Question

The magnetic flux linked with a closed coil (in Wb) varies with time \( t \) (in s) as \( \phi = 5t^2 + 4t - 2 \). If the resistance of the circuit is 14 \( \Omega \), the magnitude of induced current in the coil at \( t = 1 \) s will be:

• A. 0.5 A
• B. 1.0 A
• C. 1.5 A
• D. 2.0 A

Hint:

To calculate induced current, first find the induced emf using Faraday’s law and then use Ohm’s law to calculate the current. The negative sign indicates the direction of the current.

Answer 1

The Correct Option is B

Faraday's law of induction states that the induced electromotive force \( \mathcal{E} \) in a coil is \( \mathcal{E} = - \frac{d\phi}{dt} \), where \( \phi \) is the magnetic flux and \( \mathcal{E} \) is the induced emf.Given the magnetic flux \( \phi = 5t^2 + 4t - 2 \).Differentiating \( \phi \) with respect to time \( t \) yields \( \frac{d\phi}{dt} = \frac{d}{dt} \left( 5t^2 + 4t - 2 \right) = 10t + 4 \).Therefore, the induced emf is \( \mathcal{E} = -(10t + 4) \).At \( t = 1 \) s, the induced emf is \( \mathcal{E} = -(10(1) + 4) = -(10 + 4) = -14 \ \text{V} \).According to Ohm's law, the induced current \( I \) is \( I = \frac{\mathcal{E}}{R} \), where the resistance \( R = 14 \ \Omega \).Substituting the values, the induced current is \( I = \frac{-14}{14} = -1 \ \text{A} \).The magnitude of the induced current is \( \boxed{1.0 \ \text{A}} \).