Charge density of an uncharged sphere

Question

This post follows directly from my question about EM waves in conductors EM-wave equation in conductors with source terms

A more specific question in light of this.

Given an uncharged conductor:

We know $2$ things that are true:

$$ \vec J = \sigma\vec E $$

$$ \int \rho dV = 0 $$

In a conductor,

Maxwells equation:

$$ \vec\nabla\vec E = \rho / \epsilon_0 $$

is now set to $ \vec \nabla\vec E = 0 $

this implies $ \rho = 0 $

But, if $ \vec J = \sigma \vec E $

and $ \vec J = \rho \vec v $

which means we have no Current density inside the conductor

However using these equations for a EM wave inside a conductor, the amplitude DECAYS exponentially DUE TO current density ( energy conservation)

so my final question is: if $ \rho = 0 $ how is there a current density to effect an EM wave inside the conductor?

Answer 1

First of all an uncharged conductor implies $\rho=0$, which implies $\vec\nabla\vec E=\rho/\epsilon_0=0$. Not the other way around. Having a net-charge of $0$, or $\int_V\rho dV=0$ still allows charge separation due to electric and magnetic fields provocated in the conductor, which is nothing but current (charge flow).

Now there are still cases where the net charge $\int_V\rho dV$ is $0$ but $\rho\neq 0$. The thing is that in reality there is no such case. Or more specifically, the only cases where you have a volume without charges at all would be a vacuum and even so if there is a photon passing thrue it could split into electron and positron creating a charge distribution.

So specially when talking about a conductor, that consists of metal for example, there will be neutral atoms, which naturally have a charge distribution within them. When we write $\rho=0$, we mean there is no big amount of charged ions or electron accumulation in a specific place. There are only neutral atoms, of which the electrons (in the case of a conductor) can freely move and are thus able to create a current density.