Question

An electron gun is placed inside a long solenoid of radius $R$ on its axis. The solenoid has $n$ turns per length and carries a current $I$. The electron gun shoots an electron along the radius of the solenoid with speed $v$. If the electron does not hit the surface of the solenoid, maximum possible value of $v$ is
(all symbols have their standard meaning):

• A. $\frac{e{\mu }_{0}nIR}{m}$
• B. $\frac{e{\mu }_{0}nIR}{2m}$
• C. $\frac{e{\mu }_{0}nIR}{4m}$
• D. $\frac{2e{\mu }_{0}nIR}{m}$

Answer 1

The correct option is B $\frac{e{\mu }_{0}nIR}{2m}$

Magnetic field inside the solenoid is given by

$B={\mu }_{0}nI....\left(i\right)$
Here, $n\to$ number of turns per unit length


The path of the electron is circular with maximum possible radius $=\frac{R}{2}$
$\therefore \frac{m{V}_{max}}{qB}=\frac{R}{2}$
$\Rightarrow V_{max}=\frac{qBR}{2m}=\frac{eR\left({\mu }_{0}nI\right)}{2m}\text{(using (i))}$

Hence, $\left(B\right)$ is the correct answer.