A long magnet is cut in two parts in such a way that the ratio of their lengths is 2:1. The retio of pole strengths of both the section is

Options

A. Equal
B. In the ratio of `2:1`
C. In the ratio of `1:2`
D. In the ratio of `4:1`

Explanation

To solve the problem, we need to understand how cutting a magnet affects its pole strengths.

### Step-by-Step Solution:

1. **Understanding the Original Magnet**:
Let's denote the original magnet's length as \( L \) and its pole strength as \( P \).

2. **Cutting the Magnet**:
The magnet is cut into two parts in the ratio of 2:1. This means one part is \( \frac{2}{3}L \) and the other part is \( \frac{1}{3}L \).

3. **Pole Strength of Each Section**:
When a magnet is cut, the pole strength of each section is proportional to its length. The pole strength of a magnet is given by the formula:
\[ P = k \cdot L \]
where \( k \) is a constant that depends on the material of the magnet.

4. **Calculating Pole Strengths**:
- For the longer section (length \( \frac{2}{3}L \)):
\[ P_1 = k \cdot \frac{2}{3}L \]
- For the shorter section (length \( \frac{1}{3}L \)):
\[ P_2 = k \cdot \frac{1}{3}L \]

5. **Finding the Ratio of Pole Strengths**:
Now, we can find the ratio of the pole strengths of the two sections:
\[ \text{Ratio} = \frac{P_1}{P_2} = \frac{k \cdot \frac{2}{3}L}{k \cdot \frac{1}{3}L} = \frac{\frac{2}{3}}{\frac{1}{3}} = 2 \]

6. **Conclusion**:
Therefore, the ratio of the pole strengths of the two sections is \( 2:1 \).

### Final Answer:
The ratio of the pole strengths of both sections is \( 2:1 \).