A bar magnet of moment `'M'` is bent into a shape `'5'`. If the length of the each part is same, its new magnetic moment will be

Options

A. `(M)/(sqrt3)`
B. `(M)/(sqrt5)`
C. `(M)/(sqrt2)`
D. `(2)/(3)M`

Explanation

To solve the problem of finding the new magnetic moment of a bar magnet bent into the shape of '5', we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Initial Magnetic Moment**:
- The initial magnetic moment of the bar magnet is given as \( M \).
- The effective length of the magnet before bending is \( L \).

2. **Magnetic Moment Definition**:
- The magnetic moment \( M \) is defined as the product of the pole strength \( p \) and the length \( L \):
\[ M = p \cdot L \]
- Therefore, the pole strength can be expressed as:
\[ p = \frac{M}{L} \]

3. **Shape of the Bent Magnet**:
- The bar magnet is bent into the shape of '5', which consists of five equal segments.
- Let the length of each segment be \( \frac{L}{5} \).

4. **Finding the New Magnetic Length**:
- The new effective magnetic length \( L' \) needs to be calculated.
- The shape of '5' can be visualized as having two vertical segments and three horizontal segments.
- By applying the Pythagorean theorem, we can find the new effective length \( L' \).

5. **Calculating the Lengths**:
- The vertical segments contribute \( \frac{L}{5} \) each, and the horizontal segments also contribute \( \frac{L}{5} \) each.
- The total length of the vertical segments is:
\[ \text{Vertical Length} = \frac{L}{5} + \frac{L}{5} = \frac{2L}{5} \]
- The total horizontal length is:
\[ \text{Horizontal Length} = \frac{L}{5} + \frac{L}{5} + \frac{L}{5} = \frac{3L}{5} \]

6. **Using Pythagorean Theorem**:
- The new effective length \( L' \) can be calculated as:
\[ L' = \sqrt{\left(\frac{2L}{5}\right)^2 + \left(\frac{3L}{5}\right)^2} \]
- Simplifying this gives:
\[ L' = \sqrt{\frac{4L^2}{25} + \frac{9L^2}{25}} = \sqrt{\frac{13L^2}{25}} = \frac{L\sqrt{13}}{5} \]

7. **Calculating the New Magnetic Moment**:
- The new magnetic moment \( M' \) is given by:
\[ M' = p \cdot L' = \frac{M}{L} \cdot L' = \frac{M}{L} \cdot \frac{L\sqrt{13}}{5} \]
- This simplifies to:
\[ M' = \frac{M\sqrt{13}}{5} \]

8. **Final Result**:
- The new magnetic moment of the bent bar magnet is:
\[ M' = \frac{M\sqrt{13}}{5} \]