Continuity Equation - Electromagnetism

Electromagnetism

In electromagnetic theory, the continuity equation is an empirical law expressing (local) charge conservation. Mathematically it is an automatic consequence of Maxwell's equations, although charge conservation is more fundamental than Maxwell's equations. It states that the divergence of the current density J (in amperes per square meter) is equal to the negative rate of change of the charge density ρ (in coulombs per cubic metre),

Consistency with Maxwell's equations
One of Maxwell's equations, Ampère's law (with Maxwell's correction), 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, Gauss's law, states that substitution into the previous equation yields the continuity equation

Current is the movement of charge. 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.