Maxwell's Equations

Maxwell's Equations, formulated around 1861 by James Clerk Maxwell describe the interrelation between electric and magnetic fields.They were a synthesis of what was known about electricity and magnetism, particularly building on the work of Michael Faraday, Andre-Marie Ampere, and others. These equations predicted the existence of Electromagnetic waves, giving them properties that were recognized to be properties of light, leading to the (correct) realization that light is an electromagnetic wave. Other forms of electromagnetic waves, such as radio waves, were not known at the time, but were subsequently demonstrated by Heinrich Hertz in 1888. These equations are considered to be among the most elegant edifices of mathematical physics.

Maxwell's equations serve many purposes and take many forms. On the one hand, they are used in the solution of actual real-world problems of electromagnetic fields and radiation. On the other hand, they are the subject of admiration for their elegance. There are many T-shirts, typically obtainable on college campuses, sporting various forms of these equations.

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, B, D, and H, though the simpler forms use only E and B.

Name	differential form	integral form
Coulomb's law of electrostatics, or Gauss's Law:	<math>\nabla \cdot \mathbf{D} = \rho</math>	<math>\oint_S \mathbf{D} \cdot \mathrm{d}\mathbf{A} = \int_V \rho\, \mathrm{d}V</math>
Absence of magnetic monopoles:	<math>\nabla \cdot \mathbf{B} = 0</math>	<math>\oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0</math>
Faraday's Law of Induction:	<math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>	<math>\oint_C \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \int_S \frac{\partial\mathbf{B}}{\partial t} \cdot \mathrm{d} \mathbf{A}</math>
Ampère's Law, or the Biot-Savart Law, plus displacement current:	<math>\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}} {\partial t}</math>	<math>\oint_C \mathbf{H} \cdot \mathrm{d}\mathbf{l} = \int_S \mathbf{J} \cdot \mathrm{d} \mathbf{A} + \int_S \frac{\partial\mathbf{D}}{\partial t} \cdot \mathrm{d} \mathbf{A}</math>

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's 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 Coulomb's law of electrostatics, manipulated very elegantly (as usual) by 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:

<math>\mathrm{d}\bold{F}=0</math>

<math>\mathrm{d} * {\bold{F}}=\bold{J}</math>

where d is exterior derivative operator, * is the Hodge star operator, and F is the Faraday tensor.