A circular wire loop of radius r lies in a uniform magnetic field B, with its plane perpendicular to magnetic field. If the loop is deformed to a square shape in the same plane in time t, the emf induced in the loop is

Options

A. `(piBr^(2))/(t)(1-(pi)/(2))`
B. `(piBr^(2))/(t)(1-(pi)/(3))`
C. `(piBr^(2))/(t)(1-(pi)/(4))`
D. `(piBr^(2))/(t)(1-(pi)/(5))`

Explanation

To solve the problem of finding the induced EMF in a circular wire loop that is deformed into a square shape in a uniform magnetic field, we can follow these steps:

### Step 1: Understand the Initial and Final Areas
Initially, the area \( A_i \) of the circular loop is given by:
\[ A_i = \pi r^2 \]
where \( r \) is the radius of the circular loop.

When the loop is deformed into a square shape, we need to find the side length \( a \) of the square. The perimeter of the square must equal the circumference of the circular loop:
\[ 4a = 2\pi r \implies a = \frac{\pi r}{2} \]
Thus, the area \( A_f \) of the square is:
\[ A_f = a^2 = \left(\frac{\pi r}{2}\right)^2 = \frac{\pi^2 r^2}{4} \]

### Step 2: Calculate the Change in Magnetic Flux
The magnetic flux \( \Phi \) through the loop is given by:
\[ \Phi = B \cdot A \]
Since the plane of the loop is perpendicular to the magnetic field, the angle \( \theta \) between the magnetic field \( B \) and the area vector \( A \) is \( 0^\circ \), and thus:
\[ \Phi_i = B \cdot A_i = B \cdot \pi r^2 \]
\[ \Phi_f = B \cdot A_f = B \cdot \frac{\pi^2 r^2}{4} \]

### Step 3: Calculate the Change in Flux
The change in magnetic flux \( \Delta \Phi \) as the loop is deformed from circular to square is:
\[ \Delta \Phi = \Phi_f - \Phi_i = B \cdot \frac{\pi^2 r^2}{4} - B \cdot \pi r^2 \]
Factoring out \( B \):
\[ \Delta \Phi = B \left(\frac{\pi^2 r^2}{4} - \pi r^2\right) = B \cdot \left(\frac{\pi^2 r^2 - 4\pi r^2}{4}\right) = B \cdot \frac{(\pi^2 - 4\pi) r^2}{4} \]

### Step 4: Calculate the Induced EMF
The induced EMF \( \mathcal{E} \) is given by Faraday's law of electromagnetic induction:
\[ \mathcal{E} = -\frac{\Delta \Phi}{\Delta t} \]
Since we are given the time \( t \) for the deformation:
\[ \mathcal{E} = -\frac{B \cdot \frac{(\pi^2 - 4\pi) r^2}{4}}{t} \]
Thus, the magnitude of the induced EMF is:
\[ \mathcal{E} = \frac{B (\pi^2 - 4\pi) r^2}{4t} \]

### Final Result
The induced EMF when the circular loop is deformed into a square shape is:
\[ \mathcal{E} = \frac{B r^2 (\pi^2 - 4\pi)}{4t} \]