What is the charge per unit area in `C//m^2` of an infinite plane sheet of charge if the electric field produced by the sheet of charge has magnitude `3.0 N//C`?

Explanation

To find the charge per unit area (σ) of an infinite plane sheet of charge given the electric field (E) produced by the sheet, we can use the formula:

\[ E = \frac{\sigma}{2 \epsilon_0} \]

where:
- \( E \) is the electric field (in N/C),
- \( \sigma \) is the charge per unit area (in C/m²),
- \( \epsilon_0 \) is the permittivity of free space, approximately \( 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \).

### Step 1: Rearranging the Formula
We need to rearrange the formula to solve for σ:

\[ \sigma = 2 E \epsilon_0 \]

### Step 2: Substituting the Values
Now we can substitute the given values into the equation. We know:
- \( E = 3.0 \, \text{N/C} \)
- \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \)

Substituting these values into the equation:

\[ \sigma = 2 \times 3.0 \, \text{N/C} \times 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \]

### Step 3: Calculating σ
Now we perform the multiplication:

\[ \sigma = 6.0 \times 8.85 \times 10^{-12} \]

Calculating this gives:

\[ \sigma = 53.1 \times 10^{-12} \, \text{C/m}^2 \]

### Step 4: Final Result
Converting this to scientific notation:

\[ \sigma = 5.31 \times 10^{-11} \, \text{C/m}^2 \]

Thus, the charge per unit area of the infinite plane sheet of charge is:

\[ \sigma = 5.31 \times 10^{-11} \, \text{C/m}^2 \]

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