# 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 at the time about electricity and magnetism, particularly building on the work of Michael Faraday, Charles-Augustin Coulomb, 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 $\nabla \cdot \mathbf{E}$ and $\nabla \times \mathbf{E}$ 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.

Here are the equations expressed in differential form:

Name E and B E, B, D, and H
Coulomb's law of electrostatics, or Gauss's Law: $\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon}$ $\nabla \cdot \mathbf{D} = \rho$
Absence of magnetic monopoles: $\nabla \cdot \mathbf{B} = 0$ $\nabla \cdot \mathbf{B} = 0$
Faraday's Law of Induction: $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$ $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$
Ampère's Law, or the Biot-Savart Law,

plus displacement current:

$\nabla \times \mathbf{B} = \mu\ \mathbf{J} + \frac{1}{c^2} \frac{\partial \mathbf{E}} {\partial t}$ $\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}} {\partial t}$

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 $\rho$ denotes the free electric charge density.

The use of separate D and H fields is sometimes helpful in engineering problems involving dielectric polarizability and magnetic permeability of the materials involved. But the more elegant "pure" form removes D and H through the equations:

$\mathbf{D} = \epsilon \mathbf{E}$
$\mathbf{H} = \mathbf{B} / \mu$

where $\epsilon\,$ and $\mu\,$, often written $\epsilon_0\,$ and $\mu_0\,$, are physical constants known as the permittivity of the vacuum and the permeability of the vacuum. (The subscript zero refers to the vacuum values. By setting these constants to other values to denote the permittivity and permeability of other materials, one can solve various problems in electrodynamics and optics conveniently.)

The quantity $\epsilon \mu\,$ has units of seconds squared per meter squared. So, letting

$c = \frac{1}{\sqrt{\epsilon \mu}}\,$

we get $c\,$ as a speed. Laboratory measurements of $\epsilon\,$ and $\mu\,$ obtain a value of 3 $\times$ 108 meters per second, which is the speed of light. Careful mathematical analysis by Maxwell showed that these equations predict electromagnetic radiation at this speed.

Here are the equations expressed in integral form, though the differential versions are generally taken to be the "real" Maxwell equations. These forms can be seen to be equivalent to the differential forms through the use of the general Stokes' 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.

Name E and B E, B, D, and H
Coulomb's law: $\oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{A} = \frac{1}{\epsilon} \int_V \rho\, \mathrm{d}V$ $\oint_S \mathbf{D} \cdot \mathrm{d}\mathbf{A} = \int_V \rho\, \mathrm{d}V$
Absence of monopoles: $\oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0$ $\oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0$
Faraday's Law: $\oint_C \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \int_S \frac{\partial\mathbf{B}}{\partial t} \cdot \mathrm{d} \mathbf{A}$ $\oint_C \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \int_S \frac{\partial\mathbf{B}}{\partial t} \cdot \mathrm{d} \mathbf{A}$
Ampère / Biot-Savart Law: $\oint_C \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \mu \int_S \mathbf{J} \cdot \mathrm{d} \mathbf{A} + \frac{1}{c^2} \int_S \frac{\partial\mathbf{D}}{\partial t} \cdot \mathrm{d} \mathbf{A}$ $\oint_C \mathbf{H} \cdot \mathrm{d}\mathbf{l} = \int_S \mathbf{J} \cdot \mathrm{d} \mathbf{A} + \int_S \frac{\partial\mathbf{D}}{\partial t} \cdot \mathrm{d} \mathbf{A}$

## 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 repelling if they have like charges and attracting if unlike, is proportional to the product of the charges, and is inversely proportional to the square of the distance between them:

$F = \frac{q_1 q_2}{4 \pi \epsilon\ d^2}$

In this case, the constant defining the strength of the electric force is $4 \pi \epsilon\,$ in the denominator. More about that presently.

In SI units the charges are measured in Coulombs, the force in Newtons, the distance in Meters, and the value of $\epsilon$ is $8.854 \times 10^{-12}$ Coulombs2 per Newton meter2, or Farads per meter.

Michael Faraday reformulated the electric and magnetic forces in terms of fields He said that what was really happening was that each charge was creating an electric field (called E) that acted on the other charge. The field created by the charge q1, as observed at distance d, is

$E = \frac{q_1}{4 \pi \epsilon\ d^2}$

and points directly outward from that charge, in all directions. The force felt by charge q2 is

$F = q_2 E\,$

Now consider a sphere of radius d with the charge at the center. If $\rho$ is the charge density in Coulombs per cubic meter (Maxwell's equations are in terms of densities), the total charge in some volume is the integral, over that volume, of $\rho$.

So we have

$q = \int_V \rho\, \mathrm{d}V$

Now the field at the surface of the sphere is

$\frac{q_1}{4 \pi \epsilon\ d^2}$, or $\frac{1}{4 \pi \epsilon\ d^2} \int_V \rho\, \mathrm{d}V$

That field is directly outward, perpendicular to the sphere's surface, and is uniform over the surface. The integral of the field over the surface is $4 \pi d^2$ times that (the surface area of the sphere is $4 \pi d^2$; this is why we have the pesky factor of $4 \pi$ in various formulas; remember that d is the distance, and hence is the sphere's radius, not its diameter), so

$\int_V \frac{\rho}{\epsilon}\, \mathrm{d}V = \oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{A}$

But, by Gauss's Theorem,

$\oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{A} = \int_V \nabla \cdot \mathbf{E}\ \mathrm{d}V$

So

$\int_V \nabla \cdot \mathbf{E}\ \mathrm{d}V = \int_V \frac{\rho}{\epsilon}\, \mathrm{d}V$

Since this is true for any volume, we have

$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon}$

### Absence of Magnetic Monopoles

The second of the equations is just like the first, but for the magnetic field. The divergence of B must be the spatial density of magnetic monopoles. Since they have never been observed (though various Grand Unified Theories might allow for them), the value is zero.

This wasn't formulated initially in terms of monopoles, but was actually a statement that magnetic "lines of force" (the lines that intuitively describe the field) never end. They just circulate around various conductors carrying electric current. In contrast to this, lines of the electric field can be thought to "begin" and "end" on charged particles.

The third equation contains the result of Faraday's experiments with "electromagnetic induction"—a changing magnetic field creates an electric field, and that electric field circulates around the area experiencing the change in total magnetic flux. (Remember that the curl operator measures the extent to which a vector field runs in circles.) We won't go into the details of his experiments, except to note that he discovered that moving a coil of wire (a loop to pick up a circulating electric field) through a magnetic field (for example, by putting it on a shaft and turning it) led to the invention of electric generators, and hence made a major contribution to the industrialization of the world. Not bad for a theoretician.

### Biot-Savart Law

The Biot-Savart Law, which see, tells how flowing electric current gives rise to a magnetic field. The form that is useful to us is "Ampere's circuital law", which says that, in the vicinity of an infinitely long straight wire carrying an electric current, the magnetic field goes in circles around the wire, following a right-hand rule. The field strength is:

$B = \frac{\mu\ I}{2 \pi R}$

Now consider the path integral of the magnetic field around a circle perpendicular to the wire, with radius R, and centered at the wire. The magnetic field is parallel to that path everywhere, and the path length is $2 \pi R\,$, so:

$\oint_C \vec{B} \cdot \vec{dl} = 2 \pi R B = \mu I$

Now, by Stokes' theorem, this path integral of the B field is the same as the surface integral of the curl of the B field:

$\iint (\nabla \times \vec{B}) \cdot \vec{dA} = \oint_C \vec{B} \cdot \vec{dl} = \mu\ I = \iint \mu \vec{J} \cdot \vec{dA}$

Where $\vec{J}\,$ is the current flow density, in amperes per square meter. The integral of $\vec{J}\,$ over any surface is just the total amount of current flowing across that surface. Since the contribution of current density to the magnetic field is linear, this formula must work for any surface, so we have:

$\nabla \times \vec{B} = \mu \vec{J}$

There is another term in the fourth of Maxwell's equations, called Maxwell's displacement current. This provides for a contribution to the magnetic field from a variation, in time, of the electric field. This phenomenon is hard to exhibit experimentally, and was added by Maxwell on theoretical grounds in order to satisfy charge conservation. So the final form is:

$\nabla \times \vec{B} = \mu\ \vec{J} + \frac{1}{c^2} \frac{\partial \vec{E}} {\partial t}$

## Formulations in Relativity

Relativity provides a unification of the electric and magnetic fields into a second-rank tensor defined on four-dimensional spacetime. This is called the Faraday tensor, denoted $F^{\alpha\beta}\,$. In this formulation, Maxwell's equations are

• $F^{\alpha\beta}_{;\beta} = \mu J^\alpha\,$
• $F_{[\alpha\beta;\lambda]} = 0\,$

Where $J^\alpha\,$ is the relativistic "four-current density"—its spatial components are the usual current density $\vec{J}\,$, and its time component is the charge density $\rho\,$. The subscripted semicolon in the first equation is the "covariant gradient" of tensor calculus. The subscript with brackets in the second equation is the "antisymmetrizer".

In the language of Exterior Calculus, the equations can be rewritten even more compactly as:

• $\mathrm{d}\bold{F}=0$
• $\mathrm{d}\ *\ \bold{F} = *\ \bold{J}$

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