The ultimate guide to learn the Maxwell’s equations: Final Part

This post is the continuation of our Ultimate Guide to learn the Maxwell’s equations where you learnt these equations in the vacuum. Now we will explain the modifications required to apply them to a material medium.


Maxwell’s equations in material medium

The equations studied in our previous post were only valid in cases where the electromagnetic fields exist in the vacuum. In material medium (considering that these are linear, homogeneous, isotropic and non-dispersive), the field vectors’ relations are expressed by using the constitutive relations:



The new terms that appear in these equations are:

  • ε: dielectric constant
  • µ: magnetic permeability
  • H: magnetic excitation

Therefore, the Maxwell’s equations in material medium will be:





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The first Maxwell equation is the Gauss theorem for the electric field which states that the electric field through a enclosed surface is equal to the net charge contained on it and divided by the dielectric constant in the vacuum.

The second one is the Gauss theorem for the magnetic field which affirms that the magnetic flux through a enclosed surface is zero.

The third equation is the Faraday’s law about the electromagnetic induction phenomena which Maxwell deeper interpreted. He stated that a time changing magnetic field induces an electric field. This electric field is proportional to the magnetic flux variation rate and orthogonal to it.

The fourth one is an extension of the Ampère’s theorem about the variable electric fields. According to Maxwell, a time changing electric field induces an magnetic field. This magnetic field is proportional to the electrical flux variation rate and orthogonal to it. Therefore, a magnetic field is not only generated by an electric current, but also by any electric field variation.

In addition, in the Maxwell’s theory, the attention is paid to the medium and not in the material. Thus, a variable magnetic field induces an electric field in the presence or absence of a conducting in the medium. The conducting medium acquires the role of a test charge.

Similarly, the idea of distance forces disappear and it’s replaced by the propagation of the medium interaction as an electromagnetic wave. Moreover, by calculating the propagation speed of these electromagnetic waves, we obtain a value extraordinarily close to the speed of the light. Maxwell understood that the light is an electromagnetic perturbation, propagated through a field. This is how he unified the electromagnetism and optics theories.

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