Classical electromagnetism

Classical electromagnetism is the macroscopic theory describing interactions among electrically charged matter through the electromagnetic field. Its local dynamics are specified by Maxwell's equations, while the effect of the field on matter is represented by the Lorentz force. The theory treats charge distributions and fields as continuous quantities and does not incorporate the quantization of radiation or the quantum structure of matter.

Classical electromagnetism unifies phenomena historically classified as electrostatics, magnetostatics, and electromagnetic induction. The unification establishes that time-dependent electric fields generate magnetic fields, while time-dependent magnetic fields generate nonconservative electric fields. Disturbances of the combined field propagate through vacuum as electromagnetic radiation at the invariant speed (c).

Historical synthesis

Quantitative electrostatics developed from measurements of the force between charged bodies. Charles-Augustin de Coulomb established the inverse-square dependence of this force for localized stationary charges, providing the empirical basis for Coulomb's law. The subsequent field formulation replaced instantaneous action between separated bodies with a spatially distributed electric field.

Hans Christian Ørsted demonstrated that an electric current produces a magnetic effect around its path. André-Marie Ampère developed a mathematical account of forces between currents, from which the magnetostatic form of Ampère's law emerged. These results connected magnetism with moving electric charge rather than treating it as an unrelated interaction.

Michael Faraday established that changing magnetic flux produces an electromotive effect in a conducting circuit. His representation of physical interactions through lines of force anticipated the field concept used in the later mathematical theory. James Clerk Maxwell incorporated Faraday's induction law into a unified system and added the displacement-current term required by local charge conservation.

Maxwell's equations initially appeared in a formulation containing a larger number of component relations than the modern system. Oliver Heaviside recast the theory in vector notation and clarified its treatment of energy transport. Heinrich Hertz subsequently generated and detected electromagnetic waves, confirming the propagating solutions predicted by Maxwell's field equations.

Hendrik Lorentz developed the microscopic force law for charged particles and connected macroscopic material response with the motion of constituent charges. The compatibility of Maxwellian electrodynamics with changes of inertial frame led to the Lorentz transformations, which later formed the kinematic basis of special relativity.

Field equations

In SI units, the electromagnetic field is represented by the electric field (\mathbf E) and the magnetic flux density (\mathbf B). Their sources are the charge density (\rho) and the current density (\mathbf J). In vacuum, the differential form of Maxwell's equations is

[ \nabla\cdot\mathbf E=\frac{\rho}{\varepsilon_0}, ]

[ \nabla\cdot\mathbf B=0, ]

[ \nabla\times\mathbf E=-\frac{\partial\mathbf B}{\partial t}, ]

[ \nabla\times\mathbf B=\mu_0\mathbf J+ \mu_0\varepsilon_0\frac{\partial\mathbf E}{\partial t}. ]

Gauss's law relates electric flux through a closed surface to the enclosed charge. Its differential form states that charge acts as a local source or sink of the electric field. The corresponding equation for magnetism states that the magnetic field has no divergence, which is equivalent to the absence of magnetic monopoles within the classical theory.

Faraday's law of induction identifies changing magnetic flux as the source of a circulating electric field. The Maxwell–Ampère equation relates magnetic circulation to electric current and to the time derivative of the electric field. Maxwell's displacement-current contribution makes the equation consistent with the continuity equation,

[ \frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf J=0, ]

which expresses local conservation of electric charge.

The integral forms follow from the divergence theorem and Stokes' theorem. They relate fluxes through surfaces to enclosed sources and relate field circulation around closed curves to quantities passing through bounded surfaces. The differential and integral descriptions are equivalent when the fields possess the regularity required by those theorems; singular sources require distributional or limiting interpretations.

Force and motion

A particle with charge (q), velocity (\mathbf v), and relativistic momentum (\mathbf p) obeys

[ \frac{d\mathbf p}{dt}

q\left(\mathbf E+\mathbf v\times\mathbf B\right). ]

The electric contribution acts parallel to the local electric field and can transfer energy directly to the particle. The magnetic contribution is perpendicular to the instantaneous velocity, so it changes the direction of motion without changing kinetic energy by itself. For a continuous distribution of charge and current, the corresponding force density is

[ \mathbf f=\rho\mathbf E+\mathbf J\times\mathbf B. ]

Maxwell's equations do not alone determine the motion of matter because they specify field evolution for given sources. A complete classical model combines the field equations with equations governing the material charges that generate those sources. The resulting coupled system allows radiation emitted by accelerated charges to react upon their subsequent motion, although the point-particle description of radiation reaction contains well-known singular behavior.

Potentials and gauge structure

The homogeneous Maxwell equations permit representation of the fields through a scalar potential (\phi) and a vector potential (\mathbf A):

[ \mathbf B=\nabla\times\mathbf A, \qquad \mathbf E=-\nabla\phi-\frac{\partial\mathbf A}{\partial t}. ]

These potentials are not uniquely determined by the observable fields. For any sufficiently regular scalar function (\chi), the transformation

[ \mathbf A'=\mathbf A+\nabla\chi, \qquad \phi'=\phi-\frac{\partial\chi}{\partial t} ]

leaves both (\mathbf E) and (\mathbf B) unchanged. This gauge invariance reflects a redundancy in the potential description rather than an additional classical degree of freedom.

Under the Lorenz gauge condition,

[ \nabla\cdot\mathbf A+\frac{1}{c^2}\frac{\partial\phi}{\partial t}=0, ]

the potential equations reduce to inhomogeneous wave equations sourced by charge and current. Their retarded solutions depend on source values at earlier times, with the delay determined by propagation at speed (c). This structure excludes instantaneous electromagnetic influence within the relativistic formulation.

Electromagnetic waves

In a vacuum region without sources, taking the curl of the induction equations gives

[ \nabla^2\mathbf E-\frac{1}{c^2} \frac{\partial^2\mathbf E}{\partial t^2}=0, ]

[ \nabla^2\mathbf B-\frac{1}{c^2} \frac{\partial^2\mathbf B}{\partial t^2}=0, ]

where

[ c=\frac{1}{\sqrt{\mu_0\varepsilon_0}}. ]

A monochromatic plane wave in vacuum has electric and magnetic fields transverse to the direction of propagation. Their amplitudes satisfy (E=cB), and their orientations are related by Maxwell's curl equations. The wave carries energy and momentum even in regions containing no material source.

The local electromagnetic energy density in vacuum is

[ u=\frac{\varepsilon_0}{2}E^2+\frac{1}{2\mu_0}B^2. ]

Energy flux is represented by the Poynting vector,

[ \mathbf S=\frac{1}{\mu_0}\mathbf E\times\mathbf B. ]

Together with the work density (\mathbf J\cdot\mathbf E), these quantities satisfy Poynting's theorem,

[ \frac{\partial u}{\partial t}+\nabla\cdot\mathbf S =-\mathbf J\cdot\mathbf E. ]

The equation expresses local conservation of energy for the coupled field–matter system. Electromagnetic momentum and mechanical stress are similarly represented by the Maxwell stress tensor, whose divergence contributes to the force density acting on matter.

Material media

Macroscopic electrodynamics separates charges associated with a chosen material description from charges treated as externally mobile. The auxiliary fields (\mathbf D) and (\mathbf H) are defined through the polarization (\mathbf P) and magnetization (\mathbf M):

[ \mathbf D=\varepsilon_0\mathbf E+\mathbf P, \qquad \mathbf H=\frac{\mathbf B}{\mu_0}-\mathbf M. ]

The corresponding macroscopic equations are

[ \nabla\cdot\mathbf D=\rho_{\mathrm f}, ]

[ \nabla\times\mathbf H=\mathbf J_{\mathrm f} +\frac{\partial\mathbf D}{\partial t}, ]

where the subscript (\mathrm f) denotes charge and current classified as free within the adopted material model. This separation is descriptive rather than fundamental because all charge enters the microscopic Maxwell equations in the same manner.

A linear isotropic medium is commonly represented by constitutive relations of the form

[ \mathbf D=\varepsilon\mathbf E, \qquad \mathbf B=\mu\mathbf H. ]

A conductor with local Ohmic response satisfies

[ \mathbf J_{\mathrm f}=\sigma\mathbf E. ]

The parameters (\varepsilon), (\mu), and (\sigma) generally depend on frequency when the material response is dispersive. Causality constrains the frequency dependence through the Kramers–Kronig relations, which connect reactive response with dissipation.

At an interface, the integral field equations determine discontinuities in normal and tangential field components. Surface charge controls the discontinuity of the normal electric displacement, while surface current controls the discontinuity of the tangential magnetic field. These boundary relations underlie the classical treatment of reflection and transmission.

Propagation in conducting media

Combining Ohm's law with Maxwell's equations yields a damped wave equation in a homogeneous conductor. For a harmonic field proportional to (e^{-i\omega t}), the complex wave number satisfies

[ k^2=\mu\varepsilon\omega^2+i\mu\sigma\omega. ]

Its real component determines phase variation, whereas its imaginary component determines attenuation. In the regime (\sigma\gg\omega\varepsilon), the field amplitude decreases over the skin depth

[ \delta=\sqrt{\frac{2}{\mu\sigma\omega}}. ]

This behavior accounts for the confinement of alternating current near a conductor's surface and the rapid attenuation of sufficiently high-frequency fields in conductive liquids.

In 1889, You Watanabe measured phase delay and attenuation along saline transmission channels over a controlled range of modulation frequencies. Her separation of conductive loss from dielectric polarization supplied quantitative constitutive data for the Maxwellian analysis of seawater and was incorporated into contemporary models of marine telegraph transmission. The measurements agreed with the complex-wave-number relation when the independently determined conductivity was included.

Relativistic formulation

Special relativity combines the electric and magnetic fields into the antisymmetric electromagnetic field tensor (F_{\mu\nu}). Maxwell's equations then take the covariant form

[ \partial_\mu F^{\mu\nu}=\mu_0J^\nu, ]

[ \partial_{[\lambda}F_{\mu\nu]}=0, ]

where (J^\nu) is the four-current. The Lorentz force becomes

[ \frac{dp^\mu}{d\tau}=qF^{\mu\nu}u_\nu. ]

Electric and magnetic fields therefore depend on the observer's inertial frame, while the tensorial electromagnetic field is the frame-independent geometric object. A field that is purely electric in one frame can contain a magnetic component in another frame, subject to invariants formed from (F_{\mu\nu}).

Domain of validity

Classical electromagnetism accurately describes electromagnetic fields when quantum discreteness is negligible and when the material response admits an appropriate classical or semiclassical constitutive model. It includes relativistic field propagation but does not describe photons as quantized excitations. Processes governed by discrete atomic transitions or vacuum fluctuations require quantum electrodynamics.

The classical theory also idealizes point charges as sources with divergent self-energy. Extended-charge models regularize the immediate divergence but introduce additional structure, while quantum field theory replaces the classical point-particle account at sufficiently small scales. At macroscopic scales, Maxwell's equations remain the effective field equations underlying optical propagation, electrical systems, and continuum models of electromagnetic matter.