A magnet is suspended in a uniform magnetic field by a thin wire. On twisting the wire through half revolution, the magnet twists through `30^(@)` from the original position. How much should we rotate the wire in order to twist the magnet through `45^(@)` from its original position

Options

A. `257^(@)`
B. `252^(@)`
C. `275^(@)`
D. `127^(@)`

Explanation

To solve the problem, we will use the relationship between the angle of twist of the wire and the angle of twist of the magnet. We know from the problem statement that twisting the wire through half a revolution (180 degrees) causes the magnet to twist through 30 degrees. We need to find out how much we should rotate the wire to make the magnet twist through 45 degrees.

### Step-by-Step Solution:

1. **Understanding the relationship**:
- When the wire is twisted through 180 degrees, the magnet twists through 30 degrees.
- This gives us a ratio of the angle of twist of the wire to the angle of twist of the magnet.

2. **Setting up the ratio**:
- Let the angle of twist of the wire be \( \theta_w \) and the angle of twist of the magnet be \( \theta_m \).
- From the given information, we have:
\[ \frac{\theta_m}{\theta_w} = \frac{30^\circ}{180^\circ} = \frac{1}{6} \]

3. **Finding the required twist for 45 degrees**:
- We want the magnet to twist through \( \theta_m = 45^\circ \).
- Using the ratio we established:
\[ \frac{45^\circ}{\theta_w} = \frac{1}{6} \]
- Rearranging gives:
\[ \theta_w = 45^\circ \times 6 = 270^\circ \]

4. **Conclusion**:
- Therefore, to twist the magnet through 45 degrees from its original position, the wire should be twisted through \( 270^\circ \).

### Final Answer:
The wire should be twisted through **270 degrees**.