A bar magnet of length 3cm has points X and Y along the axis at a distance of 24 cm and 48cm on opposite ends. Ratio of magnetic field at these points will be

Options

A. 3
B. 8
C. `(1)/(2) sqrt2`
D. 4

Explanation

To solve the problem, we need to find the ratio of the magnetic fields at points X and Y along the axis of a bar magnet.

### Step-by-Step Solution:

1. **Identify the distances from the center of the magnet**:
- The length of the bar magnet is 3 cm. Therefore, the distance from the center of the magnet to point X (24 cm from one end) is:
\[ r_X = 24 \, \text{cm} - \frac{3}{2} \, \text{cm} = 24 \, \text{cm} - 1.5 \, \text{cm} = 22.5 \, \text{cm} \]
- The distance from the center of the magnet to point Y (48 cm from one end) is:
\[ r_Y = 48 \, \text{cm} - \frac{3}{2} \, \text{cm} = 48 \, \text{cm} - 1.5 \, \text{cm} = 46.5 \, \text{cm} \]

2. **Use the formula for the magnetic field along the axis of a bar magnet**:
The magnetic field \( B \) at a distance \( r \) from the center of a bar magnet is given by:
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{2M}{r^3} \]
where \( M \) is the magnetic moment of the magnet and \( \mu_0 \) is the permeability of free space.

3. **Find the ratio of the magnetic fields at points X and Y**:
The ratio of the magnetic fields \( B_X \) and \( B_Y \) at points X and Y can be expressed as:
\[ \frac{B_X}{B_Y} = \frac{r_Y^3}{r_X^3} \]

4. **Substituting the values of \( r_X \) and \( r_Y \)**:
\[ \frac{B_X}{B_Y} = \frac{(46.5)^3}{(22.5)^3} \]

5. **Calculating the ratio**:
- First, calculate the ratio of the distances:
\[ \frac{r_Y}{r_X} = \frac{46.5}{22.5} = 2.0667 \approx 2 \]
- Now, cube the ratio:
\[ \left(\frac{r_Y}{r_X}\right)^3 = 2^3 = 8 \]

6. **Conclusion**:
Therefore, the ratio of the magnetic fields at points X and Y is:
\[ \frac{B_X}{B_Y} = 8 \]

### Final Answer:
The ratio of the magnetic field at points X and Y is \( 8:1 \).