continuity equation

In continuum mechanics, the continuity equation is a statement about the inability for particles to teleport between two different points in space. In other words, each particle must take a path between two points. In particular, if \(Q_{enc} = \int_{V} \rho(r')d\tau\) or \(Q_{enc}\) is the total amount of particles inside some surface where \(\rho\) is the density:

\begin{align*} \oint_{S} \vec{J} \cdot d\vec{a} = -\frac{\partial Q_{enc}}{\partial t}, \end{align*}

or in other words, the amount that the current goes through some closed surface must be proportional to the loss of particles inside of the enclosure. This is of course because of conservation of mass (which is in and of itself conservation of energy). Using the :

\begin{align*} \int_{V}\vec{\nabla} \cdot \vec{J}d\tau = - \int_{V}\frac{\partial\rho}{\partial t}d\tau \\ \vec{\nabla} \cdot \vec{J} = -\frac{\partial\rho}{\partial t} \end{align*}