PX284 - O3 - charge

current density

considering a cylindrical conductor, with a cross-section, A, and charge density, $ρ$
the charges are travelling at a speed, $v$
so, the charge through $A$ in time, $δ t :$

Q = ρ A v δ t

the charge density:

\vec{J} = ρ \vec{v}

conservation of charge

\subset \supset \iint \vec{J} \cdot d \vec{S} = - \frac{d}{d t} Q_{encl.} = - \frac{d}{d t} ∭_{V} ρ d V

using the divergence theorem:

\begin{matrix} \oint_{S} \vec{J} \cdot d \vec{S} = ∭_{V} \vec{\nabla} \cdot \vec{J} d V = - \frac{d}{d t} ∭_{V} ρ d V \\ ∭_{V} (\frac{\partial ρ}{\partial t} + \vec{\nabla} \cdot \vec{J}) d V = 0 \\ \frac{\partial ρ}{\partial t} + \vec{\nabla} \cdot \vec{J} = 0 \end{matrix}

this is the continuity equation for the conservation of charge