PlanetPhysics: 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. Together, they completely describe classical electromagnetic phenomena, just as Newton's laws completely describe classical mechanical phenomena. All four are named after persons other than Maxwell, but Maxwell was the first to add the displacement current term to Ampère's Law, which led to the association of electromagnetic waves with light and paved the way for the discovery of special relativity. All four equations can be written in both integral and differential forms, with both forms convenient for specific problems. Note that strictly speaking these are Maxwell's equation in vacuo, with different forms for interaction with matter.

Notation

Throughout this article SI units are adopted for clarity, but the interesting mathematical aspects of the equations are independent of the constants $\mu_0$ and $\epsilon_0$ , and indeed of the physical meaning of the equations.

$\displaystyle \mathbf{E} =$ Electrical field strength, SI units Volt m $\displaystyle ^{-1}$

$\displaystyle \mathbf{B} =$ Magnetic flux density, SI units Tesla

$\displaystyle \mathbf{J} =$ Current density, SI units Ampère m $\displaystyle ^{-3}$

$\displaystyle \mathbf{\epsilon_0} =$ Permittivity of free space $\displaystyle \approx 8.85 \times 10^{-12}$ m $\displaystyle ^{-1}$

$\displaystyle \mathbf{\mu_0} =$ Permeability of free space $\displaystyle = 4\pi \times 10^7$ Henry m $\displaystyle ^{-1}$

Gauss' Law of Electrostatics

Differential form

$\displaystyle \nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$

Integral form

$\displaystyle \oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{A} = \frac {q}{\epsilon_0}$

where

is the charge enclosed in the volume bounded by the surface

Gauss' Law of Magnetostatics

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

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

This law can be interpreted as a statement of the non-existence of magnetic monopoles, a fact confirmed by all experiments to date.

Faraday's Law

Differential form

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

Ampère's Law

Differential form

$\displaystyle \nabla \times \mathbf{B} = - \mu_0 \epsilon_0 \frac{ \partial \mathbf{E}}{\partial t}$

Integral form

Properties of Maxwell's Equations

These four equations together have several interesting properties:

Lorentz invariance
Gauge invariance
Invariance under the transformation $B \rightarrow \frac{E}{c}$ , $E \rightarrow B c$

"Maxwell's equations" is owned by invisiblerhino.

(view preamble)

Also defines:

FaradaysLaw

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

Cross-references: volume, charge, special relativity, Newton's laws
There are 8 references to this object.

This is version 5 of Maxwell's equations, born on 2008-03-08, modified 2008-03-10.
Object id is 266, canonical name is MaxwellsEquations.
Accessed 1283 times total.

Classification:

Physics Classification:

40. (ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID MECHANICS)

Pending Errata and Addenda

None.

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