A surface element `vec(ds) = 5 hat(i)` is placed in an electric field `vec(E) = 4 hat(i) + 4 hat(j) + 4 hat(k)`. What is the electric flux emanting from the surface ?

Options

A. 40 unit
B. 50 unit
C. 30 unit
D. 20 unit

Answer

To find the electric flux emanating from the surface, we can use the formula for electric flux, which is given by:

\[ \Phi_E = \vec{E} \cdot \vec{A} \]

Where:
- \(\Phi_E\) is the electric flux,
- \(\vec{E}\) is the electric field vector,
- \(\vec{A}\) is the area vector.

### Step-by-Step Solution:

1. **Identify the given vectors**:
- The electric field vector is given as:
\[ \vec{E} = 4 \hat{i} + 4 \hat{j} + 4 \hat{k} \]
- The surface element vector is given as:
\[ \vec{ds} = 5 \hat{i} \]

2. **Calculate the dot product**:
The dot product \(\vec{E} \cdot \vec{ds}\) can be calculated as follows:
\[ \vec{E} \cdot \vec{ds} = (4 \hat{i} + 4 \hat{j} + 4 \hat{k}) \cdot (5 \hat{i}) \]
Using the properties of dot product:
\[ \vec{E} \cdot \vec{ds} = 4 \cdot 5 (\hat{i} \cdot \hat{i}) + 4 \cdot 0 (\hat{j} \cdot \hat{i}) + 4 \cdot 0 (\hat{k} \cdot \hat{i}) \]
Since \(\hat{j} \cdot \hat{i} = 0\) and \(\hat{k} \cdot \hat{i} = 0\), we have:
\[ \vec{E} \cdot \vec{ds} = 20 + 0 + 0 = 20 \]

3. **State the units of electric flux**:
The unit of electric flux is:
\[ \text{Newton meter}^2/\text{Coulomb} \quad (\text{N m}^2/\text{C}) \]

4. **Final answer**:
Thus, the electric flux emanating from the surface is:
\[ \Phi_E = 20 \, \text{N m}^2/\text{C} \]