An infinitely long line charge distribution produces a field of $9 × 10^4\, NC^{-1}$ at a distance of 2 cm. The linear charge density of the distribution is:

$0.1\, μC\, m^{-1}$

The correct answer is Option (2) → $0.1\, μC\, m^{-1}$

Electric field due to an infinite line charge:

$E = \frac{\lambda}{2 \pi \epsilon_0 r}$

Given $E = 9 \times 10^4\ \text{N/C}$, $r = 2\ \text{cm} = 0.02\ \text{m}$, $\epsilon_0 = 8.854 \times 10^{-12}\ \text{F/m}$

Solving for linear charge density $\lambda$:

$\lambda = 2 \pi \epsilon_0 r E = 2 \pi (8.854 \times 10^{-12}) (0.02) (9 \times 10^4)$

$\lambda \approx 1.0 \times 10^{-7}\ \text{C/m}$

Final Answer: $\lambda \approx 1.0 \times 10^{-7}\ \text{C/m}$