The ratio of magnetic fields due to a bar magnet at the two axial points `P_(1) and P_(2)` which are separated from each other by 10 cm is 25 :2 . Point `P_(1)` is situated at 10 cm from the centre of the magnet. Magnetic length of the bar magnet is (Points `P_(1) and P_(2)` are on the same side of magnet and distance of `P_(2)` from the centre is greater than distance of `P_(1)` from the centre of magnet)

Options

A. 5 cm
B. 10 cm
C. 15 cm
D. 20 cm

Answer

To solve the problem, we need to find the magnetic length of a bar magnet given the ratio of magnetic fields at two axial points \( P_1 \) and \( P_2 \) and their respective distances from the center of the magnet.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Ratio of magnetic fields at points \( P_1 \) and \( P_2 \): \( \frac{B_1}{B_2} = \frac{25}{2} \)
- Distance between \( P_1 \) and \( P_2 \): \( 10 \, \text{cm} \)
- Distance of \( P_1 \) from the center of the magnet: \( d_1 = 10 \, \text{cm} \)
- Therefore, distance of \( P_2 \) from the center of the magnet: \( d_2 = d_1 + 10 = 20 \, \text{cm} \)

2. **Magnetic Field Formula:**
The magnetic field \( B \) at an axial point from a bar magnet is given by:
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{2m}{d^3} \]
where \( m \) is the magnetic moment of the magnet and \( d \) is the distance from the center.

3. **Write the Equations for \( B_1 \) and \( B_2 \):**
- For point \( P_1 \):
\[ B_1 = \frac{\mu_0}{4\pi} \cdot \frac{2m}{(10)^3} = \frac{\mu_0}{4\pi} \cdot \frac{2m}{1000} \]
- For point \( P_2 \):
\[ B_2 = \frac{\mu_0}{4\pi} \cdot \frac{2m}{(20)^3} = \frac{\mu_0}{4\pi} \cdot \frac{2m}{8000} \]

4. **Set Up the Ratio:**
Using the ratio given:
\[ \frac{B_1}{B_2} = \frac{\frac{2m}{1000}}{\frac{2m}{8000}} = \frac{8000}{1000} = 8 \]
However, we know from the problem statement that:
\[ \frac{B_1}{B_2} = \frac{25}{2} \]
Therefore, we can equate:
\[ 8 = \frac{25}{2} \]

5. **Equate the Two Ratios:**
From the above, we can set up the equation:
\[ \frac{8000}{1000} = \frac{25}{2} \]
Cross-multiplying gives:
\[ 16000 = 2500 \]
This indicates that we need to consider the lengths involved.

6. **Using the Length of the Magnet:**
Let \( l \) be the half-length of the magnet. The magnetic field equations will also depend on \( l \):
\[ B_1 = \frac{\mu_0}{4\pi} \cdot \frac{2m}{(10)^3 - l^2} \]
\[ B_2 = \frac{\mu_0}{4\pi} \cdot \frac{2m}{(20)^3 - l^2} \]

7. **Set Up the Ratio Again:**
\[ \frac{B_1}{B_2} = \frac{(20^3 - l^2)}{(10^3 - l^2)} = \frac{25}{2} \]
Substituting \( 20^3 = 8000 \) and \( 10^3 = 1000 \):
\[ \frac{8000 - l^2}{1000 - l^2} = \frac{25}{2} \]

8. **Cross-Multiply and Solve for \( l^2 \):**
Cross-multiplying gives:
\[ 2(8000 - l^2) = 25(1000 - l^2) \]
Expanding:
\[ 16000 - 2l^2 = 25000 - 25l^2 \]
Rearranging gives:
\[ 23l^2 = 9000 \implies l^2 = \frac{9000}{23} \]

9. **Calculate \( l \):**
\[ l \approx \sqrt{391.3043} \approx 19.78 \, \text{cm} \]
Therefore, the total length \( L = 2l \approx 39.56 \, \text{cm} \).

### Final Answer:
The magnetic length of the bar magnet is approximately \( 39.56 \, \text{cm} \).