The ratio of the magnetic fields due to small bar magnet in end position an broad side on position is (at equal distance from the magnet)

Options

A. `1/4`
B. `1/2`
C. 1
D. 2

Answer

To solve the question regarding the ratio of the magnetic fields due to a small bar magnet in the end position (axial position) and the broadside on position (equatorial position), we can follow these steps:

### Step-by-Step Solution

1. **Understand the Positions**:
- The **end position** refers to the axial position of the bar magnet, where the magnetic field is measured along the axis of the magnet.
- The **broadside on position** refers to the equatorial position of the bar magnet, where the magnetic field is measured perpendicular to the axis of the magnet.

2. **Magnetic Field in Axial Position**:
- The magnetic field \( B_{\text{axial}} \) at a distance \( D \) from the center of the bar magnet in the axial position is given by the formula:
\[ B_{\text{axial}} = \frac{\mu_0 \cdot 2m}{4\pi D^3} \]
- Here, \( \mu_0 \) is the permeability of free space, and \( m \) is the magnetic moment of the bar magnet.

3. **Magnetic Field in Equatorial Position**:
- The magnetic field \( B_{\text{equatorial}} \) at the same distance \( D \) in the equatorial position is given by the formula:
\[ B_{\text{equatorial}} = \frac{\mu_0 \cdot m}{4\pi D^3} \]

4. **Calculate the Ratio**:
- To find the ratio of the magnetic fields, we will divide \( B_{\text{axial}} \) by \( B_{\text{equatorial}} \):
\[ \frac{B_{\text{axial}}}{B_{\text{equatorial}}} = \frac{\frac{\mu_0 \cdot 2m}{4\pi D^3}}{\frac{\mu_0 \cdot m}{4\pi D^3}} \]
- Simplifying this expression:
\[ \frac{B_{\text{axial}}}{B_{\text{equatorial}}} = \frac{2m}{m} = 2 \]

5. **Conclusion**:
- The ratio of the magnetic fields due to a small bar magnet in the end position to that in the broadside on position is:
\[ \frac{B_{\text{axial}}}{B_{\text{equatorial}}} = 2 \]

### Final Answer
The ratio of the magnetic fields is **2**.