Past exam of the physics 2 paper of the University of Oxford / 2013 resit / 9 / ii / Solution

For chosen positive terminal currents, define self-inductance by the coil's own flux linkage ${\Lambda }_j=L_j{I}_j$ and mutual inductance by ${\Lambda }_2=M{I}_1$ when only coil A carries current. With matching winding directions and terminal references,The reciprocal flux-linkage relation gives the same $M$ from B to A, and $M^2=L_A{L}_B$ in this ideal fully coupled long-solenoid limit. The induced emf is $-d\Lambda /dt$; passive terminal voltage drops use the opposite sign, $v_A=L_A{\dot{I}}_A+M{\dot{I}}_B$ and similarly for B.