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 :
The electric charge conservation law allows us to write :
Because the volume is steady :
Finally, the principle of charge conservation leads to :
By using Green - Ostrogradsky's law :
This result is true for any volume , so:
It is the local charge conservation law.
Attention : Local equation charge conservation law