An infinite line charge produces a field of `9xx10^(-4)NC^(-1)` at a distance of 2 cm. Calculate the linear charge density.

Options

A. `10muC//m`
B. `12muC//m`
C. `16muC//m`
D. `18muC//m`

Explanation

To solve the problem of finding the linear charge density (λ) of an infinite line charge that produces an electric field (E) of \(9 \times 10^{-4} \, \text{N/C}\) at a distance (r) of 2 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the formula for the electric field due to an infinite line charge**:
The electric field (E) produced by an infinite line charge is given by the formula:
\[ E = \frac{1}{4 \pi \epsilon_0} \cdot \frac{2\lambda}{r} \]
where:
- \(E\) is the electric field,
- \(\lambda\) is the linear charge density,
- \(r\) is the distance from the line charge,
- \(\epsilon_0\) is the permittivity of free space, approximately \(8.85 \times 10^{-12} \, \text{C}^2/(\text{N} \cdot \text{m}^2)\).

2. **Rearrange the formula to solve for linear charge density (λ)**:
We can rearrange the formula to isolate λ:
\[ \lambda = \frac{E \cdot r \cdot 4 \pi \epsilon_0}{2} \]

3. **Convert the distance from centimeters to meters**:
The distance given is 2 cm. We need to convert this to meters:
\[ r = 2 \, \text{cm} = 0.02 \, \text{m} \]

4. **Substitute the known values into the equation**:
Now we can substitute the known values into the rearranged formula:
- \(E = 9 \times 10^{-4} \, \text{N/C}\)
- \(r = 0.02 \, \text{m}\)
- \(\epsilon_0 \approx 8.85 \times 10^{-12} \, \text{C}^2/(\text{N} \cdot \text{m}^2)\)

Plugging in these values:
\[ \lambda = \frac{(9 \times 10^{-4}) \cdot (0.02) \cdot (4 \pi \cdot 8.85 \times 10^{-12})}{2} \]

5. **Calculate the value of λ**:
First, calculate \(4 \pi \epsilon_0\):
\[ 4 \pi \epsilon_0 \approx 4 \cdot 3.14 \cdot 8.85 \times 10^{-12} \approx 1.11 \times 10^{-10} \, \text{C}^2/(\text{N} \cdot \text{m}^2) \]
Now substitute this back into the equation:
\[ \lambda = \frac{(9 \times 10^{-4}) \cdot (0.02) \cdot (1.11 \times 10^{-10})}{2} \]
\[ \lambda = \frac{(1.998 \times 10^{-15})}{2} \approx 9.99 \times 10^{-16} \, \text{C/m} \]

6. **Final result**:
The linear charge density (λ) is approximately:
\[ \lambda \approx 1.0 \times 10^{-7} \, \text{C/m} \]