If the electric field given by `(5hat(i)+4hat(j)+9hat(k))`, then the electric flux through a surface of area 20 unit lying in the `yz`- plane will be

Options

A. 100 unit
B. 80 unit
C. 180 unit
D. 20 unit

Answer

To find the electric flux through a surface in an electric field, we can use the formula:

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

where \(\Phi_E\) is the electric flux, \(\vec{E}\) is the electric field vector, and \(\vec{A}\) is the area vector.

### Step-by-Step Solution:

1. **Identify the Electric Field Vector**:
The electric field is given as:
\[ \vec{E} = 5\hat{i} + 4\hat{j} + 9\hat{k} \]

2. **Determine the Area Vector**:
Since the surface lies in the \(yz\)-plane, the area vector will be perpendicular to this plane, which means it will be along the \(x\)-axis. The magnitude of the area is given as 20 units. Therefore, the area vector is:
\[ \vec{A} = 20\hat{i} \]

3. **Calculate the Dot Product**:
Now, we calculate the dot product of the electric field vector \(\vec{E}\) and the area vector \(\vec{A}\):
\[ \Phi_E = \vec{E} \cdot \vec{A} = (5\hat{i} + 4\hat{j} + 9\hat{k}) \cdot (20\hat{i}) \]

Performing the dot product:
\[ \Phi_E = (5 \cdot 20) + (4 \cdot 0) + (9 \cdot 0) = 100 + 0 + 0 = 100 \]

4. **Conclusion**:
Thus, the electric flux through the surface is:
\[ \Phi_E = 100 \text{ units} \]

### Final Answer:
The electric flux through the surface is \(100\) units.

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