Editing Continuity equation (section)

== Electromagnetism ==

{{Main|Charge conservation}}

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]] {{math|'''J'''}} (in [[amperes]] per square meter) is equal to the negative rate of change of the [[charge density]] {{math|''ρ''}} (in [[coulomb]]s per cubic meter),
<math display="block"> \nabla \cdot \mathbf{J} = - \frac{\partial \rho}{\partial t} </math>

{{math proof | title = Consistency with Maxwell's equations | proof =
One of [[Maxwell's equations]], [[Ampère's law|Ampère's law (with Maxwell's correction)]], states that
<math display="block"> \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}. </math>

Taking the divergence of both sides (the divergence and partial derivative in time commute) results in
<math display="block"> \nabla \cdot ( \nabla \times \mathbf{H} ) = \nabla \cdot \mathbf{J} + \frac{\partial (\nabla \cdot \mathbf{D})}{\partial t}, </math>
but the divergence of a curl is zero, so that
<math display="block"> \nabla \cdot \mathbf{J} + \frac{\partial (\nabla \cdot \mathbf{D})}{\partial t} = 0. </math>

But [[Gauss's law]] (another Maxwell equation), states that
<math display="block"> \nabla \cdot \mathbf{D} = \rho, </math>
which can be substituted in the previous equation to yield the continuity equation
<math display="block"> \nabla \cdot \mathbf{J} + \frac{\partial \rho}{\partial t} = 0.</math>
}}

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.

If [[magnetic monopole]]s exist, there would be a continuity equation for monopole currents as well, see the monopole article for background and the duality between electric and magnetic currents.