Maxwell’s Equations · aiwiki.page

Maxwell’s equations are four coupled equations governing electric fields and magnetic fields, their relationship to electric charge and current, and their evolution over time. They form the foundation of classical electromagnetism. Together with the Lorentz force law, they describe how electromagnetic fields act on charged matter. Named after James Clerk Maxwell, they unify electric, magnetic, and optical phenomena within a single field theory. (feynmanlectures.caltech.edu)

Historical development

Maxwell developed his theory by combining established experimental laws with the field concept associated with Michael Faraday. His decisive contribution was the displacement-current term, which extended the relationship between current and magnetic fields to time-dependent situations. His paper A Dynamical Theory of the Electromagnetic Field, published in 1865, connected electromagnetic propagation with light. The familiar four-equation vector presentation is a later reformulation rather than the exact notation of that paper. (feynmanlectures.caltech.edu)

Differential form

In the International System of Units (SI), the microscopic equations are

\nabla\cdot\mathbf E=\frac{\rho}{\varepsilon_0}, \qquad \nabla\cdot\mathbf B=0,

\nabla\times\mathbf E=-\frac{\partial\mathbf B}{\partial t}, \qquad \nabla\times\mathbf B =\mu_0\mathbf J+ \mu_0\varepsilon_0\frac{\partial\mathbf E}{\partial t}.

Here $\mathbf E$ and $\mathbf B$ are the electric field and magnetic flux density; $\rho$ and $\mathbf J$ are total charge density and current density. The constants $\varepsilon_0$ and $\mu_0$ are vacuum permittivity and permeability. These are partial differential equations: their spatial and temporal derivatives specify local relationships between fields and sources. (feynmanlectures.caltech.edu)

The four equations have distinct physical meanings:

Gauss’s law: electric charge determines the divergence of the electric field. In electrostatics, the field of an isolated point charge reproduces Coulomb’s law.
Gauss’s law for magnetism: magnetic flux through any closed surface is zero. The standard equations contain no magnetic-charge source.
Faraday’s law of induction: a changing magnetic field is associated with a circulating electric field.
Ampère–Maxwell law: electric current and a changing electric field contribute to the curl of the magnetic field. (feynmanlectures.caltech.edu)

The last equation includes the displacement current density $\varepsilon_0\partial\mathbf E/\partial t$ . Unlike conduction current, this term need not represent charge crossing a region. Between the plates of a capacitor being charged, the changing electric field supplies the contribution needed to make the magnetic circulation independent of the chosen spanning surface. (feynmanlectures.caltech.edu)

Integral form

An equivalent formulation uses integrals over surfaces and curves. For a fixed surface $S$ bounded by a curve $C$ , with compatible orientations,

\oiint_{\partial V}\mathbf E\cdot d\mathbf A =\frac{Q_V}{\varepsilon_0}, \qquad \oiint_{\partial V}\mathbf B\cdot d\mathbf A=0,

\oint_C\mathbf E\cdot d\boldsymbol\ell =-\frac{d}{dt}\int_S\mathbf B\cdot d\mathbf A,

\oint_C\mathbf B\cdot d\boldsymbol\ell =\mu_0 I_S+ \mu_0\varepsilon_0\frac{d}{dt} \int_S\mathbf E\cdot d\mathbf A.

Here $Q_V$ is the enclosed charge and $I_S$ the current through $S$ . These expressions relate flux and circulation to enclosed sources. The divergence theorem and Stokes’ theorem connect the integral and differential formulations wherever the required regularity conditions hold. Moving contours require additional treatment of their motion. (feynmanlectures.caltech.edu)

Fields in matter

For macroscopic materials, polarization and magnetization are incorporated into auxiliary fields $\mathbf D$ and $\mathbf H$ :

\nabla\cdot\mathbf D=\rho_{\mathrm f}, \qquad \nabla\times\mathbf H =\mathbf J_{\mathrm f}+\frac{\partial\mathbf D}{\partial t}.

The other two equations retain their forms in terms of $\mathbf E$ and $\mathbf B$ . Subscript $\mathrm f$ denotes free charge or current, distinguished from bound sources associated with the material. (live.ocw.mit.edu)

These equations require constitutive relations describing the material response. Simple linear, isotropic media may obey $\mathbf D=\varepsilon\mathbf E$ , $\mathbf B=\mu\mathbf H$ , and $\mathbf J=\sigma\mathbf E$ . More general responses can depend on direction, frequency, or field strength. Initial and boundary conditions, together with source specifications and material relations, define a particular electromagnetic problem. (live.ocw.mit.edu)

Waves and conservation laws

In a source-free vacuum, the equations imply

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

where $c=1/\sqrt{\mu_0\varepsilon_0}$ is the speed of light. Plane-wave solutions have electric and magnetic fields perpendicular to each other and to the propagation direction, with $E=cB$ . This identifies light as electromagnetic radiation and places radio waves, visible light, and X-rays within the same physical framework. Their wave behavior underlies classical optics, including interference and diffraction. (feynmanlectures.caltech.edu)

Taking the divergence of the Ampère–Maxwell equation gives local charge conservation:

\frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf J=0.

The equations also imply Poynting’s theorem, expressing conservation of electromagnetic energy. In vacuum, energy density and energy flux are

u=\frac{\varepsilon_0E^2}{2}+\frac{B^2}{2\mu_0}, \qquad \mathbf S=\frac{\mathbf E\times\mathbf B}{\mu_0},

with $\partial u/\partial t+\nabla\cdot\mathbf S=-\mathbf J\cdot\mathbf E$ . The right-hand side represents energy transferred from the field to charged matter. (live.ocw.mit.edu)

Relativity and scope

Maxwell’s equations retain their form under a Lorentz transformation, although observers in relative motion generally assign different electric and magnetic components to the same electromagnetic field. This compatibility is central to the theory of relativity. (feynmanlectures.caltech.edu)

The equations are classical: by themselves they do not describe individual photon detection or quantized exchanges between radiation and matter. Quantum electrodynamics supplies the quantum framework, with Maxwell’s field equations governing its classical limit. (preskill.caltech.edu)