PlanetMath: Maxwell's equations

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Maxwell's equations

(Definition)

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.

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.

$\displaystyle \mathbf{E} =$ Electric field strength

$\displaystyle \mathbf{B} =$ Magnetic flux density

Gauss' Law of Electrostatics

$\displaystyle \nabla \cdot \mathbf{E} = 0$

$\displaystyle \oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{S} = 0$

Gauss' Law of Magnetostatics

$\displaystyle \nabla \cdot \mathbf{B} = 0$

$\displaystyle \oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{S} = 0$

Faraday's Law

Differential form

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

Integral form

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

Ampère's Law

Differential form

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

Integral form

$\displaystyle \oint_C \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \int_S \frac{\partial \mathbf{E}}{\partial t} \cdot \mathrm{d} \mathbf{S}$

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.

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"Maxwell's equations" is owned by invisiblerhino.

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See Also: partial differential equation

Also defines:

Faraday's Law, Ampere's Law, Gauss' Law of Electrostatics, Gauss' Law of Magnetostatics

Attachments:: derivation of wave equation from Maxwell's equations (Derivation) by invisiblerhino
derivation of Coulomb's Law from Gauss' Law (Derivation) by invisiblerhino

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Cross-references: variables, transformation, gauge, potentials, solenoidal, irrotational, fields, properties, integral, differential form, units, wave equation, equations, integral equations, partial differential equations
There are 4 references to this entry.

This is version 25 of Maxwell's equations, born on 2008-02-26, modified 2008-04-21.
Object id is 10336, canonical name is MaxwellsEquations.
Accessed 1149 times total.

Classification:

AMS MSC:	35Q60 (Partial differential equations :: Equations of mathematical physics and other areas of application :: Equations of electromagnetic theory and optics)
	78A25 (Optics, electromagnetic theory :: General :: Electromagnetic theory, general)

Pending Errata and Addenda

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