The current density is given by:
J = N · q · u
Since the current is given by:
ΔI = ΔQ/Δt = N · q · u · an · Δs = N · q · u · Δs
Where:
N is the number of charge carries per unit.
q is the charge each one carries.
u is the velocity the charges has when they move across the surface Δs, which has a unit normal an.
Δt is the time variation.
Since N · q = ρ, the current density can also be expressed as:
J = ρ · u
