04 // Interactive Formula Compendium // PHYSICSℰ = −N · (dΦ_B / dt), where Φ_B = B · A · cos(θ)
Faraday's Law of Electromagnetic Induction & Lenz's Law Sheet
Faraday's Law of Induction states that the electromotive force (EMF) ℰ induced in an N-turn coil equals the negative rate of change of magnetic flux Φ_B = B·A cos(θ) through the coil: ℰ = −N(dΦ_B/dt), where the minus sign encodes Lenz's Law.
Formula Reference
Main Formula
ℰ = −N · (dΦ_B / dt), where Φ_B = B · A · cos(θ)
Rearranged Forms
|ℰ_avg| = N · |ΔΦ_B / Δt| (Average Induced EMF in Volts)
ℰ_motional = B · ℓ · v (Motional EMF of Moving Conductor)
ℰ_inductor = −L · (dI / dt), U_L = ½ L I² (Self-Inductance & Magnetic Energy)
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| |ℰ| | Magnitude of Induced EMF |ℰ| | V |
| N | Number of Coil Turns (N) | turns |
| dPhi | Magnetic Flux Change |ΔΦ_B| (Webers) | Wb |
| dt | Time Interval (Δt) | s |
Inputs & Parameters
[turns]
[Wb]
[s]
Calculated Result (Magnitude of Induced EMF |ℰ|)
|ℰ| =120V
Formula Notes & Background
- Define magnetic flux through a loop of area A as Φ_B = ∫ B · dA (in Webers, 1 Wb = 1 T·m²).
- From the differential Maxwell-Faraday equation ∇ × E = −∂B/∂t, apply Stokes' Theorem over a surface bounded by the wire loop: ∮ E · dl = −d/dt ∫ B · dA.
- For N tightly wound series turns, the total induced EMF is ℰ = −N dΦ_B/dt.
Step-by-Step Examples
#1: A 200-turn stator coil experiences a magnetic flux change ΔΦ_B = 0.015 Wb over Δt = 0.025 s. Find the induced EMF.
N = 200 turnsΔΦ_B = 0.015 WbΔt = 0.025 s
|ℰ| = 200 × (0.015 / 0.025) = 200 × 0.60
→ |ℰ| = 120 V
#2: Find the ideal step-down transformer secondary voltage if N_p = 1,200 turns, N_s = 120 turns, and V_p = 120 V AC.
Same dΦ_B/dt links both windings ⇒ V_s / V_p = N_s / N_p
V_s = 120 V × (120 / 1200)
→ V_s = 12.0 V AC
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Frequently Asked Questions
What does the negative sign in ℰ = −N dΦ_B/dt represent (Lenz's Law)?+
Lenz's Law states that the induced current flows in a direction whose own magnetic field opposes the change in external magnetic flux that produced it, enforcing conservation of energy.
What are the three ways to change magnetic flux Φ_B = B A cos(θ)?+
You can vary the magnetic field strength B(t), change the loop area A(t), or rotate the coil angle θ(t) = ωt (which is how AC power plant turbines generate sinusoidal voltage).
Why can't a standard transformer step up or step down steady DC voltage?+
Steady DC current creates constant magnetic flux (dΦ_B/dt = 0), inducing zero voltage in the secondary winding.