A square loop of side 44 cm is changed to a circle in time 0.5 sec with its plane normal to a magnetic field 0.6 T. The emf induced is

Options

A. 0.5 V
B. 0.4 V
C. 0.2 V
D. 0.6 V

Explanation

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

### Step 1: Convert the side length of the square loop to meters
The side of the square loop is given as 44 cm. We need to convert this into meters.

\[ \text{Side length} = 44 \, \text{cm} = 44 \times 10^{-2} \, \text{m} = 0.44 \, \text{m} \]

**Hint:** Remember to convert all measurements to the same unit system, preferably SI units.

### Step 2: Calculate the area of the square loop
The area \( A_s \) of the square loop can be calculated using the formula for the area of a square:

\[ A_s = \text{side}^2 = (0.44)^2 = 0.1936 \, \text{m}^2 \]

**Hint:** The area of a square is calculated as the length of one side squared.

### Step 3: Determine the circumference of the square loop
When the square loop is transformed into a circle, the perimeter of the square becomes the circumference of the circle. The perimeter \( P \) of the square is:

\[ P = 4 \times \text{side} = 4 \times 0.44 = 1.76 \, \text{m} \]

**Hint:** The perimeter of a square is four times the length of one side.

### Step 4: Calculate the radius of the circular loop
The circumference \( C \) of the circle is given by:

\[ C = 2\pi r \]

Setting the circumference equal to the perimeter of the square, we can solve for the radius \( r \):

\[ 1.76 = 2\pi r \implies r = \frac{1.76}{2\pi} \approx 0.28 \, \text{m} \]

**Hint:** To find the radius from the circumference, divide the circumference by \( 2\pi \).

### Step 5: Calculate the area of the circular loop
The area \( A_c \) of the circular loop can be calculated using the formula for the area of a circle:

\[ A_c = \pi r^2 = \pi (0.28)^2 \approx 0.2464 \, \text{m}^2 \]

**Hint:** The area of a circle is calculated as \( \pi \) times the radius squared.

### Step 6: 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 (0.6 T) and \( A \) is the area of the loop. The change in magnetic flux \( \Delta \Phi \) when the loop changes from square to circle is:

\[ \Delta \Phi = B \cdot (A_c - A_s) = 0.6 \cdot (0.2464 - 0.1936) \approx 0.6 \cdot 0.0528 = 0.03168 \, \text{Wb} \]

**Hint:** The change in flux is the difference between the final and initial areas multiplied by the magnetic field.

### Step 7: Calculate the induced emf
The induced emf \( E \) can be calculated using Faraday's law of electromagnetic induction:

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

Given that the time \( \Delta t = 0.5 \, \text{s} \):

\[ E = -\frac{0.03168}{0.5} \approx -0.06336 \, \text{V} \]

Since emf is a magnitude, we take the absolute value:

\[ E \approx 0.06336 \, \text{V} \]

**Hint:** The negative sign indicates the direction of the induced emf according to Lenz's law, but we are interested in the magnitude.

### Final Result
The induced emf when the square loop is changed to a circle is approximately **0.06336 V** or **63.36 mV**.

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