Inductance of solenoid ' \(L\) ' having diameter ' \(d\) '. Let \(n\) be the number of turns per unit length. The inductance per length near the middle of a solenoid is (Asume that current passes through the turns, \(\mu_0=\) Permeability of vacuum)

Concept: Using the definition of the flux associated through the surface we can obtain the inductance per unit length near the center of the solenoid.
We define selfinductance \(\mathrm{L}\) as:
\(\phi=\mathrm{Li}\)
where, \(\phi\) is the associated flux and \(\mathrm{i}\) is the current through the solenoid.
\(\phi=\mathrm{B} \cdot \mathrm{A}\)
where, \(B=\left(\mu_0\right.\) in \(), A=2 \pi \frac{d^2}{4}\) is the area.
\(\begin{aligned} & \therefore \mathrm{Li}=\left(\mu_0 \mathrm{in}\right)\left(\mathrm{n} \pi \frac{\mathrm{d}^2}{4}\right) \\ & \therefore \mathrm{L}=\frac{\mu_0 \pi \mathrm{n}^2 \mathrm{~d}^2}{4}\end{aligned}\)