Maxwell's Equations (Mathematical Form)

The four equations

Maxwell's equations are four coupled partial differential equations linking two vector fields — the electric field E(x, t) and the magnetic field B(x, t) — to two sources, the charge density ρ(x, t) and the current density J(x, t). In SI units:

∇·E   = ρ/ε₀                       (Gauss for E)
∇·B   = 0                           (Gauss for B — no monopoles)
∇×E   = −∂B/∂t                      (Faraday's induction)
∇×B   = μ₀J + μ₀ε₀ ∂E/∂t            (Ampère–Maxwell)

Two divergence equations and two curl equations. The first pair tells you where field lines start and end; the second pair tells you how the fields swirl, and how time-variation of one field generates the other.

The constants ε₀, μ₀, and c

• ε₀ ≈ 8.854 × 10⁻¹² F/m. The permittivity of free space. Sets the strength of electric fields produced by a given charge density.
• μ₀ = 4π × 10⁻⁷ H/m (formerly defined exactly; now a measured constant after the 2019 SI redefinition). The permeability of free space. Sets the strength of magnetic fields produced by a given current.
• c = 1/√(μ₀ε₀) ≈ 2.998 × 10⁸ m/s. Falls out of the vacuum form of the equations — it is the speed at which all EM disturbances propagate in vacuum.

The product μ₀ε₀ has units of (time/length)² = 1/(speed)². Plug in the SI values and you get c² to extraordinary precision. Maxwell noticed this in 1865 and identified light with electromagnetic waves.

Deriving the vacuum wave equation

In vacuum the source terms vanish: ρ = 0 and J = 0. The four equations become

∇·E = 0
∇·B = 0
∇×E = −∂B/∂t
∇×B = μ₀ε₀ ∂E/∂t

Take the curl of Faraday:

∇×(∇×E) = −∂(∇×B)/∂t = −μ₀ε₀ ∂²E/∂t²

Use the vector identity ∇×(∇×F) = ∇(∇·F) − ∇²F. Since ∇·E = 0 in vacuum, the first term drops:

−∇²E = −μ₀ε₀ ∂²E/∂t²

⟹     ∇²E − (1/c²) ∂²E/∂t² = 0       with  c = 1/√(μ₀ε₀)

Each component of E satisfies the classical wave equation with phase speed c. The same argument applied to the curl of Ampère–Maxwell gives the same wave equation for B. Plane-wave solutions E = E₀ ê e^{i(k·x − ωt)} have ω = c|k|, and the divergence conditions force E₀ ⊥ k and B₀ ⊥ k — light is a transverse wave.

Why the displacement current matters

Pre-Maxwell, Ampère's law read ∇×B = μ₀J. But taking the divergence of both sides gives 0 = μ₀ ∇·J — current must be divergenceless. The continuity equation ∂ρ/∂t + ∇·J = 0 then forces ∂ρ/∂t = 0 — charge densities can never change. Manifestly false. Charging a capacitor moves charges around; ∂ρ/∂t is non-zero on the plates.

Maxwell's fix (1861): add the term μ₀ε₀ ∂E/∂t — the displacement current. Now ∇·(∇×B) = μ₀ ∇·J + μ₀ε₀ ∂(∇·E)/∂t = μ₀ ∇·J + μ₀ ∂ρ/∂t = 0 by continuity. Consistency restored.

The displacement current is more than a bookkeeping fix. It is the term that couples ∂E/∂t back into B, which couples back into E via Faraday — the feedback loop that makes self-propagating electromagnetic waves possible.

Worked example — plane wave in vacuum

Try the ansatz E(x, t) = E₀ ŷ cos(kx − ωt), with E polarized along ŷ and travelling in +x̂. Check each Maxwell equation in vacuum.

• ∇·E = ∂E_y/∂y = 0. ✓ (no y-dependence)
• ∇×E has only z-component: ∂E_y/∂x = −E₀ k sin(kx − ωt). So ∂B_z/∂t = E₀ k sin(kx − ωt), giving B_z = −(E₀ k/ω) cos(kx − ωt). The B-field is along ẑ, perpendicular to both E and k.
• ∇·B = ∂B_z/∂z = 0. ✓
• ∇×B has y-component −∂B_z/∂x = −(E₀ k²/ω) sin(kx − ωt). Set equal to μ₀ε₀ ∂E_y/∂t = μ₀ε₀ E₀ ω sin(kx − ωt). Equating: k²/ω = μ₀ε₀ ω, so ω² = k²/(μ₀ε₀), i.e. ω = ck. ✓

All four equations satisfied iff ω = ck. The fields E and B are perpendicular to each other and to the propagation direction, with amplitude ratio |B|/|E| = 1/c.

Maxwell vs older formulations

	Coulomb (1785)	Ampère (1820)	Faraday (1831)	Maxwell (1865)
Domain	Static electric	Static magnetic	Quasi-static	Full dynamic
Form	Force law (1/r²)	Integral over wires	EMF = −dΦ/dt	4 PDEs
Predicts EM waves	No	No	No	Yes
Self-consistent	Yes (statics only)	No (continuity broken)	Partial	Yes
Speed of light	—	—	—	c = 1/√(μ₀ε₀)
Number of fields	1 (E)	1 (B)	2 (E, B)	2 (E, B) unified
Lorentz invariant	No	No	No	Yes (in covariant form)

Maxwell unified the previous laws and added the displacement current — closing a gap that no one had noticed was a gap. The result was the first relativistic field theory, decades before Einstein.

Integral form via Gauss and Stokes

Apply the divergence theorem to the two ∇· equations, and Stokes' theorem to the two ∇× equations:

∮_{∂V} E·dA = Q_enc / ε₀                 (Gauss for E)
∮_{∂V} B·dA = 0                            (Gauss for B)
∮_{∂S} E·dl = −dΦ_B/dt                     (Faraday's EMF)
∮_{∂S} B·dl = μ₀ I_enc + μ₀ε₀ dΦ_E/dt     (Ampère–Maxwell)

The integral forms are what you use to solve symmetric problems (long wires, spherical charges, infinite planes) by picking a Gaussian surface or Amperian loop that exploits the symmetry. Differential form is the foundation; integral form is the calculator.

Scalar and vector potentials

Since ∇·B = 0, B can be written as the curl of a vector potential: B = ∇×A. Plug into Faraday: ∇×(E + ∂A/∂t) = 0. So the combination is the gradient of a scalar potential: E = −∇φ − ∂A/∂t.

The two homogeneous equations (∇·B = 0 and ∇×E = −∂B/∂t) become identities once you introduce (φ, A). The two inhomogeneous equations become, after fixing the Lorenz gauge ∇·A + (1/c²) ∂φ/∂t = 0:

□φ = ρ/ε₀
□A = μ₀J             where  □ = ∇² − (1/c²) ∂²/∂t² (d'Alembertian)

Both potentials satisfy the inhomogeneous wave equation with the appropriate source. This is the form used in QED, where (φ, A) becomes the four-potential A^μ and gets quantized as the photon.

Covariant tensor form

Define the antisymmetric field-strength tensor F^{μν} = ∂^μ A^ν − ∂^ν A^μ. Its components are

F^{0i} = E_i / c        (electric)
F^{ij} = −ε^{ijk} B_k   (magnetic)

The four Maxwell equations pack into two manifestly Lorentz-invariant tensor equations:

∂_μ F^{μν} = μ₀ J^ν             (Gauss-E and Ampère–Maxwell)
∂_μ F̃^{μν} = 0                   (no monopoles and Faraday)

where F̃^{μν} = ½ ε^{μνρσ} F_{ρσ} is the Hodge dual. Different inertial observers see different E and B, but the same F^{μν} — what one observer calls "electric" another partially calls "magnetic". Einstein arrived at special relativity (1905) in part by taking this Lorentz invariance of Maxwell's equations seriously.

Where the equations show up

• Optics. Every optical phenomenon — refraction, diffraction, polarization, dispersion — is Maxwell's equations applied to wave packets in materials with ε(ω), μ(ω) replacing ε₀, μ₀.
• Antennas and radio. Hertz (1887) generated and detected radio waves using exactly the source–field structure Maxwell predicted. All wireless communication is engineering of these PDEs.
• Waveguides and fiber optics. Maxwell's equations in a bounded medium with appropriate boundary conditions; transverse-electric (TE) and transverse-magnetic (TM) modes.
• MHD and plasma physics. Maxwell coupled with fluid equations of a charged-particle gas. The basis of fusion-reactor confinement, solar physics, and astrophysical jets.
• Quantum electrodynamics (QED). The classical (φ, A) potentials are quantized; F^{μν} becomes an operator on Fock space. Maxwell is the free-photon Lagrangian.
• General relativity. In curved spacetime, partial derivatives become covariant derivatives: ∇_μ F^{μν} = μ₀ J^ν. Maxwell's equations are conformally invariant — light follows null geodesics.

Common mistakes

• Forgetting the displacement current. Using ∇×B = μ₀J alone for time-varying fields breaks continuity and kills all EM waves. The μ₀ε₀ ∂E/∂t term is mandatory whenever ∂E/∂t ≠ 0.
• Treating E and B as independent. In any region with time variation, the two fields are coupled — you cannot solve for one without the other. The "transverse" structure of light follows from the four equations together.
• Confusing Gaussian and SI units. Different textbooks scale ε₀ and μ₀ differently; the structural form of the equations is the same but factors of 4π and c migrate between sides.
• Forgetting boundary conditions across material interfaces. E_∥ and B_⊥ are continuous; D_⊥ and H_∥ have jumps given by surface charge and surface current. Snell's law and Fresnel coefficients come from these boundary conditions, not from the bulk PDEs.
• Assuming the wave equation works in source regions. The clean ∇²E − (1/c²)∂²E/∂t² = 0 form holds only where ρ = 0 and J = 0. With sources, you get an inhomogeneous wave equation with terms from ∇ρ and ∂J/∂t.