Question

The magnetic flux \(\phi\) (in weber) linked with a closed circuit of resistance \(8 \, \Omega\) varies with time (in seconds) as \(\phi = 5t^2 - 36t + 1\). The induced current in the circuit at \(t = 2 \, \text{s}\) is ______ A.

Answer 1

Correct Answer: 2

To determine the induced current in the circuit at \( t = 2 \, \text{s} \), the electromotive force (emf) is first calculated using Faraday's Law. The emf (\( \varepsilon \)) is the negative rate of change of magnetic flux (\( \phi \)) over time, expressed as \( \varepsilon = -\frac{d\phi}{dt} \). Given \( \phi = 5t^2 - 36t + 1 \), its derivative with respect to time is \( \frac{d\phi}{dt} = \frac{d}{dt}(5t^2 - 36t + 1) = 10t - 36 \).

Substituting \( t = 2 \) into the derivative yields:
\(\left.\frac{d\phi}{dt}\right|_{t=2} = 10(2) - 36 = 20 - 36 = -16 \, \text{Wb/s}.\)

The induced emf is thus calculated as \(\varepsilon = -\left(-16\right) = 16 \, \text{V}.\)

Applying Ohm's Law, \( I = \frac{\varepsilon}{R} \), with a resistance \( R = 8 \, \Omega \):
\( I = \frac{16}{8} = 2 \, \text{A}.\)

The calculated current of \( 2 \, \text{A} \) falls within the expected range of 2 to 2 A. Consequently, the induced current at \( t = 2 \, \text{s} \) is \( 2 \, \text{A}. \)