Question

A charged particle of mass $m_1$ and charge $q_1$ is revolving in a circle of radius $r$. Another charged particle of charge $q_2$ and mass $m_2$ is situated at the centre of the circle. If the velocity and time period of the revolving particle be $v$ and $T$ respectively, then :

• A. $v=\sqrt{\frac{q_1{q}_{2}r}{4\pi {ϵ}_0{m}_1}}$
• B. $v=\frac{1}{m_1}\sqrt{\frac{q_1{q}_2}{4\pi {ϵ}_{0}r}}$
• C. $T=\sqrt{\frac{16{\pi }^3{ϵ}_0{m}_1^2{r}^3}{q_1{q}_2}}$
• D. $T=\sqrt{\frac{16{\pi }^3{ϵ}_0{m}_2{r}^3}{q_1{q}_2}}$
• E. None of the above

Answer 1

Answer: (E)

None of the above

Both charges are either both positive or both negative(since answers has the product $q_1{q}_2$inside square root). Hence, circular motion is not possible. So either $q_1$ or $q_2$ should be negative.

According to the diagram,

$\frac{m_1{v}^2}{r}=\frac{1}{4\pi {ϵ}_0}\frac{q_1{q}_2}{r^2}$

$⟹v=\sqrt{\frac{1}{4\pi {ϵ}_0{m}_1}\frac{q_1{q}_2}{r}}$

On putting, $v=r\omega$ and $\omega =\frac{2\pi }{T}$, we can find out:

$T=\sqrt{\frac{16{\pi }^3{ϵ}_0{m}_1{r}^3}{q_1{q}_2}}$

Hence, none of the answers is correct.