A circular plane sheet of radius 10 cm is placed in a uniform electric field of `5xx10^(5) NC^(-1)`, making an angle of `60^(@)` with the field. Calculate electric flux through the sheet.

Options

A. `1.36xx10^(2)Nm^(2)C^(-1)`
B. `1.36xx10^(4)Nm^(2)C^(-1)`
C. `0.515xx10^(2)Nm^(2)C^(-1)`
D. `0.515xx10^(4)Nm^(2)C^(-1)`

Explanation

To calculate the electric flux through a circular plane sheet placed in a uniform electric field, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Radius of the circular sheet, \( r = 10 \, \text{cm} = 0.1 \, \text{m} \)
- Electric field strength, \( E = 5 \times 10^5 \, \text{N/C} \)
- Angle between the electric field and the normal to the surface of the sheet, \( \theta = 60^\circ \)

2. **Determine the Area of the Circular Sheet:**
The area \( A \) of a circular sheet is given by the formula:
\[ A = \pi r^2 \]
Substituting the radius:
\[ A = \pi (0.1)^2 = \pi \times 0.01 \approx 0.0314 \, \text{m}^2 \]

3. **Calculate the Angle for Electric Flux:**
The angle \( \phi \) between the electric field and the area vector of the sheet is:
\[ \phi = 90^\circ - \theta = 90^\circ - 60^\circ = 30^\circ \]

4. **Apply the Electric Flux Formula:**
The electric flux \( \Phi_E \) through the sheet is given by:
\[ \Phi_E = E \cdot A \cdot \cos(\phi) \]
Substituting the known values:
\[ \Phi_E = (5 \times 10^5) \cdot (0.0314) \cdot \cos(30^\circ) \]

5. **Calculate \( \cos(30^\circ) \):**
We know that:
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.866 \]

6. **Substitute and Calculate the Electric Flux:**
Now substituting \( \cos(30^\circ) \):
\[ \Phi_E = (5 \times 10^5) \cdot (0.0314) \cdot (0.866) \]
\[ \Phi_E \approx (5 \times 10^5) \cdot (0.0314) \cdot (0.866) \approx 1.36 \times 10^4 \, \text{N m}^2/\text{C} \]

### Final Answer:
The electric flux through the sheet is approximately:
\[ \Phi_E \approx 1.36 \times 10^4 \, \text{N m}^2/\text{C} \]

---