Question

A circular loop of radius 7 cm is placed in uniform magnetic field of 0.2 T directed perpendicular to plane of loop. The loop is converted into a square loop in 0.5 s. The EMF induced in the loop is ___ mV.

• A. 13.2
• B. 8.25
• C. 6.6
• D. 1.32

Hint:

For a given perimeter, the circle has the maximum area. Changing shape to square reduces flux.

Answer 1

The Correct Option is D

To find the induced EMF when the circular loop is converted into a square loop, we will use Faraday's law of electromagnetic induction. This law states that the EMF induced is equal to the rate of change of magnetic flux through the loop. Let's solve this step-by-step:

1. The initial shape of the loop is circular with a radius \(r = 7\, \text{cm} = 0.07\, \text{m}\).
2. Calculate the area of the circular loop:
   • Area of the circle, \(A_1 = \pi r^2\) 
   • \(A_1 = \pi \times (0.07)^2 = 0.0154\, \text{m}^2\)
3. The loop is converted into a square with the same perimeter as the circle. First, calculate the perimeter of the circle:
   • Perimeter of circle, \(P = 2\pi r\)
   • \(P = 2 \times \pi \times 0.07 = 0.44\, \text{m}\)
4. Since the perimeter of the square is the same, each side of the square will be:
   • Side of square, \(s = \frac{P}{4} = \frac{0.44}{4} = 0.11\, \text{m}\)
5. Calculate the area of the square:
   • Area of the square, \(A_2 = s^2\)
   • \(A_2 = (0.11)^2 = 0.0121\, \text{m}^2\)
6. The change in area \(\Delta A = A_2 - A_1 = 0.0121 - 0.0154 = -0.0033\, \text{m}^2\).
7. Use Faraday's law to find the induced EMF:
   • The change in magnetic flux \(\Delta \Phi = B \times \Delta A\) where \(B = 0.2\, \text{T}\).
   • \(\Delta \Phi = 0.2 \times (-0.0033) = -0.00066\, \text{Wb}\)
8. The EMF induced, \(\varepsilon = -\frac{\Delta \Phi}{\Delta t}\) where \(\Delta t = 0.5\, \text{s}\):
   • \(\varepsilon = -\frac{-0.00066}{0.5} = 0.00132\, \text{V}\)
   • Convert volts to millivolts: \(0.00132\, \text{V} = 1.32\, \text{mV}\)

Therefore, the EMF induced in the loop is \(1.32\, \text{mV}\). The correct answer is 1.32.