Faraday's Law and Maxwell's Equations
Faraday's law and Lenz's law, motional and transformer EMF, displacement current, Maxwell's equations in integral and point form, Poynting vector and basic wave propagation.
When fields change with time, electricity and magnetism become linked. Faraday's law explains generators and transformers; Maxwell's displacement current completes the picture and predicts electromagnetic waves.
Faraday's law
EMF = −N dΦ/dt
The minus sign is Lenz's law: the induced current opposes the change that caused it.
- Transformer EMF: a stationary loop in a time-varying field: −∫(∂B/∂t)·dS.
- Motional EMF: a conductor moving in a field: e = B l v (perpendicular B, l and v), or ∮(v × B)·dl.
- Sinusoidal flux Φ_m sin ωt in N turns gives an RMS EMF of 4.44 f N Φ_m.
Displacement current
Ampere's law fails for a charging capacitor. Maxwell added the displacement current density J_d = ∂D/∂t:
∇ × H = J + ∂D/∂t
Ratio of conduction to displacement current in a material: σ/(ωε). Good conductor if σ/(ωε) ≫ 1.
Maxwell's equations
| Point form | Integral form | Name |
|---|---|---|
| ∇·D = ρ_v | ∮D·dS = Q | Gauss's law |
| ∇·B = 0 | ∮B·dS = 0 | No monopoles |
| ∇ × E = −∂B/∂t | ∮E·dl = −dΦ/dt | Faraday's law |
| ∇ × H = J + ∂D/∂t | ∮H·dl = I + I_d | Ampere–Maxwell law |
Continuity: ∇·J = −∂ρ_v/∂t.
Waves and power
- In a lossless medium the fields travel as waves at v = 1/√(με) (c = 3 × 10⁸ m/s in free space).
- Intrinsic impedance η = √(μ/ε); η₀ ≈ 377 Ω (120π).
- Poynting vector P = E × H (W/m²) gives the direction and density of power flow; average for a plane wave = E₀²/(2η).
- In a good conductor fields decay with the skin depth δ = 1/√(π f μ σ).
Worked example
A 0.5 m rod moves at 10 m/s perpendicular to a 0.2 T field.
- e = Blv = 0.2 × 0.5 × 10 = 1 V.
Practice
A coil of 200 turns has a flux of 5 mWb (peak) varying at 50 Hz. What is the RMS induced EMF in V?
Show answer and explanation
Answer: 222E = 4.44fNΦ_m = 4.44 × 50 × 200 × 0.005 = 222 V.
The flux through a 100-turn coil falls uniformly from 0.04 Wb to 0 in 0.02 s. What is the magnitude of the induced EMF in V?
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Answer: 200The magnitude is NΔΦ/Δt = 100 × 0.04/0.02 = 200 V.
A plane wave in free space has E₀ = 377 V/m. What is its average power density in W/m²? (Take η₀ = 377 Ω; give 1 decimal place.)
Show answer and explanation
Answer: 188.5P_avg = E₀²/(2η₀) = 377/2 = 188.5 W/m².
Maxwell's correction to Ampere's law added:
Show answer and explanation
Answer: AWithout ∂D/∂t, Ampere's law would contradict charge continuity, for example in a charging capacitor.