Question

A circular coil of area \( 2\sqrt{2} \, \text{cm}^2 \) and resistance \( 2\Omega \) is arranged vertically in the east-west direction. A uniform magnetic field \( 0.2 \, \text{T} \) is set up across the plane in the north to east direction. Now the magnetic field is removed at a steady rate in \( 0.4 \, \text{s} \). What is the current developed in the coil?

• A. \( 0.5 \times 10^{-3} \, A \)
• B. \( 0.5 \times 10^{-4} \, A \)
• C. \( 1 \times 10^{-4} \, A \)
• D. \( 0.5 \times 10^{-5} \, A \)

Hint:

Always convert area into SI units and carefully determine angle between field and area vector in flux calculations.

Answer 1

The Correct Option is B

Step 1: Use Faraday’s Law of Electromagnetic Induction.
Induced emf is given by:
\[ \mathcal{E} = \frac{d\Phi}{dt} \]

Step 2: Magnetic flux expression.
\[ \Phi = BA\cos\theta \]
Here \( \theta = 45^\circ \) since directions are perpendicular diagonally.

Step 3: Convert area to SI units.
\[ A = 2\sqrt{2} \, \text{cm}^2 = 2\sqrt{2} \times 10^{-4} \, \text{m}^2 \]

Step 4: Calculate change in flux.
Since field becomes zero:
\[ \Delta \Phi = B A \cos 45^\circ \]
\[ = 0.2 \times 2\sqrt{2}\times 10^{-4} \times \frac{1}{\sqrt{2}} = 0.4 \times 10^{-4} \]

Step 5: Calculate induced emf.
\[ \mathcal{E} = \frac{\Delta \Phi}{\Delta t} = \frac{0.4 \times 10^{-4}}{0.4} = 1 \times 10^{-4} \, V \]

Step 6: Use Ohm’s law to find current.
\[ I = \frac{\mathcal{E}}{R} = \frac{1 \times 10^{-4}}{2} = 0.5 \times 10^{-4} \, A \]

Step 7: Final Answer.
\[ \boxed{0.5 \times 10^{-4} \, A} \]