A long magnetic needle of length 2L, magnetic moment M amd pole strength m units is broken into two pieces at the middle. The magnetic moment amd pole strength of each piece will be

Options

A. `M/2,m/2`
B. `M,m/2`
C. `M/2,m`
D. `M,m`

Explanation

To solve the problem, we need to analyze the properties of the magnetic needle before and after it is broken into two equal parts.

### Step-by-Step Solution:

1. **Understanding the Original Magnetic Needle**:
- The original magnetic needle has a length of \(2L\), magnetic moment \(M\), and pole strength \(m\).
- The magnetic moment \(M\) of a magnetic needle is given by the formula:
\[ M = m \times L \]
- Here, the total length \(L\) of the needle is \(L = 2L\).

2. **Breaking the Needle**:
- When the needle is broken into two equal parts, each part will have a length of:
\[ L' = \frac{2L}{2} = L \]

3. **Calculating the Magnetic Moment of Each Piece**:
- The magnetic moment of each piece after breaking can be calculated using the formula:
\[ M' = m \times L' \]
- Since the length of each piece is now \(L\) and the pole strength \(m\) remains unchanged, we can substitute \(L'\) into the equation:
\[ M' = m \times L = m \times \frac{L}{2} = \frac{M}{2} \]
- Therefore, the magnetic moment of each piece is:
\[ M' = \frac{M}{2} \]

4. **Determining the Pole Strength of Each Piece**:
- The pole strength \(m\) of each piece remains the same as the original pole strength of the needle because breaking the needle does not change the pole strength.
- Thus, the pole strength of each piece is:
\[ m' = m \]

5. **Final Results**:
- After breaking the needle into two equal pieces:
- The magnetic moment of each piece is \(M' = \frac{M}{2}\).
- The pole strength of each piece is \(m' = m\).

### Summary of Results:
- **Magnetic Moment of Each Piece**: \( \frac{M}{2} \)
- **Pole Strength of Each Piece**: \( m \)