Continuity equation conservation in a region S and local conservation

[lys04]

Homework Statement

What’s the differences between these two versions of the continuity equation? $\frac{dq}{dt} =-\iint_S (\vec{J}. d\vec{S})$ and $\frac{\partial q}{\partial t} =-\nabla . \vec{J}$?

Relevant Equations

$$ \frac{dq}{dt}=-\iint_S (\vec{J}. d\vec{S})$$ $$ \frac{\partial q}{\partial t} =-\nabla . \vec{J}$$

Does $\frac{dq}{dt}=-\iint_S \vec{J}.d\vec{S}$ correspond to conservation of some quantity q in a region with boundary S whereas $\frac{\partial q}{\partial t} = - \nabla . \vec{J}$ means that for any point in space the quantity q is conserved? Since the divergence measures how much of q is flowing out or in at any point.

 

[ergospherical]

The "q"s in your two equations mean different things. In the differential form, $\partial q / \partial t = -\nabla \cdot J$, the $q$ is the density of the quantity (i.e. dimensions of quantity / length^3). In the integral form, $dQ/dt = -\int J \cdot dS$, the $Q$ is the total amount of quantity inside the region with boundary S. The link between them is the divergence theorem.

 

[lys04]

Oh ok, but is there a difference in the physical interpretations of the two?

 

[renormalize]

Oh ok, but is there a difference in the physical interpretations of the two?

No, one is simply the differential (local) form and the other is the integral (global) form of charge conservation. See, e.g., https://en.wikipedia.org/wiki/Maxwell's_equations#Charge_conservation.