A long thin magnet of moment `M` is bent into a semi circle. The decrease in the magnetic moment is

Explanation

To solve the problem of finding the decrease in the magnetic moment when a long thin magnet of moment \( M \) is bent into a semicircle, we can follow these steps:

### Step 1: Understand the initial magnetic moment
The magnetic moment \( M \) of a long thin magnet is given by the formula:
\[ M = m \cdot L \]
where \( m \) is the pole strength and \( L \) is the length of the magnet.

### Step 2: Determine the radius of the semicircle
When the magnet is bent into a semicircle, the length of the magnet becomes the perimeter of the semicircle. The perimeter \( P \) of a semicircle is given by:
\[ P = \pi r \]
where \( r \) is the radius of the semicircle. Since the length of the magnet is equal to the perimeter of the semicircle, we have:
\[ L = \pi r \implies r = \frac{L}{\pi} \]

### Step 3: Calculate the new magnetic moment
The magnetic moment of the bent magnet can be calculated using the new effective length. The effective length of the semicircle is the diameter, which is \( 2r \):
\[ \text{Effective Length} = 2r = 2 \left(\frac{L}{\pi}\right) = \frac{2L}{\pi} \]
Thus, the new magnetic moment \( M' \) when the magnet is bent into a semicircle is:
\[ M' = m \cdot \text{Effective Length} = m \cdot \frac{2L}{\pi} \]

### Step 4: Find the decrease in magnetic moment
Now, we need to find the decrease in the magnetic moment, which is given by:
\[ \Delta M = M - M' = mL - m \cdot \frac{2L}{\pi} \]
Factoring out \( mL \), we get:
\[ \Delta M = mL \left(1 - \frac{2}{\pi}\right) \]

### Step 5: Conclusion
Thus, the decrease in the magnetic moment when the long thin magnet is bent into a semicircle is:
\[ \Delta M = mL \left(1 - \frac{2}{\pi}\right) \]