Equation of Continuity statements I dont understand

Question

There are two statements about Equation of Continuity in my professors notes that I don't understand.

$$ \nabla \cdot \textbf{J} = - \frac{\partial\rho}{\partial t} $$

• The Equation of Continuity can be used to show that, in some cases, the field (or flux) lines of current density close in on themselves (i.e., each flux line forms a closed loop).

• The phenomenon of the electric field E being zero in a conductor can be explained by the use of the Equation of Continuity.

The first item means that the divergence of J is sometimes non zero basically, right? So its when charge density is not constant?

Not sure how one would go about for the 2nd point though. I tried using Ohms law in point form, that $\rho = 0$ inside a conductor and $ \nabla \cdot \textbf{E} = \frac{\rho}{\epsilon}$ but I came nowhere.

$$ \textbf{J} = \sigma \textbf{E} \\ \nabla \cdot \sigma \textbf{E} = - \frac{\partial\rho}{\partial t} \\ $$

Answer 1

Imagine you bring a charge close to a conductor, there will be an electric field inside the conductor for a short period of time. But the time period where electric field present in conductor is infinitesimal. This time varies from material to material but it's around $\approx 10^{-16}s$ for metals

$$\textbf{J}=\sigma \textbf{E}$$

$$\nabla \cdot \textbf{J} = \nabla \cdot \sigma \textbf{E} =-\frac{\partial \rho}{\partial t}$$

and you know $\nabla \cdot \textbf{E} = \frac{\rho}{\varepsilon_0}$

Thus

$$\frac{d\rho}{dt}+ \frac{\sigma \rho}{\varepsilon_0}=0$$

and solution for this differential equation is

$$\rho = \rho_0 exp({-\frac{\sigma}{\varepsilon_0}t})$$

by following this derivation, you can calculate the time $t$ for charges to move to the surface and leave electric field inside conductor zero.

Edit regarding your comment

$$ \lim\limits_{t \to\infty} \rho = 0$$

so you will have $\nabla \cdot \textbf{E} = 0$, get use of divergence theorem

$$\int_{V}\nabla\cdot\textbf{E} dV = \oint_{S}\textbf{E} \textbf{dS} = 0$$

Hence $\textbf{E} = 0$