Maxwell equations

Let ρ(r,t) denote total electric charge density and J(r,t) total electric current density in the chosen vacuum-source formulation. The fields E and B obey four equations, the first two specifying divergence and the latter two specifying curl. An electric charge produces electric flux; magnetic flux has no monopole source in the classical theory; varying B produces electric circulation; J and varying E produce magnetic circulation.

A pointwise source-free region means ρ=0 and J=0 THERE, not that E and B must vanish. Waves pass through empty space because the time derivatives couple the fields. Under ordinary vacuum conditions the four equations are local laws, whereas simple static Coulomb/Biot–Savart formulas have additional source/time assumptions.

Choose a fixed volume V bounded by a closed outward-oriented surface ∂V. Apply the divergence theorem ∮∂V F·n̂dS=∫V∇·F d³r. Electric Gauss gives the total signed electric flux equal to Qinside/ε₀, where Qinside=∫Vρd³r.

For B the closed flux is exactly zero. This does NOT imply that B vanishes at the surface; inward and outward contributions can cancel. The surface must be closed to use these divergence statements, and its normal is outward by convention.

Choose a FIXED smooth open surface S with oriented closed boundary C. Its positive contour direction follows the right-hand rule about n̂. Stokes' theorem converts the local curl equations to two signed circulation equations; the time derivatives act on flux through that fixed S.

The minus sign in Faraday makes the induced E circulation oppose the INCREASE of a positive B flux for a passive loop. In Ampère–Maxwell the electric-flux term permits nonzero magnetic circulation with no conduction current through S. For a moving material contour, the emf contains u×B as developed separately in EM-03.01; do not apply the fixed-loop electric-circulation equation alone to a moving wire.

Apply the divergence operator to Maxwell–Ampère. Since ∇·(∇×B)=0, one obtains 0=μ₀∇·J+μ₀ε₀∂(∇·E)/∂t. Insert Gauss's law ∇·E=ρ/ε₀ to derive ∂ρ/∂t+∇·J=0.

This shows the four local field equations are compatible with local charge conservation; it does not replace the physical statement that charge is conserved. The detailed interpretation of integrated current at a moving boundary belongs in the following continuity chapter.

In a vacuum region with ρ=0 and J=0, Maxwell gives ∇·E=0 and ∇×B=μ₀ε₀∂E/∂t. Take the curl of ∇×E=−∂B/∂t. Commute the spatial curl with the time derivative for sufficiently regular fields, and substitute Maxwell–Ampère.

The double-curl identity ∇×(∇×E)=∇(∇·E)−∇²E reduces to −∇²E because there is no local charge. Thus E obeys the three-component homogeneous wave equation with coefficient μ₀ε₀. The elimination of ∇(∇·E) is NOT valid as written where ρ varies spatially.

Take the curl of ∇×B=μ₀ε₀∂E/∂t in the same source-free region, substitute ∇×E=−∂B/∂t, and use ∇·B=0 in the double-curl identity. The magnetic field satisfies the same wave equation.

Comparing with ∇²F−(1/v²)∂²F/∂t²=0 yields v=c=1/√(μ₀ε₀). In modern SI the vacuum light speed is fixed at c=299 792 458 m/s, while the measured μ₀ and ε₀ have consistent values. This c is the vacuum wave speed, not an automatic phase or group velocity in a dispersive material.

Use the real part of complex fields E(r,t)=E₀exp[i(k·r−ωt)] and B(r,t)=B₀exp[i(k·r−ωt)] with positive ω, source-free vacuum and nonzero k. The divergence laws give k·E₀=0 and k·B₀=0. Faraday gives ik×E₀=iωB₀, so B₀=(1/ω)k×E₀.

For |k|=ω/c, the magnetic amplitude magnitude is |B₀|=|E₀|/c. If a wave propagates toward +z with E toward +x, then B points +y: x̂×ŷ=ẑ. Reversing k reverses B for the same E orientation. This geometry follows from the SIGNED complex convention, not an unsigned arrow mnemonic.

Let n̂ point from medium/region 1 to region 2. For ordinary finite fields near an interface, a thin Gaussian pillbox and shrinking Stokes loop give the jumps of normal E, normal B, tangential E and tangential B. In a vacuum formulation with TOTAL surface charge σ and surface current K (with no singular sheet time-derivative fields), the jumps below hold with the chosen n̂.

The tangential B jump is a VECTOR cross-product condition, not a freely assignable scalar sign. Across material interfaces one often instead uses D and H with FREE σ and K and constitutive assumptions. When a singular sheet has explicit surface polarization/magnetization or time-dependent generalized sheet responses, additional effective interface terms may enter; do not blindly apply ordinary textbook jumps.

At f=1.00 THz, λ₀=c/f≈2.998×10⁻⁴ m=300 μm in vacuum. For a monochromatic plane wave with electric amplitude E₀=300 V/m, the magnetic amplitude is B₀=E₀/c≈1.00×10⁻⁶ T=1.00 μT. These are paired field amplitudes of the SAME wave, not separately chosen source fields.

This vacuum calculation does not predict a wavelength inside an arbitrary conductor, graphene device or dispersive dielectric. The corresponding material propagation depends on the frequency-dependent constitutive response and boundaries. An apparently small physical object can still violate an instantaneous quasistatic approximation when relevant delay, transport or resonance times matter.

1. State all four vacuum Maxwell equations in local form, identify the two divergence constraints and their electric/magnetic surface flux counterparts.
2. For a FIXED +z-oriented loop enclosing a uniform increasing +z magnetic flux, determine the sign of ∮E·dℓ relative to counterclockwise traversal; explain whether a global single-valued electrostatic potential alone can represent E.
3. Starting from source-free Maxwell, derive ∇²E−μ₀ε₀∂²E/∂t²=0, clearly identifying the assumption needed to eliminate ∇(∇·E).
4. For a vacuum plane wave traveling in +z with E₀=300 x̂ V/m and f=1.00 THz, determine B₀ vector, wavelength and units.
5. At a regular interface with normal from 1 to 2 and surface current K=K x̂, compute n̂×(B₂−B₁) when n̂=ẑ; describe why a scalar B₂−B₁=μ₀K is not the general boundary equation.
6. Derive continuity from vacuum Ampère–Maxwell and Gauss, and explain why J=0 in an empty region does not force ∇×B=0 during wave propagation.

EM-03.04 will develop charge continuity for fixed and moving control volumes, derive its integral flux form and connect interface-charge accumulation to measurable circuit and displacement-current balances.