Chapter 2

Maxwell's equations

Maxwell's equations

Fondamental

In Maxwell's theory, the interaction between two particles is transmitted through local variations of the electromagnetic field.

This propagation of the interaction through the electromagnetic field is possible via electromagnetic waves, which have a celerity .

To picture the interaction between two particles in the setting of a field theory, there is an image :

Two bottle caps A and B are floating on the water. They are initially steady.

A vertical oscillation of A creates oscillations of the water. The latter are transmitted locally in all directions until they reach the point B. Then point B is set in motion.

Maxwell's equations are true locally.

They express the relationship between the electromagnetic field and its sources :

  • (Gauss' law for electricity)

  • (Gauss' law for magnetism)

  • (Faraday's law of induction)

  • (Ampere's law)

  • Maxwell's equations and charge conservation :

Maxwell's equations hold the principle of charge conservation.

Indeed, if we apply the divergence to the equation about Ampere's law :

Thus :

Because it is a consequence of Maxwell's equations, it is not necessary to add charge conservation to the statements of electromagnetic.

  • Need for displacement current :

For any kind of state, let's write :

Then :

Indeed : (see the lesson "Vector calculus")

Moreover, the charge conservation principle induces :

Gauss' law gives :

Thus :

The most simple solution to this equation corresponds to the choice of a displacement current :

AttentionMaxwell's equations

  • (Gauss' law for electricity)

  • (Gauss' law for magnetism)

  • (Faraday's law of induction)

  • (Ampere's law)

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Density of current and intensity in steady-state
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Maxwell's equations in metals, skin effect