A rectangular loop of wire with sides 0.255 and 0.410 m lies in a plane perpendicular to a constant magnetic field. The magnetic field has a magnitude of 0.830 T and is directed parallel to the normal of the loop's surface (into the page/screen). In a time of 0.189 s, one-half of the loop is then folded back onto the other half, making the area zero. Determine the magnitude of the average emf induced in the loop.

Question

A rectangular loop of wire with sides 0.255 and 0.410 m lies in a plane perpendicular to a constant magnetic field. The magnetic field has a magnitude of 0.830 T and is directed parallel to the normal of the loop's surface (into the page/screen). In a time of 0.189 s, one-half of the loop is then folded back onto the other half, making the area zero. Determine the magnitude of the average emf induced in the loop.

BlackTom AI · Accepted Answer

Faraday’s law: the induced emf (magnitude) equals the rate of change of magnetic flux, |ε| = |ΔΦ/Δt|.

1) Relevant concepts
- Magnetic flux Φ = B·A·cosθ. Here B = 0.830 T, initial area A = (0.255 m)(0.410 m), and the field is parallel to the loop normal so cosθ = 1.  
- Folding half the loop onto the other half reduces the effective area to zero, so final flux is zero.  
- Average induced emf magnitude = |Φ_final − Φ_initial| / Δt.

2) Step-by-step reasoning and calculations
- Compute initial area: A = 0.255 m × 0.410 m = 0.10455 m^2.
- Initial magnetic flux: Φ_i = B·A = 0.830 T × 0.10455 m^2.
  Calculate: 0.10455 × 0.83 = 0.0867765 Wb (we keep a few extra digits for accuracy).
- Final flux after folding: Φ_f = 0 (area becomes zero).
- Change in flux magnitude: |ΔΦ| = |Φ_f − Φ_i| = |0 − 0.0867765| = 0.0867765 Wb.
- Time interval: Δt = 0.189 s.
- Average induced emf magnitude: |ε| = |ΔΦ| / Δt = 0.0867765 Wb / 0.189 s ≈ 0.4593 V.

3) Rounding and final value
- Rounding to two significant figures consistent with given data gives |ε| ≈ 0.46 V.

4) Why this answer is correct
- The calculation directly applies Faraday’s law using the exact area change to zero; sign (negative in Faraday’s law) only gives direction of induced emf, while the question asks for magnitude.

5) Why other choices are incorrect
- 0.87 V is roughly twice the correct value (would correspond to either twice the flux change or half the time). 1.7 V and 2.11 V are much larger and inconsistent with the computed ΔΦ and Δt. “None of these” is wrong because one option (0.46 V) matches the correct computation.

6) Conclusion
- The magnitude of the average emf induced in the loop is 0.46 V, matching the provided answer.

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