Obtain an expression for the Mutual inductance (M) of two long co-axially solenoids.

1. Consider two long solenoids S₁ & S₂ each of length (l) and area (A)
2. Let N₁, N₂ are no of turns and i₁, i₂ are currents through two solenoids S₁ & S₂
 

3. The magnetic field along the solenoid S₁ is
${\mathrm{B}}_1=\frac{{\mathrm{\mu }}_0{\mathrm{N}}_1{\mathrm{i}}_1}{\mathrm{l}}.....\left(1\right)$
4. Mutual inductance of S₁ w.r.t S₂
$\begin{array}{l}{\mathrm{M}}_{12}=\frac{{\mathrm{N}}_2{\mathrm{\varphi }}_2}{{\mathrm{i}}_1}=\frac{{\mathrm{N}}_2{\mathrm{B}}_1\mathrm{A}}{{\mathrm{i}}_1}\\ =\frac{{\mathrm{N}}_2{\mathrm{\mu }}_0{\mathrm{N}}_1{\mathrm{i}}_1\mathrm{A}}{{\mathrm{i}}_1\mathrm{Al}}=\frac{{\mathrm{\mu }}_0{\mathrm{N}}_1{\mathrm{N}}_2\mathrm{A}}{\mathrm{l}}\dots \dots \left(2\right)\end{array}$
5. The magnetic field along the solenoid S₂ is
${\mathrm{B}}_2=\frac{{\mathrm{\mu }}_0{\mathrm{N}}_2{\mathrm{i}}_2}{\mathrm{l}} ......\left(3\right)$
6. Mutual inductance S₂ with respect to S₁ is
$\begin{array}{l}{\mathrm{M}}_{21}=\frac{{\mathrm{N}}_1{\mathrm{\varphi }}_1}{{\mathrm{i}}_2}=\frac{{\mathrm{N}}_1{\mathrm{B}}_2\mathrm{A}}{{\mathrm{i}}_2}\\ =\frac{{\mathrm{N}}_1{\mathrm{\mu }}_0{\mathrm{N}}_2{\mathrm{i}}_2\mathrm{A}}{{\mathrm{i}}_2\mathrm{l}}=\frac{{\mathrm{\mu }}_0{\mathrm{N}}_1{\mathrm{N}}_2\mathrm{A}}{1}......\left(4\right)\end{array}$
From eq (2) and (4) inductance of two solenoids is $\mathrm{M}=\frac{{\mathrm{\mu }}_0{\mathrm{N}}_1{\mathrm{N}}_2\mathrm{A}}{\mathrm{l}}$