A long solenoid carries a sinusoidally alternating current. A small, single-turn rectangular coil is placed coaxially inside the solenoid. Explain why an alternating EMF is induced in the rectangular coil, and explain why the peak EMF induced in the coil would increase if the frequency of the alternating current in the solenoid were doubled while its peak current were kept constant.
Written & reviewed by James Millett — Biology (Imperial College London), PGCE Science (University of Cambridge).
Model answer (4 marks)
A sinusoidal current in the solenoid produces a magnetic field B(t)=B_0 sin(ωt) that varies continuously with time, so the magnetic flux Φ through the rectangular coil changes with time.
By Faraday’s law, ε=−dΦ/dt, the changing flux linkage (NΦ, N=1) induces an EMF in the coil. Because dΦ/dt is sinusoidal, the induced EMF is also sinusoidal and has the same frequency as the solenoid current.
If the frequency f is doubled (ω=2πf), the time‑rate of change of the flux linkage is doubled: dΦ/dt∝ω. Since the peak EMF ε_max=NBAω, keeping B_0 and N constant, ε_max∝ω, so the peak EMF is twice as large.
By Faraday’s law, ε=−dΦ/dt, the changing flux linkage (NΦ, N=1) induces an EMF in the coil. Because dΦ/dt is sinusoidal, the induced EMF is also sinusoidal and has the same frequency as the solenoid current.
If the frequency f is doubled (ω=2πf), the time‑rate of change of the flux linkage is doubled: dΦ/dt∝ω. Since the peak EMF ε_max=NBAω, keeping B_0 and N constant, ε_max∝ω, so the peak EMF is twice as large.
Examiner tips
- Use the phrase ‘magnetic flux changes with time’ to show understanding of the source of EMF.
- State Faraday’s law explicitly (ε=−dΦ/dt) and note the coil has N=1.
- Show the proportionality ε_max∝ω and explain why doubling f doubles ε_max.
- Mention that B_0 and N remain unchanged to justify the proportionality.
Common mistakes
- Failing to state that the magnetic flux through the coil changes with time.
- Using the wrong sign or omitting the negative sign in Faraday’s law.
- Assuming the induced EMF is independent of frequency or confusing current with magnetic field amplitude.
Mark scheme (4 marks)
- The alternating current in the solenoid produces a magnetic field whose magnitude (and hence flux through the rectangular coil) changes continuously with time.
- By Faraday's law, an EMF is induced in the coil because the rate of change of magnetic flux linkage through the coil is non-zero, producing an alternating EMF at the same frequency.
- Doubling the frequency means the magnetic flux (and flux linkage) varies twice as rapidly with time, so the rate of change of flux linkage is doubled.
- Because peak EMF is proportional to the rate of change of flux linkage (EMF_peak = NBAω, where ω = 2πf), doubling the frequency doubles the peak EMF.
Key terms in this question
alternating EMF · solenoid · coaxial
Related
- All IB DP Physics Higher Level (2023 syllabus) revision notes →
- How to answer a "Explain" question →
- Decode the mark scheme abbreviations →
More Induction (HL only) questions
- Explain why a straight conducting rod moving at constant velocity through a unif…
- A circular conducting loop is held stationary in a region where a uniform magnet…
- Explain how Lenz's law is consistent with the principle of conservation of energ…
- Explain how the phenomenon of self-inductance arises in a coil carrying a changi…