The Equation of Continuity

The archetypal example of a differential equation which implies a conservation law is the equation of continuity, which, in its differential form says that

$${\partial\rho\over\partial t} + \nabla\cdot{\bi j} = 0$$ (2.33)

where $\rho$ represents a density and $\bi{j}$ a current density. The density and current density could be of mass, charge, energy or something else which is conserved. Here we shall use the words charge and current for convenience. We consider here the 1D form for simplicity. The equation is derived by considering space as divided into sections of length $\delta x$. The change in the total charge in a section is equal to the total current coming in (going out) through its ends

$${\partial\over\partial t}\int_{x_i}^{x_{i+1}} \rho\,\d x = -\int \bi{j}\cdot\d S_{i},$$ (2.34)

It is therefore useful to re-express the differential equation in terms of the total charge in the section and the total current coming in through each face, so that we obtain a discrete equation of the form

$${\partial\over\partial t} Q_j = \bi{I}_{j-1,j} - \bi{I}_{j,j+1}$$ (2.35)

where $Q_j$ represents the total charge in part $j$ and $\bi{I}_{j,j+1}$ is the current through the boundary between parts $j$ and $j+1$. This takes care of the spatial part. What about the time derivative? We can express the physics thus:

The change in the charge in a cube is equal to the total charge which enters through its faces.