**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 / ε_{o}

∇ • **B** = 0

∇ x **E** = - ∂**B**/∂t

∇ x **B** = μ_{o}**J** + μ_{o}ε_{o}∂**E**/∂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, μ_{o}and ε_{o} 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 ∫_{surf}**E** • d**a** = Q_{enc}/ ε_{o}. By the divergence theorem ∫_{ surf}**E** • d**a** = ∫_{vol} ∇ • **E** dτ. Also Q_{enc} = ∫_{vol} ρ dτ, hence the first Maxwell equation ∇ • **E** = ρ / ε_{o}.

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** • d**l** = - dΦ/dt , where Φ is the magnetic flux. By Stokes’ theorem ∫ **E** • d**l** = ∫_{surf} ∇ x **E**, and Φ = ∫_{surf} **B** • d**a**, so

∇ x **E** = d**B**/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** • d**l** = μ_{o}I_{enc}. Again, by Stokes’ Theorem,

**B** • d**l** = ∫_{surf} ∇ x **B**, and I_{enc} = ∫_{surf} **J** dτ, so ∇ x **B** = μ_{o} **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** = μ_{o}ε_{o}∂**E**/∂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):

∇^{2}**E** = μ_{o}ε_{o} ∂^{2}**E**/∂t^{2}

and

∇^{2}**B** = μ_{o}ε_{o} ∂^{2}**B**/∂t^{2}

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 / √ μ_{o}ε_{o}, and sure enough, with the appropriate values 1 / √ μ_{o}ε_{o} = 3x10^{8} = 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** = ρ_{f}

∇ • **B** = 0

∇ x **E** = - ∂**B**/∂t

∇ x **H** = **J**_{f} + ∂**D**/∂t

where ρ_{f} is the position dependent *free charge* density (that charge which is unaffected by the properties of the material), and **J**_{f} 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**)/ε_{o}, 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**)*μ_{o}, 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 / ε_{o}

∇ • **B** = 0

∇ x **E** = - ∂**B**/∂t

∇ x **B** = μ_{o}**J** + μ_{o}ε_{o}∂**E**/∂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.

^{2}Our 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.

^{3}we have:

∇ • **E** = 0

∇ • **B** = 0

∇ x **E** = - ∂**B**/∂t

∇ x **B** = μ_{o}ε_{o}∂**E**/∂t

Taking the curl of the left side of third equation:

∇ x (∇ x **E**) = ∇(∇ • **E**) - ∇^{2}**E** (an easily proven result from vector calculus – no physics here)

= - ∇^{2}**E** (because ∇ • **E** = 0 from the second equation)

meanwhile, taking the curl of the right side:

∇ x (-∂**B**/∂t) = -∂/∂t (∇ x **B**)

= - μ_{o}ε_{o}∂^{2}**E**/∂t^{2} (by the fourth equation)

so ∇^{2}**E** = μ_{o}ε_{o}∂^{2}**E**/∂t^{2}

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