Question

A particle of charge $-1\text{C}$ and mass $0.2\text{g}$ moves in a circle of radius $1\text{m}$ around an infinite long line charge of linear charge density $+1\text{nC/m}$. The time period of motion is

• A. $\frac{\pi }{150}\text{s}$
• B. $\frac{\pi }{75}\text{s}$
• C. $\frac{\pi }{100}\text{s}$
• D. $\frac{\pi }{60}\text{s}$

Answer 1

The correct option is A $\frac{\pi }{150}\text{s}$

Given:
$q=-1\text{C};m=0.2\text{g};r=1\text{m};\lambda =+1\text{nC/m}$

Let $v$ be the speed of particle moving in the circle.

Here, the electric force $\left(F_e\right)$ acting on charge $-1\text{C}$ due to infinite line charge will act as the required centripetal force.

So, $F_e=m{\omega }^{2}r$

$\Rightarrow qE=m{\omega }^{2}r$

$\Rightarrow q\times \frac{2k\lambda }{r}=m{\omega }^{2}r[\because E=\frac{2k\lambda }{r}]$

$\Rightarrow \omega =\sqrt{\frac{2k\lambda q}{m{r}^2}}$


Now, time period, $T=\frac{2\pi }{\omega }$

$\Rightarrow T=2\pi \times \sqrt{\frac{m{r}^2}{2k\lambda q}}$

$\Rightarrow T=2\times \pi \times \sqrt{\frac{0.2\times {10}^{-3}\times 1^2}{2\times 9\times {10}^9\times 1\times {10}^{-9}\times 1}}$

$\Rightarrow T=\frac{\pi }{150}\text{s}$

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