The field due to a magnet at a distance `R~ from the centre of the magnet is proportional

Options

A. `R^(2)`
B. `R^(3)`
C. `1//R^(2)`
D. `1//r^(3)`

Answer

To solve the question regarding the magnetic field due to a magnet at a distance \( R \) from the center of the magnet, we will analyze the magnetic field in two different scenarios: at an axial point and at an equatorial point.

### Step-by-Step Solution:

1. **Understanding the Magnetic Moment**:
- The magnetic moment \( m \) of a magnet is a measure of its strength and orientation in a magnetic field.

2. **Magnetic Field at an Axial Point**:
- For a point located along the axis of the magnet at a distance \( r \) from its center, the magnetic field \( B_1 \) can be expressed as:
\[ B_1 = \frac{\mu_0}{4\pi} \cdot \frac{m}{r^3} \]
- Here, \( \mu_0 \) is the permeability of free space.

3. **Magnetic Field at an Equatorial Point**:
- For a point located on the equatorial line of the magnet at the same distance \( r \), the magnetic field \( B_2 \) is given by:
\[ B_2 = \frac{\mu_0}{4\pi} \cdot \frac{2m}{r^3} \]

4. **Proportionality Analysis**:
- In both cases, we observe that the magnetic field \( B \) is inversely proportional to the cube of the distance \( r \):
- For axial point: \( B_1 \propto \frac{1}{r^3} \)
- For equatorial point: \( B_2 \propto \frac{1}{r^3} \)

5. **Conclusion**:
- Therefore, the magnetic field due to a magnet at a distance \( R \) from its center is proportional to \( \frac{1}{R^3} \).

### Final Answer:
The field due to a magnet at a distance \( R \) from the center of the magnet is proportional to \( \frac{1}{R^3} \).

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