If the electric flux through a surface of area `100 m^(2)` lying in thé `x-y` plane is `p sqrt(q) V m`, then find `p q .` Given `vec(E) w i+sqrt(2) hat(j)+sqrt(3) hat(k)`

Answer

To solve the problem, we need to calculate the electric flux through a surface in the x-y plane given the electric field vector and the area of the surface.

### Step-by-Step Solution:

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

2. **Identify the Area Vector**:
Since the surface lies in the x-y plane, the area vector \(\vec{A}\) will be perpendicular to the surface and directed along the z-axis. Therefore, we can express the area vector as:
\[ \vec{A} = A \hat{k} = 100 \hat{k} \, \text{m}^2 \]

3. **Calculate the Electric Flux**:
The electric flux \(\Phi_E\) through the surface is given by the dot product of the electric field vector and the area vector:
\[ \Phi_E = \vec{E} \cdot \vec{A} \]
Substituting the values:
\[ \Phi_E = (\hat{i} + \sqrt{2} \hat{j} + \sqrt{3} \hat{k}) \cdot (100 \hat{k}) \]
Since only the k-component contributes to the dot product:
\[ \Phi_E = 100 \cdot \sqrt{3} \]

4. **Express the Flux in the Given Form**:
We are given that the electric flux can be expressed as:
\[ \Phi_E = p \sqrt{q} \, \text{V m} \]
From our calculation:
\[ \Phi_E = 100 \sqrt{3} \]
This implies:
\[ p = 100 \quad \text{and} \quad q = 3 \]

5. **Calculate \(p \cdot q\)**:
Now, we need to find the product \(p \cdot q\):
\[ p \cdot q = 100 \cdot 3 = 300 \]

### Final Answer:
The value of \(p \cdot q\) is \(300\).