A circular coil of radius r carries I. If the same coil is now changed into a square loop carrying the same current, magnitude of its magnetic moment will

Options

A. increase
B. decrease
C. remain constant
D. changes depending upon the value of r

Answer

To solve the problem of finding the change in the magnetic moment when a circular coil is transformed into a square loop while maintaining the same current, we can follow these steps:

### Step 1: Understand the Magnetic Moment Formula
The magnetic moment \( m \) of a coil is given by the formula:
\[ m = n \cdot I \cdot A \]
where:
- \( n \) = number of turns (assumed to be 1 in this case),
- \( I \) = current flowing through the coil,
- \( A \) = area of the loop.

### Step 2: Calculate the Magnetic Moment of the Circular Coil
For a circular coil of radius \( r \), the area \( A \) is:
\[ A = \pi r^2 \]
Thus, the magnetic moment \( m \) for the circular coil becomes:
\[ m = 1 \cdot I \cdot \pi r^2 = I \pi r^2 \]

### Step 3: Convert the Circular Coil to a Square Loop
When the circular coil is transformed into a square loop, the length of wire remains the same. The circumference of the circular coil is:
\[ C = 2\pi r \]
For a square loop, if the side length is \( a \), the perimeter is:
\[ P = 4a \]
Setting the two equal gives:
\[ 2\pi r = 4a \implies a = \frac{\pi r}{2} \]

### Step 4: Calculate the Area of the Square Loop
The area \( A \) of the square loop is:
\[ A = a^2 = \left(\frac{\pi r}{2}\right)^2 = \frac{\pi^2 r^2}{4} \]

### Step 5: Calculate the Magnetic Moment of the Square Loop
Using the area calculated, the magnetic moment \( m' \) for the square loop is:
\[ m' = 1 \cdot I \cdot \frac{\pi^2 r^2}{4} = I \frac{\pi^2 r^2}{4} \]

### Step 6: Compare the Magnetic Moments
Now we compare \( m' \) with \( m \):
\[ \frac{m'}{m} = \frac{I \frac{\pi^2 r^2}{4}}{I \pi r^2} = \frac{\frac{\pi^2}{4}}{\pi} = \frac{\pi}{4} \]
This shows that:
\[ m' = m \cdot \frac{\pi}{4} \]
Since \( \frac{\pi}{4} < 1 \), it indicates that the magnetic moment of the square loop is less than that of the circular coil.

### Conclusion
The magnitude of the magnetic moment decreases when the circular coil is changed into a square loop while maintaining the same current.