The distance of two points on the axis of a magnet from its centre is `10 cm` and `20 cm` repectively. The ratio of magnatic intensity at these points is `12.5 : 1`. The length of the megnet will be

Options

A. 5cm
B. 25cm
C. 10cm
D. 20cm

Answer

To find the length of the magnet based on the given distances and the ratio of magnetic intensities, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Distance from the center of the magnet to point 1, \( r_1 = 10 \, \text{cm} \)
- Distance from the center of the magnet to point 2, \( r_2 = 20 \, \text{cm} \)
- Ratio of magnetic intensities, \( \frac{B_1}{B_2} = \frac{12.5}{1} \)

2. **Write the Formula for Magnetic Intensity on the Axis of a Magnet:**
The magnetic intensity \( B \) at a distance \( r \) from the center of a magnet is given by:
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{2M}{r^2 - \left(\frac{L}{2}\right)^2} \]
where \( M \) is the magnetic moment and \( L \) is the length of the magnet.

3. **Set Up the Ratio of Magnetic Intensities:**
Using the formula, we can express the ratio of the magnetic intensities at the two points:
\[ \frac{B_1}{B_2} = \frac{\frac{2M}{r_1^2 - \left(\frac{L}{2}\right)^2}}{\frac{2M}{r_2^2 - \left(\frac{L}{2}\right)^2}} = \frac{r_2^2 - \left(\frac{L}{2}\right)^2}{r_1^2 - \left(\frac{L}{2}\right)^2} \]

4. **Substitute the Known Values:**
Substitute \( r_1 = 10 \, \text{cm} \) and \( r_2 = 20 \, \text{cm} \) into the ratio:
\[ \frac{12.5}{1} = \frac{20^2 - \left(\frac{L}{2}\right)^2}{10^2 - \left(\frac{L}{2}\right)^2} \]
This simplifies to:
\[ 12.5 = \frac{400 - \left(\frac{L}{2}\right)^2}{100 - \left(\frac{L}{2}\right)^2} \]

5. **Cross Multiply to Solve for \( L \):**
Cross multiplying gives:
\[ 12.5(100 - \left(\frac{L}{2}\right)^2) = 400 - \left(\frac{L}{2}\right)^2 \]
Expanding this results in:
\[ 1250 - 12.5\left(\frac{L}{2}\right)^2 = 400 - \left(\frac{L}{2}\right)^2 \]

6. **Rearranging the Equation:**
Rearranging gives:
\[ 1250 - 400 = 12.5\left(\frac{L}{2}\right)^2 - \left(\frac{L}{2}\right)^2 \]
Simplifying further:
\[ 850 = (12.5 - 1)\left(\frac{L}{2}\right)^2 \]
\[ 850 = 11.5\left(\frac{L}{2}\right)^2 \]

7. **Solve for \( \left(\frac{L}{2}\right)^2 \):**
\[ \left(\frac{L}{2}\right)^2 = \frac{850}{11.5} \]
\[ \left(\frac{L}{2}\right)^2 \approx 73.913 \]

8. **Calculate \( \frac{L}{2} \):**
\[ \frac{L}{2} \approx \sqrt{73.913} \approx 8.6 \, \text{cm} \]

9. **Find the Length of the Magnet \( L \):**
\[ L = 2 \times 8.6 \approx 17.2 \, \text{cm} \]

### Final Answer:
The length of the magnet is approximately \( 17.2 \, \text{cm} \).