A bar magnet of magnetic moment `'M'` is bent in the form of an arc which makes angle `60^(0)`. The percentage change in the magnetic moment is

Options

A. `9%`increase
B. `9%`Decrease
C. `4.5%`Decrease
D. `4.5%` Increase

Explanation

To solve the problem of finding the percentage change in the magnetic moment of a bar magnet bent into an arc of 60 degrees, we can follow these steps:

### Step 1: Understand the initial conditions
The initial magnetic moment \( M \) of the bar magnet is given. We denote the pole strength as \( m \) and the effective length of the magnet as \( l \). The magnetic moment can be expressed as:
\[ M = m \cdot l \]

### Step 2: Determine the new effective length after bending
When the bar magnet is bent into an arc that subtends an angle of \( 60^\circ \), we need to find the new effective length of the magnet. The arc length \( L \) corresponding to this angle can be calculated using the formula:
\[ L = \frac{\theta}{360^\circ} \times 2\pi r \]
where \( \theta = 60^\circ \).

### Step 3: Calculate the radius of the arc
From the above formula, we can rearrange to find the radius \( r \):
\[ L = \frac{60}{360} \times 2\pi r = \frac{1}{6} \times 2\pi r = \frac{\pi r}{3} \]
Thus, we have:
\[ r = \frac{3L}{\pi} \]

### Step 4: Find the new magnetic moment
After bending, the new effective length \( l' \) is the chord length corresponding to the arc. In an equilateral triangle formed by the two ends of the arc and the center, the effective length can be calculated as:
\[ l' = 2r \sin\left(\frac{60^\circ}{2}\right) = 2r \sin(30^\circ) = 2r \cdot \frac{1}{2} = r \]
Substituting \( r \):
\[ l' = \frac{3L}{\pi} \]
The new magnetic moment \( M' \) is given by:
\[ M' = m \cdot l' = m \cdot \frac{3L}{\pi} \]

### Step 5: Calculate the percentage change in magnetic moment
Now we can find the percentage change in magnetic moment:
\[ \text{Percentage Change} = \frac{M' - M}{M} \times 100 \]
Substituting \( M' \) and \( M \):
\[ \text{Percentage Change} = \frac{\left(m \cdot \frac{3L}{\pi}\right) - (m \cdot l)}{m \cdot l} \times 100 \]
This simplifies to:
\[ \text{Percentage Change} = \frac{\frac{3L}{\pi} - l}{l} \times 100 \]

### Step 6: Substitute \( L \) with \( l \)
Since \( L = l \) (the original length of the magnet), we can substitute:
\[ \text{Percentage Change} = \frac{\frac{3l}{\pi} - l}{l} \times 100 = \left(\frac{3}{\pi} - 1\right) \times 100 \]

### Step 7: Calculate the numerical value
Using \( \pi \approx 3.14 \):
\[ \text{Percentage Change} = \left(\frac{3}{3.14} - 1\right) \times 100 \approx (0.955 - 1) \times 100 \approx -4.45\% \]

Thus, the percentage change in the magnetic moment is approximately \( -4.46\% \).

### Final Answer
The percentage change in the magnetic moment is approximately \( 4.46\% \) decrease.

---