Introduction to Maxwell's Equations
Over the previous six parts we built up four separate laws: Gauss's law (Part 2), Gauss's law for magnetism (Part 3), Faraday's law (Part 4) and Ampère's law with Maxwell's displacement current (Part 5 and Part 6). Together they are Maxwell's equations — the complete description of classical electromagnetism. With the Lorentz force law and the material properties of the medium, they explain every electric, magnetic and optical phenomenon outside quantum physics.
For the RF engineer, Maxwell's equations are the foundation under every design rule of thumb. Transmission-line theory, Smith charts, antenna patterns, waveguide modes and S-parameters are all solutions or consequences of these four equations under particular conditions. This article presents them together, explains what each one means, and shows how they fit into a single system.
Historical Development of Maxwell's Equations
| Year | Who | Contribution |
|---|---|---|
| 1785 | Charles-Augustin de Coulomb | Inverse-square law for the force between charges |
| 1813 / 1835 | Lagrange, Gauss | Divergence theorem and the flux law for electric fields (Gauss's work was published posthumously in 1867) |
| 1820 | Hans Christian Ørsted | Currents deflect compass needles — electricity and magnetism are linked |
| 1820–1826 | Biot, Savart, Ampère | Magnetic field of currents; Ampère's circuital law |
| 1831 | Michael Faraday (and Joseph Henry) | Electromagnetic induction; the concept of field lines |
| 1861–1865 | James Clerk Maxwell | Displacement current; unified theory; predicted electromagnetic waves travelling at the speed of light (A Dynamical Theory of the Electromagnetic Field, 1865) |
| 1873 | Maxwell | A Treatise on Electricity and Magnetism |
| 1884–1885 | Oliver Heaviside (also Hertz and Gibbs) | Rewrote Maxwell's twenty equations in compact vector notation — the four equations used today |
| 1887–1888 | Heinrich Hertz | Generated and detected radio waves, confirming Maxwell's prediction |
| 1895–1901 | Bose, Marconi, Popov and others | First practical radio and microwave experiments and transatlantic transmission |
Maxwell's original formulation used twenty equations in twenty unknowns, written component by component and partly in terms of potentials. It was Heaviside — a self-taught telegraph engineer — who condensed them into the four vector equations below. Heaviside also introduced much of transmission-line theory, so the equations RF engineers use, and the way they apply them to cables, both trace back to him.
Overview of the Four Maxwell's Equations
| # | Name | Integral form | Differential form |
|---|---|---|---|
| 1 | Gauss's law | ∮ 𝐃·d𝐀 = Q_enc | ∇·𝐃 = ρ_v |
| 2 | Gauss's law for magnetism | ∮ 𝐁·d𝐀 = 0 | ∇·𝐁 = 0 |
| 3 | Faraday's law | ∮ 𝐄·d𝐥 = −d/dt ∫ 𝐁·d𝐀 | ∇ × 𝐄 = −∂𝐁/∂t |
| 4 | Ampère–Maxwell law | ∮ 𝐇·d𝐥 = I_enc + d/dt ∫ 𝐃·d𝐀 | ∇ × 𝐇 = 𝐉 + ∂𝐃/∂t |
There is a clear structure. Equations 1 and 2 are divergence equations: they describe the sources of the fields (charges for 𝐃, none for 𝐁). Equations 3 and 4 are curl equations: they describe how the fields circulate, and they couple the electric and magnetic fields to each other through time derivatives. In static situations the time derivatives vanish and electricity and magnetism separate; when fields vary in time they are inseparable.
First Equation: Gauss's Law for Electricity
Integral Form
Differential Form
Physical Interpretation
Electric charges are the sources and sinks of electric flux. The net flux leaving any closed surface counts the free charge inside; at each point, the divergence of 𝐃 equals the local charge density. In charge-free regions — the dielectric of a cable, free space around an antenna — ∇·𝐃 = 0 and every flux line that enters a small volume leaves it.
Practical Example
Charge on a PCB Power–Ground Plane Pair
Problem: A 3.3 V power plane sits 0.1 mm above its ground plane in FR-4 (ε_r = 4.4). Find the field in the dielectric and the surface charge density on the planes.
That charge is the plane capacitance at work: C/A = ε/d ≈ 39 pF per cm². Thin, high-ε_r plane pairs are used as distributed decoupling capacitors that work up to frequencies where discrete capacitors are ineffective.
Second Equation: Gauss's Law for Magnetism
Integral Form
Differential Form
Physical Interpretation
There are no magnetic charges. Magnetic field lines always close on themselves, so as much magnetic flux enters any closed surface as leaves it. Magnetic flux is therefore conserved along any "tube" of field lines — through a transformer core, around a toroid, through a gap — which is the basis of magnetic circuit analysis.
Practical Example
Flux Through a Gapped Ferrite Core
Problem: A ferrite core (μ_r = 2000) with a small air gap carries a flux of 50 μWb. Core and gap both have a cross-section of 0.5 cm². Find B and H in the core and in the gap.
B is continuous across the gap (equation 2), but H jumps by a factor of μ_r. Almost all the magnetomotive force NI is "spent" across the gap, which is why a tiny gap controls the inductance and stores most of the energy. (A real ferrite saturates near 0.3–0.5 T, so this core would need fewer ampere-turns — the numbers just illustrate the principle.)
Third Equation: Faraday's Law
Integral Form
Differential Form
Physical Interpretation
A time-varying magnetic field creates an electric field that circulates around it. The EMF around any loop equals the rate of decrease of the magnetic flux through it, and the minus sign (Lenz's law) makes induced currents oppose the change. In static conditions the right-hand side is zero, 𝐄 is conservative, and voltage between two points is path-independent. With changing fields this is no longer true.
Practical Example
Mains Hum Induced in a Ground Loop
Problem: Two pieces of audio/RF equipment are connected by a shielded cable, and both are also grounded through their mains leads, forming a loop of 0.5 m². A nearby transformer produces a 50 Hz field of 10 μT rms through the loop. Find the induced voltage.
That is small but easily audible in a sensitive audio input, and the loop also picks up harmonics up to kilohertz, which induce proportionally more voltage. Breaking the loop (single-point grounding, isolation transformer) or shrinking its area removes the problem — Faraday's law at work.
Fourth Equation: Ampere-Maxwell Law
Integral Form
Differential Form
Physical Interpretation
Magnetic fields circulate around two kinds of source: conduction currents (moving charge) and displacement currents (changing electric flux). The displacement term makes the total current continuous — through capacitors and through empty space — and, together with Faraday's law, allows electromagnetic waves to propagate.
Practical Example
Displacement Current Through a 1 pF Capacitor at 1 GHz
Problem: A 1 pF capacitor (or a 1 pF parasitic capacitance between two traces) has 1 V peak at 1 GHz across it. Find the current through it and its reactance. Compare with 1 MHz.
A picofarad is negligible at 1 MHz but at 1 GHz it is a 159 Ω path — comparable to circuit impedances. This displacement current through parasitic capacitance is the origin of capacitive crosstalk and of pad and package parasitics in RF ICs.
Comparison of the Four Maxwell's Equations
| Equation | Operator | Source term | Describes | Static limit | Key RF use |
|---|---|---|---|---|---|
| Gauss (E) | Divergence | Charge density ρ_v | Where 𝐃 begins and ends | Coulomb's law | Capacitance, boundary conditions on conductors |
| Gauss (B) | Divergence | None (0) | 𝐁 has no sources | Same | Flux continuity, magnetic circuits |
| Faraday | Curl | −∂𝐁/∂t | How 𝐄 circulates | ∇ × 𝐄 = 0 (conservative 𝐄) | Induction, coupling, half of wave propagation |
| Ampère–Maxwell | Curl | 𝐉 + ∂𝐃/∂t | How 𝐇 circulates | Ampère's law | Inductance, currents on lines and antennas, other half of wave propagation |
The equations are almost symmetric between electricity and magnetism. The asymmetries — a charge density in equation 1 but zero in equation 2, a current 𝐉 in equation 4 but nothing equivalent in equation 3 — both reflect the absence of magnetic charges. In source-free regions (no ρ_v, no 𝐉) the symmetry is complete apart from a sign: ∇ × 𝐄 = −∂𝐁/∂t and ∇ × 𝐇 = +∂𝐃/∂t. That opposite sign is what makes the coupled fields oscillate and propagate rather than grow without limit.
Integral Form vs. Differential Form
The two forms carry the same physics and are converted into each other by two theorems of vector calculus:
| Integral form | Differential form | |
|---|---|---|
| Describes | Totals over surfaces, loops and volumes | Relationships at each point in space |
| Best for | Symmetric hand calculations; problems with discontinuities (surfaces, edges) | Deriving wave equations; numerical solvers (FEM, FDTD) |
| Handles interfaces | Directly — gives boundary conditions (Part 9) | Needs separate boundary conditions where fields jump |
| Typical use | Field of a coax, toroid, sphere; Faraday EMF of a coil | Wave equation (Part 8), waveguide modes, HFSS / CST meshes |
Electromagnetic Constitutive Relations
Maxwell's equations contain four field vectors — 𝐄, 𝐃, 𝐇, 𝐁 — plus the current density 𝐉. To solve them we need to know how a material responds to the fields. For linear, isotropic, homogeneous materials, the constitutive relations are:
Electric Permittivity
Permittivity ε describes how strongly a material polarises in an electric field. A higher ε_r stores more electric energy for a given field, increases capacitance, slows waves (v = c/√ε_r) and shrinks wavelengths — which is how high-ε_r ceramics make compact antennas and filters.
| Material | ε_r (approx., at RF) | tan δ (approx.) | Typical use |
|---|---|---|---|
| Vacuum / air | 1.0 / 1.0006 | 0 | Reference; air-line coax |
| PTFE (Teflon) | 2.1 | 0.0002–0.001 | Microwave coax, laminates |
| Polyethylene | 2.25 | 0.0002 | RG-58/RG-213 coax dielectric |
| Rogers RO4350B | 3.48 | 0.0037 | RF/microwave PCBs |
| FR-4 | 4.2–4.6 | 0.015–0.02 | Standard PCBs |
| Alumina (Al₂O₃) | 9.8 | 0.0001 | Thin-film microwave substrates |
| Water (distilled) | ≈ 78–80 (falls at GHz) | high at GHz | Why microwave ovens heat food |
Magnetic Permeability
Permeability μ describes how strongly a material responds to a magnetic field. Almost all RF dielectrics and conductors have μ_r ≈ 1. Magnetic materials are used where flux must be concentrated or guided: ferrites (μ_r ≈ 10–15,000, low loss up to hundreds of MHz depending on grade) for chokes, baluns and transformers; powdered iron (μ_r ≈ 10–100) for RF inductors; and silicon steel (μ_r of several thousand) for low-frequency transformers. Permeability of magnetic materials falls and loss rises with frequency, so each material has a usable frequency range.
Electrical Conductivity
Conductivity σ (S/m) sets how much conduction current flows for a given field. Good conductors confine RF currents to a thin skin depth δ = 1/√(πfμσ) at their surface (Part 8).
| Material | σ (S/m) | Skin depth at 1 GHz |
|---|---|---|
| Silver | 6.3 × 10⁷ | 2.0 μm |
| Copper | 5.8 × 10⁷ | 2.1 μm |
| Gold | 4.1 × 10⁷ | 2.5 μm |
| Aluminium | 3.5–3.8 × 10⁷ | 2.6–2.7 μm |
| Seawater | ≈ 4 | (lossy dielectric at 1 GHz — see Part 6) |
| FR-4, PTFE | ≈ 0 (losses via tan δ) | — |
How Maxwell's Equations Work Together
The four equations are not independent facts but a tightly connected system:
- The divergence equations are built into the curl equations. Take the divergence of Faraday's law: ∇·(∇ × 𝐄) = 0 = −∂(∇·𝐁)/∂t. So if ∇·𝐁 = 0 holds at one instant, Faraday's law keeps it true forever. Likewise, the divergence of the Ampère–Maxwell law combined with the continuity equation keeps Gauss's law true.
- Charge conservation is a consequence, not an extra assumption. As shown in Part 6, ∇·(Ampère–Maxwell) plus Gauss's law gives ∇·𝐉 = −∂ρ_v/∂t.
- The two curl equations generate waves. Taking the curl of one and substituting the other eliminates either 𝐄 or 𝐇, giving a wave equation with speed 1/√(με). This is derived in Part 8.
- Interfaces follow from the integral forms. Applying the four integral laws to vanishingly thin pillboxes and loops across a boundary gives the four boundary conditions of Part 9.
- Energy is conserved. Combining the curl equations gives Poynting's theorem: the power flowing out of a region through the Poynting vector 𝐒 = 𝐄 × 𝐇 equals the decrease of stored energy minus the power dissipated. 𝐒 (W/m²) is the power density of a radio wave.
Summary
- Maxwell's four equations — Gauss's law, Gauss's law for magnetism, Faraday's law and the Ampère–Maxwell law — are the complete laws of classical electromagnetism.
- The divergence equations describe sources: electric charge for 𝐃, no magnetic charge for 𝐁.
- The curl equations couple 𝐄 and 𝐇 through time variation and lead directly to electromagnetic waves.
- Integral forms suit symmetric hand calculations and interfaces; differential forms suit wave derivations and numerical solvers.
- The constitutive relations 𝐃 = ε𝐄, 𝐁 = μ𝐇, 𝐉 = σ𝐄 bring in the materials — dielectrics, magnetic cores and conductors.
In Part 8: Deriving the Electromagnetic Wave Equation, we combine the two curl equations to obtain the wave equation and calculate the speed, wavelength and impedance of electromagnetic waves in different media.
Put It Into Practice
Use the NEXT ANTENNAS calculators to apply the ideas from this article to real designs.