Question

The magnetic flux through a coil perpendicular to its plane is varying according to the relation \( \phi = 5t^3 + 4t^2 + 2t - 5 \). If the resistance of the coil is \( 5 \, \Omega \), then the induced current through the coil at \( t = 2 \, \text{sec} \) will be:

• A. \( 15.6 \, \text{A} \)
• B. \( 16.6 \, \text{A} \)
• C. \( 17.6 \, \text{A} \)
• D. \( 18.6 \, \text{A} \)

Hint:

Always differentiate the magnetic flux \( \phi \) with respect to time \( t \) to calculate the induced EMF, and use \( i = \frac{e}{R} \) to determine the current.

Answer 1

The Correct Option is A

Faraday's Law defines the induced EMF (\( e \)) as: \[e = \left| \frac{d\phi}{dt} \right|.\]Given the magnetic flux: \[\phi = 5t^3 + 4t^2 + 2t - 5.\]Differentiating \( \phi \) with respect to \( t \) yields: \[e = \left| \frac{d\phi}{dt} \right| = \left| 15t^2 + 8t + 2 \right|.\]Calculating \( e \) at \( t = 2 \, \text{sec} \): \[e = 15(2)^2 + 8(2) + 2 = 15(4) + 16 + 2 = 60 + 16 + 2 = 78 \, \text{V}.\]Ohm's Law is applied to determine the induced current \( i \): \[i = \frac{e}{R}.\]Substituting \( R = 5 \, \Omega \): \[i = \frac{78}{5} = 15.6 \, \text{A}.\] Final Answer:\[\boxed{15.6 \, \text{A}}.\]