An air-core solenoid having a diameter of 4cm and of length 60cm is wound with 4000 turns. find the inductance of the solenoid. what will be the inductance of the solenoid if it has an iron core of relative permeability of 400?

Solution:

Length(l)=60cm=0.6m

Radius (r)=$\frac{4}{2}$cm=$2\times {10}^{-2}$m

${\mu }_r$=400

Inductance of solenoid,

$\begin{array}{l}L={\mu }_0{N}^2\frac{Area}{Length}\\ L={\mu }_0{N}^2\frac{\pi r^2}{l}\\ L=4\pi \times {10}^{-7}\times (4000{)}^2\times \frac{\pi \times {(2\times {10}^{-2})}^2}{0.6}\\ \therefore L=0.042\end{array}$

Now

Inductance of solenoid when it has an iron core of relative permeability is 

$L^{\prime }={\mu }_{r}L=400\times 0.042=16.84H$