# 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.

 $\mathbf{E}=\mbox{Electric field strength}$
 $\mathbf{B}=\mbox{Magnetic flux density}$

## 0.2 Gauss’ Law of Electrostatics

 $\nabla\cdot\mathbf{E}=0$
 $\oint_{S}\mathbf{E}\cdot\mathrm{d}\mathbf{S}=0$

## 0.3 Gauss’ Law of Magnetostatics

 $\nabla\cdot\mathbf{B}=0$
 $\oint_{S}\mathbf{B}\cdot\mathrm{d}\mathbf{S}=0$

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

Integral form

 $\oint_{C}\mathbf{E}\cdot\mathrm{d}\mathbf{l}=-\frac{\mathrm{d}}{\mathrm{d}t}% \left(\int_{S}\mathbf{B}\cdot\mathrm{d}\mathbf{S}\right)$

## 0.5 Ampère’s Law

Differential form

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

Integral form

 $\oint_{C}\mathbf{B}\cdot\mathrm{d}\mathbf{l}=\int_{S}\frac{\partial\mathbf{E}}% {\partial t}\cdot\mathrm{d}\mathbf{S}$

## 0.6 Properties of Maxwell’s Equations

These four equations together have several interesting properties:

• Lorentz invariance

• The fields $\mathbf{E}$ and $\mathbf{B}$ 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