Maxwell's equations for electrical systems

Introduction

Maxwell's equations are the fundamental laws of electromagnetism.

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

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

General form of the equations

The general form of Maxwell's equations in domains without moving bodies is the following:

Maxwell-Gauss : (1)

Maxwell-Faraday : (2)

Conservation of magnetic flux density : (3)

Maxwell-Ampere: (4)

Other equations

With these equations, the following constitutive laws of materials are added:

Characteristics of the conducting media : (5)

Characteristics of the magnetic media : (6)

Characteristics of the dielectric media : (7)

where :

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

Separation

In the case of low frequency AC fields , the equations in the electric 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 electric fields, and
• another set of equations for the magnetic fields.

This separation of the electrical and magnetic systems depends on: materials, work frequencies, size of the study domain. It is usually possible for technical devices working in the range of frequency f < 1 to 10 GHz.

Form of equations for an electrical system

For an electrical system, we suppose that the magnetic field does not modify the current distribution in the conductors. This assumption remains true as long as the frequency does not exceed a certain limit. This results in the null value of the term in equation (2).

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

Maxwell-Gauss : (1)

Maxwell-Faraday : (2)

Maxwell-Ampere : (4)

Characteristic of the conducting media : (5)

Characteristic of the dielectric media : (7)