Maxwell's Equations and Electromagnetic Waves
This course has built four separate laws, one unit-family at a time: Gauss's Law for electric fields (Unit 202), its magnetic counterpart implicit in Unit 207, Faraday's Law (Unit 208), and Ampere's Law (Unit 207). Maxwell collected all four, added one missing piece (displacement current, needed for logical consistency), and showed the result predicts something dramatic: self-sustaining waves of oscillating E and B fields, needing no medium at all, traveling at a speed built entirely from ε₀ and μ₀ — a speed that turned out to be exactly the speed of light.
What you'll learn
- State Maxwell's four equations and describe what each one means physically.
- Explain the concept of displacement current and why Ampere's Law needed it.
- State the relationship between E and B in an electromagnetic wave, and derive the wave speed c=1/√(μ₀ε₀).
- Calculate the intensity of an electromagnetic wave from its field amplitude.
- Relate frequency and wavelength across the electromagnetic spectrum.
- Describe, qualitatively, how accelerating charges produce electromagnetic radiation.
1. Maxwell's Four Equations
This course has developed four laws separately, one at a time: Gauss's Law for E (Unit 202), an implicit Gauss's Law for B (Unit 207's field-line-circles-close-on-themselves geometry), Faraday's Law (Unit 208), and Ampere's Law (Unit 207). Maxwell's contribution was recognizing these four belong together as a single, complete description of electromagnetism — plus one crucial fix.
Ampere's Law as originally written broke down for situations like a charging capacitor's gap: no actual current crosses the gap, yet a magnetic field is observed there anyway. Maxwell's fix, the displacement current term μ₀ε₀dΦ_E/dt, treats a changing electric field as equivalent to a current for the purpose of generating B — restoring consistency, and, unexpectedly, opening the door to self-sustaining waves.
2. Electromagnetic Waves
Combined, Faraday's Law and the (now-complete) Ampere-Maxwell Law describe something no single one of the four equations alone predicts: a changing E field creates a circulating B field (Ampere-Maxwell); that changing B field creates a circulating E field (Faraday); which, being itself now changing, creates more changing B — the fields sustain each other, propagating outward as a wave, with no charges, currents, or medium needed anywhere along the way.
In this wave, E and B are perpendicular to each other and to the direction the wave travels, locked together by E=cB, and the wave speed works out to c=1/√(μ₀ε₀) — built entirely from electric and magnetic constants, yet numerically identical to the independently measured speed of light.
3. Intensity and the Electromagnetic Spectrum
An EM wave's intensity — average power delivered per unit area — is I=½cε₀E_max², proportional to the square of the field amplitude, the same square-law relationship power has to amplitude for any wave carrying energy.
Every electromagnetic wave, from long-wavelength radio waves through visible light to short-wavelength gamma rays, obeys c=fλ and travels at exactly the same speed c in vacuum — they are one phenomenon at different frequencies, not a family of distinct effects, which is the unifying insight this entire unit (and in some sense, this entire course) has been building toward.
Key equations
- ∮E·dA=Q_enc/ε₀, ∮B·dA=0 — Maxwell's first two equations: electric field lines begin and end on charge (Gauss's Law), while magnetic field lines never begin or end anywhere — they always close on themselves, which is the precise statement that no magnetic monopoles exist.
- ∮E·dl=−dΦ_B/dt, ∮B·dl=μ₀I_enc+μ₀ε₀dΦ_E/dt — Faraday's Law (a changing B field creates a circulating E field) and the Ampere-Maxwell Law (a changing E field, the displacement current term, creates a circulating B field just as a real current does) — together, these two equations are what let E and B sustain each other without any charges or currents present at all.
- c = 1/√(μ₀ε₀) — The speed predicted for an electromagnetic wave, built entirely from the electric and magnetic constants — numerically equal to the already-measured speed of light, the discovery that revealed light itself is an electromagnetic wave.
- E = cB — In an electromagnetic wave, the E and B amplitudes are always locked in this ratio — the two fields aren't independent, they're two aspects of one traveling disturbance.
- I = ½cε₀E_max², c=fλ — The average intensity of an electromagnetic wave from its field amplitude, and the universal relationship linking frequency and wavelength for any EM wave, from radio to gamma rays, all traveling at the same c in vacuum.