A wire carrying a current i is first bent in the form of a square of side a and placed at right angle to a uniform magnetic field of induction B. The work done in changing its shape into a circle is

Options

A. A) `ia^2 B ( pi +2)`
B. B) `ia^2 B ( pi -2)`
C. C) `ia^2 B ((4)/(pi )-1)`
D. D) `ia^2 B (1- (4)/(pi))`

Answer

To solve the problem of calculating the work done in changing the shape of a wire from a square to a circle while carrying a current \( i \) in a magnetic field \( B \), we can follow these steps:

### Step 1: Determine the Area of the Square
The wire is initially bent into a square shape with each side of length \( a \). The area \( A_s \) of the square is given by:
\[ A_s = a^2 \]

### Step 2: Calculate the Magnetic Moment of the Square
The magnetic moment \( \mu_s \) of the square loop can be calculated using the formula:
\[ \mu_s = i \times A_s = i \times a^2 \]

### Step 3: Determine the Circumference of the Square
The total length of the wire, which is also the perimeter of the square, is:
\[ P_s = 4a \]

### Step 4: Find the Radius of the Circle
When the wire is reshaped into a circle, the circumference of the circle must equal the length of the wire:
\[ C = 2\pi r = 4a \]
From this, we can solve for the radius \( r \):
\[ r = \frac{4a}{2\pi} = \frac{2a}{\pi} \]

### Step 5: Calculate the Area of the Circle
The area \( A_c \) of the circular loop is given by:
\[ A_c = \pi r^2 = \pi \left(\frac{2a}{\pi}\right)^2 = \pi \times \frac{4a^2}{\pi^2} = \frac{4a^2}{\pi} \]

### Step 6: Calculate the Magnetic Moment of the Circle
The magnetic moment \( \mu_c \) of the circular loop is:
\[ \mu_c = i \times A_c = i \times \frac{4a^2}{\pi} \]

### Step 7: Calculate the Change in Magnetic Moment
The change in magnetic moment \( \Delta \mu \) when changing from the square to the circle is:
\[ \Delta \mu = \mu_c - \mu_s = \left(i \times \frac{4a^2}{\pi}\right) - \left(i \times a^2\right) \]
Factoring out \( i \):
\[ \Delta \mu = i \left(\frac{4a^2}{\pi} - a^2\right) = i a^2 \left(\frac{4}{\pi} - 1\right) \]

### Step 8: Calculate the Work Done
The work done \( W \) in changing the shape of the wire is given by the change in energy associated with the change in magnetic moment:
\[ W = \Delta \mu \times B = i a^2 \left(\frac{4}{\pi} - 1\right) B \]

### Final Result
Thus, the work done in changing the shape of the wire from a square to a circle is:
\[ W = i a^2 B \left(\frac{4}{\pi} - 1\right) \]