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Maxwell’s Equations: The Synthesis

You’ll be able to

Four equations, the whole course

Everything in this course compresses into Maxwell’s equations. Gauss’s law: ∮E·dA = Q_enc/ε₀ — charges are sources of E fields (Unit 1). Gauss’s law for magnetism: ∮B·dA = 0 — no magnetic monopoles exist; field lines of B always close on themselves. Faraday’s law: ∮E·dl = −dΦ_B/dt — a changing magnetic flux creates a circulating electric field (this unit). Ampère–Maxwell law: ∮B·dl = μ₀I_enc + μ₀ε₀·dΦ_E/dt — both currents and changing electric flux create circulating magnetic fields (Unit 5, completed here).

Maxwell’s equations (integral form)
∮→E·d→A = Q_enc/ε₀ ∮→B·d→A = 0 ∮→E·d→l = −dΦ_B/dt ∮→B·d→l = μ₀I_enc + μ₀ε₀·dΦ_E/dt
Two flux laws (what makes fields start and end) and two circulation laws (what makes fields curl). The final term is Maxwell’s displacement current.

Why Ampère’s law needed repair

Ampère’s law as Unit 5 left it has a hole. Apply it to a charging capacitor: an Amperian loop around the wire can bound a flat surface pierced by the wire (I_enc = I) or a bulging surface passing between the plates (I_enc = 0). Same loop, two answers — contradiction. Maxwell noticed what does cross the gap: a growing electric flux. His fix, the displacement current I_d = ε₀·dΦ_E/dt, restores consistency and carries a stunning implication: a changing electric field creates a magnetic field, the mirror image of Faraday’s law.

Worked example

Show that the displacement current between the plates of a charging parallel-plate capacitor exactly equals the conduction current I in the wire.

  1. 1.Field between plates of charge q and area A (Unit 1 result via Gauss’s law): E = σ/ε₀ = q/(ε₀A).
  2. 2.Electric flux through a surface spanning the gap: Φ_E = E·A = q/ε₀.
  3. 3.Displacement current: I_d = ε₀·dΦ_E/dt = ε₀·(1/ε₀)·dq/dt = dq/dt.
  4. 4.But dq/dt is precisely the conduction current I delivering charge to the plate. So I_d = I.
  5. 5.Interpretation: the “current” is seamless — real current in the wire hands off to displacement current in the gap, and Ampère–Maxwell gives the same B for every surface bounded by the loop.
Answer: I_d = ε₀·dΦ_E/dt = dq/dt = I — the displacement current in the gap equals the wire’s conduction current

The payoff: light

With both cross-generation terms in place, the field equations feed each other: a changing B makes a circulating E (Faraday), and that changing E makes a circulating B (Maxwell). In empty space — no charges, no currents — this loop can sustain itself as a traveling electromagnetic wave: E and B perpendicular to each other and to the direction of travel, regenerating one another as they go. The equations even fix the speed, from the two constants you have used all course: v = 1/√(μ₀ε₀). Maxwell computed it, recognized the measured speed of light, and concluded that light is an electromagnetic wave — the greatest unification in physics before relativity.

Worked example

Compute 1/√(μ₀ε₀) using μ₀ = 4π × 10⁻⁷ T·m/A and ε₀ = 8.85 × 10⁻¹² C²/(N·m²), and interpret the result.

  1. 1.Multiply the constants: μ₀ε₀ = (4π × 10⁻⁷)(8.85 × 10⁻¹²) = (1.257 × 10⁻⁶)(8.85 × 10⁻¹²) = 1.11 × 10⁻¹⁷ s²/m².
  2. 2.Take the square root: √(1.11 × 10⁻¹⁷) = 3.34 × 10⁻⁹ s/m.
  3. 3.Invert: v = 1/(3.34 × 10⁻⁹) = 3.00 × 10⁸ m/s.
  4. 4.This is exactly the measured speed of light c. Two constants calibrated from tabletop experiments — Coulomb forces between charges and magnetic forces between currents — conspire to give the speed of every radio wave, light beam, and X-ray in vacuum.
Answer: c = 1/√(μ₀ε₀) = 3.00 × 10⁸ m/s — electricity and magnetism together predict the speed of light
On the exam

Know each Maxwell equation by claim, not just symbol: Gauss (E) — charges make diverging E fields; Gauss (B) — no monopoles, B lines close; Faraday — changing Φ_B makes circulating E; Ampère–Maxwell — currents and changing Φ_E make circulating B. Exam questions ask “which equation forbids magnetic monopoles?” or “which term did Maxwell add, and why?” far more often than they ask you to compute with them.

Tip

Notice the near-symmetry of the four equations — and the asymmetries that remain: Q_enc appears in Gauss’s law for E but has no magnetic partner (no monopoles), and μ₀I_enc has no electric partner (no magnetic “current”). If magnetic monopoles were ever discovered, the missing terms would slot straight in.

Checkpoint

What did Maxwell add to the equations of electromagnetism?

Checkpoint

What sustains an electromagnetic wave as it travels through empty space?

Answer the 2 checkpoints as you read.

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