Question

A long solenoid has $1000$ turns per unit length, radius $0.5\text{m}$ and carries a current $2\text{A}$ is kept in gravity free region. From its axis at a distance $0.2\text{m}$, an electron is placed. If the current in the solenoid is suddenly switched off, find the velocity attained by the electron.

• A. $4.41\times {10}^7{\text{ms}}^{-1}$
• B. $8.82\times {10}^7{\text{ms}}^{-1}$
• C. $2.2\times {10}^7{\text{ms}}^{-1}$
• D. $1.12\times {10}^7{\text{ms}}^{-1}$

Answer 1

The correct option is A $4.41\times {10}^7{\text{ms}}^{-1}$

Magnetic field induction inside a solenoid is given by,

$B={\mu }_{0}nI$

If the duration in which the field drops to zero after switching off is $t$, then the electric field induced at distance $x$ inside the solenoid is given by,

$E=\frac{x}{2}\frac{dB}{dt}=\frac{x}{2}\frac{({\mu }_{0}nI-0)}{t}=\frac{{\mu }_{0}xnI}{2t}$

Due to induced electric field, impulse on the electron in this duration is given as,

$J=Ft=\left(qE\right)t=mv$

$\Rightarrow v=\frac{qEt}{m}$

$\Rightarrow v=\frac{q}{m}\left(\frac{{\mu }_{0}nIx}{2}\right)=\frac{{\mu }_{0}nIxq}{2m}$

From the data given in the question,

$v=\frac{4\pi \times {10}^{-7}\times 1000\times 2\times 0.2\times 1.6\times {10}^{-19}}{2\times 9.1\times {10}^{-31}}$

$\therefore v=4.41\times {10}^7{\text{ms}}^{-1}$

Hence, $\left(\text{A}\right)$ is the correct answer.