A square loop of side `22 cm` is changed to a circle in time `0.4 sec` with its plane normal to a magnetic field `0.2 T`. The emf induced is

Options

A. `+6.6 m v`
B. `-6.6 m v`
C. `+13.2 m v`
D. `-13.2 m v`

Explanation

To solve the problem, we need to calculate the induced electromotive force (emf) when a square loop is transformed into a circle in a magnetic field. Here’s a step-by-step solution:

### Step 1: Calculate the area of the square loop
The side length of the square loop is given as \(22 \, \text{cm}\).

\[ \text{Area of the square} (A_s) = \text{side}^2 = (22 \, \text{cm})^2 = 484 \, \text{cm}^2 \]

### Step 2: Convert the area to square meters
Since we need the area in square meters for our calculations:

\[ A_s = 484 \, \text{cm}^2 \times \left(\frac{1 \, \text{m}^2}{10000 \, \text{cm}^2}\right) = 0.0484 \, \text{m}^2 \]

### Step 3: Calculate the radius of the circle
The perimeter of the square loop is equal to the circumference of the circle formed.

\[ \text{Perimeter of the square} = 4 \times \text{side} = 4 \times 22 \, \text{cm} = 88 \, \text{cm} \]

The circumference of the circle is given by:

\[ \text{Circumference of the circle} = 2 \pi r \]

Setting the two equal gives:

\[ 88 \, \text{cm} = 2 \pi r \]

Solving for \(r\):

\[ r = \frac{88 \, \text{cm}}{2 \pi} = \frac{88}{6.28} \approx 14 \, \text{cm} \]

### Step 4: Calculate the area of the circle
Now we calculate the area of the circle:

\[ \text{Area of the circle} (A_c) = \pi r^2 = \pi (14 \, \text{cm})^2 = \pi \times 196 \, \text{cm}^2 \approx 615.75 \, \text{cm}^2 \]

Converting this to square meters:

\[ A_c = 615.75 \, \text{cm}^2 \times \left(\frac{1 \, \text{m}^2}{10000 \, \text{cm}^2}\right) \approx 0.061575 \, \text{m}^2 \]

### Step 5: Calculate the change in magnetic flux
The magnetic flux (\(\Phi\)) through the loop is given by:

\[ \Phi = B \cdot A \]

Where \(B\) is the magnetic field strength (0.2 T). The change in flux (\(\Delta \Phi\)) as the loop changes from a square to a circle is:

\[ \Delta \Phi = B \cdot (A_c - A_s) = 0.2 \, \text{T} \cdot (0.061575 \, \text{m}^2 - 0.0484 \, \text{m}^2) \]

Calculating the difference in area:

\[ \Delta A = A_c - A_s = 0.061575 \, \text{m}^2 - 0.0484 \, \text{m}^2 \approx 0.013175 \, \text{m}^2 \]

Then, substituting back into the flux equation:

\[ \Delta \Phi = 0.2 \, \text{T} \cdot 0.013175 \, \text{m}^2 \approx 0.002635 \, \text{Wb} \]

### Step 6: Calculate the induced emf
The induced emf (\( \mathcal{E} \)) is given by the rate of change of magnetic flux:

\[ \mathcal{E} = -\frac{\Delta \Phi}{\Delta t} \]

Where \(\Delta t = 0.4 \, \text{s}\):

\[ \mathcal{E} = -\frac{0.002635 \, \text{Wb}}{0.4 \, \text{s}} \approx -0.0065875 \, \text{V} \approx -6.59 \, \text{mV} \]

### Final Answer
The induced emf is approximately \(-6.59 \, \text{mV}\).

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