An infinite line charge produces an electric field of `9xx10^(4)N//C` at a distance of 2 cm. what is the linear charge density?

Options

A. `10^(-5)C//m`
B. `10^(-6)C//m`
C. `10^(-7)C//m`
D. `10^(-8)C//m`

Explanation

To find 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 use the formula for the electric field due to an infinite line charge:

\[ E = \frac{\lambda}{2 \pi \epsilon_0 r} \]

Where:
- \(E\) is the electric field,
- \(\lambda\) is the linear charge density,
- \(\epsilon_0\) is the permittivity of free space (\(8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2\)),
- \(r\) is the distance from the line charge.

### Step 1: Convert the distance from cm to meters
Given that \(r = 2 \, \text{cm}\), we convert this to meters:

\[ r = 2 \, \text{cm} = 2 \times 10^{-2} \, \text{m} \]

**Hint:** Always convert units to the SI system for consistency.

### Step 2: Substitute the known values into the electric field formula
We can rearrange the formula to solve for \(\lambda\):

\[ \lambda = E \cdot 2 \pi \epsilon_0 r \]

Now substituting the known values:

\[ E = 9 \times 10^4 \, \text{N/C} \]
\[ \epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \]
\[ r = 2 \times 10^{-2} \, \text{m} \]

### Step 3: Calculate \(\lambda\)
Now we can plug in the values into the equation:

\[ \lambda = (9 \times 10^4) \cdot (2 \pi) \cdot (8.85 \times 10^{-12}) \cdot (2 \times 10^{-2}) \]

Calculating step by step:

1. Calculate \(2 \pi \approx 6.2832\).
2. Calculate \(9 \times 10^4 \times 6.2832 \approx 5.654 \times 10^5\).
3. Calculate \(5.654 \times 10^5 \times 8.85 \times 10^{-12} \approx 5.007 \times 10^{-6}\).
4. Finally, multiply by \(2 \times 10^{-2}\):

\[ \lambda \approx 5.007 \times 10^{-6} \times 2 \approx 1.0014 \times 10^{-5} \, \text{C/m} \]

### Step 4: Round to significant figures
Rounding to significant figures, we find:

\[ \lambda \approx 1.0 \times 10^{-5} \, \text{C/m} \]

### Final Answer
Thus, the linear charge density is:

\[ \lambda \approx 1.0 \times 10^{-7} \, \text{C/m} \]

**Hint for Step 4:** When rounding, consider the significant figures based on the precision of the given values.