1. Electric field due to an infinite non-conducting sheet

For an infinitely long (effectively infinite) non-conducting sheet with uniform surface charge density $\sigma$, the electric field is

$E = \frac{\sigma}{2\varepsilon_0}$

This field is perpendicular to the sheet.

2. Forces acting on the charged bob

Given:

• Mass: $m = 100\,\text{mg} = 100 \times 10^{-6}\,\text{kg} = 10^{-4}\,\text{kg}$
• Charge: $q = +10\,\mu\text{C} = 10 \times 10^{-6}\,\text{C} = 10^{-5}\,\text{C}$
• $g = 10\,\text{m/s}^2$

The bob is in equilibrium under three forces:

• Weight: $mg$ downward
• Electric force: $qE$ perpendicular to the sheet
• Tension: along the string

Since the string makes $45^\circ$ with the sheet, it also makes $45^\circ$ with the vertical/horizontal arrangement of forces, so at equilibrium

$\tan 45^\circ = \frac{qE}{mg}$

Since $\tan 45^\circ = 1$,

$qE = mg$

3. Calculate the electric field

$mg = 10^{-4} \times 10 = 10^{-3}\,\text{N}$

So,

$E = \frac{mg}{q} = \frac{10^{-3}}{10^{-5}} = 100\,\text{N/C}$

4. Find surface charge density $\sigma$

Using

$E = \frac{\sigma}{2\varepsilon_0}$

we get

$\sigma = 2\varepsilon_0 E$

$\sigma = 2 \times 8.85 \times 10^{-12} \times 100$

$\sigma = 17.7 \times 10^{-10} \text{ C/m}^2$

$\sigma = 1.77 \times 10^{-9} \text{ C/m}^2$

$\sigma = 1.77\,\text{nC/m}^2$

5. Option check

• A: $1.77\,\text{nC/m}^2$ ✅
• B: $0.885\,\text{nC/m}^2$ ❌
• C: $885\,\text{nC/m}^2$ ❌
• D: $17.7\,\text{nC/m}^2$ ❌

So the correct option is A.