Question

A very long straight solenoid carries a current $I$. The cross-sectional area of the solenoid is equal to $S$, the number of turns per unit length is equal to $n$. The flux of the vector $\overrightarrow{B}$ through the end plane of the solenoid is $\frac{{\Phi }_0}{x}$. Find $x$

Answer 1

Magnetic field at the end of solenoid$=\overrightarrow{B}=\frac{{\mu }_{0}ni}{2}$
The magnetic field$=\varphi =\overrightarrow{B}.\overrightarrow{S}$
$=\frac{{\mu }_{0}ni}{2}S\cos 0$
$=\frac{\left({\mu }_{0}niS\right)}{2}$$\varphi =\frac{{\varphi }_0}{2}$
${\varphi }_0\to$flux at the center of solenoid
$\Rightarrow \varphi =\frac{{\varphi }_0}{x}=\frac{{\varphi }_0}{2}$
$\Rightarrow x=2$