Maxwell’s equations

For a long time, the theory of magnetism lacked an exact mathematical description. It was not until 1864 that James Clerk Maxwell completely explained the phenomena in a physics context. The four Maxwell equations he discovered still form the basis of electrodynamics today. Essentially, Maxwell's equations describe how large the electric and magnetic fields and therefore the corresponding forces are when certain charge or current distributions are present. Maxwell recognised that electric and magnetic phenomena are not independent of each other. Thus, an electric field in motion also gives rise to magnetic fields. You can learn more about the history of magnets in our guide.

In an electromagnetic wave, time-varying electric and magnetic fields influence each other. The extension of Maxwell's vacuum equations to Maxwell's equations in matter also takes into account the phenomena of electric polarisation and magnetisation and can therefore also describe the propagation of electric and magnetic fields in matter.

Notation

Maxwell's equations use a mathematical differential operator, also known as the 'derivative vector'. Its symbol is a triangle standing on one point:

\( \vec{\nabla}=\left(\begin{array}{c} \partial/\partial{x} & & \partial/\partial{y} & & \partial/\partial{z} \end{array}\right) \),

where \(\partial/\partial{x}\) denotes the partial differentiation with respect to the variable x.

This allows the proportion of "field lines emanating from a point", e.g., the electric field \(\vec{E}\), to be described using the so-called divergence of a field (\(\nabla\cdot\vec{E}\)). On the other hand, closed loops of field lines, so-called vortices, are possible. These are distinguished by using rotation (\(\nabla\times\vec{E}\)).

The four time-independent Maxwell equations and their statements

The time-independent Maxwell equations describe the course of the electric fields (\(\vec{E}\)) and the magnetic flux density (\(\vec{B}\)) for given static charges ρ and currents \(\vec{j}\) in a vacuum or as an approximate in air space:

\(1) \nabla\cdot\vec{E} = \frac\rho\epsilon_0\)
\(2) \nabla{\times{\vec{E}}} = 0\)
\(3) \nabla\cdot\vec{B} = 0\)
\(4) \nabla{\times{\vec{B}}} =\mu_0\cdot\vec{j}\)
ε₀ denotes the dielectric constant of the vacuum and μ₀ the magnetic permeability of the vacuum.

In concrete terms, the statements of these equations can be thought of as follows:

1) Field lines emanate from charges. Charges are therefore the sources (positive charges) or sinks (negative charges) of the electric field. These field sources are characterised by the divergence. The strength of the electric field caused by a charge is proportional to the charge.

2) However, the electric field has no vortices at rest. The vortices are calculated using the rotation described above.

3) Magnetic flux density, on the other hand, has no sources. There are no "magnetic monopoles", i.e. no physical object from which magnetic field lines would simply emanate.

4) Instead, currents cause vortices of the magnetic flux density and thus also the magnetic field. The strength of the magnetic field is proportional to the enclosed current.

The four time-dependent Maxwell equations

In addition to the aforementioned phenomena, the time-dependent Maxwell equations also take into account time-varying electric and magnetic fields. The temporal change of a field is characterised by a dot. This symbolises the derivative with respect to time. In the case of an electric field,\(\dot{\vec{E}}=\frac{d}{dt}\vec{E}\) denotes the change in the electric field over time. The time-dependent Maxwell equations in a vacuum are thus as follows:

\(1) \nabla\cdot\vec{E} = \frac\rho\epsilon_0\)
\(2) \nabla{\times{\vec{E}}}+\dot{\vec{B}} = 0\)
\(3) \nabla\cdot\vec{B} = 0\)
\(4) \nabla{\times{\vec{B}}} =\mu_0\cdot\vec{j}+\frac1{c^2}\dot{\vec{E}}\)
According to equation 2), a time-varying magnetic flux density therefore causes additional vortices in the electric field. A time-varying electric field (equation 4) in turn causes additional vortices in the magnetic field. For example, equations 2) and 4) can be used to determine the behaviour of electromagnetic waves. Value c is the speed of light, which is linked to the constants ε₀ and μ₀ as follows:

\(\epsilon_0\mu_0=\frac{1}{c^2}\).

The introduction of material-specific parameters is necessary to describe the propagation of electric and magnetic fields in matter. In matter, electric fields cause electric polarisation and magnetic fields cause magnetisation. The time-dependent Maxwell equations in matter take this into account as follows:

\(1) \nabla\cdot\vec{E} = \frac\rho\epsilon_0-\nabla\cdot\frac{\vec{P}}{\epsilon_0}\)
\(2) \nabla{\times{\vec{E}}}+\dot{\vec{B}} = 0\)
\(3) \nabla\cdot\vec{B} = 0\)
\(4) \nabla{\times{\vec{B}}} =\frac{1}{c^2}\dot{\vec{E}}+\mu_0\dot{\vec{P}}+\mu_0\nabla\times\vec{M}+\mu_0\cdot\vec{j}\)
According to equation 1), the sources of the electric field are therefore not only real charges ρ, but also the polarisation \(\vec{P}\). The polarisation depends on the material-specific dielectricity (polarisability).

According to equation 4), the vortices of the magnetic flux density are caused by currents \(\vec{j}\), time-varying electric fields (including polarisation) and magnetisation \(\vec{M}\). Since the magnetisation depends on the material-specific magnetic permeability constant μ information is contained in Maxwell’s 4th equation via \(\vec{M}\), which states how the material can be magnetised in external fields and influences the magnetic flux density.