In a region of space, the electric field is given by `vecE = 8hati + 4hatj + 3hatk`. The electric flux through a surface of area 100 units in the xy plane is

Options

A. 800 units
B. 300 units
C. 400 units
D. 1500 units

Explanation

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

### Step 1: Identify the Electric Field and Surface Area
The electric field is given as:
\[ \vec{E} = 8\hat{i} + 4\hat{j} + 3\hat{k} \]
The surface area is given as 100 units in the xy-plane.

### Step 2: Determine the Surface Area Vector
Since the surface is in the xy-plane, the normal vector to this surface will point in the z-direction. Thus, the surface area vector \(\vec{A}\) can be represented as:
\[ \vec{A} = 100\hat{k} \]

### Step 3: Calculate the Electric Flux
The electric flux \(\Phi\) through the surface is given by the dot product of the electric field \(\vec{E}\) and the area vector \(\vec{A}\):
\[ \Phi = \vec{E} \cdot \vec{A} \]
Substituting the values, we have:
\[ \Phi = (8\hat{i} + 4\hat{j} + 3\hat{k}) \cdot (0\hat{i} + 0\hat{j} + 100\hat{k}) \]

### Step 4: Perform the Dot Product
Calculating the dot product:
\[ \Phi = 8 \cdot 0 + 4 \cdot 0 + 3 \cdot 100 = 0 + 0 + 300 = 300 \]

### Step 5: State the Final Answer
Thus, the electric flux through the surface is:
\[ \Phi = 300 \text{ units} \]

### Summary of Steps:
1. Identify the electric field and surface area.
2. Determine the surface area vector.
3. Calculate the electric flux using the dot product.
4. Perform the dot product calculation.
5. State the final answer.