Let `r` be the distance of a point on the axis of a magnetic dipole from its centre. The magnetic field at such a point is proportional to

Options

A. `(1)/(r)`
B. `(1)/(r^(2))`
C. `(1)/(r^(3))`
D. none of these

Explanation

To solve the problem, we need to determine how the magnetic field \( B \) at a point on the axis of a magnetic dipole varies with the distance \( r \) from the center of the dipole.

### Step-by-Step Solution:

1. **Understanding the Magnetic Field of a Dipole**:
The magnetic field \( B \) at a point on the axis of a magnetic dipole is given by the formula:
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{m}{r^3} \cdot (1 + 3 \cos^2 \theta) \]
where:
- \( \mu_0 \) is the permeability of free space,
- \( m \) is the magnetic dipole moment,
- \( r \) is the distance from the center of the dipole,
- \( \theta \) is the angle between the dipole axis and the line connecting the dipole to the point where the field is being measured.

2. **Considering the Point on the Axis**:
For a point located on the axis of the dipole, \( \theta = 0^\circ \). Therefore, \( \cos^2 \theta = 1 \). Substituting this into the equation simplifies it:
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{m}{r^3} \cdot (1 + 3 \cdot 1) = \frac{\mu_0}{4\pi} \cdot \frac{m}{r^3} \cdot 4 \]
Thus, we can rewrite the equation as:
\[ B = \frac{4\mu_0 m}{4\pi r^3} = \frac{\mu_0 m}{\pi r^3} \]

3. **Analyzing the Relationship**:
From the simplified expression, we see that the magnetic field \( B \) is inversely proportional to the cube of the distance \( r \):
\[ B \propto \frac{1}{r^3} \]

4. **Conclusion**:
Therefore, the magnetic field at a point on the axis of a magnetic dipole is proportional to \( \frac{1}{r^3} \).

### Final Answer:
The magnetic field \( B \) at a point on the axis of a magnetic dipole is proportional to \( \frac{1}{r^3} \).