Question

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

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

Answer 1

The correct option is B

$\frac{Re{\mu }_{0}in}{2m}$

Step 1. Given Data:

Radius of solenoid $=R$

Number of turns in the solenoid $=n$

Current carrying the solenoid $=i$

Step 2. Finding the magnetic force

Magnetic field inside the solenoid, $B={\mu }_{0}ni$ [ Where, $n$ is number of turns, $i$ is current inside the solenoid, ${\mu }_0$ is the permeability of free space.]

Magnetic force, $F=qvB$ [Where, $q=e$ is the charge, $v$ is the velocity, $B$is magnetic field]

$F=ev{\mu }_{0}ni$

Step 3. Finding the maximum velocity

In a magnetic field, the charge will move on a circular path. Magnetic force is equal to the centripetal force, $F_c$

$F=F_c$

$ev{\mu }_{0}ni=\frac{m{v}^2}{R}$ [Where, $R$ is the radius of the circular path, $m$ is the mass.] [$eq\left(1\right)$]

Maximum possible radius of electron is $\frac{R}{2}$ because electron move along the axis which is half of the radius.

From $eq\left(1\right)$,

$ev{\mu }_{0}ni=\frac{m{v}^2}{{\displaystyle \frac{R}{2}}}$

$v=\frac{Re{\mu }_{0}ni}{2m}$

Therefore, the maximum velocity will be $\frac{\mathbf{R}\mathbf{e}{\mathbf{\mu }}_{\mathbf{0}}\mathbf{n}\mathbf{i}}{\mathbf{2}\mathbf{m}}$

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