Maxwell’s equations

Maxwell’s equations are a set of four partial differential equationsMathworldPlanetmath first combined by James Clerk Maxwell. They may also be written as integral equationsMathworldPlanetmath. Two other important equations, the electromagnetic wave equation and the equation of conservation of charge, may be derived from them.

0.1 Notation

As this article considers merely the mathematical aspects of the equations, natural units have been used throughout. For their use in physics see any classical electromagnetism textbook.

𝐄=Electric field strength
𝐁=Magnetic flux density

0.2 Gauss’ Law of Electrostatics


0.3 Gauss’ Law of Magnetostatics


0.4 Faraday’s Law

Differential formMathworldPlanetmath


Integral form


0.5 Ampère’s Law

Differential form


Integral form


0.6 Properties of Maxwell’s Equations

These four equations together have several interesting properties:

  • Lorentz invariance

  • The fields 𝐄 and 𝐁 may be Helmholtz decomposed into irrotational and solenoidal potentials. A gauge transformation in these variables does not affect the values of the fields.

Title Maxwell’s equations
Canonical name MaxwellsEquations
Date of creation 2013-03-22 17:51:34
Last modified on 2013-03-22 17:51:34
Owner invisiblerhino (19637)
Last modified by invisiblerhino (19637)
Numerical id 28
Author invisiblerhino (19637)
Entry type Definition
Classification msc 35Q60
Classification msc 78A25
Related topic PartialDifferentialEquation
Defines Faraday’s Law
Defines Ampere’s Law
Defines Gauss’ Law of Electrostatics
Defines Gauss’ Law of Magnetostatics