Chapter 2

Maxwell's equations

Local charge conservation law


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 :

Local charge conservation law

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.

AttentionLocal equation charge conservation law

RemarqueLocal 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.

Learning program
Density of current and intensity in steady-state