In a certain region of space, there exists a uniform electric field of value ` 2xx 10^2hatkVm^(-1)`. A rectangular coil of dimension `10 cm xx 20 cm ` is placed in the xy plane. The electric flux through the coil is

Options

A. zero
B. 30 V m
C. 40 V m
D. 50 V m

Explanation

To find the electric flux through the rectangular coil placed in the xy-plane in a uniform electric field, we can follow these steps:

### Step 1: Understand the Given Information
We are given:
- Electric field \( \vec{E} = 2 \times 10^2 \hat{k} \, \text{V/m} \)
- Dimensions of the rectangular coil: \( 10 \, \text{cm} \times 20 \, \text{cm} \)

### Step 2: Convert Dimensions to Meters
Convert the dimensions of the coil from centimeters to meters:
- Length = \( 10 \, \text{cm} = 0.1 \, \text{m} \)
- Width = \( 20 \, \text{cm} = 0.2 \, \text{m} \)

### Step 3: Calculate the Area of the Coil
The area \( A \) of the rectangular coil can be calculated as:
\[ A = \text{Length} \times \text{Width} = 0.1 \, \text{m} \times 0.2 \, \text{m} = 0.02 \, \text{m}^2 \]

### Step 4: Determine the Angle Between Electric Field and Area Vector
The electric field is directed along the \( z \)-axis (in the \( \hat{k} \) direction), and since the coil is in the xy-plane, the area vector \( \vec{A} \) of the coil is perpendicular to the electric field. Therefore, the angle \( \theta \) between the electric field and the area vector is \( 90^\circ \).

### Step 5: Calculate the Electric Flux
The electric flux \( \Phi_E \) through the coil is given by the formula:
\[ \Phi_E = \vec{E} \cdot \vec{A} = E \cdot A \cdot \cos(\theta) \]
Since \( \theta = 90^\circ \), \( \cos(90^\circ) = 0 \). Thus:
\[ \Phi_E = E \cdot A \cdot 0 = 0 \]

### Conclusion
The electric flux through the rectangular coil is \( 0 \, \text{V m} \) because the electric field is perpendicular to the area vector of the coil.

---