Maxwell's Equations
Maxwell's equations are a set of four coupled partial differential equations that describe how electric and magnetic fields are generated and altered by electric charges, currents, and by each other, and together predict the existence of electromagnetic waves travelling at the speed of light.
Overview: the four equations (integral form)
| # | Name | What it says |
|---|---|---|
| 1 | Gauss's Law | Electric charge produces a diverging electric field |
| 2 | Gauss's Law for Magnetism | There are no magnetic monopoles — magnetic field lines always close on themselves |
| 3 | Faraday's Law of Induction | A changing magnetic field creates a circulating electric field |
| 4 | Ampère-Maxwell Law | Electric current and a changing electric field both create a circulating magnetic field |
1. Gauss's Law
| Symbol | Name | SI Unit |
|---|---|---|
| E | Electric field | V/m |
| ρ | Electric charge density | C/m³ |
| ε₀ | Vacuum permittivity | F/m |
Conditions of validity: Holds in vacuum and, with the substitution of the material's permittivity, inside dielectric media. Valid at all field strengths encountered in classical electromagnetism; at extremely high field strengths approaching the Schwinger limit, quantum electrodynamic corrections become relevant.
Derivation sketch (from Coulomb's law): Starting from Coulomb's law for a point charge, F = kq₁q₂/r², and defining the electric field as force per unit test charge, integrating the resulting field over any closed surface enclosing a charge distribution and applying the divergence theorem yields ∇·E = ρ/ε₀ directly — Gauss's law is mathematically equivalent to Coulomb's law, just expressed in differential rather than force form.
2. Gauss's Law for Magnetism
| Symbol | Name | SI Unit |
|---|---|---|
| B | Magnetic field | tesla (T) |
Conditions of validity: Holds universally in all classical electromagnetism experiments performed to date. No isolated magnetic monopole (a particle with net "magnetic charge") has ever been observed, though some grand unified theories predict their existence at energy scales far beyond current experimental reach.
3. Faraday's Law of Induction
| Symbol | Name | SI Unit |
|---|---|---|
| E | Electric field | V/m |
| B | Magnetic field | tesla (T) |
| t | Time | seconds (s) |
Conditions of validity: Valid for all classical (non-quantum) electromagnetic phenomena. This is the physical principle behind every electric generator, transformer, and induction motor: a changing magnetic flux through a loop of wire induces an electromotive force around that loop.
4. Ampère-Maxwell Law
| Symbol | Name | SI Unit |
|---|---|---|
| B | Magnetic field | tesla (T) |
| J | Electric current density | A/m² |
| μ₀ | Vacuum permeability | N/A² |
| E | Electric field | V/m |
Historical significance: The original Ampère's circuital law included only the current term (μ₀J). Maxwell's crucial 1861–1862 addition of the second term — the "displacement current," μ₀ε₀(∂E/∂t) — was necessary for mathematical self-consistency (charge conservation) and had the remarkable consequence of predicting self-sustaining electromagnetic waves, before any such wave had been observed experimentally.
Derivation of the electromagnetic wave equation
Taking the curl of Faraday's law, ∇ × (∇ × E) = −∂/∂t(∇ × B), and substituting the Ampère-Maxwell law (in vacuum, with J = 0) for ∇ × B, together with the vector identity ∇ × (∇ × E) = ∇(∇·E) − ∇²E and Gauss's law (∇·E = 0 in vacuum), yields:
This is the standard wave equation, with wave speed v = 1/√(μ₀ε₀). Substituting CODATA values for μ₀ and ε₀ gives v ≈ 299,792,458 m/s — exactly the measured speed of light. This numerical coincidence, discovered by Maxwell in 1862, was the first evidence that light itself is an electromagnetic wave.
Worked example
Confirm the wave-speed prediction: using μ₀ = 1.25663706127 × 10⁻⁶ N/A² and ε₀ = 8.8541878188 × 10⁻¹² F/m (CODATA 2022), compute v = 1/√(μ₀ε₀).
μ₀ε₀ = 1.25663706127 × 10⁻⁶ × 8.8541878188 × 10⁻¹² ≈ 1.11265 × 10⁻¹⁷
v = 1/√(1.11265 × 10⁻¹⁷) ≈ 2.998 × 10⁸ m/s — matching c to within measurement precision, as expected since ε₀ is now defined self-consistently with the exact value of c.
Related equations
- Electromagnetism domain hub [PENDING FULL BUILD]
- Speed of light, vacuum permittivity, vacuum permeability (Constants Reference)
- Optics & Photonics — light as an electromagnetic wave [PENDING FULL BUILD]
Primary Sources
- Maxwell, J.C. (1865). "A Dynamical Theory of the Electromagnetic Field." Philosophical Transactions of the Royal Society of London, 155, 459–512.
- Maxwell, J.C. (1873). A Treatise on Electricity and Magnetism. Oxford: Clarendon Press. Public domain.
- Griffiths, D.J. (2017). Introduction to Electrodynamics, 4th ed. Cambridge University Press, Ch. 7.
- Jackson, J.D. (1998). Classical Electrodynamics, 3rd ed. Wiley, Ch. 6.