Electric flux through a surface are `10 m^(2)` lying in the xy plane is `(ifvecE= hati + sqrt2hatj + sqrt3 hatk N//C)`

Options

A. 100
B. 141.4
C. 17.32
D. 200

Answer

To solve the problem of calculating the electric flux through a surface in the xy-plane given the electric field vector, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Electric Field Vector**:
The electric field is given as:
\[ \vec{E} = \hat{i} + \sqrt{2} \hat{j} + \sqrt{3} \hat{k} \, \text{N/C} \]

2. **Determine the Surface Area and Orientation**:
The surface area is given as \( A = 10 \, m^2 \) and it lies in the xy-plane. The normal vector to the surface (which is perpendicular to the surface) in the xy-plane is:
\[ \hat{n} = \hat{k} \]

3. **Calculate the Dot Product**:
To find the electric flux, we need to calculate the dot product of the electric field vector and the normal vector:
\[ \vec{E} \cdot \hat{n} = \left( \hat{i} + \sqrt{2} \hat{j} + \sqrt{3} \hat{k} \right) \cdot \hat{k} \]
Since the dot product only considers the component of \(\vec{E}\) in the direction of \(\hat{n}\):
\[ \vec{E} \cdot \hat{n} = 0 + 0 + \sqrt{3} = \sqrt{3} \]

4. **Calculate the Electric Flux**:
The electric flux \( \Phi \) through the surface is given by:
\[ \Phi = \vec{E} \cdot \hat{n} \cdot A \]
Substituting the values we have:
\[ \Phi = \sqrt{3} \cdot 10 = 10\sqrt{3} \]

5. **Calculate the Numerical Value**:
To find the numerical value of \( 10\sqrt{3} \):
\[ \sqrt{3} \approx 1.732 \]
Therefore:
\[ \Phi \approx 10 \times 1.732 = 17.32 \, \text{N m}^2/\text{C} \]

### Conclusion:
The electric flux through the surface is approximately \( 17.32 \, \text{N m}^2/\text{C} \). Thus, the correct answer is option number 3.

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