A uniform magnetic field exists in the space `B=B_(1)hati+B_(2)hatj-B_(3)hatk`. Find the magnetic flux through an area S, if the area S is in yz - plane.

Options

A. `B_(2)S`
B. `B_(1)S`
C. `B_(3)S`
D. `ZERO`

Explanation

To solve the problem of finding the magnetic flux through an area \( S \) in the yz-plane when a uniform magnetic field is given by \( \mathbf{B} = B_1 \hat{i} + B_2 \hat{j} - B_3 \hat{k} \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Area Vector**:
Since the area \( S \) is in the yz-plane, the area vector \( \mathbf{A} \) will be perpendicular to this plane. The direction of the area vector in the yz-plane is along the x-axis. Thus, we can express the area vector as:
\[ \mathbf{A} = A \hat{i} \]
where \( A \) is the magnitude of the area.

2. **Identify the Magnetic Field Vector**:
The magnetic field vector is given as:
\[ \mathbf{B} = B_1 \hat{i} + B_2 \hat{j} - B_3 \hat{k} \]

3. **Calculate the Magnetic Flux**:
The magnetic flux \( \Phi \) through the area \( S \) is given by the dot product of the magnetic field vector \( \mathbf{B} \) and the area vector \( \mathbf{A} \):
\[ \Phi = \mathbf{B} \cdot \mathbf{A} \]
Substituting the expressions for \( \mathbf{B} \) and \( \mathbf{A} \):
\[ \Phi = (B_1 \hat{i} + B_2 \hat{j} - B_3 \hat{k}) \cdot (A \hat{i}) \]

4. **Perform the Dot Product**:
The dot product can be calculated as follows:
\[ \Phi = B_1 A (\hat{i} \cdot \hat{i}) + B_2 A (\hat{j} \cdot \hat{i}) - B_3 A (\hat{k} \cdot \hat{i}) \]
Since \( \hat{j} \cdot \hat{i} = 0 \) and \( \hat{k} \cdot \hat{i} = 0 \), we have:
\[ \Phi = B_1 A \]

5. **Final Result**:
Therefore, the magnetic flux through the area \( S \) is:
\[ \Phi = B_1 A \]
The unit of magnetic flux is Tesla meter squared (T·m²).