Faraday’s Law Application: A 100-turn coil has flux per turn φ(t) = (t^3 − 2t) mWb. Find the magnitude of the induced emf at t = 4 s.

Introduction / Context:Induced electromotive force (emf) in a coil arises from time-varying magnetic flux linkage. This question tests direct application of Faraday’s law to a given flux function, including careful handling of units (mWb to Wb) and differentiation with respect to time.

Given Data / Assumptions:

• Number of turns, N = 100.
• Flux per turn: φ(t) = (t^3 − 2t) mWb.
• Time of interest: t = 4 s.
• Sign convention: magnitude of emf is requested.

Concept / Approach:

Faraday’s law for a coil: e(t) = −N * dφ/dt. Magnitude is |e(t)|. Convert milli-weber to weber when computing volts: 1 mWb = 10^-3 Wb.

Step-by-Step Solution:

Verification / Alternative check:

Dimensional check: Wb/s multiplied by turns gives volts; numerical scaling by 10^-3 aligns with milli-prefix, confirming 4.6 V.

Why Other Options Are Wrong:

• 46 mV and 56 mV: off by a factor of 1000 due to ignoring mWb → Wb conversion.
• 5.6 V and 0.46 V: arithmetic/scale errors.

Common Pitfalls:

• Forgetting to convert mWb to Wb.
• Missing the factor of N turns, or confusing flux per turn with total linkage.

Final Answer:

4.6 V