Question

A tightly wound solenoid of radius $‘a’$ and length $‘l’$ has n turns per unit length. It carries an electric current i. Magnetic field at a distance $\frac{l}{4}$ from one of the end (inside the solenoid on its axis) is $B=\frac{\left(\mu \right)_{0} n i \left(\sqrt{5} + 3\right)}{\sqrt{K}}$ for $l=4a$ . Then find the value of K.

Answer 1

Hint: $B=\frac{\mu _{0} n i}{2} \, \left[s i n \theta _{1} + s i n \theta _{2}\right]$

$\Rightarrow \, B=\frac{\mu _{0} n i}{2}\left[\frac{1}{\sqrt{2}} + \frac{3}{\sqrt{10}}\right]$
$\Rightarrow B=\frac{\mu_0 n i(\sqrt{5}+3)}{\sqrt{40}}$
$\therefore \, K=40$