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

Answer

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

### Step-by-Step Solution:

1. **Understanding the Initial Conditions**:
- The length of the magnetic needle is \(2L\).
- The magnetic moment of the needle is \(M\).
- The pole strength is \(m\).

2. **Breaking the Needle**:
- The needle is broken into two equal pieces at the middle. Therefore, each piece will have a length of \(L\).

3. **Pole Strength**:
- The pole strength of a magnet is defined as the strength of each pole. When the needle is broken, the pole strength of each piece remains unchanged. Thus, the pole strength of each piece is still:
\[ m' = m \]

4. **Magnetic Moment**:
- The magnetic moment \(M\) of a magnet is given by the formula:
\[ M = m \cdot l \]
where \(l\) is the length of the magnet.
- Initially, for the whole needle:
\[ M = m \cdot (2L) \]
- After breaking the needle into two pieces, the length of each piece is \(L\). Therefore, the magnetic moment \(M'\) of each piece can be calculated as:
\[ M' = m \cdot L \]

5. **Calculating the New Magnetic Moment**:
- Since the original magnetic moment \(M\) was \(m \cdot (2L)\), we can express the magnetic moment of each piece as:
\[ M' = \frac{M}{2} \]

6. **Final Results**:
- For each piece:
- Pole Strength \(m' = m\)
- Magnetic Moment \(M' = \frac{M}{2}\)

### Summary of Results:
- The pole strength of each piece remains \(m\).
- The magnetic moment of each piece is \(\frac{M}{2}\).