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

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 \(4 \, \text{cm}\), we can follow these steps:

### Step 1: Write 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:
\[ E = \frac{\lambda}{2 \pi \epsilon_0 r} \]
where:
- \(E\) is the electric field,
- \(\lambda\) is the linear charge density,
- \(\epsilon_0\) is the permittivity of free space (\(\epsilon_0 \approx 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2\)),
- \(r\) is the distance from the line charge.

### Step 2: Rearrange the formula to solve for linear charge density (λ).
To find \(\lambda\), we rearrange the formula:
\[ \lambda = 2 \pi \epsilon_0 E r \]

### Step 3: Convert the distance from centimeters to meters.
The distance \(r\) is given as \(4 \, \text{cm}\). We need to convert this to meters:
\[ r = 4 \, \text{cm} = 4 \times 10^{-2} \, \text{m} \]

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

Substituting these values:
\[ \lambda = 2 \pi (8.85 \times 10^{-12}) (9 \times 10^4) (4 \times 10^{-2}) \]

### Step 5: Calculate the value of λ.
Calculating this step-by-step:
1. Calculate \(2 \pi \approx 6.2832\).
2. Calculate \(8.85 \times 10^{-12} \times 9 \times 10^4 \approx 7.965 \times 10^{-7}\).
3. Multiply by \(4 \times 10^{-2}\):
\[ \lambda \approx 6.2832 \times 7.965 \times 10^{-7} \times 4 \times 10^{-2} \]
\[ \lambda \approx 2 \times 10^{-7} \, \text{C/m} \]

### Final Answer:
The linear charge density \(\lambda\) is approximately:
\[ \lambda \approx 2 \times 10^{-7} \, \text{C/m} \]

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