Maxwell's equations in differential form

Intuition [Beginner]

Electricity and magnetism look like two phenomena. A static charge produces an electric force on other charges; a moving charge — a current — produces a magnetic force on other moving charges. For two centuries after Coulomb (1785) these were studied as separate sciences. Maxwell's accomplishment in the 1860s was to write down four equations, now called Maxwell's equations, that together describe both as facets of a single field — the electromagnetic field — and to predict that disturbances of this field propagate through empty space as waves moving at a specific speed. That speed turned out, on calculation, to equal the measured speed of light. Light is electromagnetic radiation.

Two vector fields, $E$ and $B$, fill space and time. $E$ is the electric field: charges push and pull on each other through it. $B$ is the magnetic field: moving charges and currents wrap around each other through it. Maxwell's equations are four rules about how these fields are sourced, and how they interact when they change in time.

In words rather than symbols, the four rules are:

1. Gauss's law for $E$. Electric field lines start and end on electric charges. The total electric flux out of any closed surface counts the charge enclosed (up to a fixed constant). Positive charges are sources of $E$; negative charges are sinks.

2. No magnetic monopoles. Magnetic field lines have no starting or ending points. They form closed loops. The total magnetic flux out of any closed surface is zero. There are no isolated north or south poles — only dipoles.

3. Faraday's law of induction. A changing magnetic field produces a circulating electric field. If you push a magnet through a loop of wire, an electric field appears in the wire that drives current around the loop. The induced $E$ wraps around the changing $B$ in the opposite sense.

4. Ampère-Maxwell law. A circulating magnetic field is produced both by electric currents and by a changing electric field. The first part is Ampère's contribution (1820s); the second part — a changing $E$ creates $B$ even with no current — is Maxwell's own correction (1865), and it is what makes electromagnetic waves possible.

The two italicised lines in (3) and (4) are mirrors of each other. A changing $B$ makes a circulating $E$; a changing $E$ makes a circulating $B$. Once you have a moving wave of $E$, it produces a moving $B$, which produces $E$, and so on indefinitely. The disturbance propagates. Solving the equations in empty space gives a wave that travels at speed $$ c ;=; \frac{1}{\sqrt{\mu_0 \varepsilon_0}}, $$ where ${\mu }_0$ and ${\epsilon }_0$ are constants measurable in tabletop experiments with currents and capacitors. Maxwell computed this number and it matched the speed of light. The identification of light with electromagnetic radiation was the first major unification in physics.

The deepest payoff is what the equations imply about frames of reference. A pure electric field in one frame becomes a mixture of electric and magnetic in another frame moving relative to the first. A pure magnetic field becomes a mixture too. The split between $E$ and $B$ is observer-dependent; only the combined electromagnetic field is frame-independent. This is the physical reason special relativity sits inside Maxwell's theory: the equations are already relativistic before Einstein wrote them down. Einstein's 1905 paper was titled On the Electrodynamics of Moving Bodies because that was the puzzle Maxwell's equations forced.

The formal symbolic statement is in the Formal definition section at the Intermediate tier. The four words-of-rules above are the Beginner-tier statement.

Visual [Beginner]

Picture a positive charge sitting at the origin. Electric field lines radiate outward from it in every direction, like the spokes of a wheel — drawing more lines where the field is stronger, fewer where it is weaker. Gauss's law says the total number of lines leaving any closed surface around the charge equals the charge inside (times a constant). The lines have to come from somewhere; that somewhere is the charge.

Now picture a long straight wire carrying a current. Magnetic field lines circle the wire — concentric circles in the plane perpendicular to the wire, with direction given by the right-hand rule (thumb along the current, fingers curl in the direction of $B$). The lines never start or stop; they close on themselves. Wrap a closed surface around any region: the same number of magnetic field lines enter as leave. That is the no-monopoles law.

Now picture a bar magnet being pushed into a circular loop of wire. As the magnet enters, the magnetic flux through the loop increases. Faraday's law says: an electric field is induced around the loop, circulating in the direction that would drive a current opposing the change (Lenz's rule). Pull the magnet out and the induced $E$ reverses. The induced circulating $E$ does not start or end on charges — it is a non-conservative field, the kind that powers electric generators.

Finally, the most subtle picture. A capacitor charging up between two plates: current flows through the wires, charge accumulates on the plates, and an electric field builds up between them. There is no current in the gap between the plates. But the changing $E$ between the plates produces exactly the same circulating $B$ as a current would — Maxwell's displacement current. This is the term that closes the system and lets disturbances propagate as waves.

Put the four pictures together. A wave of $E$ oscillating in the $y$-direction, travelling in the $x$-direction, drives by Faraday a wave of $B$ oscillating in the $z$-direction, which drives by Ampère-Maxwell a wave of $E$ in the $y$-direction, and so on. The pair propagates together at speed $c$. That picture is light.

Worked example [Beginner]

A plane electromagnetic wave travels in the $+x$-direction in empty space. Its electric field points in the $+y$-direction and oscillates as $$ E_y(x, t) ;=; E_0 \cos(kx - \omega t), $$ where $E_0$, $k$, and $\omega$ are positive numbers. Find the magnetic field that goes with it and identify the wave speed.

Step 1. Faraday's law in words: a changing $B$ produces a circulating $E$. The reverse reading (Ampère-Maxwell with no current) says a changing $E$ produces a circulating $B$. For a plane wave moving in $+x$, the right-hand rule pairs $E$ in the $+y$-direction with $B$ in the $+z$-direction (so that $E\times B$ points in the direction of propagation $+x$).

Step 2. Guess the form $B_z(x,t)=B_0\cos (kx-\omega t)$ with the same phase as $E_y$ (this is the standard plane-wave pairing in vacuum; the full justification comes from the wave equation in the Intermediate tier).

Step 3. The Ampère-Maxwell law in vacuum, in plain words: the circulation of $B$ around a small loop equals ${\mu }_0{\epsilon }_0$ times the rate of change of $E$-flux through the loop. Taking a small rectangular loop in the $xy$-plane and tracking signs carefully, the law forces $$ B_0 ;=; \frac{E_0}{c}, \qquad \omega ;=; c k, \qquad c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}. $$ (The Intermediate-tier derivation of the wave equation produces this relation cleanly; here it is stated as a quoted consequence.)

Step 4. Plug in numbers. The measured values are ${\epsilon }_0\approx 8.85\times 10^{-12}F/m$ and ${\mu }_0=4\pi \times 10^{-7}H/m$, giving $$ c ;=; \frac{1}{\sqrt{(8.85 \times 10^{-12})(1.26 \times 10^{-6})}} ;\approx; 3.00 \times 10^{8}\ \mathrm{m/s}. $$ This is the speed of light. The wave's $E$ and $B$ are perpendicular to each other and both perpendicular to the direction of propagation; they oscillate in phase; and the ratio of their amplitudes is fixed at $E_0/B_0=c$.

What this tells us: starting from the qualitative rules in the Intuition section, the wave speed and the geometric structure of the wave (transverse, $E\perp B$, $|E|/|B|=c$) drop out as forced consequences. Maxwell's calculation in the 1860s was this calculation, run on the freshly-completed set of four equations: out came the speed of light, with no input from optics.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Choose Cartesian coordinates on ${\mathbb{R}}^3$ and take $E,B:{\mathbb{R}}^3\times \mathbb{R}\to {\mathbb{R}}^3$ to be smooth vector-valued functions of position and time. Let $\rho :{\mathbb{R}}^3\times \mathbb{R}\to \mathbb{R}$ be the charge density and $J:{\mathbb{R}}^3\times \mathbb{R}\to {\mathbb{R}}^3$ the current density. In SI units, with ${\epsilon }_0$ the vacuum permittivity and ${\mu }_0$ the vacuum permeability, Maxwell's equations are $$ \nabla \cdot E ;=; \frac{\rho}{\varepsilon_0}, \qquad \nabla \cdot B ;=; 0, $$ $$ \nabla \times E ;=; -\frac{\partial B}{\partial t}, \qquad \nabla \times B ;=; \mu_0 J + \mu_0 \varepsilon_0 \frac{\partial E}{\partial t}. $$ The first two are the homogeneous-source and the no-monopoles equations (Gauss for $E$, Gauss for $B$); the second two are the induction and the Ampère-Maxwell equations.

Charge conservation is a consequence, not an independent input. Take the divergence of Ampère-Maxwell: $$ \nabla \cdot (\nabla \times B) ;=; 0 ;=; \mu_0 \nabla \cdot J + \mu_0 \varepsilon_0 \frac{\partial}{\partial t}(\nabla \cdot E) ;=; \mu_0 \nabla \cdot J + \mu_0 \frac{\partial \rho}{\partial t}, $$ which gives the continuity equation $$ \frac{\partial \rho}{\partial t} + \nabla \cdot J ;=; 0. $$ The displacement-current term ${\mu }_0{\epsilon }_0{\partial }_{t}E$ in Ampère-Maxwell is exactly what makes this work: without it, charge conservation would have to be added as a separate axiom, and the four equations would be inconsistent on time-dependent sources.

Wave equation in vacuum. Set $\rho =0$, $J=0$. Take the curl of Faraday and use $\nabla \times (\nabla \times E)=\nabla (\nabla \cdot E)-{\nabla }^{2}E=-{\nabla }^{2}E$ (using Gauss for $E$). Substitute Ampère-Maxwell on the right: $$ \nabla^2 E - \mu_0 \varepsilon_0 \frac{\partial^2 E}{\partial t^2} ;=; 0. $$ Identical reasoning for $B$ gives the same equation. So in vacuum, each Cartesian component of $E$ and $B$ obeys the homogeneous wave equation with propagation speed $c=1/\sqrt{{\mu }_0{\epsilon }_0}$. Maxwell's identification of light as an electromagnetic wave is precisely the numerical agreement $c\approx 3\times 10^8$ m/s between this calculated speed and the measured speed of light.

Potentials. The homogeneous Maxwell equations $\nabla \cdot B=0$ and $\nabla \times E+{\partial }_{t}B=0$ are integrability conditions. The first says $B$ is divergence-free; locally there exists a vector field $A$ with $B=\nabla \times A$, the magnetic vector potential. Substituting into Faraday, $\nabla \times (E+{\partial }_{t}A)=0$, so locally $E+{\partial }_{t}A=-\nabla \phi$ for some scalar electric potential $\phi$: $$ E ;=; -\nabla \varphi - \frac{\partial A}{\partial t}, \qquad B ;=; \nabla \times A. $$ The potentials are not unique. The transformation $$ \varphi ;\to; \varphi - \frac{\partial \chi}{\partial t}, \qquad A ;\to; A + \nabla \chi $$ for any smooth function $\chi (x,t)$ leaves $E$ and $B$ unchanged. This is the gauge transformation, and the freedom to choose $\chi$ is gauge invariance.

Covariant formulation. Introduce the Minkowski metric ${\eta }_{\mu \nu }=\mathrm{diag}(+,-,-,-)$ on ${\mathbb{R}}^{1,3}$, spacetime coordinates $x^{\mu }=(ct,x,y,z)$ with index $\mu =0,1,2,3$, and the four-potential $A^{\mu }=(\phi /c,A)$. The electromagnetic field tensor (or Faraday tensor) is the antisymmetric two-index tensor $$ F_{\mu\nu} ;:=; \partial_\mu A_\nu - \partial_\nu A_\mu, $$ with $A_{\mu }={\eta }_{\mu \nu }{A}^{\nu }$ lowering the index. Computing components, $$ F_{0i} ;=; -E_i / c, \qquad F_{ij} ;=; -\varepsilon_{ijk} B_k \quad (i, j, k = 1, 2, 3), $$ where ${\epsilon }_{ijk}$ is the totally antisymmetric Levi-Civita symbol with ${\epsilon }_{123}=+1$. The 6 independent components of the antisymmetric $F_{\mu \nu }$ in 4 dimensions are exactly the 3 components of $E$ plus the 3 components of $B$. The split between $E$ and $B$ depends on the choice of timelike direction; the combined object $F_{\mu \nu }$ is a Lorentz tensor.

The four-current is $J^{\mu }=(c\rho ,J)$. Maxwell's equations become two tensor equations: $$ \partial_\mu F^{\mu\nu} ;=; \mu_0 J^\nu, \qquad \partial_{[\rho} F_{\mu\nu]} ;=; 0, $$ where square brackets denote antisymmetrisation. The first encodes Gauss for $E$ and Ampère-Maxwell; the second encodes Gauss for $B$ and Faraday. Charge conservation ${\partial }_{\mu }{J}^{\mu }=0$ follows from antisymmetry of $F^{\mu \nu }$ in the first equation.

Gauge invariance in covariant form is $$ A_\mu ;\to; A_\mu + \partial_\mu \chi, $$ which leaves $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$ unchanged because partial derivatives commute.

Differential-forms preview. The field tensor $F_{\mu \nu }$ is the component expression of a 2-form on Minkowski space, $$ F ;=; \tfrac{1}{2} F_{\mu\nu}, dx^\mu \wedge dx^\nu ;\in; \Omega^2(\mathbb{R}^{1,3}). $$ The four-potential is a 1-form $A=A_{\mu }d{x}^{\mu }\in {\Omega }^1({\mathbb{R}}^{1,3})$, and the relation $F=dA$ is the exterior-calculus statement of $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$. The two Maxwell tensor equations become two differential-form equations, $dF=0$ and $d\star F=\star J$, where $\star$ is the Hodge star of the Minkowski metric and $J=J_{\mu }d{x}^{\mu }$ is the current 1-form. The full development belongs to the Master tier.

Counterexamples to common slips

• The displacement current ${\mu }_0{\epsilon }_0{\partial }_{t}E$ in Ampère-Maxwell is not an electric current of physical charges. It is a term whose presence in the equation is forced by the requirement that the system be self-consistent under charge conservation. Maxwell introduced it on theoretical grounds (1865) before it could be experimentally tested. Hertz's 1887 experiments verifying electromagnetic waves were the experimental confirmation.

• Gauge transformations are not a symmetry of the fields $E$ and $B$ — those are invariant by construction. They are a redundancy of the description in terms of potentials $(\phi ,A)$. Two gauge-equivalent potentials give identical $E$, $B$, and identical physics. The redundancy becomes a real symmetry only when potentials are coupled to charged matter (where the wavefunction also transforms, $\psi \to e^{ie\chi /\hslash }\psi$).

• The Minkowski sign convention $\eta =\mathrm{diag}(+,-,-,-)$ chosen here is mostly-minus. The mostly-plus convention $\eta =\mathrm{diag}(-,+,+,+)$ used by Misner-Thorne-Wheeler and most GR texts flips the sign of $F_{0i}=-E_i/c$ to $F_{0i}=+E_i/c$ and several derived expressions accordingly. Within this unit, mostly-minus is in force throughout; cross-citations may need a sign translation.

• Maxwell in matter (with polarization $P$ and magnetization $M$) introduces auxiliary fields $D={\epsilon }_{0}E+P$ and $H=B/{\mu }_0-M$; the macroscopic equations $\nabla \cdot D={\rho }_{\mathrm{free}}$ and $\nabla \times H=J_{\mathrm{free}}+{\partial }_{t}D$ look formally identical but distinguish free and bound sources. The microscopic (vacuum) form stated here is the fundamental theory; the macroscopic form is a constitutive averaging — see Jackson Ch. 6.

Key theorem with proof [Intermediate+]

Theorem (Wave equation and the speed of light). In a source-free region of vacuum, every Cartesian component of $E$ and $B$ satisfies the homogeneous wave equation $$ \Box \psi ;=; 0, \qquad \Box ;:=; \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2, $$ with propagation speed $c=1/\sqrt{{\mu }_0{\epsilon }_0}$. Plane-wave solutions are transverse ($E$ and $B$ perpendicular to the direction of propagation), $E\perp B$, with the amplitudes related by $|E|=c|B|$ in SI units. The combination $E\times B$ points in the direction of propagation.

Proof. In a source-free region, $\rho =0$ and $J=0$. Maxwell's equations reduce to $$ \nabla \cdot E = 0, \quad \nabla \cdot B = 0, \quad \nabla \times E = -\partial_t B, \quad \nabla \times B = \mu_0 \varepsilon_0 \partial_t E. $$

Take the curl of Faraday and apply the vector-calculus identity $\nabla \times (\nabla \times E)=\nabla (\nabla \cdot E)-{\nabla }^{2}E$: $$ \nabla(\nabla \cdot E) - \nabla^2 E ;=; -\partial_t(\nabla \times B). $$ The first term vanishes by Gauss for $E$. Substitute Ampère-Maxwell for $\nabla \times B$: $$

• \nabla^2 E ;=; -\partial_t(\mu_0 \varepsilon_0 \partial_t E) ;=; -\mu_0 \varepsilon_0 \partial_t^2 E,$$thatis,$$ \nabla^2 E - \mu_0 \varepsilon_0 \partial_t^2 E ;=; 0. $$ Setting $c^2=1/({\mu }_0{\epsilon }_0)$ this is $\square E=0$ component by component. The same argument applied to $B$ — take the curl of Ampère-Maxwell, use Gauss for $B$, substitute Faraday — gives $\square B=0$.

For the transversality and amplitude statements, substitute a plane-wave ansatz $E(x,t)=E_0{e}^{i(k\cdot x-\omega t)}$, $B(x,t)=B_0{e}^{i(k\cdot x-\omega t)}$ with constant vector amplitudes $E_0,B_0\in {\mathbb{C}}^3$ and wave vector $k\in {\mathbb{R}}^3$, $\omega \in \mathbb{R}$. The wave equation forces ${\omega }^2=c^2|k{|}^2$, i.e., $\omega =c|k|$.

Gauss for $E$ gives $ik\cdot E_0=0$, so $E_0\perp k$ — the field is transverse. Faraday gives $ik\times E_0=i\omega B_0$, i.e., $B_0=(k\times E_0)/\omega$. So $B_0\perp E_0$ and $B_0\perp k$, with magnitude $|B_0|=|k||E_0|/\omega =|E_0|/c$. The pair $(E_0,B_0,k)$ forms a right-handed orthogonal triple. Computing $E\times B$ for real fields shows the Poynting vector $S:=(E\times B)/{\mu }_0$ points along $+k$ — the direction of energy propagation. ∎

Corollary. The speed of light $c=1/\sqrt{{\mu }_0{\epsilon }_0}$ is a property of the vacuum, independent of the motion of any observer. The corollary is the postulate of special relativity; in Maxwell's theory it appears as a theorem. Reconciling its frame-independence with Galilean kinematics is what forced Einstein in 1905 to revise the structure of space and time.

Worked example: covariant Lorentz force at intermediate level

A particle of charge $e$ and four-momentum $p^{\mu }$ in an external electromagnetic field $F_{\mu \nu }$ experiences the Lorentz force in covariant form $$ \frac{dp^\mu}{d\tau} ;=; e F^{\mu\nu} u_\nu, $$ where $\tau$ is the particle's proper time and $u^{\nu }=d{x}^{\nu }/d\tau$ is its four-velocity. Splitting into time and spatial components recovers the familiar non-relativistic forms.

The spatial part. With $u^{\nu }=\gamma (c,v)$ (where $\gamma =(1-v^2/c^2{)}^{-1/2}$) and using $F^{i0}=E^i/c$, $F^{ij}=-{\epsilon }^{ijk}{B}_k$, the spatial Lorentz-force component evaluates to $$ \frac{dp^i}{d\tau} ;=; e F^{i0} u_0 + e F^{ij} u_j ;=; e(E^i/c)(\gamma c) + e(-\varepsilon^{ijk} B_k)(-\gamma v_j) ;=; \gamma e (E + v \times B)^i. $$ Using $d\tau =dt/\gamma$ converts $d{p}^i/d\tau$ to $\gamma d{p}^i/dt$, the $\gamma$ cancels, and the result is $dp/dt=e(E+v\times B)$ — the textbook Lorentz force. The time component yields the power equation $d{E}_{\mathrm{kin}}/dt=ev\cdot E$. The Lorentz-force law is one tensor equation; the textbook three-vector decomposition is the $\eta =(+,-,-,-)$ projection.

Exercises [Intermediate+]

Differential-forms formulation [Master]

The intermediate-tier presentation already foreshadows it: the field tensor $F_{\mu \nu }$ is the component expression of a 2-form. The full differential-forms picture, due to Cartan in spirit and made canonical in physics by Misner-Thorne-Wheeler (1973), reorganises Maxwell's theory so that the four equations become two equations, gauge invariance becomes exact-form ambiguity, and the magnetic-monopole question becomes a question about $H^2(M)$.

Let $(M,g)$ be a smooth pseudo-Riemannian 4-manifold with Lorentzian signature; for the rest of this section $M={\mathbb{R}}^{1,3}$ with the Minkowski metric $\eta =\mathrm{diag}(+,-,-,-)$, though the formulation extends without change to curved spacetime once ${\partial }_{\mu }$ is replaced by the Levi-Civita covariant derivative.

The electromagnetic 4-potential is a 1-form $$ A ;=; A_\mu, dx^\mu ;\in; \Omega^1(M). $$ The Faraday 2-form is $$ F ;:=; dA ;\in; \Omega^2(M), $$ with components $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$ as in the Intermediate tier. The exterior derivative $d$ is the one introduced in 03.04.04.

The Hodge star $\star :{\Omega }^k(M)\to {\Omega }^{n-k}(M)$ on an oriented pseudo-Riemannian $n$-manifold of signature $(p,q)$ is the linear isomorphism characterised, for $\alpha ,\beta \in {\Omega }^k$, by $$ \alpha \wedge \star \beta ;=; \langle \alpha, \beta\rangle, d\mathrm{vol}, $$ where $⟨\cdot ,\cdot ⟩$ is the metric-induced inner product on ${\Lambda }^k{T}^{\ast }M$ and $d\mathrm{vol}$ is the metric volume form. In Minkowski signature $\star \star =(-1{)}^{k(n-k)+1}\mathrm{id}$ on $k$-forms (the extra $-1$ relative to the Riemannian case comes from the negative determinant); in particular, on 2-forms in 4D Lorentzian, $\star \star =-\mathrm{id}$.

The current 1-form is $J=J_{\mu }d{x}^{\mu }$ where $J^{\mu }=(c\rho ,\mathbf{J})$. The Hodge dual current $\star J$ is a 3-form. Maxwell's equations in differential-form notation are $$ \boxed{;; dF ;=; 0, \qquad d\star F ;=; \mu_0,\star J. ;;} $$

The first equation, $dF=0$, is automatic from $F=dA$ because $d^2=0$ (03.04.04). Equivalently, in any local chart it is the Bianchi identity ${\partial }_{[\rho }{F}_{\mu \nu ]}=0$, which packs in Faraday's law and Gauss for $B$. The second equation, $d\star F={\mu }_0\star J$, contains Gauss for $E$ and Ampère-Maxwell. Charge conservation follows from $d^2=0$: applying $d$ to $d\star F={\mu }_0\star J$ gives $0=d\star J$, equivalently ${\partial }_{\mu }{J}^{\mu }=0$.

Two equations replace four. Both are coordinate-free. The split into "electric" and "magnetic" parts of $F$ requires a choice of timelike direction — i.e., an observer — and is therefore frame-dependent; the geometric statement is observer-independent.

Gauge invariance is exact-form ambiguity. The transformation $A\mapsto A+d\chi$ for $\chi \in {\Omega }^0(M)$ leaves $F=dA$ unchanged: $d(A+d\chi )=dA+d^2\chi =dA$. The space of physically distinct connections is the quotient ${\Omega }^1(M)/d{\Omega }^0(M)$. On a contractible domain (e.g., ${\mathbb{R}}^{1,3}$ minus an open ball), the Poincaré lemma states that every closed 1-form is exact; on a topologically nontrivial domain, the quotient ${\Omega }^1/d{\Omega }^0$ has a $H^1(M;\mathbb{R})$-worth of additional content. The Aharonov-Bohm effect (1959) is the physical realisation of this: a loop encircling a solenoid in which $B=0$ along the loop's path picks up a phase $e\oint A/\hslash$ that is a topological invariant ($H^1$-class) of the configuration.

Magnetic monopoles as cohomology. The relation $F=dA$ globally requires $F$ to be exact, which on a smooth manifold $M$ implies $[F]=0$ in $H_{\mathrm{dR}}^2(M;\mathbb{R})$. A magnetic monopole at the origin produces a Coulomb-like $B$-field on $M={\mathbb{R}}^3\setminus \{0\}$ (3D space minus a point); the integral of the corresponding $F$ over a small sphere around the origin is the (constant times the) magnetic charge, and the 2-sphere is the generator of $H^2({\mathbb{R}}^3\setminus \{0\};\mathbb{Z})\cong \mathbb{Z}$. If this integral is non-zero, $F$ is not globally exact — there is no $A$ defined on all of $M$ satisfying $F=dA$.

The Dirac quantisation condition (1931) recovers a $\mathrm{U}(1)$ connection by allowing $A$ to be defined only on overlapping charts, with the two patches related on the overlap by a $\mathrm{U}(1)$-valued gauge transformation $g_{\alpha \beta }:U_{\alpha }\cap U_{\beta }\to \mathrm{U}(1)$. The transition function defines a line bundle on $M$, and the magnetic charge becomes the first Chern class $c_1\in H^2(M;\mathbb{Z})$ of that bundle, cross-referencing 03.06.04. Wu and Yang (1975) made this picture mathematically precise. The original Dirac quantisation $eg/(2\pi \hslash )\in \mathbb{Z}$ becomes the integrality of the Chern class. The absence of magnetic monopoles in nature is therefore a statement about a topological invariant of the electromagnetic field bundle: that bundle carries $c_1=0$ in our universe, so $F$ admits a globally-defined potential $A$.

Maxwell as a $\mathrm{U}(1)$ gauge theory. From the bundle viewpoint, $A$ is a connection on a $\mathrm{U}(1)$ principal bundle 03.05.07 and $F$ is its curvature 03.05.09. The gauge group is the abelian Lie group $\mathrm{U}(1)$; the corresponding Lie algebra is $\mathfrak{u}(1)\cong i\mathbb{R}$, abelian, so the curvature simplifies to $F=dA$ (the bracket term $\frac{1}{2}[A,A]$ that appears in non-abelian Yang-Mills theory vanishes). The non-abelian generalisation, replacing $\mathrm{U}(1)$ by $\mathrm{SU}(N)$ or any compact Lie group, is the Yang-Mills theory of 03.07.05; electromagnetism is the abelian case, the simplest member of the family.

Lagrangian formulation and Noether currents [Master]

The electromagnetic Lagrangian density is $$ \mathcal{L} ;=; -\frac{1}{4\mu_0}, F_{\mu\nu} F^{\mu\nu} ;-; J^\mu A_\mu, $$ a Lorentz-scalar function on the space of 4-potentials. The action $S={\int }_M\mathcal{L}{d}^{4}x$ is invariant under the gauge transformation $A_{\mu }\to A_{\mu }+{\partial }_{\mu }\chi$ when ${\partial }_{\mu }{J}^{\mu }=0$ (since the variation $\delta S=-\int J^{\mu }{\partial }_{\mu }\chi d^{4}x=\int ({\partial }_{\mu }{J}^{\mu })\chi d^{4}x$ after integration by parts vanishes for conserved currents). In differential-form notation $$ S ;=; -\frac{1}{2\mu_0}\int_M F \wedge \star F ;-; \int_M A \wedge \star J. $$

The Euler-Lagrange equation ${\partial }_{\mu }(\partial \mathcal{L}/\partial ({\partial }_{\mu }{A}_{\nu }))-\partial \mathcal{L}/\partial A_{\nu }=0$ evaluates to ${\partial }_{\mu }{F}^{\mu \nu }-{\mu }_0{J}^{\nu }=0$, which is exactly the inhomogeneous Maxwell equation. The homogeneous Maxwell equation $dF=0$ is automatic from $F=dA$ and does not arise from the variational principle — it is a constraint built into the choice of potential as the fundamental variable. This asymmetry is a hint that the variational picture privileges $A$ over $F$, foreshadowing the role of the connection in quantum gauge theory.

Stress-energy tensor. The translation symmetry of $\mathcal{L}$ in Minkowski coordinates yields, via Noether, a conserved current — the stress-energy tensor $$ T^{\mu\nu} ;=; \frac{1}{\mu_0}\left( F^{\mu\alpha} F^\nu{}\alpha - \tfrac{1}{4} \eta^{\mu\nu} F{\alpha\beta} F^{\alpha\beta} \right). $$ The symmetric (Belinfante-Rosenfeld-improved) form is what appears here; the canonical Noether current is the gauge-non-invariant precursor, and the improvement adds a divergence-free total derivative to symmetrise. The interpretations:

• $T^{00}=\frac{1}{2}({\epsilon }_0|E{|}^2+|B{|}^2/{\mu }_0)$ is the energy density of the electromagnetic field.
• $T^{0i}/c=(E\times B/{\mu }_0{)}^i=S^i$ is the Poynting vector, the energy flux.
• $T^{ij}$ is the Maxwell stress tensor, the momentum flux (and hence the source of radiation pressure).

Conservation ${\partial }_{\mu }{T}^{\mu \nu }=-F^{\nu }{}_{\alpha }{J}^{\alpha }$. In a source-free region the stress-energy is divergence-free, and the field carries energy and momentum independently of any matter; when $J$ is present, the divergence of $T^{\mu \nu }$ matches the Lorentz force on the matter, so total energy-momentum (field plus matter) is conserved. This is the Noether realisation of energy-momentum conservation for the electromagnetic field; in general relativity, $T^{\mu \nu }$ becomes the source of the Einstein field equations, and the Maxwell-Einstein system is the simplest coupled gauge-gravity theory with non-vanishing matter source.

Other Noether currents. Lorentz invariance gives the conserved angular-momentum tensor $M^{\mu \nu \rho }=x^{\mu }{T}^{\nu \rho }-x^{\nu }{T}^{\mu \rho }$. Gauge invariance — when restored to a global symmetry by promoting $\chi$ to a constant — gives the conserved electric four-current $J^{\mu }$ (this is the easy half of Noether; the converse — that any conserved current is the Noether current of a global symmetry — is the second Noether theorem applied to gauge symmetries). Conformal invariance of vacuum electromagnetism (the action $\int F\wedge \star F$ is conformally invariant in $n=4$ but only there) gives an additional 15 conserved currents corresponding to the conformal group $\mathrm{SO}(2,4)$; this conformal invariance underlies the masslessness of the photon.

Connections [Master]

• Exterior derivative 03.04.04 is the operator that makes $F=dA$ and the two Maxwell equations $dF=0$, $d\star F=\star J$ definable. The identity $d^2=0$ produces Bianchi automatically and charge conservation as a direct corollary.

• Differential forms 03.04.02 supply the ambient algebra of ${\Omega }^k(M)$ and the wedge product needed to define $F$, the Lagrangian density $F\wedge \star F$, and the topological pairings $\int F\wedge F$ that classify monopoles.

• Stokes' theorem 03.04.05 turns the differential Maxwell equations into the integral Gauss/Faraday/Ampère laws. The flux-and-circulation textbook statements are the integral form of $dF=0$ and $d\star F=\star J$ over compact submanifolds with boundary.

• Principal bundles and connections 03.05.01, 03.05.07 are the global framework in which $A$ is a connection and $F$ its curvature. Maxwell is the abelian $\mathrm{U}(1)$ case; the magnetic-monopole topology classifies $\mathrm{U}(1)$-bundles by $H^2(M;\mathbb{Z})$.

• Curvature 03.05.09 is the geometric object the field-strength $F$ realises. In abelian gauge theory the bracket term $[A,A]$ vanishes and curvature is just $dA$; this is the simplification specific to $\mathrm{U}(1)$.

• Chern classes 03.06.04 classify $\mathrm{U}(1)$-bundles by $H^2(M;\mathbb{Z})$ via $c_1=[F/(2\pi )]$. The Dirac quantisation of magnetic charge is the integrality of $c_1$.

• Chern-Weil homomorphism 03.06.06 is the general theorem behind $c_1=[F/(2\pi )]$ — invariant polynomials in the curvature represent characteristic classes.

• Yang-Mills action 03.07.05 is the non-abelian generalisation. Replace $\mathrm{U}(1)$ by $\mathrm{SU}(N)$ or another compact Lie group; the Lagrangian $-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }$ becomes $-\frac{1}{4}\mathrm{tr}(F_{\mu \nu }{F}^{\mu \nu })$ with $F=dA+A\wedge A$, and the Euler-Lagrange equations are the Yang-Mills equations.

• Special relativity [10.05.NN] (pending) develops the Lorentz transformation laws and the covariant formulation of mechanics that this unit's tensor $F_{\mu \nu }$ presumes. The historical order — Maxwell first, special relativity later — is reversed in pedagogy because the relativistic formulation makes Maxwell's structure visible.

• Covariant electrodynamics [10.06.NN] (pending) develops in full what this unit foreshadows: $F_{\mu \nu }$, $A^{\mu }$, the Lorentz force in covariant form, and the energy-momentum tensor as the substrate for radiation theory.

• EM Lagrangian and Noether [10.09.NN] (pending) develops the action principle for $A_{\mu }$ in full, including the proper handling of gauge invariance as a redundancy (via Faddeev-Popov / BRST) and the connection to QED.

Historical & philosophical context [Master]

Maxwell's A Dynamical Theory of the Electromagnetic Field (1865) and the consolidated Treatise of 1873 ^{[Maxwell-Treatise]} presented the four equations in their original, prolix form — twenty equations in twenty unknowns, written in quaternions and components, with the electromagnetic field a state of stress in a hypothetical ether. The compact vector form $\nabla \cdot E$, $\nabla \times B$, etc., is due to Heaviside and Gibbs in the 1880s. The covariant tensor form $F_{\mu \nu }$ was introduced by Minkowski (1908) shortly after Einstein's 1905 paper On the electrodynamics of moving bodies, where the postulate of frame-independence of $c$ was made explicit and the Lorentz transformations of $E$ and $B$ derived from it. Maxwell's original equations were already Lorentz-covariant — Einstein's contribution was to recognise this as the physical statement that simultaneity, length, and duration must be revised, not the equations.

The differential-forms formulation $dF=0$, $d\star F=\star J$ has multiple roots. Cartan (1922) introduced the exterior calculus and applied it to electromagnetism in his Leçons sur les invariants intégraux; Schouten and Synge extended it through the 1930s. The version now standard in physics — and presented in this unit — was canonised by Misner-Thorne-Wheeler's Gravitation (1973) ^[MTW], which made the forms language the default for relativistic electrodynamics. The cohomological reading (no monopoles = $H^2$ statement) was made explicit by Wu and Yang (1975) ^{[Wu-Yang-1975]} in the framework of nonintegrable phase factors; the bundle-theoretic interpretation built on Dirac (1931) ^[Dirac-1931], whose magnetic-monopole quantisation argument is the historical bridge between gauge theory and topology.

The substantival-vs-relational debate about the electromagnetic field — is $F$ a real entity in the world, or a calculational device organising relations between charged matter? — has been a focus of philosophy of physics since the late 19th century. The Lorentz-covariant formulation made it untenable to treat $E$ and $B$ as separate physical entities; the differential-forms picture sharpened the question to whether $F$ or $A$ is the fundamental object. The Aharonov-Bohm effect (1959), where charged particles are affected by a vector potential $A$ in regions where $F=0$, is the empirical wedge: it indicates that $A$ is not merely a calculational device, but a globally-defined connection (modulo gauge) is. The bundle-theoretic resolution — neither $A$ alone nor $F$ alone, but the gauge-equivalence class of connections, equivalently a principal $\mathrm{U}(1)$-bundle with connection — is the modern stance.

Bibliography [Master]

Primary literature (cite when used; not all currently in reference/):

• Maxwell, J. C., A Dynamical Theory of the Electromagnetic Field, Phil. Trans. Roy. Soc. 155 (1865), 459-512. [Need to source — originator paper.]
• Maxwell, J. C., A Treatise on Electricity and Magnetism, 2 vols. (Clarendon, 1873; reprinted Dover 1954).
• Einstein, A., "Zur Elektrodynamik bewegter Körper", Ann. Phys. 322 (1905), 891-921.
• Minkowski, H., "Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern", Nachr. Ges. Wiss. Göttingen (1908), 53-111.
• Dirac, P. A. M., "Quantised singularities in the electromagnetic field", Proc. Roy. Soc. A 133 (1931), 60-72.
• Aharonov, Y. & Bohm, D., "Significance of electromagnetic potentials in the quantum theory", Phys. Rev. 115 (1959), 485-491.
• Wu, T. T. & Yang, C. N., "Concept of nonintegrable phase factors and global formulation of gauge fields", Phys. Rev. D 12 (1975), 3845-3857.
• Cartan, É., Leçons sur les invariants intégraux (Hermann, 1922).
• Misner, C. W., Thorne, K. S. & Wheeler, J. A., Gravitation (Freeman, 1973), Ch. 3-4.
• Jackson, J. D., Classical Electrodynamics, 3rd ed. (Wiley, 1998).
• Landau, L. D. & Lifshitz, E. M., The Classical Theory of Fields, 4th ed. (Pergamon, 1980).
• Griffiths, D. J., Introduction to Electrodynamics, 4th ed. (Cambridge, 2013).
• Zangwill, A., Modern Electrodynamics (Cambridge, 2013).
• Purcell, E. M. & Morin, D. J., Electricity and Magnetism, 3rd ed. (Cambridge, 2013).
• Tong, D., Lectures on Electromagnetism (DAMTP Cambridge lecture notes).

---

Wave 1 physics seed unit (§10 EM+SR branch), produced 2026-05-18 (per docs/plans/PHYSICS_PLAN.md §5 — agent-drafted-plus-editorial pattern for §13, §10, §11, §12 seeds). All three cross-domain hooks_out targets are proposed; no chem/bio/phil seed unit yet exists to receive confirmed promotion. Status remains draft pending Tyler's review and the §11 Next-Actions retro per PHYSICS_PLAN.