The Maxwell Equations
This chapter unifies electrodynamics: electromagnetic induction, the displacement current, the full Maxwell system, its conservation laws, and the potentials through which it is most efficiently formulated (the Lagrangian formulation itself is developed in Generalized Classical Field Theory). The statics of Electrostatics and Magnetostatics treated \(\vect{E}\) and \(\vect{B}\) as separate fields with separate sources; here the time derivatives couple them. Faraday's discovery that a changing magnetic flux drives a current supplies one coupling from experiment; Maxwell's displacement current supplies the other from a consistency argument—and turns the system into one that propagates waves at a speed fixed by two electrical constants. That the speed came out equal to the measured speed of light is the single most consequential coincidence in classical physics, and its experimental confirmation by Hertz is treated in Experiments: Electromagnetism.
The Maxwell Equations: all derivations of this chapter are pending.
Electromagnetic induction
Faraday's discovery
[Reserved: the induction of transient currents by switching a neighbouring circuit, by moving a magnet, and by moving the circuit itself, reported in the First Series of Faraday's Experimental Researches in Electricity [Faraday:1832]; Henry's independent observation of induction and of self-induction [Henry:1832]; electromotive force and the flux rule; full experimental detail in Experiments: Electromagnetism.]
A current flows in a closed conducting circuit whenever — and only while — the magnetic flux threading it changes. A steady current in a neighbouring circuit produces nothing; making or breaking that current produces a transient. So does moving a magnet through the circuit, and so does moving the circuit itself while the magnet stands still [Faraday:1832] [Henry:1832]. The electromotive force driving the current is the rate at which the flux changes,
with \(S\) any surface bounded by the circuit, and it does not matter which of the three ways the change was brought about.
The flux rule: derive it from the Maxwell–Faraday equation together with the Lorentz force acting on the charge carriers of a moving circuit, using the transport theorem for a surface whose boundary moves with the circuit, and show that the two physically distinct mechanisms — an induced electric field in the transformer case, a magnetic force on carried charge in the motional case — combine into the single total rate of change of the flux.
Lenz's law
[Reserved: the sign of the induced current opposes the change of flux that produces it [Lenz:1834]; its reading as a stability statement and its connection with energy conservation, made precise once Poynting's theorem is available in Section 60.4; eddy currents as its macroscopic manifestation.]
The induced current always circulates in the sense whose own magnetic flux opposes the change that produced it [Lenz:1834]. A magnet pushed towards a closed loop is resisted; the same magnet withdrawn is held back; a sheet of copper swung between the poles of a magnet is braked as though moving through a fluid, though nothing touches it. The rule fixes the sign in Equation (60.1) and is what makes that minus sign an experimental statement rather than a convention.
Derivation. Suppose the opposite sign, and let a magnet approach a loop of resistance \(R\). The induced current would then produce a flux reinforcing the change, so the force it exerts on the magnet would act along the magnet's motion. The magnet would accelerate; the flux would change faster; the electromotive force and the current would grow with it; and the loop would dissipate \(\mathcal{E}^{2}/R\) at an increasing rate while the magnet's kinetic energy also increased. Both the mechanical energy and the Joule heat would rise without anything supplying them, which no experiment has ever exhibited. The only consistent sign is therefore the observed one: the induced current opposes the change, the force on the magnet resists its motion, and the work done against that force is exactly the energy dissipated in the loop. The local form of the same bookkeeping is Poynting's theorem (Section 60.4.2), which turns this argument from a consistency requirement into an identity of the field equations.
∎The Maxwell–Faraday equation
[Reserved: the local form \(\vect{\nabla}\times\vect{E}=-\pp\vect{B}/\pp t\) abstracted from the flux rule [Faraday:1832] [Maxwell:1865]; the two distinct physical cases hiding in one flux rule—motional emf (magnetic force on carried charges) versus transformer emf (induced electric field)—whose unification is relativistic and is completed in Lorentz Transformations.]
The displacement current and the complete system
The inconsistency of the static circuital law
[Reserved: Ampère's law of Magnetostatics requires \(\vect{\nabla}\cdot\vect{J}=0\) and therefore fails for open circuits —the charging capacitor as the canonical case; charge conservation, \(\pp\rho/\pp t+\vect{\nabla}\cdot\vect{J}=0\), as the constraint the corrected law must respect.]
Maxwell's dynamical theory
[Reserved: the displacement current \(\epsilon_{0}\,\pp\vect{E}/\pp t\), introduced by Maxwell first within the molecular-vortex model of On Physical Lines of Force [Maxwell:1861] and then rederived, free of mechanical scaffolding, in A Dynamical Theory of the Electromagnetic Field [Maxwell:1865]; the complete system of field equations, its rearrangement in the Treatise [Maxwell:1873], and the counting of its independent equations.]
The Heaviside vector form
[Reserved: the reduction of Maxwell's original twenty equations in twenty variables to the four vector equations used ever since, carried out by Heaviside in 1885 in the Electrician series reprinted in his Electrical Papers [Heaviside:1892]; the modern SI statement of the system [Jackson:1999] [BIPM:2019], boxed here as the reference form for the rest of the treatise.]
Potentials and gauge freedom
Scalar and vector potentials
[Reserved: the homogeneous pair of Maxwell equations is solved identically by \(\vect{B}=\vect{\nabla}\times\vect{A}\) and \(\vect{E}=-\vect{\nabla}\phi-\pp\vect{A}/\pp t\), extending the static potentials of Electrostatics and Magnetostatics; the inhomogeneous pair becomes two coupled second-order equations for \((\phi,\vect{A})\) [Maxwell:1865].]
Gauge transformations
[Reserved: the invariance of the fields under \(\vect{A}\to\vect{A}+\vect{\nabla}\chi\), \(\phi\to\phi-\pp\chi/\pp t\); the Lorenz condition, due to L. V. Lorenz [Lorenz:1867]—the Danish physicist, not H. A. Lorentz, a misattribution the literature keeps alive—which decouples the potential equations into wave equations; the Coulomb gauge; gauge freedom as the classical seed of the gauge principle of Generalized Classical Field Theory.]
Conservation laws
Charge conservation
[Reserved: the continuity equation as an identity of the Maxwell system—the displacement current makes conservation automatic rather than assumed [Maxwell:1865]; global charge conservation and its experimental precision.]
The total electric charge of an isolated system never changes. Charge appears and disappears only in pairs of opposite sign, and no process has ever been seen to create or destroy a net amount of it. The sharpest test is the stability of the electron: as the lightest charged particle it has nothing charged to decay into, so any decay would destroy charge outright, and no such decay has been observed — the resulting bounds on its mean life exceed the age of the universe by many orders of magnitude [Navas:2024]. Locally the statement is a continuity equation,
Derivation. Take the divergence of the Ampère–Maxwell law \(\vect{\nabla}\times\vect{B} =\mu_{0}\vect{J}+\mu_{0}\epsilon_{0}\,\pp\vect{E}/\pp t\). The left-hand side is the divergence of a curl and vanishes identically, so
and substituting Gauss's law \(\vect{\nabla}\cdot\vect{E}=\rho/ \epsilon_{0}\) gives Equation (60.2) after dividing by \(\mu_{0}\). Integrating over a volume and applying the divergence theorem turns it into the global statement that the charge inside a region changes only by the current crossing its boundary.
The step is worth reading backwards, because it is the whole reason for the displacement current. Without that term the same operation yields \(\vect{\nabla}\cdot\vect{J}=0\), which is false for any circuit that is open — a capacitor being charged — and it is the demand that charge be conserved, not any new measurement, that forced the correction (Section 60.2.1). Conservation is thereby an identity of the completed system rather than an extra assumption imposed on it.
∎Poynting's theorem
[Reserved: the local energy balance of the electromagnetic field, with energy density \(u=(\epsilon_{0}E^{2}+B^{2}/\mu_{0})/2\) and flux \(\vect{S}=\vect{E}\times\vect{B}/\mu_{0}\), derived by Poynting [Poynting:1884]; where the energy of a circuit actually flows; the ambiguity of \(\vect{S}\) up to a curl and why no experiment inside electrodynamics resolves it.]
The stress tensor and field momentum
[Reserved: the Maxwell stress tensor [Maxwell:1873], the momentum density \(\epsilon_{0}\vect{E}\times\vect{B}\), and the momentum balance of field plus matter; radiation pressure predicted by Maxwell [Maxwell:1873] and measured by Lebedev [Lebedew:1901] and by Nichols and Hull [Nichols:1903]; angular momentum of the field, with its mechanical detection in circularly polarized light by Beth [Beth:1936]; the covariant packaging of these densities awaits Generalized Classical Field Theory.]
Light falling on a surface pushes it. The effect was predicted from the field momentum [Maxwell:1873] and measured, against enormous thermal backgrounds, with torsion balances in evacuated vessels by Lebedev [Lebedew:1901] and by Nichols and Hull [Nichols:1903]. At normal incidence the pressure on a perfectly absorbing surface is
with \(I\) the incident irradiance, and twice that on a perfect reflector. Electromagnetic energy therefore carries mechanical momentum, and the field is a dynamical object rather than a bookkeeping device.
Derivation. The electromagnetic momentum density is \(\vect{g}=\epsilon_{0}\vect{E}\times\vect{B}=\vect{S}/c^{2}\), with \(\vect{S}=\vect{E}\times\vect{B}/\mu_{0}\) the Poynting vector (Section 60.4.2) and \(\vect{g}\) the momentum entry of the Maxwell stress tensor (Section 60.4.3). For a plane wave in vacuum \(\abs{\vect{S}}=uc\), where \(u\) is the energy density, so \(\abs{\vect{g}}=u/c\): the wave carries one unit of momentum for every \(c\) units of energy.
Momentum arrives at an absorbing surface at the rate at which the wave delivers it, that is, the momentum density times the speed at which it advances:
since \(I=\avg{\abs{\vect{S}}}=\avg{u}c\) and all quantities are here understood as time averages. A perfect reflector reverses the momentum instead of absorbing it, so the momentum transferred per unit time is doubled and so is the pressure.
∎A circularly polarized beam carries angular momentum about its own direction of propagation, and delivers it to matter as a measurable mechanical torque. A birefringent half-wave plate suspended on a fine quartz fibre reverses the handedness of a beam passing through it, and is observed to experience a steady torque of the sense and magnitude that the transfer of angular momentum requires [Beth:1936].
Angular momentum of the electromagnetic field: derive the angular momentum density from the field momentum density, and show that for a circularly polarized beam of angular frequency \(\omega\) the ratio of angular momentum to energy is \(1/\omega\), so that reversing the handedness transfers twice that amount per unit energy.
The electromagnetic wave prediction
The wave equation and the speed $c$
[Reserved: in vacuum the system yields wave equations for \(\vect{E}\) and \(\vect{B}\) with speed \(1/\sqrt{\mu_{0}\epsilon_{0}}\); Weber and Kohlrausch's measurement of the ratio of electrical units [Weber:1856], Maxwell's identification of that ratio with the measured speed of light, and his conclusion that light is an electromagnetic disturbance [Maxwell:1865]; the wave-equation machinery is that of Partial Differential Equations.]
A speed can be extracted from measurements that involve no light at all. Comparing the charge on a Leyden jar, determined electrostatically from the force between charged bodies, with the same charge determined electrodynamically from the deflection it produces on discharge through a galvanometer, gives a ratio of units with the dimensions of a velocity — and that velocity came out equal, within the accuracy of the measurement, to the speed of light determined optically [Weber:1856]. Electromagnetic disturbances were later generated directly, and found to propagate through air, to reflect, to refract in a prism of pitch, to be polarized by a grid of wires, and to travel at a speed consistent with the optical value [Hertz:1888]. Light is an electromagnetic wave [Maxwell:1865], and in vacuum both fields obey
Derivation. In a region free of charge and current take the curl of the Maxwell–Faraday equation and exchange the order of the space and time derivatives on the right:
the last step by the Ampère–Maxwell law with \(\vect{J}=0\). On the left use the identity \(\vect{\nabla}\times(\vect{\nabla}\times\vect{E}) =\vect{\nabla}\left(\vect{\nabla}\cdot\vect{E}\right)-\nabla^{2} \vect{E}\), in which the first term vanishes because \(\vect{\nabla}\cdot\vect{E}=0\) where \(\rho=0\). What remains is Equation (60.6); the identical manipulation starting from the Ampère–Maxwell law gives the same equation for \(\vect{B}\).
The propagation speed is therefore fixed by \(\mu_{0}\) and \(\epsilon_{0}\) alone — two constants measured with magnets, currents and charged spheres — and it is the coincidence of that number with the optical speed of light that identifies the two phenomena. Note that the displacement current is indispensable: delete \(\mu_{0}\epsilon_{0}\,\pp\vect{E}/\pp t\) and the right-hand side of Equation (60.7) vanishes, leaving no wave.
∎Hertz's confirmation
[Reserved: the generation, propagation, reflection and interference of electromagnetic waves in air by Hertz [Hertz:1888], closing the prediction; standing waves and the measured wave speed; full apparatus-level treatment in Experiments: Electromagnetism, and the developed optics of the waves themselves in Electromagnetic Waves and Optics.]
Structure of the theory
Relativity of the field decomposition
[Reserved: the split of the electromagnetic field into \(\vect{E}\) and \(\vect{B}\) depends on the observer; the asymmetry of explanation for induction (magnet moving versus circuit moving) that opens Einstein's 1905 paper [Einstein:1905a]; forward references to Lorentz Transformations and Minkowski Space and Its Symmetries, where the field strength is assembled into a single tensor.]
Duality and the absence of magnetic charge
[Reserved: the vacuum system is symmetric under the rotation of \((\vect{E},c\vect{B})\) into one another; sources break the symmetry because no magnetic charge is observed—the empirical record is that of Magnetostatics; what a monopole term would do to the equations [Dirac:1931].]
Limits and the domain of validity
[Reserved: the quasi-static limits recovering Electrostatics and Magnetostatics; circuit theory as a limit; linearity and superposition, exact classically and modified only at the quantum level (photon–photon scattering, treated with QED in Quantum Electrodynamics and Renormalization); classical electrodynamics as the low-photon-number, high-occupancy limit examined in Quantum Optics and the Photon.]