Suppose a uniformly charged wall provides a uniform electric field of $2\times10^4\ \text{N C}^{-1}$ normal to it. A charged particle of mass $2\ \text{g}$ suspended through a silk thread of length $20\ \text{cm}$ remains at rest at a distance of $10\ \text{cm}$ from the wall. Then the charge on the particle will be $\dfrac{1}{\sqrt x}\ \mu\text{C}$, where $x = $ ______. [use $g = 10\ \text{m s}^{-2}$]
Numerical value type. Enter your answer.
Answer: 3
The thread hangs from a point on the wall. With the particle $10\ \text{cm}$ from the wall on a $20\ \text{cm}$ thread,
$$\sin\theta = \frac{10}{20} \Rightarrow \theta = 30^\circ$$
Balance of forces: $\tan\theta = \dfrac{qE}{mg}$.
$$q = \frac{mg\tan30^\circ}{E} = \frac{2\times10^{-3}\times10}{\sqrt3\times2\times10^4} = \frac{10^{-6}}{\sqrt3}\ \text{C} = \frac{1}{\sqrt3}\ \mu\text{C}$$
So $x = 3$.
Solution by Sreeraj P, M.Sc Physics