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 r1 and r2 (r2 >> r1). Total numbers of turns in the two solenoids are N1 and N2, respectively.
Mutual inductance is defined as the current induced in a coil in proximity when the flux change is produced by the same coil.
Given: -
Two long co-axial solenoids each of length l. The radius of the inner solenoid S1 by r1 and the number of turns per unit length by n1. The corresponding quantities for the outer solenoid S2 are r2 and n2, respectively. Let N1 and N2 be the total number of turns of coils S2 and S2, respectively.
Derivation: -
When a current I2 is set up through S2, it in turn sets up a magnetic flux through S1. Let us denote it by F1. The corresponding flux linkage with solenoid S1 is
N1 
= M12I2 ….(a)
M12 is called the mutual inductance of solenoid S1 with respect to solenoid S2. It is also referred to as the coefficient of mutual induction. The magnetic field due to the current I2 in S2 is,
B = μ0n2I2
The resulting flux linkage with coil S1 is,
N1 
= (n1l) (
r12) (
n2I2)
N1 
= 
n1n2 
r12l I2
where n1l is the total number of turns in solenoid S1. Thus, from Eq. (a), we can write,
Conclusion: -
The mutual inductance is 