The total current flowing out of a volume $v$ is equal to the flux of the current density $J$ through the surface $S$, which in turn is equal to the rate of decrease of charge enclosed in $v$
$$ \nabla \cdot J = -\frac{\delta \rho _v}{\delta t} $$
If the volume charge density within an elemental volume $\Delta v$ is equal to the current flowing out of it, $\nabla \cdot J = 0$:
$$ \oint _S J \cdot ds = 0 $$
Kirchoff’s current law: algebraic sum of all the currents flowing out of a junction is zero
