Maxwell's equations for magnetic systems


Maxwell's equations are the fundamental laws of electromagnetism.

They relate the density of the electrical charges q and the density of the electrical current in a domain to the fields which result from it:

  • the electrical field strength and the electrical flux density
  • the magnetic flux density and the magnetic field strength

General form

The general form of Maxwell's equations is the following:

Maxwell-Gauss: (1)

Maxwell-Faraday: (2)

Gauss law for magnetism: (3)

Maxwell-Ampère: (4)

Other equations

The following constitutive laws of materials are added to the previous equations:

Characteristics of the conducting media: (5)

Characteristics of the magnetic media: (6)

Characteristics of the dielectrical media: (7)


  • σ is the conductivity of the material (in S)
  • μ is the permeability (in H/m)
  • ε is the permittivity (in F/m)


In the case of low frequency , the equations of the electrical fields and and the equations of the magnetic fields and can be decoupled.

Thus, there are Maxwell's equations for the electrical systems and Maxwell's equations for the magnetic systems, respectively:

  • a set of equations for the electrical fields , and
  • another set of equations for the magnetic fields.

This separation depends on materials, frequency, and on the dimension of the study domain. The decoupling of Maxwell equations in AC applications is usually possible for those technical devices working in the range of frequency f < 1 to 10 GHz.

Form of Maxwell equations for a magnetic system

For a magnetic system , we use the hypothesis of the quasi-static state and we neglect the time variation of the electric flux density D (the displacement currents are neglected). The hypothesis of quasi-static state remains true as long as the frequency does not exceed a certain limit. This results in the null value of the term in equation 4.

Thus, the equations can be written in the following way:

Maxwell-Faraday: (2)

Gauss law for magnetism: (3)

Maxwell-Ampere: (4)

Characteristic of the conducting media: (5)

Characteristic of the magnetic media: (6)