### Induction

The magnetic field is the sum of the fields produced by the two coils separately:
$B=4\pi K I_1N_1\mu /l+4\pi KI_2N_2\mu /l ^$
The flux through the coil is $\Phi=BSN$, so that
$\Phi_1= (4\pi KN_1^2S\mu /l)I_1 + (4\pi KN_1N_2S\mu /l) I_2, \\ \Phi_2= (4\pi KN_1N_2S\mu /l)I_1 + (4\pi KN_2^2S\mu /l) I_2$
and comparing with $\Phi_i=\sum_j L_{ij}I_j$
we get $L_{11}=(4\pi KN_1^2S\mu /l), \quad L_22=(4\pi KN_2^2S\mu /l), \quad L_{12}=L_{21}=(4\pi KN_1N_2S\mu /l) ^$