Question

The magnetic flux linked with a coil satisfies the relation $ \phi = (4t^2 + 6t + 9) \, \text{Wb} $, where $ t $ is time in seconds. The emf induced in the coil at $ t = 2 $ seconds is

• A. 22 V
• B. 18 V
• C. 16 V
• D. 40 V

Hint:

To calculate the induced emf, differentiate the magnetic flux \( \phi \) with respect to time \( t \) and evaluate it at the given time.

Answer 1

The Correct Option is A

The emf induced in a coil is given by Faraday's law of electromagnetic induction: \[ \mathcal{E} = -\frac{d\phi}{dt} \] where: 
- \( \mathcal{E} \) is the induced emf, 
- \( \phi \) is the magnetic flux. The magnetic flux is given by: \[ \phi = (4t^2 + 6t + 9) \, \text{Wb} \] To find the induced emf, we differentiate \( \phi \) with respect to \( t \): \[ \frac{d\phi}{dt} = \frac{d}{dt} \left( 4t^2 + 6t + 9 \right) \] Differentiating term by term: \[ \frac{d\phi}{dt} = 8t + 6 \] Now, substituting \( t = 2 \) seconds into the equation: \[ \frac{d\phi}{dt} = 8(2) + 6 = 16 + 6 = 22 \, \text{V} \] 
Therefore, the induced emf at \( t = 2 \) seconds is: \[ \mathcal{E} = 22 \, \text{V} \] 
Thus, the correct answer is: \( \text{(A) 22 V} \)