A bar magnet of magnetic moment `M_(1)` is axially cut into two equal parts. If these two pieces are arranged perpendiucular to each other, the resultant magnetic moment is `M_(2)`.
Then the vale of `(M_(1))/(M_(2))` is

Options

A. `1/(2sqrt2)`
B. 1
C. `1/sqrt2`
D. `sqrt2`

Answer

To solve the problem, we need to find the ratio \( \frac{M_1}{M_2} \), where \( M_1 \) is the magnetic moment of the original bar magnet and \( M_2 \) is the resultant magnetic moment when the two equal parts of the magnet are arranged perpendicular to each other.

### Step-by-Step Solution:

1. **Understanding the Original Magnet:**
- Let the magnetic moment of the original bar magnet be \( M_1 \).
- The magnetic moment is given by the product of the pole strength \( m \) and the length \( L \) of the magnet:
\[ M_1 = m \cdot L \]

2. **Cutting the Magnet:**
- When the bar magnet is cut into two equal parts, each part will have half the pole strength:
\[ m' = \frac{m}{2} \]
- The length of each part remains the same, which is \( L \).

3. **Magnetic Moment of Each Half Magnet:**
- The magnetic moment of each half magnet \( M' \) is:
\[ M' = m' \cdot L = \frac{m}{2} \cdot L = \frac{M_1}{2} \]

4. **Arranging the Magnets Perpendicular to Each Other:**
- When the two half magnets are arranged perpendicular to each other, we can calculate the resultant magnetic moment \( M_2 \) using the Pythagorean theorem:
\[ M_2 = \sqrt{(M')^2 + (M')^2} \]
- Substituting \( M' = \frac{M_1}{2} \):
\[ M_2 = \sqrt{\left(\frac{M_1}{2}\right)^2 + \left(\frac{M_1}{2}\right)^2} = \sqrt{2 \left(\frac{M_1}{2}\right)^2} = \sqrt{2} \cdot \frac{M_1}{2} = \frac{M_1}{\sqrt{2}} \]

5. **Finding the Ratio \( \frac{M_1}{M_2} \):**
- Now we can find the ratio:
\[ \frac{M_1}{M_2} = \frac{M_1}{\frac{M_1}{\sqrt{2}}} = \sqrt{2} \]

### Final Answer:
The value of \( \frac{M_1}{M_2} \) is \( \sqrt{2} \).