# Conservation of bound charge? [closed]

It is intuitively obvious that free charge is conserved in vaccum. But when is bound charge conserved? I tried to find this by formula:

The charge conservation law in microscopic view (Assuming there is zero displacement current):
$$\nabla \cdot \vec{J} = - \frac{\partial \rho}{\partial t}$$ Assuming free charges remain free and bound charges remain bound:
$$\nabla \cdot \vec{J_b} = - \frac{\partial \rho_b}{\partial t}$$ Decomposing the bound current yields:
$$\nabla \cdot (\nabla \times \vec{M} + \frac{\partial \vec{P}}{\partial t}) = - \frac{\partial \rho_b}{\partial t}$$ Since divergence through curl is zero:
$$\nabla \cdot \frac{\partial \vec{P}}{\partial t} = - \frac{\partial \rho_b}{\partial t}$$ Assuming polarization is continuous in space and time:
$$\frac{\partial}{\partial t}(\nabla \cdot \vec{P}) = - \frac{\partial \rho_b}{\partial t}$$ Optionally, assuming polarization and bound charge are initially zero: $$\nabla \cdot \vec{P} = -\rho_b$$

I can't think of any actual situation that demonstrates this. For example, in a capacitor, free charges turn bound, or vice versa, so this isn't a case.

Is there any actual situation that demonstrates this?

## closed as unclear what you're asking by Aaron Stevens, Jon Custer, user191954, Norbert Schuch, hyportnexNov 19 '18 at 0:14

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• You are asking for an example of when $\nabla\cdot\vec P=-\rho_b$? – Aaron Stevens Nov 11 '18 at 5:15
• @AaronStevens Yes. – Dannyu NDos Nov 11 '18 at 5:35
• In a uniformly polarized object, $\rho_b=0$ – Aaron Stevens Nov 11 '18 at 5:39
• @AaronStevens And $\nabla \cdot \vec{P} = 0$. This is a trivial example and not quite interesting... – Dannyu NDos Nov 11 '18 at 5:41
• Well you never said that example wasn't allowed :) – Aaron Stevens Nov 11 '18 at 5:42

Intuitively, bound charge ($$\rho_b$$) can be thought as a charge caught in a specific point of an object. For example, atomic nuclei. On the contrary, free charge ($$\rho_f$$) is a charge that can move freely around. For example, free electrons, or amorphous ions. In this sense, it is even more obvious that bound charge is conserved.
$$\rho_b$$ can change in time if the object itself restricting the bound charges moves. In this case, we introduce the concept of polarization ($$\vec{P}$$) and have the equation $$\frac{\partial \rho_b}{\partial t} = - \frac{\partial}{\partial t}(\nabla \cdot \vec{P})$$. (This is a "demonstration" of the desired situation.)
In the dielectric of a capacitor, since the dielectric always has zero total charge inside, $$\rho_b = 0$$ anytime and everywhere. There's only free charges that builds up in both plates. So "In a capacitor, free charges turn bound, or vice versa" is a misconception. It should be "In a capacitor, free current turns into displacement currect, or vice versa."