Science Fair Project Encyclopedia
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,
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.
where ρ is density, t is time, and u is fluid velocity.
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