An infinite line charge produce a field of `7.182xx10^(8)NC^(-1)` at a distance of 2 cm. The linear charge density is

Options

A. `7.27xx10^(-4)Cm^(-1)`
B. `7.98xx10^(-4)Cm^(-1)`
C. `7.11xx10^(-4)Cm^(-1)`
D. `7.04xx10^(-4)Cm^(-1)`

Explanation

To find the linear charge density (λ) of an infinite line charge that produces a given electric field (E) at a certain distance (R), 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 (in N/C),
- \( \lambda \) is the linear charge density (in C/m),
- \( \epsilon_0 \) is the permittivity of free space (\( \epsilon_0 = 8.854 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \)),
- \( R \) is the distance from the line charge (in meters).

### Step-by-Step Solution:

**Step 1: Convert the distance from centimeters to meters.**
- Given \( R = 2 \, \text{cm} = 0.02 \, \text{m} \).

**Step 2: Rearrange the formula to solve for λ.**
- From the electric field equation, we can express λ as:
\[ \lambda = E \cdot 2 \pi \epsilon_0 R \]

**Step 3: Substitute the known values into the equation.**
- Given:
- \( E = 7.182 \times 10^8 \, \text{N/C} \)
- \( R = 0.02 \, \text{m} \)
- \( \epsilon_0 = 8.854 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \)

Now substituting these values into the equation:
\[ \lambda = (7.182 \times 10^8) \cdot (2 \pi \cdot 8.854 \times 10^{-12}) \cdot (0.02) \]

**Step 4: Calculate the value of λ.**
- First, calculate \( 2 \pi \cdot 8.854 \times 10^{-12} \):
\[ 2 \pi \cdot 8.854 \times 10^{-12} \approx 5.577 \times 10^{-11} \]

- Now substitute this back into the equation:
\[ \lambda = (7.182 \times 10^8) \cdot (5.577 \times 10^{-11}) \cdot (0.02) \]

- Performing the multiplication:
\[ \lambda = 7.182 \times 10^8 \cdot 5.577 \times 10^{-11} \cdot 0.02 \]
\[ \lambda = 7.182 \times 5.577 \times 0.02 \times 10^{-3} \]
\[ \lambda \approx 7.98 \times 10^{-4} \, \text{C/m} \]

### Final Answer:
The linear charge density \( \lambda \) is approximately:
\[ \lambda \approx 7.98 \times 10^{-4} \, \text{C/m} \]

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