An infinite line charge produces a field of `4.5xx10^(4) N//C` at a distance 2 m from it. The linear charge density is

Options

A. `5 mu C//m`
B. `6 mu C//m`
C. `4 mu C//m`
D. `7 mu C//m`

Answer

To find the linear charge density (λ) of an infinite line charge that produces an electric field (E) of \(4.5 \times 10^4 \, \text{N/C}\) at a distance (r) of 2 m from it, we can use the formula for the electric field due to an infinite line charge:

\[ E = \frac{1}{4\pi \epsilon_0} \cdot \frac{2\lambda}{r} \]

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 \(8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2\).

### Step 1: Rearrange the formula to solve for λ
We can rearrange the formula to isolate λ:

\[ \lambda = \frac{E \cdot r \cdot 4\pi \epsilon_0}{2} \]

### Step 2: Substitute the known values
Now we can substitute the known values into the equation. We know:
- \(E = 4.5 \times 10^4 \, \text{N/C}\)
- \(r = 2 \, \text{m}\)
- \(\epsilon_0 \approx 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2\)

Substituting these values gives:

\[ \lambda = \frac{(4.5 \times 10^4) \cdot (2) \cdot (4\pi \cdot 8.85 \times 10^{-12})}{2} \]

### Step 3: Calculate the constant
First, we calculate \(4\pi \cdot 8.85 \times 10^{-12}\):

\[ 4\pi \approx 12.5664 \]
\[ 4\pi \cdot 8.85 \times 10^{-12} \approx 1.113 \times 10^{-10} \, \text{C}^2/\text{N m}^2 \]

### Step 4: Substitute back into the equation
Now substitute this value back into the equation for λ:

\[ \lambda = \frac{(4.5 \times 10^4) \cdot (2) \cdot (1.113 \times 10^{-10})}{2} \]

### Step 5: Simplify the equation
The 2s cancel out:

\[ \lambda = (4.5 \times 10^4) \cdot (1.113 \times 10^{-10}) \]

### Step 6: Perform the multiplication
Calculating this gives:

\[ \lambda \approx 5.0065 \times 10^{-6} \, \text{C/m} \]

### Step 7: Express in microcoulombs
Converting to microcoulombs:

\[ \lambda \approx 5 \times 10^{-6} \, \text{C/m} = 5 \, \mu\text{C/m} \]

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

\[ \lambda = 5 \, \mu\text{C/m} \]

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