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What follows is a survey of the various forms that these equations take, beginning with the most utilitarian and progressing to the most elegant. Which form you prefer depends on your scientific outlook, and perhaps your taste in T-shirts. The various <math>\nabla \cdot \mathbf{E}</math> and <math>\nabla \times \mathbf{E}</math> symbols appearing in some of the equations are the [[divergence]] and [[curl]] operators, respectively.
What follows is a survey of the various forms that these equations take, beginning with the most utilitarian and progressing to the most elegant. Which form you prefer depends on your scientific outlook, and perhaps your taste in T-shirts. The various <math>\nabla \cdot \mathbf{E}</math> and <math>\nabla \times \mathbf{E}</math> symbols appearing in some of the equations are the [[divergence]] and [[curl]] operators, respectively.
They are usually formulated as four equations (but later we will see some particularly elegant versions with only two), and the equations are usually expressed in ''differential form'', that is, as [[Partial Differential Equations]] involving the divergence and curl operators. They can also be expressed with integrals. They are often expressed in terms of four vector fields: E, D, B, and H, though the simpler form uses only E and B.
They are usually formulated as four equations (but later we will see some particularly elegant versions with only two), and the equations are usually expressed in ''differential form'', that is, as [[Partial Differential Equations]] involving the divergence and curl operators. They can also be expressed with integrals. They are often expressed in terms of four vector fields: E, B, D, and H, though the simpler forms use only E and B.
{| class="wikitable" border="1" cellpadding="8" cellspacing="0"
{| class="wikitable" border="1" cellpadding="8" cellspacing="0"
In these, '''E''' denotes the [[electric field]], '''B''' denotes the [[magnetic field]], '''D''' denotes the ''electric displacement field'', and '''H''' denotes the ''magnetic field strength'' or ''auxiliary field''. '''J''' denotes the free [[current density]], and <math>\rho</math> denotes the free [[electric charge density]].
In these, '''E''' denotes the [[electric field]], '''B''' denotes the [[magnetic field]], '''D''' denotes the ''electric displacement field'', and '''H''' denotes the ''magnetic field strength'' or ''auxiliary field''. '''J''' denotes the free [[current density]], and <math>\rho</math> denotes the free [[electric charge density]].
==Integral Form==
Let's dispose of the integral form first. The integral forms can be seen to be equivalent to the differential forms thorough the use of the general [[Stoke%27s Theorem]]. The form known as ''Gauss's Theorem'' (k=3) takes care of the equations involving the divergence, and the form commonly known as just ''Stokes' Theorem'' (k=2) takes care of those involving the curl.
We will say nothing further about the equations in integral form. The differential versions are the "real" Maxwell equations.
==What the Four Equations mean==
===Coulomb's Law===
The first equation is just [[C. A. Coulomb|Coulomb]]'s law of electrostatics, manipulated very elegantly (as usual) by [[Michael Faraday|Faraday]] and [[Gauss]]. Coulomb's law simply says that the electric force between two charged particles acts in the direction of the line between them, is attracting if they have like charges and repelling if unlike, is proportional to the product of the charges, and is inversely proportional to the square of the distance between them:
:<math>F = \frac{q_1 q_2}{4 \pi \epsilon\ d^2}</math>
==Other Formulations==
In the language of [[Exterior Calculus]], Maxwell's equations can be rewritten much more compactly as:
In the language of [[Exterior Calculus]], Maxwell's equations can be rewritten much more compactly as:
:<math>\mathrm{d}\bold{F}=0</math>
:<math>\mathrm{d}\bold{F}=0</math>