A bar magnet of moment M is bent to the letter "L" shape a position that divides the length in the ratio `1:2`.Its new magnetic moment becomes

Options

A. `(3M)/(5)`
B. `sqrt(5)/(3)M`
C. `(5)/(6)M`
D. `sqrt(3)/(5)M`

Explanation

To find the new magnetic moment of the bent bar magnet shaped like the letter "L", we can follow these steps:

### Step 1: Understand the Original Magnetic Moment
The original magnetic moment \( M \) of the bar magnet is given. The magnetic moment is defined as:
\[ M = m \cdot L \]
where \( m \) is the magnetic strength and \( L \) is the length of the magnet.

### Step 2: Determine the Lengths of the Segments
Since the bar magnet is bent into an "L" shape and the length is divided in the ratio \( 1:2 \), we can denote the total length of the magnet as \( L \). Thus, we can define:
- Length of the shorter segment \( L_1 = \frac{L}{3} \)
- Length of the longer segment \( L_2 = \frac{2L}{3} \)

### Step 3: Calculate the Magnetic Moments of Each Segment
The magnetic moment of each segment can be calculated as follows:
- For the shorter segment:
\[ M_1 = m \cdot L_1 = m \cdot \frac{L}{3} = \frac{M}{3} \]
- For the longer segment:
\[ M_2 = m \cdot L_2 = m \cdot \frac{2L}{3} = \frac{2M}{3} \]

### Step 4: Combine the Magnetic Moments
Since the two segments are at right angles to each other, we can find the resultant magnetic moment using the Pythagorean theorem:
\[ M_{\text{net}} = \sqrt{M_1^2 + M_2^2} \]
Substituting the values we found:
\[ M_{\text{net}} = \sqrt{\left(\frac{M}{3}\right)^2 + \left(\frac{2M}{3}\right)^2} \]

### Step 5: Simplify the Expression
Calculating the squares:
\[ M_{\text{net}} = \sqrt{\frac{M^2}{9} + \frac{4M^2}{9}} = \sqrt{\frac{5M^2}{9}} = \frac{M \sqrt{5}}{3} \]

### Step 6: Final Result
Thus, the new magnetic moment of the bent bar magnet is:
\[ M_{\text{net}} = \frac{M \sqrt{5}}{3} \]

### Conclusion
The final answer is:
\[ \text{New Magnetic Moment} = \frac{M \sqrt{5}}{3} \]