Electric flux through a surface of area `100 m^(2)` lying in the plane in the xy plane is (in V-m) if `E=hat(i)sqrt(2)hat(j)+sqrt(3)hat(k)`:

Options

A. 100
B. 141.4
C. 173.2
D. 200

Explanation

To find the electric flux through a surface of area \(100 \, m^2\) lying in the xy-plane when the electric field is given as \(\mathbf{E} = \hat{i} + \sqrt{2} \hat{j} + \sqrt{3} \hat{k}\), we can follow these steps:

### Step 1: Understand the electric flux formula
The electric flux \(\Phi\) through a surface is given by the formula:
\[ \Phi = \int \mathbf{E} \cdot d\mathbf{A} \]
where \(\mathbf{E}\) is the electric field vector and \(d\mathbf{A}\) is the differential area vector.

### Step 2: Identify the area vector
Since the surface lies in the xy-plane, the area vector \(d\mathbf{A}\) will be in the direction of the z-axis. Thus, we can express it as:
\[ d\mathbf{A} = dA \hat{k} \]
where \(dA\) is the area element. For a surface area of \(100 \, m^2\), we can write:
\[ \mathbf{A} = 100 \hat{k} \]

### Step 3: Calculate the dot product
Now, we need to calculate the dot product \(\mathbf{E} \cdot \mathbf{A}\):
\[ \mathbf{E} = \hat{i} + \sqrt{2} \hat{j} + \sqrt{3} \hat{k} \]
\[ \mathbf{A} = 100 \hat{k} \]
The dot product is given by:
\[ \mathbf{E} \cdot \mathbf{A} = (\hat{i} + \sqrt{2} \hat{j} + \sqrt{3} \hat{k}) \cdot (100 \hat{k}) = 100 \cdot \sqrt{3} \]
The terms involving \(\hat{i}\) and \(\hat{j}\) will vanish since they are perpendicular to \(\hat{k}\).

### Step 4: Calculate the electric flux
Now we can calculate the electric flux:
\[ \Phi = 100 \cdot \sqrt{3} \]

### Step 5: Numerical evaluation
Calculating \(100 \cdot \sqrt{3}\):
\[ \sqrt{3} \approx 1.732 \]
Thus,
\[ \Phi \approx 100 \cdot 1.732 = 173.2 \, \text{V-m} \]

### Final Answer
The electric flux through the surface is approximately:
\[ \Phi \approx 173.2 \, \text{V-m} \]