight coupling, $k \approx 1$).