Maxwell equations

In physics, the Maxwell equations are the mathematical equations that describe the interrelationship between electric and magnetic fields. They tell us how these fields depend on electric charges and electric currents. The equations are named after the Scottish physicist James Clerk Maxwell, who published them (in a somewhat old-fashioned notation) in 1865. The Maxwell equations are still held to be valid, even in quantum electrodynamics where the electromagnetic fields are reinterpreted as quantum mechanical operators satisfying canonical commutation relations.

Among physicists, the Maxwell equations take a place of equal importance as Newton's equation F=ma, Einstein's equation E=mc², and Schrödinger's equation Hψ=Eψ. Yet, in the eyes of the general, well-educated, public, Clerk Maxwell does not have the same fame as the other three physicists. This is somewhat surprising, because the applications of Maxwell's equations have far-reaching impact on society. Maxwell was the first to see that his equations predict electromagnetic waves. Without knowledge and understanding of these waves we would not have radio, radar, television, cell phones, global positioning systems, etc. Maybe, the lack of fame of the Maxwell's equations is due to the fact that they cannot be caught in a simple iconic equation like E=mc². In modern textbooks Maxwell's equations are presented as four fairly elaborate vector equations, involving abstract mathematical notions as curl and divergence.

There are two sets of Maxwell's equations, the most basic ones are the microscopic equations, which describe the electric field E, the magnetic field B, charges, and currents, in vacuum. That is, there is no other ponderable matter in the system than accounted for in the equations. The other set is known as the macroscopic equations. Here it is assumed that there is a continuous medium present (air, for instance) that is polarizable and magnetizable. In this case two additional vectors, P (the polarization vector of the medium) and M (the magnetization vector of the medium) play a role. It will be discussed below that it is convenient to replace those two vectors by two auxiliary vectors, the dielectric displacement D and the magnetic field H. The Dutch physicist Lorentz has shown that the macroscopic equations can be derived from the microscopic equations by an averaging of electric and magnetic dipoles over the medium. In that sense the microscopic equations are the most basic. On the other hand, given the macroscopic equations, one simply retrieves the microscopic equations by putting P = M = 0.

Microscopic equations

The fields E and B depend on time t and position r, for brevity this dependence is not shown explicitly in the equations. The first two Maxwell equations do not depend on charges or currents. In SI units they read,

${\begin{aligned}{\boldsymbol {\nabla }}\cdot \mathbf {B} =0&\Longleftrightarrow \iint _{\mathrm {closed} \atop \mathrm {surface} }\mathbf {B} \cdot d\mathbf {S} =0&&\mathrm {(Gauss'\;law)} \\{\boldsymbol {\nabla }}\times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}&\Longleftrightarrow \oint _{C}\mathbf {E} \cdot d\mathbf {l} =-{\frac {d}{dt}}\iint _{S}\mathbf {B} \cdot d\mathbf {S} &&\mathrm {(Faraday's\;law)} \\\end{aligned}}$

The first Maxwell equation, given in differential form, is converted to the magnetic Gauss' law, an integral equation, by integrating over a volume V and applying the divergence theorem. The closed surface integrated over is the surface enveloping V. The second Maxwell equation is converted into Faraday's law by integrating left- and right-hand side over a surface S bounded by a contour C and applying Stokes' theorem.

Let ρ(r) be an electric charge density and J(r) be an electric current density, both quantities enter the second set of Maxwell equations (again in SI units)

${\begin{aligned}{\boldsymbol {\nabla }}\cdot \mathbf {E} ={\frac {\rho (\mathbf {r} )}{\epsilon _{0}}}&\Longleftrightarrow \iint _{\mathrm {closed} \atop \mathrm {surface} }\mathbf {E} \cdot d\mathbf {S} ={\frac {Q_{\mathrm {tot} }}{\epsilon _{0}}}&&\mathrm {(Gauss'\;law)} \\{\boldsymbol {\nabla }}\times \mathbf {B} =\mu _{0}\mathbf {J} (\mathbf {r} )+\epsilon _{0}\mu _{0}{\frac {\partial \mathbf {E} }{\partial t}}&\Longleftrightarrow \oint _{C}\mathbf {B} \cdot d\mathbf {l} =\mu _{0}i+\epsilon _{0}\mu _{0}{\frac {d}{dt}}\iint _{\mathrm {closed} \atop \mathrm {surface} }\mathbf {E} \cdot d\mathbf {S} &&\mathrm {(Ampere's\;law)} \\\end{aligned}}$

The third Maxwell equation is converted to the electrostatic Gauss' law by integrating over a volume V and applying the divergence theorem. The closed surface integrated over is the surface enveloping V. The fourth Maxwell equation is converted into Ampère's law by integrating left- and right-hand side over a surface S bounded by a contour C and applying Stokes' theorem. Here we need to add the historical note that André-Marie Ampère had not seen the necessity of the second term on the right-hand side containing the time derivative of the electric field, the so-called displacement current. Ampère formulated his law for the conduction current i only, which is correct if the displacement is zero. It was Clerk Maxwell who recognized the need of the displacement current.

The electric constant ε₀ and the magnetic constant μ₀ are peculiar to the use of SI units. Their product satisfies

$\epsilon _{0}\,\mu _{0}={\frac {1}{c^{2}}},$

where c is the speed of light.

The microscopic Maxwell equations in Gaussian units do not contain the electric and magnetic constant, but c instead, they read

${\begin{aligned}{\boldsymbol {\nabla }}\cdot \mathbf {B} &=0\\{\boldsymbol {\nabla }}\times \mathbf {E} &=-{\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}}\\{\boldsymbol {\nabla }}\cdot \mathbf {E} &=4\pi \,\rho (\mathbf {r} )\\{\boldsymbol {\nabla }}\times \mathbf {B} &={\frac {4\pi }{c}}\mathbf {J} (\mathbf {r} )+{\frac {1}{c}}{\frac {\partial \mathbf {E} }{\partial t}}\\\end{aligned}}$

The factor 4π arises here because the Gaussian system of units is not rationalized, in contrast to the SI system.

Macroscopic equations

In SI units one defines

$\mathbf {D} \equiv \epsilon _{0}\mathbf {E} +\mathbf {P} \qquad \mathbf {H} \equiv {\frac {1}{\mu _{0}}}\mathbf {B} -\mathbf {M} ,$

where P is the polarization and M is the magnetization of the medium. The somewhat asymmetric definition of the electric and magnetic auxiliary fields has a historical origin. For many years the field H was seen as the analogue of E and B was an auxiliary field, analogous to D. Later, when it became clear that there are no magnetic charges and currents, the field B was considered to be basic. The same historic circumstance explains that B is usually called magnetic induction and H magnetic field. The first two macroscopic Maxwell equations (the ones that do not contain charge and current densities) are exactly the same as the microscopic equations, the other two equations are very similar but in terms of the auxiliary fields,

${\begin{aligned}{\boldsymbol {\nabla }}\cdot \mathbf {D} &=\rho (\mathbf {r} )\\{\boldsymbol {\nabla }}\times \mathbf {H} &=\mathbf {J} (\mathbf {r} )+{\frac {\partial \mathbf {D} }{\partial t}}\\\end{aligned}}$