Chapter 2

# 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)

• (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 :

### Attention : Maxwell's equations

• (Gauss' law for electricity)

• (Gauss' law for magnetism)