Keeping r constant a graph is plotted by varying both 0 and the potential V due to the magnetic dipole.Which is a correct graph in the figure given below?

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining the correct graph of potential \( V \) versus angle \( \theta \) for a magnetic dipole while keeping \( r \) constant, we can follow these steps:

### Step 1: Understand the Formula for Magnetic Potential
The potential \( V \) due to a magnetic dipole at a point in space is given by the formula:
\[ V = \frac{\mu_0}{4\pi} \cdot \frac{m \cos \theta}{r^2} \]
where:
- \( \mu_0 \) is the permeability of free space,
- \( m \) is the magnetic dipole moment,
- \( r \) is the distance from the dipole,
- \( \theta \) is the angle between the dipole moment and the line connecting the dipole to the point of interest.

### Step 2: Analyze the Relationship Between \( V \) and \( \theta \)
From the formula, we see that \( V \) is directly proportional to \( \cos \theta \):
\[ V \propto \cos \theta \]
This means that as \( \theta \) varies, \( V \) will vary in a manner that follows the cosine function.

### Step 3: Determine the Characteristics of the Cosine Function
The cosine function has the following characteristics:
- It starts at 1 when \( \theta = 0 \),
- It decreases to 0 when \( \theta = 90^\circ \),
- It becomes negative as \( \theta \) approaches \( 180^\circ \),
- It is periodic with a period of \( 360^\circ \).

### Step 4: Identify the Correct Graph
Since \( V \) is proportional to \( \cos \theta \), the graph of \( V \) versus \( \theta \) will resemble the graph of \( \cos \theta \) but scaled by the factor \( \frac{\mu_0 m}{4\pi r^2} \). The graph will not start from zero; instead, it will start from a positive value when \( \theta = 0 \).

### Step 5: Conclusion
Based on the characteristics of the cosine function and the relationship derived from the formula, the correct graph will be the one that resembles the cosine function starting from a positive value.

Thus, the correct answer is option **B**, which represents the graph of \( V \) versus \( \theta \).

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