Question

What are Faraday’s laws of electromagnetic induction? A wire is placed in a magnetic field of $100 \, T$, with its perpendicular plane in the form of a circle of radius $10 \, cm$. If the wire is pulled in the same plane in $0.1 \, s$, so as to give it the form of a square, then find the average induced e.m.f. produced in the loop.

Hint:

Always compare areas when shape changes in magnetic field — emf depends on change in flux, not initial shape.

Answer 1

Step 1: Faraday’s laws.
1. Whenever magnetic flux linked with a circuit changes, an emf is induced in it. 2. The magnitude of induced emf is equal to the rate of change of flux: \[ e = - \frac{d\Phi}{dt}. \]
Step 2: Initial flux (circular loop).
Area of circle: \[ A_1 = \pi r^2 = \pi (0.1)^2 = 0.0314 \, m^2. \] Flux: \[ \Phi_1 = B A_1 = 100 \times 0.0314 = 3.14 \, Wb. \]
Step 3: Final flux (square loop).
Perimeter of circle = perimeter of square. \[ 2 \pi r = 4a \quad \Rightarrow \quad a = \frac{\pi r}{2}. \] \[ a = \frac{3.14 \times 0.1}{2} = 0.157 \, m. \] Area of square: \[ A_2 = a^2 = (0.157)^2 \approx 0.0247 \, m^2. \] Flux: \[ \Phi_2 = B A_2 = 100 \times 0.0247 = 2.47 \, Wb. \]
Step 4: Change in flux.
\[ \Delta \Phi = \Phi_1 - \Phi_2 = 3.14 - 2.47 = 0.67 \, Wb. \]
Step 5: Average emf.
\[ e = \frac{\Delta \Phi}{\Delta t} = \frac{0.67}{0.1} = 6.7 \, V. \] Correction: Wait! Magnetic field is $100 T$, so flux values must be: \[ \Phi_1 = 100 \times 0.0314 = 3.14 \, Wb, \quad \Phi_2 = 100 \times 0.0247 = 2.47 \, Wb. \] \[ \Delta \Phi = 0.67 \, Wb. \] \[ e = \frac{0.67}{0.1} = 6.7 \, V. \]
Step 6: Conclusion.
The average induced emf is $6.7 \, V$.