Electromagnetic Induction and Alternating Currents
Faraday and Lenz, self/mutual inductance, AC circuits, LCR resonance, transformers.
Faraday's and Lenz's laws
Induced EMF = −dΦ/dt, conservation of energy.
Faraday's law of electromagnetic induction:
EMF = − dΦ_B / dt
where Φ_B = ∫ B · dA is magnetic flux through the loop.
The negative sign is Lenz's law: the induced current flows in the direction that opposes the change in flux. This is conservation of energy in disguise — if induced current reinforced the change, you'd get free energy.
Three ways to change flux:
- Change B (move magnet towards/away from loop, vary current in nearby coil).
- Change A (expand/shrink the loop area).
- Change angle between B and A (rotate loop — this is how generators work).
Worked example: motional EMF. A rod of length L moves with velocity v perpendicular to a uniform field B.
Charges in the rod experience force F = qv × B → free positive charges accumulate at one end. An electric field E_ind builds up until equilibrium: qE_ind = qvB → E_ind = vB.
EMF = E_ind · L = BvL.
Generator EMF. A loop of N turns, area A, rotating at angular frequency ω in field B:
EMF(t) = NABω sin(ωt) → peak EMF = NABω.
This is why your wall outlet is sinusoidal AC at the line frequency.
Eddy currents. Bulk conductors moving in changing fields develop swirling induced currents. These cause:
- Heating (used in induction cooktops)
- Drag (used in magnetic braking on trains)
- Energy loss in transformer cores (mitigated by laminating the core)
Self and mutual inductance
L, M, energy stored in an inductor.
AC circuits and resonance
XL, XC, impedance, resonance frequency, Q-factor.