Define mutual inductance between two long coaxial solenoids. Find out the expression for the mutual inductance of inner solenoid of length l having the radius `r_(1)` and the number of turns `n_(1)` per unit length due to the second other solenoid of same length and `n_(2)` number of turns per unit length.

Explanation

Definitaioin of self inductance and its SI unit.
Derivation of expression for mutual inductance.
Self inductance of a coil equals, the magnitude of the magnetic flux, linked with it, when a unit current flows through it.
Alternaively
Self inductance, of a coil, equals the magnitude of the emf induced in it, when the current in the coil, is changing at a unit rate.
SI unit: henry/ (webre/ampere)/(ohm second)

When current `I_(2)` is passed through coil `S_(2)` it in turns set up a magnetic flux through
`S_(1)=bar(omega)_(1)=(n_(1)l)(theta r_(1)^(2))(B_(2))`
`bar(omega)_(1)=(n_(1)I)(pir_(1)^(2))(mu_(0)n_(2)I_(2))`
`bar(omega)_(1)=mu_(0)n_(1)n_(2)I_(2)pir_(1)^(2)lI_(2)`
But `bar(omega)_(1)=M_(12)I_(2)`
`impliesM_(12)=mu_(0)n_(1)n_(2)pir_(1)^(2)l`