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Continuity equation

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,


One of Maxwell's equations states that


Taking the divergence of both sides results in

but the divergence of a curl is zero, so that

Another one of Maxwell's equations states that
Substitute this into equation (1) to obtain
which is the continuity equation.


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, the continuity equation is an equation of conservation of mass. Its form is

See the Euler equations.