Maxwell's Equations in Differential Form

The Great Unification of the 19th Century

In 1865, James Clerk Maxwell published a system of equations that unified electricity, magnetism, and light into a single theory. Before Maxwell, these seemed like three separate phenomena of nature. After him, it became clear: light is an electromagnetic wave, and the predictions of the theory astonishingly coincided with experiment.

Maxwell's equations are written with the tools of vector analysis: divergence ∇· and curl ∇×. They allow the laws to be stated locally—in every point of space.

The Four Maxwell Equations

Gauss’s Law for the Electric Field:

∇·E = ρ/ε₀

Read as: "the divergence of the electric field E is proportional to the density of electric charge ρ." Physically: electric field lines "flow out" of charges. Charge is the source of the E-field.

Symbols: E — electric field vector (V/m); ρ — volumetric charge density (C/m³); ε₀ = 8.85 × 10⁻¹² F/m — permittivity of vacuum.

Gauss’s Law for the Magnetic Field:

∇·B = 0

Read as: "the divergence of the magnetic field B is zero." Physically: magnetic monopoles do not exist—magnetic field lines are always closed. There is no magnetic "charge."

Faraday’s Law:

∇×E = −∂B/∂t

Read as: "the curl of E equals minus the rate of change of B." A changing magnetic field produces a vortex electric field. This is the operating principle of a transformer: alternating current in the primary winding creates a changing B, which induces EMF in the secondary winding.

Ampère–Maxwell Law:

∇×B = μ₀(J + ε₀ ∂E/∂t)

Read as: "the curl of the magnetic field is created by currents (J) and displacement currents (ε₀ ∂E/∂t)." Maxwell’s addition—the displacement current ε₀ ∂E/∂t—is crucial. Without it, the system is inconsistent for nonstationary charges. With it, electromagnetic wave propagation is predicted.

Symbols: B — magnetic induction vector (T); μ₀ = 4π × 10⁻⁷ H/m — magnetic constant; J — current density vector (A/m²).

Consequence: Wave Equations

In vacuum (ρ = 0, J = 0) Maxwell's equations lead to the wave equations:

∇²E − (1/c²) ∂²E/∂t² = 0
∇²B − (1/c²) ∂²B/∂t² = 0

Electromagnetic wave velocity: c = 1/√(μ₀ε₀). Substitute the numbers: c = 1/√(4π·10⁻⁷ × 8.85·10⁻¹²) ≈ 3 × 10⁸ m/s — this is the speed of light! Maxwell predicted that light is an electromagnetic wave.

Plane wave: E = E₀ cos(kx − ωt) ŷ, B = (E₀/c) cos(kx − ωt) ẑ. Here, k = ω/c — wave vector, λ = 2π/k — wavelength. E and B are perpendicular to each other and to the direction of propagation x.

Numerical Example: Electrostatics of a Point Charge

Charge q at the origin: ρ = q δ³(r). From ∇·E = ρ/ε₀ (integrating over a sphere of radius r): 4πr² E = q/ε₀ → E = q/(4πε₀r²) — Coulomb’s law! The Maxwell equation reproduces the known result.

Electromagnetic Potentials and Gauge Invariance

Instead of E and B, it is convenient to introduce potentials: B = ∇×A (vector potential) and E = −∇φ − ∂A/∂t. Automatically: ∇·B = 0 (since ∇·(∇×A) = 0) and ∇×E = −∂B/∂t ✓.

Gauge invariance: replacing A → A + ∇χ, φ → φ − ∂χ/∂t (χ — arbitrary function) does not change E and B. This is a fundamental symmetry, generalized in quantum electrodynamics to U(1) gauge symmetry.

Real World Application: Cellular Communication and Wi-Fi

The LTE standard operates at frequencies 700–2600 MHz. Wavelengths are 11.5 cm–43 cm. The transmitter antenna is an electric dipole, creating an oscillating E-field, which, through Maxwell's equations, generates B-field and so on—a wave propagates. Wi-Fi at 2.4 GHz: wavelength 12.5 cm. Penetration through walls is determined by attenuation (dielectric losses in materials)—a direct consequence of Maxwell's equations in a medium.

Gauge symmetry and the Standard Model: Gauge invariance of Maxwell's equations is a particular case of U(1) symmetry. The Standard Model of particle physics is built on three gauge groups: U(1) (electromagnetism), SU(2) (weak interaction), SU(3) (strong interaction). Maxwell’s equations are the prototype on which all three theories are constructed.

Maxwell’s Equations in Modern Technologies

Maxwell's equations are the foundation of all electromagnetic technology. Design of antennas, waveguides, and radio-frequency circuits is carried out using numerical solutions of these equations by the finite-difference time-domain (FDTD) and finite element methods. Wireless communications—from AM/FM radio to satellite communication and Wi-Fi standards—require precise understanding of the propagation of electromagnetic waves. In medical diagnostics, MRI uses variable magnetic fields to excite nuclear spins, and analysis of the field in tissues is carried out based on macroscopic Maxwell equations taking into account the dielectric permittivity of biological media. Optical fibers, which transmit internet traffic, use total internal reflection—a consequence of Maxwell boundary conditions at the glass–air interface. In laser technology, stimulated emission is described as interaction between a quantum oscillator and an electromagnetic field, and semiconductor lasers for fiber-optic data transmission operate precisely on the basis of Maxwell's equations in an active medium. Quantum electrodynamics extends Maxwell’s equations to the quantum level, but in the macroscopic limit returns to the classical four equations.

Assignment: (a) From ∇×B = μ₀(J + ε₀∂E/∂t) and ∇·E = ρ/ε₀ derive the continuity equation: ∂ρ/∂t + ∇·J = 0. (b) Monochromatic wave E = E₀ sin(kz − ωt) x̂. Find B. Check all 4 Maxwell equations in vacuum. (c) Calculate the Poynting vector S = (E×B)/μ₀ — averaged energy flux. For E₀ = 1 V/m: find the wave intensity.