Question 9 of 311

A small bob of mass 100 mg and charge $+10 \mu \mathrm{C}$ is connected to an insulating string of length 1 m. It is brought near to an infinitely long nonconducting sheet of charge density ' $\sigma$ ' as shown in figure. If string subtends an angle of $45^{\circ}$ with the sheet at equilibrium the charge density of sheet will be :
(Given, $\varepsilon_0=8.85 \times 10^{-12} \frac{\mathrm{~F}}{\mathrm{~m}}$ and acceleration due to gravity, $g=10 \mathrm{~m} / \mathrm{s}^2$)

A$0.885 \mathrm{nC} / \mathrm{m}^2$B$17.7 \mathrm{nC} / \mathrm{m}^2$C$885 \mathrm{nC} / \mathrm{m}^2$D$1.77 \mathrm{nC} / \mathrm{m}^2$

Show answer and solution▾

Answer: (D) $1.77 \mathrm{nC} / \mathrm{m}^2$

$\begin{aligned} & \mathrm{qE}=\mathrm{mg} \\ & \mathrm{q}\left[\frac{\sigma}{2 \varepsilon_0}\right]=\mathrm{mg} \\ & \sigma=\frac{2 \varepsilon_0 \mathrm{mg}}{\mathrm{q}} \\ & \sigma=\frac{2 \times 8.85 \times 10^{-12} \times 100 \times 10^{-6} \times 10}{10 \times 10^{-6}} \\ & \sigma=17.7 \times 10^{-10} \mathrm{C} / \mathrm{m}^2 \\ & \sigma=1.77 \mathrm{nC} / \mathrm{m}^2\end{aligned}$