Maxwell's equations are part of a gauge theory, implying the existence of a certain group of symmetries. In this case the group is abelian, namely U(1). This is also known as the circle group, i.e. the group of complex numbers of magnitude 1 under multiplication. Maxwell's equations are described in terms of a vector potential or connection 1-form A (depending on who you ask - let's call it a 1-form). Then there is an associated curvature F = dA where "d" is the antisymmetrised derivative on differential forms. (You could also think of this as an antisymmetric (0,2) tensor field). Maxwell's equations are: dF = 0 (vacuous as dd = 0) and d*F = 4pi*j, where "j" is the electromagnetic current and "*" is the Hodge star. The six components of F may be thought of, viewed in terms of their spatial transformations once we pick a preferred time direction, as a vector E (the electric field) and a pseudovector B (the magnetic field). These are related by a duality transformation. We are free to change our particles by multiplying by a phase from U(1), which corresponds to transforming A by a gradient (i.e. A -> A + df). This gives us an abelian Yang-Mills theory. Nonabelian gauge groups, like SU(2), lead to nonabelian Yang-Mills theories which look roughly the same but with commutators.

Maxwell’s Equations succinctly contain the basis of our theoretical knowledge of classical electrodynamics, that is electricity, magnetism, and electromagnetic radiation. They do so by relating the electric and magnetic fields to their sources and to each other. The statement of a coherent theory of electrodynamics as contained in the equations is considered one of the greatest triumphs of physics.

The equations, in all their glory, are:

∇ • E = ρ / εo

∇ • B = 0

∇ x E = - ∂B/∂t

∇ x B = μoJ + μoεoE/∂t

where E and B are the Electric and Magnetic fields, respectively, ρ is the position dependent density of electric charge, J is the position dependent density of electric current, μoand εo are constants determined experimentally, ∇ • V is the divergence of a vector field, and ∇ x V is the curl of a vector field.1

These equations were first summarized as such by the Scottish physicist James Clerk Maxwell in 1860,2 although most of their content individually was already known.

The first equation is Gauss’ Law in differential form. Gauss’ Law describes the static electric field in terms of electric charges, by giving a formula for the component of the electric field perpendicular to the surface of a boundary added over the entire boundary in terms of the total charge contained within the boundary. It states ∫surfE • da = Qenc/ εo. By the divergence theorem surfE • da = ∫vol ∇ • E dτ. Also Qenc = ∫vol ρ dτ, hence the first Maxwell equation ∇ • E = ρ / εo.

The second equation has no special name but is a statement of an apparent fact, known before Maxwell’s time, that there are no point sources of the magnetic field. All magnetic fields results from electric currents.

The third equation is Faraday’s Law in differential form. Faraday’s law describes the way changing magnetic fields produce electric fields, by giving the component of the electric field along the direction around a closed path summed over the path in terms of the change in the magnetic flux through the path, in symbols
E • dl = - dΦ/dt , where Φ is the magnetic flux. By Stokes’ theoremE • dl = ∫surf ∇ x E, and Φ = ∫surf B • da, so
∇ x E = dB/dt.

The fourth equation consists of two terms. The first is Ampere’s Law, again in differential form. Ampere’s Law describes the static magnetic field in terms of the electric currents that are its source, by giving the component of the magnetic field along the direction around a closed path summed over the path in terms of the current passing through the path, in symbols ∫ B • dl = μoIenc. Again, by Stokes’ Theorem,
B • dl = ∫surf ∇ x B, and Ienc = ∫surf J dτ, so ∇ x B = μo J

However, Ampere’s Law is complete only for the magnetostatic case, where the currents, or the electric fields that produce them, are not changing. The second term in the fourth equation is Maxwell’s ‘correction’ to Ampere’s Law, and describes changes to the magnetic field, or more precisely the curl of the magnetic field, in terms of the changing electric field.

As shown, Maxwell’s Equations relate the electric and magnetic fields to their sources, charges and currents, and to each other. With the addition of the Lorentz force law, F = Q (E + v x B), which gives the force on a charge due to the electric and magnetic fields, thus relating current to the electric and magnetic fields, everything we know about classical electricity and magnetism is compactly stated.

Maxwell’s Equations are also important in that that they show the nature of electromagnetic radiation to be wavelike oscillations of the electric and magnetic fields. This is contained in the following way: In regions where there is no charge or current (ρ=0 and J=0), for instance in a vacuum, the equations become:

∇ • E = 0
∇ • B = 0
∇ x E = - ∂B/∂t
∇ x B = μoεoE/∂t

If we uncouple these partial differential equations by applying the curl to the third and fourth equations, we obtain the result (see footnote 3):

2E = μoεo2E/∂t2
and
2B = μoεo2B/∂t2

These have the familiar form of wave equations. Evidently, the electric and magnetic fields spread and oscillate in a wave-like manner. We know from the study of waves in mechanics that the speed of a wave given by the above equations is 1 / √ μoεo, and sure enough, with the appropriate values 1 / √ μoεo = 3x108 = c, the speed of light.

We also know from the four equations, which relate the directional derivatives of the E and B fields to each other, that the fields must oscillate in phase, must be mutually perpendicular and perpendicular to the direction of propagation, and that the maximum intensity of the electric field must be c times the maximum intensity of the magnetic field, all of which are observed phenomena of electromagnetic radiation.4

In materials that are responsive to electric and magnetic fields, with dielectric phenomena and diamagnetism or paramagnetism, Maxwell’s equations are sometimes expressed in the following equivalent form:

∇ • D = ρf
∇ • B = 0
∇ x E = - ∂B/∂t
∇ x H = Jf + ∂D/∂t

where ρf is the position dependent free charge density (that charge which is unaffected by the properties of the material), and Jf is the position dependent free current density (that current which is unaffected by the properties of the material). D is called the electric displacement and H is called ‘the H.’ Both D and H are called ‘auxillary fields’.

The reason for this different notation is that within materials that are responsive to electric and magnetic fields, charges and currents are set up in the presence of fields that tend to negate or augment these fields, depending on properties of the material. It is handy, then, to have the auxillary fields as a measure of what the fields would be in the absence of the responses from the material, for only a knowledge of the free charges and currents are required to calculate them. Then the actual fields can be calculated from the auxillary fields if specific properties of the responses are known. In particular, in this formulation, E = (D-P)/εo, where P is the polarization per unit volume, and is determined by the electric charge pattern set up by the material’s response (the so called bound charge), and B = (H+M)*μo, where M is the magnetization and is determined by the current pattern set up by the material’s response (the so-called bound current).

The importance of Maxwell’s Equations is stated on a clever T-shirt that says:

And God said:

∇ • E = ρ / εo
∇ • B = 0
∇ x E = - ∂B/∂t
∇ x B = μoJ + μoεoE/∂t

Translation: “Let there be light”

1 The divergence and curl are quantities from vector calculus. The divergence is can be thought of as a measure of how closely a vector field radiates from a single point, and the curl is in a way a measure of how much a vector field curls around a point. The character ∇ is a directional derivative operator, in symbols (i ∂/∂x, j ∂/∂y, k ∂/∂z), and is pronounced ‘del,’ or ‘nabla’ by some freaks.

2Our current compact vector notation hadn’t been devised yet, so Maxwell would have actually written these equations out in terms of each of the vector components. In that notation, four equations become many.

3we have:
∇ • E = 0
∇ • B = 0
∇ x E = - ∂B/∂t
∇ x B = μoεoE/∂t

Taking the curl of the left side of third equation:
∇ x (∇ x E) = ∇(∇ • E) - ∇2E (an easily proven result from vector calculus – no physics here)
= - ∇2E (because E = 0 from the second equation)

meanwhile, taking the curl of the right side:
∇ x (-∂B/∂t) = -∂/∂t (∇ x B)
= - μoεo2E/∂t2 (by the fourth equation)

so ∇2E = μoεo2E/∂t2

4 The derivation of these properties is straight forward from wave mechanics and vector calculus.

The thing I find most interesting about Maxwell's Equations is that they can be derived from Coulomb's Law (which gives the force two charges exert on each other), Einstein's Special Theory of Relativity, and a few highly-plausible assumptions (actually I only remember one from when I worked through this--that total electric charge is an invariant. Hardly an assumption!).

So how can it be that all those crazy equations above can be derived from Coulomb's Law. It would take a great deal of my time to work this out in detail--I can point to a book by Melvin Schwartz entitled Principles of Electrodynamics for some guidance, but I recall that his analysis was incomplete. The basic idea is this. Charge and potential are relativistically four-vectors: electric charge is coupled with three vectorial current components and electric potential is coupled with the three vectorial components of magnetic potential. For analysis on this, see a book on relativity (I recommend Gabriel Bergmann's Theory of Relativity). As an example of what I'm talking about, think about how the Lorentz Force between two idle charges would look if you started moving relative to the idle charges. Think about dV/dx--how does it change? Notice how dx's become dt's! and V's become magnetic vectorial components. You get forces like dB/dt! This is really fascinating if you work through it but I have to go back to work. I might come back and work through this but if you're interested, do it yourself--it's rewarding.

Some quick thoughts. Einstein developed relativity after Maxwell's Equations were developed. But relativity could have been developed simply based on analysis of light and its invariant speed, not knowing that light was an electromagnetic wave. From that, from charge conservation, and from Coulomb's Law, Maxwell's Equations could have been developed. Imagine the joy of intentionally discovering Faraday's Law, and even Ampere's Law intentionally.

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