# Maxwell's equations

# Local charge conservation law

### Fondamental :

Let us consider a volume delimited by a closed surface (set in the referential of the study).

Let be the charge density in the setting.

The total charge included in the volume at the moment is equal to :

The electric charge conservation law allows us to write :

Consequently :

Because the volume is steady :

Finally, the principle of charge conservation leads to :

By using Green - Ostrogradsky's law :

So :

This result is true for any volume , so:

It is the local charge conservation law.

### Attention : Local equation charge conservation law

### Remarque : Local conservation review

**This type of equation is frequently found when we assess a scalar extensive magnitude, which, in lack of source, obeys to a conservation principle. **

Electromagnetic energy conservation (Poynting's vector)

Equation of heat diffusion (transport phenomenon)