Continuity Equation
Since charge is conserved, current density must satisfy a continuity equation. Here is a derivation from first principles.
The net flow out of some volume V (which can have an arbitrary shape but fixed for the calculation) must equal the net change in charge held inside the volume:
where ρ is the charge density, and dA is a surface element of the surface S enclosing the volume V. The surface integral on the left expresses the current outflow from the volume, and the negatively signed volume integral on the right expresses the decrease in the total charge inside the volume. From the divergence theorem:
Hence:
This relation is valid for any volume, independent of size or location, which implies that:
and this relation is called the continuity equation.
Read more about this topic: Current Density
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