Assembly of Maxwell's Equations

Derivation

Assembly of Maxwell's Equations

D-162 Home PU-204 Threads fields · symmetry Depends on Faraday's Law and the Flux Rule, The Displacement Current from Charge Conservation, Gauss's Law from Coulomb's Law

Statement

Taking the four independently derived field laws — Gauss's law for the electric field, Gauss's law for magnetism, Faraday's law of induction, and the Ampere–Maxwell law with displacement current — we collect them into a single coupled set, and show that each law's integral form and differential form are exactly equivalent (via the divergence and curl theorems) on smooth field configurations. The assembled set closes on itself: given the sources ρ and J, the equations determine E and B up to boundary and initial data, and they are internally consistent with charge conservation.

Why it matters

The individual laws were each abstracted from a distinct class of experiment: Coulomb attraction, the absence of magnetic monopoles, induction by a changing flux, and the magnetic effect of currents. Assembling them exposes what no single law shows — that electric and magnetic fields are not two subjects but one, dynamically locked together. A time-varying B sources E (Faraday) and a time-varying E sources B (displacement current), so the pair can sustain each other in empty space. That mutual sourcing is exactly the mechanism of electromagnetic radiation.

The set is also the first fully relativistic theory of physics, written down decades before relativity. Its invariance under Lorentz transformations (not Galilean) is built in, which is why light's speed is frame-independent. Everything from antennas to optics to quantum electrodynamics starts here.

Assumptions

The fields E and B are the complete local description.If hidden variables or nonlocal action-at-a-distance were operative, the local differential form would not capture the physics and the field-only closure would fail.

Charge is the only source; charge is locally conserved.Drop conservation and the Ampere–Maxwell law becomes internally contradictory (its divergence forces ∂ρ/∂t + ∇·J = 0). Consistency of the set requires continuity.

No magnetic monopoles exist.If isolated magnetic charge ρ_m were found, ∇·B = 0 would gain a source term μ₀ρ_m and Faraday's law would gain a magnetic-current term, restoring electric–magnetic duality.

The fields are smooth enough to apply the divergence and Stokes theorems.At genuine discontinuities (surface charges, surface currents, material interfaces) the differential form holds only in a distributional sense; the physical content passes to boundary/jump conditions instead.

The description is macroscopic/classical: c is finite and ℏ is negligible.Drop finiteness of c and the displacement-current coupling vanishes, decoupling the fields (magnetostatics + electrostatics). Restore ℏ and one needs quantum electrodynamics; Maxwell's equations survive as the expectation-value / classical limit.

Derivation

∫_∂V E·dA = Q_enc/ε₀ = (1/ε₀) ∫_V ρ dV

Gauss's law, imported from the Coulomb-field derivation (prior result: gauss-law-from-coulomb). Enclosed charge written as a volume integral of density. A

∫_∂V E·dA = ∫_V (∇·E) dV ⇒ ∇·E = ρ/ε₀

Divergence theorem converts the flux integral to a volume integral; equality for every V forces equality of integrands. B

∫_∂V B·dA = 0 ⇒ ∇·B = 0

Experimentally no isolated magnetic charge exists: net magnetic flux through any closed surface vanishes. Divergence theorem then gives the local form, valid for arbitrary V. B

∮_∂S E·dl = − d/dt ∫_S B·dA

Faraday's law, imported from the flux-rule derivation (prior result: faraday-law-from-flux-rule). EMF around a fixed loop equals minus the rate of change of magnetic flux through it. A

∫_S (∇×E)·dA = − ∫_S (∂B/∂t)·dA ⇒ ∇×E = −∂B/∂t

Stokes' theorem on the left; for a stationary surface the total time derivative passes inside as a partial derivative on the right. Equality for every S forces equality of integrands. B

∮_∂S B·dl = μ₀I_enc + μ₀ε₀ d/dt ∫_S E·dA

Ampere's law with the displacement-current term added (prior result: displacement-current-necessity). The extra term is mandatory for consistency with charge conservation. A

∫_S (∇×B)·dA = ∫_S [μ₀J + μ₀ε₀ ∂E/∂t]·dA ⇒ ∇×B = μ₀J + μ₀ε₀ ∂E/∂t

Stokes' theorem plus I_enc = ∫_S J·dA; equality for every S forces the local form. B

∇·(∇×B) = 0 = μ₀(∇·J) + μ₀ε₀ ∂(∇·E)/∂t ⇒ ∂ρ/∂t + ∇·J = 0

Take the divergence of step 7. The identity ∇·(∇×·) ≡ 0 and substitution of ∇·E = ρ/ε₀ (step 2) yield the continuity equation — a derived consistency check, not an extra postulate. This closes the set. C

Result

∇·E = ρ/ε₀ ∇·B = 0 ∇×E = −∂B/∂t ∇×B = μ₀J + μ₀ε₀ ∂E/∂t

Reading. Two scalar (divergence) equations fix the sources of the fields: electric field lines begin and end on charge; magnetic field lines never end. Two vector (curl) equations govern the dynamics: a changing magnetic field circulates the electric field, and currents plus a changing electric field circulate the magnetic field. The divergence pair are constraints on initial data; the curl pair are evolution equations. Together with the Lorentz force F = q(E + v×B) they constitute all of classical electromagnetism.

Units check. [∇·E] = (V·m⁻¹)/m = V·m⁻²; [ρ/ε₀] = (C·m⁻³)/(F·m⁻¹) = C·m⁻²/F = V·m⁻² ✓. For the last law, [∇×B] = T·m⁻¹; [μ₀J] = (T·m·A⁻¹)(A·m⁻²) = T·m⁻¹ ✓; [μ₀ε₀∂E/∂t] = (s²·m⁻²)(V·m⁻¹·s⁻¹) = V·s·m⁻³ = T·m⁻¹ ✓ (since T = V·s·m⁻²). All four terms share dimension — and μ₀ε₀ has units s²·m⁻², i.e. 1/c².

Limiting cases

Electrostatics (∂/∂t = 0, J = 0): reduces to ∇·E = ρ/ε₀, ∇×E = 0. Field is a pure gradient, E = −∇φ, Poisson's equation.
Magnetostatics (∂/∂t = 0): ∇·B = 0, ∇×B = μ₀J. The two field problems fully decouple — no electromagnetic coupling survives when nothing changes in time.
Vacuum, no sources (ρ = 0, J = 0): the curl equations feed back, giving ∇²E = μ₀ε₀ ∂²E/∂t² — a wave equation with speed c = 1/√(μ₀ε₀).
Slowly varying (quasistatic): keep Faraday but drop the displacement current where L/c « timescale; recovers standard circuit and eddy-current theory.

Breaks when

Fields become strong enough for vacuum nonlinearity (E approaching the Schwinger limit ≈ 1.3×10¹⁸ V·m⁻¹): virtual electron–positron pairs make the vacuum polarizable, photons scatter off photons, and the linear Maxwell equations must be replaced by the Euler–Heisenberg effective (nonlinear) theory.
Quantum / single-photon regime: when field energies are comparable to ℏω per mode, fields become operator-valued and exhibit shot noise, entanglement, and vacuum fluctuations that classical Maxwell cannot describe. QED is required; Maxwell survives only as the coherent-state / expectation-value limit.
Inside polarizable/magnetizable media without constitutive relations: the vacuum forms above still hold for total charge and current, but become practically useless until closed by material relations D = εE, B = μH — which fail for nonlinear, dispersive, or hysteretic materials.
At field discontinuities (surface charge/current, sharp interfaces): the differential forms are ill-defined pointwise and must be replaced by jump/boundary conditions on the normal and tangential components.

Failure modes

Dropping the displacement current. Writing ∇×B = μ₀J in a time-varying problem violates charge conservation and destroys wave solutions — the single most consequential omission.
Sign error in Faraday's law. Losing the minus sign inverts Lenz's law and yields runaway (unphysical) feedback instead of the stabilizing back-reaction.
Confusing total vs. partial time derivative. The integral Faraday law uses d/dt of the flux (includes moving-boundary "motional" contribution); the differential law uses ∂B/∂t. Equating them for a moving loop double-counts or drops the v×B term.
Treating the four laws as independent. Given the two curl equations plus initial data, the two divergence equations are preserved in time automatically — they are constraints, not independent evolution equations.
Using enclosed quantities without a consistent surface. In the Ampere–Maxwell integral form, S must be any surface bounded by the same loop ∂S; picking a capacitor-spanning vs. wire-piercing surface without the displacement term gives contradictory I_enc.
Forgetting μ₀ε₀ = 1/c². Leads to dimensionally inconsistent wave equations and mis-estimated propagation speeds.

Discussion

The deepest feature of the assembled set is that the two divergence equations are not independent dynamical laws but constraints. If ∇·B = 0 holds at one instant, taking the divergence of Faraday's law gives ∂(∇·B)/∂t = −∇·(∇×E) = 0, so it holds forever. Likewise ∇·E = ρ/ε₀ is preserved by the Ampere–Maxwell law because charge is conserved. The genuine evolution content lives in the two curl equations; the divergence equations only need to be imposed on the initial data. This structure is exactly what a modern formulation makes manifest.

Written in four-vector language the eight scalar equations collapse to two: ∂_μF^μν = μ₀J^ν (the two source equations, Gauss and Ampere–Maxwell) and ∂_[λF_μν] = 0 (the two sourceless equations, no-monopoles and Faraday, expressible as dF = 0). The antisymmetric field tensor F^μν packages E and B into one geometric object; a Lorentz boost mixes them, which is why what one observer calls a pure electric field another calls a mix of electric and magnetic. The frame-dependence of the E/B split is not a defect of the theory but its core relativistic content.

More abstractly, the sourceless pair dF = 0 says the two-form F is closed, so locally F = dA for a potential one-form A (the four-potential); this is the origin of gauge freedom A → A + dχ. The source pair follows from an action, δ∫(−¼F_μνF^μν − J^μA_μ)d⁴x = 0. Gauge invariance of that action forces ∂_μJ^μ = 0 — charge conservation is Noether's theorem for the U(1) symmetry. Thus the very structure that looked like a lucky consistency check in step 8 is a symmetry principle, and this U(1) gauge idea, generalized to non-abelian groups, becomes the template for the entire Standard Model.

Common misconceptions. (i) That Maxwell "derived" all four equations — he added the displacement-current term to a pre-existing set and thereby unified them; the credit is for the completion, not the whole. (ii) That the equations require a medium (the "ether") to propagate — the vacuum forms need no medium, and their frame-independent c is precisely what killed the ether. (iii) That ∇·B = 0 is a definition — it is an empirical statement (no monopoles found) that could be falsified tomorrow.

Worked examples

Charging capacitor: displacement current equals conduction current. Parallel plates, area A = 25 cm², charging current I = 2.0 A.

Between the plates J = 0, so Ampere–Maxwell reduces to ∇×B = μ₀ε₀∂E/∂t. We verify the enclosed displacement current matches I. B

E = σ/ε₀ = Q/(ε₀A) ⇒ ∂E/∂t = I/(ε₀A)

Uniform field of an ideal capacitor; differentiate, using I = dQ/dt. Symbols first. A

I_d = ε₀ (dΦ_E/dt) = ε₀A(∂E/∂t) = ε₀A · I/(ε₀A) = I

Substitute; the areas and ε₀ cancel exactly, proving continuity of current across the gap. B

∂E/∂t = (2.0)/[(8.85×10⁻¹²)(25×10⁻⁴)] = 9.0×10¹³ V·m⁻¹·s⁻¹

Numbers in SI: ε₀A = 2.21×10⁻¹⁴ F·m; divide. A

I_d = 2.0 A = I, ∂E/∂t = 9.0×10¹³ V·m⁻¹·s⁻¹

Reading. The magnetic field circling the gap is exactly what it would be if the conduction current continued straight through — displacement current stitches the circuit together. Without the extra term, Ampere's law would give two different answers for surfaces through the wire vs. through the gap.

Units check. [I/(ε₀A)] = A/(F·m⁻¹·m²) = A/(F·m) = (C·s⁻¹)/(C·V⁻¹·m) = V·m⁻¹·s⁻¹ ✓.

Speed of light from the vacuum constants. Given μ₀ = 4π×10⁻⁷ T·m·A⁻¹ and ε₀ = 8.854×10⁻¹² F·m⁻¹, extract c from the sourceless equations.

With ρ = 0, J = 0, take the curl of Faraday's law and substitute Ampere–Maxwell to isolate a wave operator. B

∇×(∇×E) = ∇(∇·E) − ∇²E = −∇²E = −∂(∇×B)/∂t = −μ₀ε₀ ∂²E/∂t²

Vector identity with ∇·E = 0; substitute ∇×B = μ₀ε₀∂E/∂t. Symbols only. C

∇²E = (1/c²) ∂²E/∂t² with c = 1/√(μ₀ε₀)

Identify the coefficient of a standard wave equation; the propagation speed is read off directly. B

c = 1/√[(4π×10⁻⁷)(8.854×10⁻¹²)] = 1/√(1.1127×10⁻¹⁷) = 2.998×10⁸ m·s⁻¹

Numbers: product = 1.1127×10⁻¹⁷ s²·m⁻²; reciprocal square root. A

c = 2.998×10⁸ m·s⁻¹

Reading. The speed of light is fixed entirely by two constants measured in static electric and magnetic experiments — no optics needed. This numerical coincidence is what told Maxwell that light is an electromagnetic wave.

Units check. [μ₀ε₀] = (T·m·A⁻¹)(F·m⁻¹) = s²·m⁻², so [1/√(μ₀ε₀)] = m·s⁻¹ ✓.

Problems

(A) Flux through a closed surface. A point charge q = 3.0 nC sits inside a closed irregular surface. What is the net electric flux? What is the net magnetic flux from a nearby bar magnet through the same surface?
Solution
By Gauss's law Φ_E = q/ε₀ = (3.0×10⁻⁹)/(8.854×10⁻¹²) = 339 N·m²·C⁻¹, independent of surface shape. By ∇·B = 0, the net magnetic flux through any closed surface is exactly 0 — the magnet has no enclosed magnetic charge.
(B) Constraint preservation. Show explicitly that if ∇·B = 0 holds at t = 0, Faraday's law guarantees it holds for all t.
Solution
Take the divergence of ∇×E = −∂B/∂t. The left side is ∇·(∇×E) ≡ 0 identically. So 0 = −∂(∇·B)/∂t, i.e. ∂(∇·B)/∂t = 0. Thus ∇·B is constant in time at every point; if it is zero initially it stays zero. The Gauss-for-magnetism law is a constraint propagated by the dynamics, not an independent equation.
(B) Displacement current dominance. In a region with conductivity σ and a field oscillating at angular frequency ω, at what frequency does the displacement current density equal the conduction current density in copper (σ = 5.96×10⁷ S·m⁻¹, ε ≈ ε₀)?
Solution
Conduction: J_c = σE. Displacement: J_d = ε₀∂E/∂t, magnitude ωε₀E for E ∝ e^iωt. Equal when ωε₀ = σ, i.e. ω = σ/ε₀ = (5.96×10⁷)/(8.854×10⁻¹²) = 6.73×10¹⁸ rad·s⁻¹, or f = ω/2π ≈ 1.07×10¹⁸ Hz (X-ray range). Below this — i.e. essentially always for copper — conduction dominates, which is why metals are good conductors up to X-ray frequencies.
(B) Ampere–Maxwell, two surfaces. A 1.5 A current charges a capacitor. For an Amperian loop encircling the wire, compute ∮B·dl using (a) a surface pierced by the wire, (b) a surface passing between the plates. Show they agree.
Solution
(a) Surface pierced by wire: I_enc = 1.5 A, displacement current 0, so ∮B·dl = μ₀(1.5) = (4π×10⁻⁷)(1.5) = 1.88×10⁻⁶ T·m. (b) Surface between plates: conduction current 0, but I_d = ε₀dΦ_E/dt = dQ/dt = 1.5 A (from worked example 1). So ∮B·dl = μ₀(1.5) = 1.88×10⁻⁶ T·m. They agree exactly — that agreement is the reason the displacement term is mandatory.
(C) Monopole-symmetric equations. Suppose magnetic charge density ρ_m and magnetic current J_m existed. Write the modified four equations and show the required magnetic continuity equation.
Solution
The symmetrized set: ∇·E = ρ_e/ε₀; ∇·B = μ₀ρ_m; ∇×E = −μ₀J_m − ∂B/∂t; ∇×B = μ₀J_e + μ₀ε₀∂E/∂t. Take the divergence of the third (Faraday) equation: 0 = −μ₀∇·J_m − ∂(∇·B)/∂t = −μ₀∇·J_m − μ₀∂ρ_m/∂t. Hence ∂ρ_m/∂t + ∇·J_m = 0 — magnetic charge would be conserved, exactly mirroring electric continuity. The set is then invariant under the duality rotation (E → cB, cB → −E).