A flat coil having 40 turns, each one of cross-sectional area 12.0 cm2, is oriented with its plane perpendicular to a uniform magnetic field. The field varies steadily from 0.00 T to 1.20 T in 20.0 ms. What emf is induced in the coil during this time?

Question

BlackTom AI · Accepted Answer

To determine the induced emf in a coil, we use Faraday’s law for a multi-turn coil: |ε| = N A (dB/dt), where N is the number of turns, A is the cross-sectional area of the coil, and dB/dt is the rate of change of the magnetic field.

First, convert the given area to square meters: A = 12.0 cm^2 = 12.0 × 10^-4 m^2 = 1.20 × 10^-3 m^2.

Next, find dB/dt from the field change: B goes from 0.00 T to 1.20 T in Δt = 20.0 ms = 0.020 s, so dB/dt = ΔB/Δt = (1.20 T) / (0.020 s) = 60 T/s.

Now plug in the values: N = 40, A = 1.20 × 10^-3 m^2, dB/dt = 60 T/s.

ε = N A dB/dt = 40 × (1.20 × 10^-3) × 60 = 40 × 0.0012 × 60.
0.0012 × 60 = 0.072, and 0.072 × 40 = 2.88 V.

Option-by-option comments:
- 0.288 V: This would correspond to a tenfold smaller result, perhaps if A or dB/dt were divided by 10. The actual area or rate of change here is ten times larger than this value, so this is too small by a factor of 10.
- 4.88 V: This is noticeably larger than the correct value; it could come from miscomputing either N, A, or dB/dt (e.g., using 0.012 m^2 for A or 70 T/s for dB/dt). The given numbers do not support this precise product.
- 0 V: A nonzero change in magnetic flux is occurring (B changes by 1.20 T), and the coil has a nonzero area with finite turns, so the emf cannot be zero unless the flux change were zero or perfectly canceled, which it is not. Hence this is incorrect.
- 8.02 V: This is roughly triple the correct value, which could stem from multiplying dB/dt by 3 or miscounting turns or area. The precise calculation with the provided data does not yield this value.
- 2.88 V: This matches the calculated emf using ε = N A dB/dt with N = 40, A = 1.20 × 10^-3 m^2, dB/dt = 60 T/s. The units are volt, and the magnitude is consistent with the given numbers.

In summary, the correct approach uses Faraday’s law for a coil, converting units properly, calculating the rate of change of magnetic flux through the coil, and multiplying by the number of turns. The only option that aligns with the computed value is 2.88 V, while the others reflect common arithmetic missteps or misinterpretations of the quantities involved.

A flat coil having 40 turns, each one of cross-sectional area 12.0 cm2, is oriented with its plane perpendicular to a uniform magnetic field. The field varies steadily from 0.00 T to 1.20 T in 20.0 ms. What emf is induced in the coil during this time?单项选择题

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