If the electric field is given by `3 hat(i) + 2 hat(j) + 6 hat(k)` . Find the electric flux through a surface area 20 unit lying in xy plane

Options

A. Zero
B. 60 unit
C. 40 unit
D. 120 unit

Answer

To find the electric flux through a surface area in the xy-plane given the electric field, we can follow these steps:

### Step 1: Understand the Electric Field and Area Vector
The electric field is given as:
\[ \mathbf{E} = 3 \hat{i} + 2 \hat{j} + 6 \hat{k} \]
The surface area is in the xy-plane, which means the area vector (\(\mathbf{A}\)) will be perpendicular to this plane. Therefore, the area vector can be expressed as:
\[ \mathbf{A} = 20 \hat{k} \]
where 20 is the area of the surface lying in the xy-plane.

### Step 2: Calculate the Electric Flux
The electric flux (\(\Phi_E\)) through a surface is given by the dot product of the electric field vector and the area vector:
\[ \Phi_E = \mathbf{E} \cdot \mathbf{A} \]
Substituting the values of \(\mathbf{E}\) and \(\mathbf{A}\):
\[ \Phi_E = (3 \hat{i} + 2 \hat{j} + 6 \hat{k}) \cdot (20 \hat{k}) \]

### Step 3: Perform the Dot Product
Using the properties of the dot product:
\[ \Phi_E = 3 \hat{i} \cdot (20 \hat{k}) + 2 \hat{j} \cdot (20 \hat{k}) + 6 \hat{k} \cdot (20 \hat{k}) \]
Since \(\hat{i} \cdot \hat{k} = 0\) and \(\hat{j} \cdot \hat{k} = 0\), we only need to calculate:
\[ \Phi_E = 6 \hat{k} \cdot (20 \hat{k}) = 6 \times 20 \]

### Step 4: Calculate the Result
\[ \Phi_E = 120 \]

### Final Answer
The electric flux through the surface area is:
\[ \Phi_E = 120 \text{ units} \]

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