The ratio of magnetic fields on the axial line of a long magnet at distance of `10cm` and `20cm` is `12.5.1.` The length of the magnet is

Options

A. `5cm`
B. `10cm`
C. `10m`
D. `15 m`

Explanation

To solve the problem of finding the length of a long magnet given the ratio of magnetic fields at two points on its axial line, we can follow these steps:

### Step-by-Step Solution

1. **Understand the Magnetic Field Formula**:
The magnetic field \( B \) at a distance \( d \) from the center of a long magnet can be expressed as:
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{2M}{d^2 - l^2} \]
where \( M \) is the magnetic moment, \( l \) is the half-length of the magnet, and \( d \) is the distance from the center of the magnet.

2. **Set Up the Problem**:
We have two points:
- Point P at a distance of \( 10 \, \text{cm} = 0.1 \, \text{m} \)
- Point Q at a distance of \( 20 \, \text{cm} = 0.2 \, \text{m} \)

The ratio of the magnetic fields at these points is given as:
\[ \frac{B_P}{B_Q} = \frac{12}{5.1} \]

3. **Write the Magnetic Field Equations**:
For point P:
\[ B_P = \frac{\mu_0}{4\pi} \cdot \frac{2M}{(0.1)^2 - l^2} \]
For point Q:
\[ B_Q = \frac{\mu_0}{4\pi} \cdot \frac{2M}{(0.2)^2 - l^2} \]

4. **Form the Ratio**:
Substitute the expressions for \( B_P \) and \( B_Q \) into the ratio:
\[ \frac{B_P}{B_Q} = \frac{\frac{\mu_0}{4\pi} \cdot \frac{2M}{(0.1)^2 - l^2}}{\frac{\mu_0}{4\pi} \cdot \frac{2M}{(0.2)^2 - l^2}} = \frac{(0.2)^2 - l^2}{(0.1)^2 - l^2} \]
This simplifies to:
\[ \frac{(0.2)^2 - l^2}{(0.1)^2 - l^2} = \frac{12}{5.1} \]

5. **Cross Multiply**:
Cross-multiplying gives:
\[ 5.1 \left((0.2)^2 - l^2\right) = 12 \left((0.1)^2 - l^2\right) \]

6. **Substitute Values**:
Substitute \( (0.2)^2 = 0.04 \) and \( (0.1)^2 = 0.01 \):
\[ 5.1(0.04 - l^2) = 12(0.01 - l^2) \]

7. **Expand and Rearrange**:
Expanding both sides:
\[ 0.204 - 5.1l^2 = 0.12 - 12l^2 \]
Rearranging gives:
\[ 12l^2 - 5.1l^2 = 0.204 - 0.12 \]
\[ 6.9l^2 = 0.084 \]

8. **Solve for \( l^2 \)**:
\[ l^2 = \frac{0.084}{6.9} \approx 0.01217 \]
Taking the square root:
\[ l \approx \sqrt{0.01217} \approx 0.1103 \, \text{m} \]

9. **Calculate the Length of the Magnet**:
The total length of the magnet \( L \) is:
\[ L = 2l \approx 2 \times 0.1103 \approx 0.2206 \, \text{m} \approx 22.06 \, \text{cm} \]

### Final Answer:
The length of the magnet is approximately **22.06 cm**.