Electromagnetic Induction
Everything so far in this course has been static: steady charges, steady currents, unchanging fields. Faraday's Law breaks that stillness open — a *changing* magnetic flux through a loop induces an EMF, EMF=−dΦ_B/dt, whether that change comes from a moving conductor, a changing field, or a changing area. Lenz's Law (the minus sign, made physical) says the induced current always opposes the change that created it — not the field itself, the *change* — which is really just energy conservation wearing a directional hat.
What you'll learn
- Calculate induced EMF using Faraday's Law for a changing magnetic flux.
- Apply Lenz's Law to determine the direction of an induced current.
- Calculate motional EMF for a conductor moving through a magnetic field.
- Analyze a simple AC generator based on a rotating loop in a uniform field.
- Calculate the EMF induced by a magnetic field changing in time through a stationary loop.
- Explain the physical origin of Lenz's Law in terms of energy conservation.
1. Faraday's Law
Magnetic flux, Φ_B=∫B·dA, is defined exactly as electric flux was in Unit 202 — but where a changing *electric* flux was never physically relevant in this course, a changing *magnetic* flux is the entire subject here. Faraday's Law states the induced EMF around a loop equals the negative rate of change of flux through it: EMF=−dΦ_B/dt.
Crucially, it's the rate of change that matters, not the flux's size. A large, constant flux induces nothing; a small, rapidly-changing flux can induce a substantial EMF.
2. Lenz's Law and Motional EMF
Lenz's Law supplies the minus sign's physical meaning: the induced current always flows in whatever direction opposes the change in flux that caused it — not the flux itself, the change. This isn't an extra rule bolted onto Faraday's Law; it's a direct consequence of energy conservation (if induced currents reinforced the change instead, you'd get runaway energy from nothing).
A rod sliding along conducting rails in a magnetic field is a clean, concrete case: as it slides, it sweeps out changing area, EMF=BLv, viewable either as a Faraday's-Law flux-change or directly as the qv×B magnetic force separating charge within the moving rod.
3. AC Generators
A coil of N turns and area A, rotating at constant angular speed ω in a uniform field B, has flux Φ=BA cos(ωt) — sinusoidally varying purely from the changing angle between the coil and the field, with no change in field strength required at all. Differentiating gives the generator's EMF: EMF=NBAω sin(ωt).
This is the working principle behind essentially all large-scale electricity generation — mechanical rotation (from steam, water, wind, or any other source) converted directly into an alternating EMF via nothing more than Faraday's Law applied to a spinning loop.
Key equations
- EMF = −dΦ_B/dt — Faraday's Law — the induced EMF depends only on how fast flux is changing, not on how large the flux itself is at any instant.
- EMF = NdΦ_B/dt (magnitude, N turns) — For a coil of N loops, each loop contributes its own EMF, and they add — total EMF is N times a single loop's.
- EMF = BLv — Motional EMF — a conducting rod sliding along rails in a magnetic field, viewed either as changing enclosed area (Faraday) or as the magnetic force on the rod's free charges (qv×B) doing the separating.
- EMF = NBAω sin(ωt) — The EMF of an AC generator — a coil rotating at constant angular speed in a uniform field, which is exactly how large-scale electricity generation works.