Question

The magnetic flux (in weber) linked with a coil of resistance 10 Ω is varying with respect to time \( t \) as \( \phi = 4t^{2} + 2t + 1 \). Then the current in the coil at time \( t = 1 \) second is

• A. 0.5 A
• B. 2 A
• C. 1.5 A
• D. 1 A
• E. 2.5 A

Hint:

Remember that emf is the \textbf{rate of change} of flux. If flux is a function of time, you must differentiate it before plugging in the specific time value.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
According to Faraday's Law of Electromagnetic Induction, a change in magnetic flux linked with a coil induces an electromotive force (emf). The induced current can then be calculated using Ohm's Law (\( I = V/R \)).
Step 2: Key Formula or Approach:
1. Induced emf: \( \varepsilon = -\frac{d\phi}{dt} \) (we use the magnitude \( |\varepsilon| \) for current calculation).
2. Ohm's Law: \( I = \frac{|\varepsilon|}{R} \).
Step 3: Detailed Explanation:
1. Given flux equation: \( \phi = 4t^2 + 2t + 1 \).
2. Differentiate \( \phi \) with respect to \( t \) to find induced emf: \[ |\varepsilon| = \frac{d}{dt}(4t^2 + 2t + 1) \] \[ |\varepsilon| = 8t + 2 \] 3. Find the emf at \( t = 1 \) second: \[ |\varepsilon|_{t=1} = 8(1) + 2 = 10 \text{ V} \] 4. Calculate the current using resistance \( R = 10 \ \Omega \): \[ I = \frac{10 \text{ V}}{10 \ \Omega} = 1 \text{ A} \]
Step 4: Final Answer:
The current in the coil at \( t = 1 \) s is 1 A.