The ability of production of induced emf in one coil, due to varying current in the neighbouring coil is called mutual inductance.
Magnetic flux, Φ = MI Where, M is called coefficient of mutual inductionMutual Inductance of Two Long Solenoids Consider two long solenoids S1 and S2 of same length l, such that solenoid S2 surrounds solenoid S1 completely.Φ21 = M21I1
Where, M21 is the coefficient of mutual induction of the two solenoidsMagnetic field produced inside solenoid S1 on passing current through it,B1 = μ0n1I1Magnetic flux linked with each turn of solenoid S2 will be equal to B1 times the area of cross-section of solenoid S1.Magnetic flux linked with each turn of the solenoid S2 = B1A
Therefore, total magnetic flux linked with the solenoid S2,Φ21 = B1A × n2l = μ0n1I1× A× n2lΦ21 = μ0n1n2lAI1∴ M21 = μ0n1n2AlSimilarly, the mutual inductance between the two solenoids, when current is passed through solenoid S2 and induced emf is produced in solenoid S1, is given byM12 = μ0n1n2Al∴M12 = M21 = M (say)
Hence, coefficient of mutual induction between the two long solenoids