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.

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 intensity, \( E = 5 \times 10^5 \, \text{N/C} \)
- Angle between the electric field and the normal to the surface of the sheet, \( \theta = 60^\circ \)

2. **Calculate the Area of the Circular Sheet:**
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Substituting the value of \( r \):
\[ A = \pi (0.1)^2 = \pi \times 0.01 \, \text{m}^2 \]

3. **Determine the Angle for Electric Flux Calculation:**
The angle between the electric field and the area vector (normal to the surface) is \( 30^\circ \) because the area vector is perpendicular to the sheet. Thus:
\[ \phi = E \cdot A \cdot \cos(30^\circ) \]

4. **Calculate the Electric Flux:**
Using the formula for electric flux:
\[ \phi = E \cdot A \cdot \cos(30^\circ) \]
Substitute the values:
\[ \phi = (5 \times 10^5) \cdot (\pi \times 0.01) \cdot \cos(30^\circ) \]
We know that \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \):
\[ \phi = (5 \times 10^5) \cdot (\pi \times 0.01) \cdot \frac{\sqrt{3}}{2} \]

5. **Simplify the Expression:**
\[ \phi = (5 \times 10^5) \cdot \left(\frac{\pi \times 0.01 \sqrt{3}}{2}\right) \]
\[ \phi = (5 \times 10^5) \cdot (0.005 \pi \sqrt{3}) \]
\[ \phi = 2.5 \times 10^3 \pi \sqrt{3} \]

6. **Calculate the Numerical Value:**
Using \( \pi \approx 3.14 \) and \( \sqrt{3} \approx 1.732 \):
\[ \phi \approx 2.5 \times 10^3 \times 3.14 \times 1.732 \]
\[ \phi \approx 2.5 \times 10^3 \times 5.441 \approx 1.36 \times 10^4 \, \text{N m}^2/\text{C} \]

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