A small bob of mass $100\ \text{mg}$ and charge $+10\ \mu\text{C}$ is connected to an insulating string of length $1\ \text{m}$. It is brought near an infinitely long non-conducting sheet of charge density $\sigma$ as shown in the figure. If the string subtends an angle of $45^\circ$ with the sheet at equilibrium, the charge density of the sheet will be:
(Given $\varepsilon_0 = 8.85\times10^{-12}\ \text{F/m}$ and $g = 10\ \text{m/s}^2$)
Answer: (D) $1.77\ \text{nC/m}^2$
The bob is in equilibrium under tension $T$, weight $mg$ (down) and the electric force $qE$ (horizontal, away from the sheet), where $E = \dfrac{\sigma}{2\varepsilon_0}$ for a non-conducting sheet.
$$T\cos45^\circ = mg,\qquad T\sin45^\circ = qE \Rightarrow qE = mg\tan45^\circ = mg$$
$$\sigma = \frac{2\varepsilon_0 mg}{q} = \frac{2\times8.85\times10^{-12}\times(100\times10^{-6})\times10}{10\times10^{-6}}$$
$$\sigma = 1.77\times10^{-9}\ \text{C/m}^2 = 1.77\ \text{nC/m}^2$$
(Note $100\ \text{mg} = 10^{-4}\ \text{kg}$.)
Solution by Sreeraj P, M.Sc Physics