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 due to an infinitely long line charge at a radial distance \( r \) is derived using Gauss's Law.
Step 2: Key Formula or Approach:
The formula for the electric field is:
\[ E = \frac{\lambda}{2\pi \epsilon_0 r} \]
Alternatively, using \( k = \frac{1}{4\pi \epsilon_0} \):
\[ E = \frac{2k\lambda}{r} \]
Step 3: Detailed Explanation:
Given values:
\( E = 9 \times 10^4 \, \text{N/C} \)
\( r = 2 \, \text{cm} = 0.02 \, \text{m} \)
\( k = 9 \times 10^9 \, \text{Nm}^2/\text{C}^2 \)
Using the formula:
\[ 9 \times 10^4 = \frac{2 \times 9 \times 10^9 \times \lambda}{0.02} \]
Divide both sides by \( 9 \times 10^4 \):
\[ 1 = \frac{2 \times 10^5 \times \lambda}{0.02} \]
\[ 0.02 = 2 \times 10^5 \times \lambda \]
\[ \lambda = \frac{0.02}{2 \times 10^5} = 0.01 \times 10^{-5} \]
\[ \lambda = 10^{-7} \, \text{C/m} \]
Converting to $\mu$C/m:
\[ \lambda = 10^{-7} \times 10^6 \, \mu\text{C/m} = 0.1 \, \mu\text{C/m} \]
Step 4: Final Answer:
The linear charge density is 0.1 $\mu$C m$^{-1}$.