Maxwell equations

In physics, the Maxwell equations are the mathematical equations that describe the interrelationship between electric and magnetic fields. Further, the equations tell us how these fields are created by electric charges and electric currents. They 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 considered 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 the existence of 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 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 in vacuum the electric field E and the magnetic field B, together with their sources (charge- and current- densities). That is, it is assumed that there is no other ponderable matter in the system than the charges and currents 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 showed in the 1890s 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.

It is generally assumed that the Maxwell equations, together with the Lorentz force and Newton's second law, determine completely the (classical, i.e., non-quantum) behavior of a system of moving electric charges. However, their solution is in general very difficult and requires some sort of iterative procedure, because moving charges generate fields (electric as well as magnetic) and these fields give a Lorentz force that acting on the charges, through Newton's equation, changes the motion of the charges. The changed motion gives different fields and a new Lorentz force that acts differently on the charges, and so on.

Microscopic equations

The vector 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; Q_tot is the total electric charge contained in 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 A.-M. 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 current is zero. It was J. 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 the dielectric displacement D and the magnetic field H,

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

where P is the polarization and M is the magnetization of the medium. The somewhat asymmetric definition of the electric and magnetic auxiliary fields D and H 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 the basic field. 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 corresponding microscopic equations, the other two equations are very similar but are in terms of the auxiliary fields,

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

Some textbooks introduce the auxiliary fields D and H also for the vacuum. They follow easily from the defining equations by putting P = M = 0. Clearly, in the vacuum D is almost equal to E and H to B, so that there is no good reason to introduce these fields for the vacuum.

In Gaussian units the auxiliary fields in the vacuum are even identical to the basic fields, because in Gaussian units D and H are defined as

$\mathbf {D} \equiv \mathbf {E} +4\pi \,\mathbf {P} ,\qquad \mathbf {H} \equiv \mathbf {B} -4\pi \,\mathbf {M} ,$

so that in vacuum indeed D ≡ E and H ≡ B. The second set of Maxwell equations in Gaussian units is as follows,

${\begin{aligned}{\boldsymbol {\nabla }}\cdot \mathbf {D} &=4\pi \rho (\mathbf {r} )\\{\boldsymbol {\nabla }}\times \mathbf {H} &={\frac {4\pi }{c}}\mathbf {J} (\mathbf {r} )+{\frac {1}{c}}{\frac {\partial \mathbf {D} }{\partial t}}.\\\end{aligned}}$