Define the term .mutual inductance. between the two coils. Obtain the expression for mutual inductance of a pair of long coaxial solenoids each of length `l` and radii `r_(1)` and `r_(2) (r_(2) gt gt r_(1))`. Total number of turns in the two solenoids are `N_(1)` and `N_(2)` respectively.

Explanation

Mutual inductance, between a pair of coils, equals the magnetic flux, linked with one of them due to a change of unit current flowing in the other.
Alternatively :
The mutual inductance, for a pair of coils, equals the emf induced, in one of them, when the current in the other coil is changing at a unit rate.

Let a current `I_(2)` flow through the outer coil.
The magnetic field due to this current
` = mu_(0) (mu_(2))/(l) xx I_(2)`
The resulting magnetic flux linked with the inner coil `= phi_(12) = N_(1).(mu_(0) (mu_(2))/(l) xx l_(2)) xx pi r_(1)^(2)`
` = ( mu_(0) (N_(1) N_(2))/(l), pi r_(1)^(2))l_(2)`
`M_(12)I_(2)`
`:.M_(12) = mu_(0)(N_(1)N_(2))/(l) . pi r_(1)^(2)`