# Maxwell's equations **Maxwell's equations** are the four coupled partial differential equations that, with the [[Lorentz_force|Lorentz force]] law, constitute classical [[Electromagnetic_radiation|electromagnetism]]. They say that [[Electric_charge|electric charge]] is the source of the [[Electric_field|electric field]], that no magnetic charge exists, that a changing [[Magnetic_field|magnetic field]] drives a circulating electric field, and that currents *and* changing electric fields drive a circulating magnetic field.[^up2-ch16] The last clause is Maxwell's own addition, and it is what makes the set self-consistent and makes light fall out of it. In the microsim below the reader watches a plane electromagnetic wave travel through vacuum in three dimensions, with E and B drawn as two orthogonal field planes — perpendicular to each other, perpendicular to the direction of travel, and exactly in phase, so that `E = c·B` everywhere and always. One control sets the frequency on a logarithmic axis from 10³ to 10²⁰ Hz. The wavelength readout `lambda = c/f` shrinks as the frequency climbs and a marker slides along the [[Electromagnetic_spectrum|electromagnetic spectrum]] from radio through visible to gamma, the geometry unchanged throughout: the same four equations, the same `c = 1/sqrt(mu0·eps0) = 3.00×10⁸ m/s`, twenty-three decades of frequency. A second readout gives the [[Energy|energy]] flux `S = E·B/mu0`. Removing the displacement-current term collapses the wave, which is the demonstration the sim exists to make. On the [[Physics]] flagship this article serves Part I — History, the section *Maxwell's equations and light* (row P9): the hinge of the spine, with mechanics and separate electrical phenomena before it and optics, [[Special_relativity|relativity]] and [[Photon|photons]] after. ## Summary The four equations come in two equivalent presentations. The microscopic form uses only E and B and takes the total charge and current as sources, including the charges bound inside matter. The macroscopic form hides those bound charges inside two auxiliary fields, D and H, leaving only free charge and current explicit. Both are exact; the second is useful because the bound contributions in ordinary matter are neither known nor wanted in detail.[^up2-ch16] ### Microscopic version in SI units In vacuum or in matter treated atom by atom, the four read `div E = rho/eps0` ([[Gauss's_law|Gauss's law]]), `div B = 0` (Gauss's law for magnetism), `curl E = −∂B/∂t` ([[Faraday's_law_of_induction|Faraday's law]]) and `curl B = mu0·J + mu0·eps0·∂E/∂t` (the [[Ampère's_circuital_law|Ampère]]–Maxwell law).[^up2-ch16] Here ρ is total charge density and J total current density, with ε₀ = 8.854×10⁻¹² F/m and μ₀ = 4π×10⁻⁷ H/m. The eight scalar equations determine the six components of E and B given the sources and boundary conditions. ### Macroscopic version in SI units The matter form replaces two of the four: `div D = rho_f` and `curl H = J_f + ∂D/∂t`, with `D = eps0·E + P` and `H = B/mu0 − M`, for P the polarization and M the [[Magnetization|magnetization]].[^up2-ch16] The other two are unchanged, because neither has a source term to split. The price of the simplification is that D and H stay undetermined until constitutive relations are supplied. ## History of the equations The ingredients were experimental and separate. [[Coulomb's_law|Coulomb]] gave the inverse-square force between charges; Ørsted and Ampère the magnetic effect of currents; [[Michael_Faraday|Faraday]] induction and the picture of lines of force filling space. [[James_Clerk_Maxwell|Maxwell]] set out in *On Physical Lines of Force* (1861–62) to model that picture mechanically, with spinning cells and idler wheels, and found consistency required a further term: a current-like contribution from a changing electric field even where no charge moves.[^maxwell1862] The consequence was immediate. The resulting equations supported transverse waves whose speed, computed from the ratio of electromagnetic to electrostatic units measured by Weber and Kohlrausch, came out equal within experimental error to the measured speed of light. Maxwell concluded that light consists of transverse undulations of the same medium responsible for electric and magnetic phenomena.[^maxwell1862] He dropped the mechanical scaffolding in *A Dynamical Theory of the Electromagnetic Field* (1865) and restated the theory in the *Treatise* of 1873.[^maxwell1865][^treatise] Maxwell wrote twenty equations in twenty variables. The compact set of four used today is due to Oliver Heaviside, who recast them in the 1880s using the vector calculus he was developing for the purpose, with Heinrich Hertz arriving independently at a similar form.[^heaviside] Hertz then did the decisive experiment, generating and detecting the predicted waves between 1886 and 1889 and showing they reflected, refracted and interfered like light.[^hertz] ## Conceptual descriptions Each equation has a plain statement that survives the loss of the notation, and together they say what an electromagnetic field can and cannot do. ### Gauss's law Electric field lines begin on positive charge and end on negative charge, and the net flux of E through any closed surface is the enclosed charge divided by ε₀.[^up2-ch7] [[Coulomb's_law|Coulomb's]] inverse square is the special case of a point charge and a concentric sphere; the general statement applies to any surface at all, which is why it is the fastest route to the field of a charged plane, line or shell. Its content is that charge is the source of E. ### Gauss's law for magnetism The magnetic flux through any closed surface is zero: `div B = 0`. Field lines never begin or end, only close on themselves, and cutting a bar magnet in half yields two magnets rather than an isolated north pole.[^up2-ch12] This is the one equation of the four that is a pure statement of absence — no magnetic charge has ever been observed — and the one most often examined for exceptions. ### Faraday's law A magnetic field changing in time is encircled by an electric field: `curl E = −∂B/∂t`. Integrated around a circuit this is the flux rule, `EMF = −dPhi/dt`, and the minus sign is [[Lenz's_law|Lenz's law]] — the induced effect opposes its cause.[^up2-ch13] The induced E field is not conservative, so it is not the gradient of a potential, and "voltage" loses its path-independence inside a region of changing flux. ### Ampère–Maxwell law A magnetic field circulates around a current — [[Ampère's_circuital_law|Ampère's]] original law — and also around a changing electric field, which is Maxwell's addition.[^up2-ch16] The standard argument for its necessity is a capacitor being charged: an Ampèrian loop spanned by a flat surface encloses the wire's current, while the same loop spanned by a surface bulging between the plates encloses none. Without the extra term the law gives two answers for one loop; with `mu0·eps0·∂E/∂t` supplying exactly the missing amount, the two agree. ## Microscopic formulation in terms of electric and magnetic fields (in vacuum version) Setting ρ and J to zero leaves the vacuum equations, the ones the microsim solves. The two divergence equations then say E and B are both transverse — no component along the direction of travel — and the two curl equations couple them so neither can change without the other. No freedom remains to choose the relative amplitude or phase: both are fixed by the equations. In the Gaussian unit system the same physics is written with ε₀ and μ₀ absorbed and c explicit, which makes the symmetry between E and B more visible at the cost of a less direct link to laboratory units. ## Relationship between differential and integral formulations Each equation has two faces. The differential form is a statement at a point: divergence measures how much field springs from an infinitesimal volume, curl how much it circulates around an infinitesimal loop. The integral form is a statement about a finite surface or loop: flux through a closed surface, circulation around a closed curve. The divergence theorem converts between the first pair and Stokes's theorem between the second, so nothing is added or lost.[^up2-ch7] Which face is useful depends on the symmetry available. Where the field is constant over a well-chosen surface, the integral form gives the answer in one line — this is how Gauss's law yields the field of a sphere or a plane. Absent that symmetry, the differential form feeds a numerical solver. ## Charge conservation Charge conservation is not a fifth equation: it is already inside the four. Taking the divergence of the Ampère–Maxwell law, and using the fact that the divergence of a curl vanishes identically, gives `div J + ∂rho/∂t = 0` — the [[Continuity_equation|continuity equation]], which says the current flowing out of any closed region equals the rate at which the charge inside it falls.[^up2-ch16] Charge is conserved not merely globally but locally: it cannot vanish here and reappear there without something crossing the space between. Historically the implication ran the other way. It was the incompatibility of the original Ampère law with charge conservation in a circuit containing a capacitor that forced the displacement term, so the continuity equation is as much the motivation for the fourth equation as its consequence. ## Vacuum equations, electromagnetic waves and speed of light Take the curl of Faraday's law in empty space, substitute the Ampère–Maxwell law, and each field satisfies the same [[Wave_equation|wave equation]] with propagation speed `c = 1/sqrt(mu0·eps0)`.[^up2-ch16] Putting in ε₀ = 8.854×10⁻¹² F/m and μ₀ = 4π×10⁻⁷ H/m gives 2.998×10⁸ m/s (derived): the measured [[Speed_of_light|speed of light]]. Nothing about light was assumed in the derivation; two constants from electrostatics and magnetostatics produced it. The [[Plane_wave|plane-wave]] solution is what the sim animates. E and B are perpendicular to each other and to the direction of travel, in phase, amplitudes locked by `E = c·B` — so the magnetic amplitude in SI units is smaller by about 3×10⁸, which is why the electric field does nearly all the work on matter. The frequency slider changes only the wavelength, through `lambda = c/f`: 300 m at 1 MHz, about 600 nm in the green, 3 pm at 10²⁰ Hz (derived). The geometry on screen is identical at every setting, and that invariance is the point — radio, light and gamma rays are one phenomenon in three ranges. The energy readout is the Poynting vector, `S = E·B/mu0`, the power crossing unit area. A worked case anchors it: sunlight at the Earth's orbit delivers about 1,349 W/m² on the [[Black-body_radiation|blackbody]] estimate in the Portal Books, which corresponds to a root-mean-square electric field near 713 V/m and a magnetic field near 2.4 µT (derived).[^ochsner-solar] Ordinary laboratory magnitudes, worth seeing beside the extraordinary speed. ## Macroscopic formulation in terms of displacement and magnetizing fields (in matter version) Inside matter, an applied field displaces bound charge and aligns molecular currents. The bound charge density `rho_b = −div P` and bound current density `J_b = curl M + ∂P/∂t` are real sources, but counting them individually is hopeless. The macroscopic equations absorb them into D and H, leaving only free charge and current in view.[^up2-ch16] The move is exact, not an approximation; what it costs is closure. Closure comes from constitutive relations linking D to E and H to B. For a linear, isotropic, non-dispersive medium these are `D = eps·E` and `B = mu·H`, the wave speed becomes `v = 1/sqrt(mu·eps)`, and the [[Refractive_index|refractive index]] is `n = sqrt(eps·mu/(eps0·mu0))`. Real materials are less obliging: ε depends on frequency (dispersion, and therefore the rainbow), on field strength in nonlinear optics, and on direction in a crystal. [[Superconductivity|Superconductors]] and [[Plasma_(physics)|plasmas]] need different relations again. ## Alternative formulations The same content can be packaged several ways. Writing `B = curl A` and `E = −grad phi − ∂A/∂t` satisfies two of the four identically and reduces the set to two equations for the potentials, at the cost of a gauge freedom — A and φ are not unique — which is a nuisance classically and essential quantum-mechanically. In [[Special_relativity|relativistic]] notation the fields become one antisymmetric tensor and the four equations become two, making [[Lorentz_transformation|Lorentz]] covariance manifest rather than accidental.[^cline-rel] A formulation in differential forms compresses them further and generalises to curved spacetime, and a Lagrangian formulation derives them from an action principle.[^cline-field] ## Solutions Few problems admit closed-form solutions, and those that do are the ones every course teaches: plane waves, the fields of static charge and steady current, waveguides and cavities with simple cross-sections, radiation from an oscillating dipole. The general linear problem is solved with Green's functions, which give the retarded potentials — the field at a point now, built from the sources at the earlier times light needed to arrive from each. Everything else is numerical: finite-difference time-domain methods step the two curl equations forward in alternating half-steps on interleaved grids. ## Overdetermination of Maxwell's equations Counted naively there are eight scalar equations for six unknown field components, which looks like two too many. The excess is only apparent: the two divergence equations are constraints on the initial data rather than independent evolution equations. Taking the time derivative of `div B = 0` and using Faraday's law shows that if it holds at one instant it holds forever, and the same argument applied to `div E = rho/eps0` works provided charge is conserved. The dynamics is carried by the two curl equations, which preserve the constraints. ## Maxwell's equations as the classical limit of QED Classical electromagnetism is not the final theory of the field. In quantum electrodynamics the field is quantized and its excitations are [[Photon|photons]]; Maxwell's equations emerge as the classical limit, holding when photon numbers are large and the state is coherent.[^up3-photons] The limit is excellent: corrections such as light-by-light scattering, in which two photons interact through a virtual electron loop, are negligible at laboratory field strengths, and the classical equations stay exact for engineering purposes across the whole spectrum the sim spans. Where the classical description does fail is at the level of individual quanta. The [[Photoelectric_effect|photoelectric effect]] cannot be explained by a continuous wave of any intensity, and that failure — not any flaw in the four equations — is what opened the twentieth century. ## Variations The equations have been probed for modifications, and the probes are informative even when they find nothing. ### Magnetic monopoles Adding a magnetic charge density would make `div B = rho_m` and put a magnetic current in Faraday's law, restoring a symmetry the equations otherwise lack. Dirac showed in 1931 that even one monopole anywhere would force electric charge to be quantized — an argument that explains an otherwise unexplained fact, and the main motivation for searching.[^dirac1931] None has been found, and `div B = 0` stands. ### Axions Axion electrodynamics adds a term coupling a hypothetical light pseudoscalar field to the product of E and B, which mixes the source equations and would rotate the polarization of light crossing a magnetic field.[^wilczek1987] The same coupling appears in the surface physics of topological insulators, so the formalism has uses whether or not the particle exists. ### Other dimensions Maxwell's equations are not tied to three spatial dimensions. In the language of differential forms they generalize to any number, with the field a two-form and the source a current; the observable consequence is that the static field of a point charge falls as 1/r^(d−2), so the familiar inverse square is a statement about d = 3. Tests of that law at short range are also tests of the dimensionality of space. ## See also - [[Electromagnetic_radiation]] - [[Electromagnetic_spectrum]] - [[Speed_of_light]] - [[James_Clerk_Maxwell]] - [[Faraday's_law_of_induction]] - [[Gauss's_law]] - [[Ampère's_circuital_law]] - [[Lorentz_force]] ## Explanatory notes Three points of usage recur. *Displacement current* is not a current: nothing flows, and the name records only that the term enters the Ampère–Maxwell law where a current density does. *Microscopic* and *macroscopic* label two presentations of one theory, neither an approximation to the other; the difference is whether bound charge is shown or absorbed into D and H. And the unit system changes symbols, not physics: this page uses SI, so ε₀ and μ₀ appear where a Gaussian text shows 4π and c. Full citations are under References. ## References [^up2-ch16]: Sanny, Jeff; Ling, Samuel, et al. (2016). *University Physics Volume 2*. OpenStax, CC BY. Chapter 16, "Electromagnetic Waves", pp. 669–710 (page to pin) — Maxwell's equations as a set, the displacement-current term and the capacitor argument, the plane-wave solution with E ⊥ B ⊥ k and E = c·B, the wave speed c = 1/√(μ₀ε₀), and the Poynting vector. Portal Book 078, https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2. The text extraction of Portal Book 078 lost nearly every displayed equation; every equation quoted on this page from Volume 2 is the standard, unambiguous form supplied here and should be verified against the PDF pages before the page numbers are frozen. [^up2-ch7]: *University Physics Volume 2* (2016), Chapter 6, "Gauss's Law", pp. 231–278 (page to pin), for the integral and differential statements of Gauss's law, the divergence theorem and the symmetric worked cases (sphere, line, plane). Portal Book 078. Same extraction caveat. [^up2-ch12]: *University Physics Volume 2* (2016), Chapter 12, "Sources of Magnetic Fields", pp. 513–556 (page to pin), for the Biot–Savart and Ampère laws and the absence of magnetic charge. Portal Book 078. Same extraction caveat. [^up2-ch13]: *University Physics Volume 2* (2016), Chapter 13, "Electromagnetic Induction", pp. 557–602 (page to pin), for Faraday's and Lenz's laws in circuit and field form. Portal Book 078. Same extraction caveat. [^up3-photons]: Sanny, Jeff; Ling, Samuel, et al. (2016). *University Physics Volume 3*. OpenStax, CC BY. Chapter 6, "Photons and Matter Waves", pp. 241–294 (page to pin), for the photon and the limits of the classical wave description. Portal Book 079, https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3. Volume 3's extraction likewise lost most displayed equations and worked numbers; nothing numerical is taken from it here. [^ochsner-solar]: Sub-manual 10 (*Earth and Energy*), §5.1, computing from the unworked problems in Ochsner, Tyson (2019), *Rain or Shine*, p. 283 (Portal Book 119, https://open.umn.edu/opentextbooks/textbooks/rain-or-shine): the Sun at ε = 0.990 emits 6.27×10⁷ W/m² over a radius of 6.96×10⁸ m, which is 3.81×10²⁶ W, and at 1.50×10¹¹ m gives 1,349 W/m². The conversion of that flux into field amplitudes — E_rms ≈ 713 V/m and B_rms ≈ 2.4 µT through ⟨S⟩ = E_rms²/(μ₀·c) and E = c·B — is derived here and appears in no book. So are c = 2.998×10⁸ m/s from ε₀ and μ₀, and the wavelengths 300 m, ≈600 nm and 3 pm. [^maxwell1862]: Maxwell, James Clerk (1861–1862). "On Physical Lines of Force." *Philosophical Magazine*, fourth series, in four parts (volume and page ranges to pin). Part III introduces the displacement term and computes the wave speed from the electromagnetic-to-electrostatic unit ratio measured by Wilhelm Weber and Rudolf Kohlrausch, concluding that light consists of transverse undulations of the same medium that carries electric and magnetic effects. [^maxwell1865]: Maxwell, James Clerk (1865). "A Dynamical Theory of the Electromagnetic Field." *Philosophical Transactions of the Royal Society of London* 155: 459–512. [^treatise]: Maxwell, James Clerk (1873). *A Treatise on Electricity and Magnetism*, 2 vols. Oxford: Clarendon Press. [^heaviside]: Heaviside, Oliver (1892). *Electrical Papers*, 2 vols. London: Macmillan. The collection in which Heaviside's vector-calculus reformulation of Maxwell's theory, developed from 1884, was gathered; Heinrich Hertz reached a comparable four-equation form independently in the same decade. [^hertz]: Hertz, Heinrich (1893). *Electric Waves: Being Researches on the Propagation of Electric Action with Finite Velocity through Space*, trans. D. E. Jones. London: Macmillan. The collected papers of the 1886–1889 experiments generating, detecting, reflecting and refracting the predicted waves. [^cline-rel]: Cline, Douglas (2018). *Variational Principles in Classical Mechanics*, revised 2nd ed. Chapter 17, "Relativistic mechanics", pp. 481–508 (page to pin), for four-vector notation and the manifestly covariant treatment of electromagnetism. Portal Book 073, https://open.umn.edu/opentextbooks/textbooks/variational-principles-in-classical-mechanics [^cline-field]: Cline (2018), *Variational Principles in Classical Mechanics*, Chapter 16, "Analytical formulations for continuous systems", pp. 463–480 (page to pin), for Lagrangian and Hamiltonian field theory, the setting in which Maxwell's equations follow from an action principle. Portal Book 073. [^dirac1931]: Dirac, Paul A. M. (1931). "Quantised Singularities in the Electromagnetic Field." *Proceedings of the Royal Society of London A* 133 (821): 60–72. [^wilczek1987]: Wilczek, Frank (1987). "Two Applications of Axion Electrodynamics." *Physical Review Letters* 58 (page to pin). The paper introducing the θ·E·B coupling and its optical and condensed-matter consequences. ## Further reading - Sanny, Jeff; Ling, Samuel, et al., *University Physics Volume 2* (2016) — Chapters 6, 12, 13 and 16 cover the four equations in the order this page takes them, with the worked geometries. Portal Book 078, CC BY. - Cline, Douglas, *Variational Principles in Classical Mechanics*, revised 2nd ed. (2018) — Chapters 16 and 17 for the field-theoretic and covariant formulations. Portal Book 073. - Ochsner, Tyson, *Rain or Shine* (2019), Chapter on radiation — the solar flux used above as the worked case for the Poynting vector. Portal Book 119. ### Historical publications - Maxwell, James Clerk, "On Physical Lines of Force" (1861–62) — the mechanical model, the displacement term, and the identification of light. - Maxwell, James Clerk, "A Dynamical Theory of the Electromagnetic Field" (1865) — the theory without the scaffolding. - Maxwell, James Clerk, *A Treatise on Electricity and Magnetism* (1873) — the full statement in twenty equations. - Heaviside, Oliver, *Electrical Papers* (1892) — the reduction to four vector equations. - Hertz, Heinrich, *Electric Waves* (1893) — the experimental confirmation. ## External links ### Modern treatments - [*University Physics Volume 2*](https://openstax.org/books/university-physics-volume-2) — OpenStax, CC BY; Chapter 16 is the electromagnetic-waves chapter this page follows (Portal Book 078) - [*Variational Principles in Classical Mechanics*](https://open.umn.edu/opentextbooks/textbooks/variational-principles-in-classical-mechanics), Douglas Cline — the covariant and Lagrangian formulations (Portal Book 073) ### Other - [*Rain or Shine*](https://open.umn.edu/opentextbooks/textbooks/rain-or-shine), Tyson Ochsner — the radiation chapter behind the solar-flux worked case (Portal Book 119) - Historical scans, archives and lecture series are listed in the Wikipedia pair's *External links*; none is reproduced here until its URL has been checked. <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Maxwell's_equations.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Maxwell's equations* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Maxwell's_equations.html" data-title="Maxwell's equations"></div> *Built from `MICROSIM_GUIDE/specs/sims/Maxwell's_equations.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Maxwell's_equations) : [Wikitube](https://en.wikitube.io/wiki/Maxwell's_equations) · pinned revision [1373547428](https://en.wikipedia.org/w/index.php?oldid=1373547428) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. 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