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

The induced emf \( \mathcal{E} \) in a coil is given by Faraday's law of induction: \[ \mathcal{E} = - \frac{d\phi}{dt} \] where: - \( \phi \) is the magnetic flux, - \( \mathcal{E} \) is the induced emf. Given: \[ \phi = 5t^2 + 4t - 2 \] Now, differentiate \( \phi \) with respect to time \( t \) to find the induced emf: \[ \frac{d\phi}{dt} = \frac{d}{dt} \left( 5t^2 + 4t - 2 \right) = 10t + 4 \] Thus, the induced emf is: \[ \mathcal{E} = -(10t + 4) \] At \( t = 1 \) s: \[ \mathcal{E} = -(10(1) + 4) = -(10 + 4) = -14 \ \text{V} \] The induced current \( I \) is given by Ohm's law: \[ I = \frac{\mathcal{E}}{R} \] where: - \( R = 14 \ \Omega \) is the resistance. Substitute the values: \[ I = \frac{-14}{14} = -1 \ \text{A} \] The magnitude of the induced current is: \[ \boxed{1.0 \ \text{A}} \]