A coil of wire containing N turns is in an external magnetic field that is perpendicular to the plane of the coil, and itis steadily changing. Under these circumstances, an emf ε is induced in the coil. If the rate of change of the magnetic field and the number of turns in the coil are now doubled (but nothing else changes), what will be the induced emf in the coil?

Question

A coil of wire containing N turns is in an external magnetic field that is perpendicular to the plane of the coil, and itis  steadily changing. Under these circumstances, an emf ε is induced in the coil. If the rate of change of the magnetic field and the number of turns in the coil are now doubled (but nothing else changes), what will be the induced emf in the coil?

BlackTom AI · Accepted Answer

To analyze the induced emf, recall Faraday’s law in its simplest form for a single loop: ε = -N (dΦ/dt). When the magnetic field is perpendicular to the coil, the magnetic flux Φ = B A, so dΦ/dt = A (dB/dt). Therefore, ε = -N A (dB/dt).

Now apply the changes described: the rate of change of the magnetic field dB/dt is doubled, and the number of turns N is also doubled. The new emf ε' becomes ε' = -(2N) A (2 dB/dt) = 4 [ -N A (dB/dt) ] = 4 ε.

Option-by-option analysis:
- It remains unchanged.: If dB/dt and N both doubled, the emf cannot stay the same because ε is proportional to both N and dB/dt. This option ignores the amplification from both factors.
- ε/4: This would imply the emf decreases to a quarter, which contradicts the direct scaling with N and dB/dt; both are increased, so this is incorrect.
- ε/2: This suggests a twofold decrease, but the combined doubling of two independent factors (N and dB/dt) should increase emf, not reduce it. This phrasing miscounts the multiplicative effect.
- 4ε: This matches the derived relation ε' = 4 ε, since the emf scales linearly with N and with dB/dt, and both are doubled, giving a quadrupling.
- 2ε: This would be the result if only one of the two factors was doubled, or if there were some cancellation; however, both factors contribute, so this answer underestimates the effect.

In summary, the correct reasoning shows that the emf scales with the product of N and dB/dt, so doubling both yields a fourfold increase in the induced emf.

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