Question

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

• A. $0.01\ \mu\text{C m}^{-1}$
• B. $0.1\ \mu\text{C m}^{-1}$
• C. $1\ \mu\text{C m}^{-1}$
• D. $2\ \mu\text{C m}^{-1}$

Hint:

Always convert distances to meters (SI units) before plugging them into electrostatics formulas to avoid power-of-ten errors!

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The electric field produced by an infinitely long straight wire with linear charge density $\lambda$ at a distance $r$ is given by Gauss's Law.

Step 2: Identifying the Formula and Values:
\[ E = \frac{\lambda}{2\pi\epsilon_0 r} = \frac{2k\lambda}{r} \] Where $k = 9 \times 10^9\ \text{Nm}^2/\text{C}^2$, $E = 9 \times 10^4\ \text{N/C}$, and $r = 0.02\text{ m}$.

Step 3: Calculation:
\[ 9 \times 10^4 = \frac{2 \times 9 \times 10^9 \times \lambda}{0.02} \] \[ 10^4 = \frac{10^9 \times \lambda}{0.01} \] \[ \lambda = \frac{10^4 \times 10^{-2}}{10^9} = 10^{-7}\ \text{C/m} \] Converting to micro-coulombs: $10^{-7}\ \text{C/m} = 0.1 \times 10^{-6}\ \text{C/m} = 0.1\ \mu\text{C/m}$.

Step 4: Final Answer:
The linear charge density is $0.1\ \mu\text{C m}^{-1}$.