Maxwell’s equations

Maxwell’s equations are a set of four partial differential equations first combined by James Clerk Maxwell. They may also be written as integral equations. 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.

$$𝐄=\text{Electric field strength}$$

$$𝐁=\text{Magnetic flux density}$$

0.2 Gauss’ Law of Electrostatics

$$\nabla \cdot 𝐄=0$$

$${\oint }_S𝐄\cdot d𝐒=0$$

0.3 Gauss’ Law of Magnetostatics

$$\nabla \cdot 𝐁=0$$

$${\oint }_S𝐁\cdot d𝐒=0$$

0.4 Faraday’s Law

Differential form

$$\nabla \times 𝐄=-\frac{\partial 𝐁}{\partial t}$$

Integral form

$${\oint }_C𝐄\cdot d𝐥=-\frac{\mathrm{d}}{\mathrm{d}t}\left({\int }_S𝐁\cdot d𝐒\right)$$

0.5 Ampère’s Law

Differential form

$$\nabla \times 𝐁=\frac{\partial 𝐄}{\partial t}$$

Integral form

$${\oint }_C𝐁\cdot d𝐥={\int }_S\frac{\partial 𝐄}{\partial t}\cdot d𝐒$$

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