The magnetic flux linked with a vector area `vec(A)` in a uniform magnetic field `vec(B)` is

Options

A. `BxxA`
B. AB
C. `B.A`
D. `(B)/(A)`

Explanation

To find the magnetic flux linked with a vector area \(\vec{A}\) in a uniform magnetic field \(\vec{B}\), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Magnetic Flux**:
Magnetic flux (\(\Phi_B\)) is defined as the product of the magnetic field (\(\vec{B}\)) and the area (\(\vec{A}\)) through which the field lines pass. It quantifies the number of magnetic field lines passing through a given area.

2. **Formula for Magnetic Flux**:
The magnetic flux linked with an area vector in a magnetic field is given by the dot product of the magnetic field vector and the area vector. The formula is:
\[ \Phi_B = \vec{B} \cdot \vec{A} \]

3. **Dot Product Explanation**:
The dot product takes into account both the magnitude of the vectors and the angle between them. It is defined as:
\[ \vec{B} \cdot \vec{A} = |\vec{B}| |\vec{A}| \cos(\theta) \]
where \(\theta\) is the angle between the magnetic field vector and the area vector.

4. **Conclusion**:
From the above understanding, we can conclude that the magnetic flux linked with the vector area \(\vec{A}\) in a uniform magnetic field \(\vec{B}\) is given by:
\[ \Phi_B = \vec{B} \cdot \vec{A} \]

5. **Select the Correct Option**:
Among the given options, the correct answer is:
\[ \text{Option 3: } \vec{B} \cdot \vec{A} \]