Derive formula for mutual inductance for two very long coaxial solenoids. Also discuss reciprocity theorem.

Explanation

Consider figure which shows two long coaxial solenoids each of length l. We denote the radius of the inner solenoid `S_1` by `r_1` and the number of turns per unit length by `n_1`. The corresponding quantities for the outer solenoid `S_2` are `r_2` and `n_2` respectively. Let `N_1` and `N_2` be the total number of turns of coils `S_1` and `S_2` respectively.

When a current `I_2` is set up through `S_2`, it in turn sets up a magnetic flux through `S_1`. Let us denote it by `Phi_1`.
The corresponding flux linkage with solenoid `S_1` is
`therefore N_1 Phi_1 = M_12 I_2` ....(1)
`M_12` is called the mutual inductance of solenoid `S_1` with respect to solenoid `S_2`. It is also referred to as the coefficient of mutual induction.
`N_1 Phi_1=N_1A_1B_2`
`N_1 Phi_1=(n_1l)(pir_1^2)(mu_0n_2I_2)`
`=mu_0n_1n_2 pir_1^2 pi_2`...(2)
Where `n_1l` is the total number of thuns in solenoid `S_1` . Thus, from equation (1) and (2) , `M_12=mu_0 n_1n_2 pir_1^2l`...(3)
Note that we neglected the edge effects and considered the magnetic field `mu_0n_2I_2` to be uniform throughout the length and width of the solenoid `S_2`.
We now consider the reverse case. Current `I_1` is passed through the solenoid `S_1` and the flux linkage with coil `S_2` is,
`N_2 Phi_2=M_21 I_1`..(4)
`M_21` is called the mutual inductance of solenoid `S_2` with respect to solenoid `S_1`.
The flux due to the current `I_1` in `S_1` can be assumed to be confined solely inside `S_1` since the solenoids are very long. Thus, flux linkage with solenoid `S_2` is
`N_2 Phi_2=N_2A_1B_1`
`N_2Phi_2 =(n_2I)(pir_1^2)(mu_0n_1I_1)`
`=mu_0 n_1n_2pir_1^2 pi_1`....(5)
Where `n_2I` is the total number of turns of `S_2`. From equation (4) to (5),
`M_21=mu_0n_1n_2pir_1^2l`...(6)
Using equation (3) and (6), we get
`M_12=M_21=M` (say)...(7)
Equation (7) is known as reciprocity theorem.
We have considered air as a medium in solenoid. Instead of air medium of relative permeability `mu` is considered then
`M=mun_1n_2pir^2l`
`mu=mu_0mu_r=(mu_0mu_r)n_1n_2pir^2l`