§ 16. Faraday's Law & Lenz's Rule in 3D (Grade 9–11)
Generation of induced current in a coil by a moving magnet & Lenz's rule. The induced electromotive force in a closed circuit is equal and opposite in sign to the time rate of change of magnetic flux through the surface bounded by the circuit.
Theory Summary & Scientific Mechanism
Discovered by Michael Faraday in 1831, electromagnetic induction generates an electromotive force (induced EMF ℰᵢ) around a closed conducting loop whenever the magnetic flux Φ = ∫ B· dS threading the loop changes with time.
By Faraday’s Law for an N-turn coil, ℰᵢ = -N(dΦ)/(dt), driving an induced current Iᵢ = (ℰᵢ)/(R) through resistance R. The minus sign encodes Lenz’s Law: the induced current always circulates in a direction whose own magnetic field Bᵢnd opposes the flux change ΔΦ that produced it.
Thus, plunging a magnet toward the coil ((dΦ)/(dt) > 0) induces a repulsive pole that brakes the magnet; pulling it away ((dΦ)/(dt) < 0) induces an attractive pole; and holding the magnet stationary inside ((dΦ)/(dt) = 0) yields zero current. Mechanical work done against Lenz’s braking force converts directly into Joule electrical power P = Iᵢ² R.
Everyday Analogy
A bicycle dynamo, wireless phone charger, and power-plant turbines: moving a magnet through a coil creates electric current! Stop the magnet and the current vanishes instantly, only a changing magnetic flux induces voltage.
Interactive Question & Derivation
A bar magnet is rapidly pushed North-pole first (N) into a closed copper solenoid. What determines the magnitude of the induced current in the coil, and how is the coil's induced magnetic field Bᵢnd directed?
- A: The current depends only on how strong a magnet rests stationary inside the coil
- B: Induced EMF is proportional to the rate of flux change ℰᵢ = -N(dΦ_B)/(dt), and by Lenz's rule Bᵢnd points opposite to the magnet's field to oppose its approach
- C: The coil's induced field always pulls an approaching magnet inward
Study Hint: By Lenz’s law, the induced current opposes the motion that causes it (braking the magnet), and the EMF scales with the rate of flux change |dΦ/dt|.
Proof & Explanation: By Faraday's law, an induced current arises only when the magnetic flux Φ_B through the coil turns changes over time (dΦ_B/dt ≠ 0). Higher magnet oscillation frequency ω and turn count N produce a larger induced EMF ℰᵢ. The minus sign expresses Lenz's rule: the induced current generates its own magnetic field Bᵢnd that repels the approaching magnet and attracts the receding magnet.
Key Equations & 3D Simulation Parameters
ℰᵢ(t) = -N (dΦ_B)/(dt), I(t) = (ℰᵢ(t))/(R_coil), Pₑl = (ℰᵢ²)/(R_coil)
- Coil turn count N: 4 … 20 (default: 12)
- Magnet field B₀: 0.6 … 2.5 T (default: 1.6 T)
- Magnet oscillation freq. ω: 0.8 … 4 rad/s (default: 2.2 rad/s)
- Circuit resistance R: 1 … 5 Ω (default: 2 Ω)