A "bar" magnet of magnetic moment `M` is bent in `'uu'` shape such that all the parts are of equal lengths. Then new magnetic moment is

Options

A. `M//3`
B. `2M`
C. `sqrt3M`
D. `3sqrt3M`

Explanation

To solve the problem of finding the new magnetic moment of a bar magnet bent into a 'U' shape, we can follow these steps:

### Step 1: Understand the Initial Magnetic Moment
The magnetic moment \( M \) of a bar 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: Analyze the U-shaped Configuration
When the bar magnet is bent into a 'U' shape, the total length remains the same, but the configuration changes. If the original length of the magnet is \( L \), when bent into a 'U' shape, each arm of the 'U' will have a length of \( \frac{L}{3} \).

### Step 3: Determine the New Distance Between the Poles
In the 'U' shape, the distance \( D \) between the north pole and the south pole becomes the length of one arm of the 'U', which is:
\[ D = \frac{L}{3} \]

### Step 4: Calculate the New Magnetic Moment
The new magnetic moment \( M' \) for the bent magnet can be calculated using the same formula for magnetic moment:
\[ M' = m \cdot D \]
Substituting \( D \) with \( \frac{L}{3} \):
\[ M' = m \cdot \frac{L}{3} \]

### Step 5: Relate the New Magnetic Moment to the Original
Since the original magnetic moment \( M \) is \( m \cdot L \), we can express \( m \) as:
\[ m = \frac{M}{L} \]
Now substituting this into the equation for \( M' \):
\[ M' = \left(\frac{M}{L}\right) \cdot \frac{L}{3} \]
This simplifies to:
\[ M' = \frac{M}{3} \]

### Final Answer
Thus, the new magnetic moment \( M' \) after bending the bar magnet into a 'U' shape is:
\[ M' = \frac{M}{3} \]