Continuity equation
Note that all the examples given below express the same idea (i.e. they are all really examples of the same concept, which happens to be the (stronger) local form of a conservation law).
In electromagnetic theory, the continuity equation is derived from two of Maxwell's equations. It states that the divergence of the current density is equal to the negative rate of change of the charge density,
- <math> \nabla \cdot \mathbf{J} = - {\partial \rho \over \partial t} <math>
Derivation
One of Maxwell's equations states that
- <math> \nabla \times \mathbf{H} = \mathbf{J} + {\partial \mathbf{D} \over \partial t} <math>.
Taking the divergence of both sides results in
- <math> \nabla \cdot \nabla \times \mathbf{H} = \nabla \cdot \mathbf{J} + {\partial \nabla \cdot \mathbf{D} \over \partial t} <math>,
but the divergence of a curl is zero, so that
- <math> \nabla \cdot \mathbf{J} + {\partial \nabla \cdot \mathbf{D} \over \partial t} = 0 \qquad \qquad (1) <math>.
Another one of Maxwell's equations states that
- <math> \nabla \cdot \mathbf{D} = \rho <math>.
Substitute this into equation (1) to obtain
- <math> \nabla \cdot \mathbf{J} + {\partial \rho \over \partial t} = 0 <math>,
which is the continuity equation.
Interpretation
Current density is the movement of charge density. The continuity equation says that if charge is moving out of a differential volume (i.e. divergence of current density is positive) then the amount of charge within that volume is going to decrease, so the rate of change of charge density is negative. Therefore the continuity equation amounts to a conservation of charge.
In fluid dynamics, a continuity equation is an equation
of conservation of mass. Its differential form is
- <math> {\partial \rho \over \partial t} + \nabla \cdot (\rho \mathbf{u}) = 0 <math>.
where <math> \rho <math> is density, t is time, and u is fluid velocity.
See also: Euler equations, incompressible fluid.
In quantum mechanics, the conservation of probability also yields a continuity equation, mark P(x,t) as probability density and write
- <math> \nabla \cdot \mathbf{j} = { \partial \over \partial t} P(x,t) <math>
where J is Schrödinger equation, probability density.
