Doubling the time a loop of wire takes to rotate in an electrical generator produces what effect on the induced EMF, assuming all other factors remain constant?

Question

BlackTom AI · Accepted Answer

Faraday’s law of electromagnetic induction: emf = -N dΦ/dt, so the induced emf is proportional to the rate of change of magnetic flux through the loop.

1) Relevant knowledge points and concepts
- Magnetic flux Φ through a loop in a uniform field is Φ = B A cos(θ) (B = field magnitude, A = loop area, θ = angle between field and loop normal).
- If the loop rotates at angular speed ω, θ = ωt and Φ(t) = B A cos(ωt).
- Faraday’s law gives emf(t) = -N dΦ/dt = N B A ω sin(ωt) (amplitude Emax = N B A ω). Thus the emf amplitude is directly proportional to ω.
- Angular frequency ω is related to the rotation period T by ω = 2π/T.

2) Step-by-step reasoning
- Let the initial period be T and initial angular frequency ω = 2π/T.
- The initial emf amplitude is Emax = N B A ω = N B A (2π/T).
- If the time to rotate (the period) is doubled to T' = 2T, the new angular frequency is ω' = 2π/T' = 2π/(2T) = ω/2.
- The new emf amplitude is Emax' = N B A ω' = N B A (ω/2) = (1/2) N B A ω = Emax/2.

3) Intermediate calculations shown
- ω = 2π/T
- ω' = 2π/(2T) = ω/2
- Emax' = N B A ω' = N B A (ω/2) = Emax/2

4) Why the correct answer is correct
- Doubling the rotation period halves the angular speed, cutting the rate of change of flux in half; since emf ∝ dΦ/dt ∝ ω, the induced emf is halved.

5) Why other options are incorrect
- (a) No change: incorrect because ω changes when the period changes, so dΦ/dt changes.
- (c) 2 times as much: opposite effect; doubling period reduces ω, it does not increase emf.
- (d) 4 times as much: no basis—doubling time cannot increase emf by a factor of four.

6) Conclusion
- Therefore the induced emf becomes half as much; answer b. The final answer matches: "b. The induced emf is half as much."

Doubling the time a loop of wire takes to rotate in an electrical generator produces what effect on the induced EMF, assuming all other factors remain constant?单项选择题

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