The continuity equation is a fundamental statement of local charge conservation in electromagnetism. It relates the time rate of change of charge density to the divergence of current density, ensuring that electric charge is neither created nor destroyed.
The continuity equation can also be expressed in integral form by applying the divergence theorem.
Starting from the differential form,
∂t∂ρ+∇⋅J=0,
integrate over a fixed volume V:
∫V∂t∂ρdV+∫V∇⋅JdV=0.
Using the divergence theorem on the second term gives:
dtd∫VρdV+∮∂VJ⋅dA=0,
where ∂V is the boundary surface of the volume V.
This states that the rate of change of total charge inside volume V plus the net outward current through the surface is zero, a direct statement of charge conservation.