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Difficulty: MediumElectromagnetic Induction

A flat circular coil consisting of 100100 turns and having a cross-sectional area of 4.0×10−3 m24.0 \times 10^{-3}\text{ m}^2 is positioned perpendicular to a uniform magnetic field of 0.50 T0.50\text{ T}. If the direction of the magnetic field is completely reversed in a time interval of 0.02 s0.02\text{ s}, what is the magnitude of the average electromotive force (e.m.f.) induced in the coil?

  1. A
    10 V10\text{ V}
  2. 20 V20\text{ V}Answer
  3. C
    0.20 V0.20\text{ V}
  4. D
    40 V40\text{ V}

Answer

20 V20\text{ V}
According to Faraday's law of electromagnetic induction, the magnitude of induced electromotive force equals the rate of change of magnetic flux linkage. Because the magnetic field direction is completely reversed, the change in field magnitude is ΔB=0.50−(−0.50)=1.0 T\Delta B = 0.50 - (-0.50) = 1.0\text{ T}. Multiplying by the area (4.0×10−3 m24.0 \times 10^{-3}\text{ m}^2) and turn count (100100), then dividing by the duration (0.02 s0.02\text{ s}) gives 20 V20\text{ V}.

Step-by-Step Solution

1
Determine the change in magnetic flux density (ΔB\Delta B)
ΔB=Bf−Bi=−0.50 T−0.50 T=−1.0 T\Delta B = B_f - B_i = -0.50\text{ T} - 0.50\text{ T} = -1.0\text{ T}, so ∣ΔB∣=1.0 T|\Delta B| = 1.0\text{ T}
Reversing the magnetic field direction changes the sign of the flux density relative to the coil area.
2
Calculate the total change in magnetic flux linkage (ΔΦtotal\Delta \Phi_{total})
ΔΦtotal=N⋅A⋅∣ΔB∣=100×(4.0×10−3 m2)×1.0 T=0.40 Wb-turns\Delta \Phi_{total} = N \cdot A \cdot |\Delta B| = 100 \times (4.0 \times 10^{-3}\text{ m}^2) \times 1.0\text{ T} = 0.40\text{ Wb-turns}
The total magnetic flux linkage is proportional to the number of turns, the surface area, and the net field change.
3
Apply Faraday's law of electromagnetic induction to find the induced e.m.f. (E\mathcal{E})
E=ΔΦtotalΔt=0.40 Wb-turns0.02 s=20 V\mathcal{E} = \frac{\Delta \Phi_{total}}{\Delta t} = \frac{0.40\text{ Wb-turns}}{0.02\text{ s}} = 20\text{ V}
The induced e.m.f. magnitude is equal to the rate of change of magnetic flux linkage.

Key Concept

Faraday's Law of Electromagnetic Induction
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Topics

Electromagnetic Induction

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