Question

An infinite line charge produces a field of \(9 \times 10^4 \, \text{NC}^{-1}\) at a distance of 2 cm. The linear charge density is

• A. \( 10^{-7} \, \text{C/cm} \)
• B. \( 1.5 \times 10^{-7} \, \text{C/cm} \)
• C. \( 10^{-8} \, \text{C/cm} \)
• D. \( 1.5 \times 10^{-8} \, \text{C/cm} \)

Hint:

For an infinite line charge, the electric field is directly proportional to the linear charge density.

Answer 1

The Correct Option is A

The electric field due to a line charge is given by the formula: \[ E = \frac{2k\lambda}{r} \] Where: - \( E = 9 \times 10^4 \, \text{NC}^{-1} \) is the electric field, - \( \lambda \) is the linear charge density, - \( r = 2 \, \text{cm} = 0.02 \, \text{m} \) is the distance from the line charge, - \( k = 9 \times 10^9 \, \text{Nm}^2/\text{C}^2 \) is Coulomb’s constant. Rearranging to find \( \lambda \): \[ \lambda = \frac{E r}{2k} \] Substituting the values: \[ \lambda = \frac{(9 \times 10^4) \times (0.02)}{2 \times (9 \times 10^9)} = 10^{-7} \, \text{C/cm} \] Thus, the linear charge density is \( \boxed{10^{-7} \, \text{C/cm}} \).