Confusion on Continuity of Current and Free Charge in Conductor

[wzy75]

I know there must be something wrong with the following derivation based on Maxwell's equations but could not figure out what is wrong. The derivation deals with continuity of current and free charge in a conductor in general.

Continuity of current says that,

$\nabla\cdot \textbf{J}=-\frac{\partial \rho_v}{\partial t}$. (1)

However, for currents in conductor, the current density and the electric field is related as

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

Using the relation between $\textbf{E}$ and $\textbf{D}$ ,

$\textbf{E}=\textbf{D}/\epsilon$,

we have

$\nabla\cdot \textbf{J}=\sigma\nabla\cdot\textbf{E}=\frac{\sigma}{\epsilon} \nabla\cdot\textbf{D}=\frac{\sigma}{\epsilon}\rho_v$. (2)

Comparing (1) and (2) gives an equation on free charge $\rho_v$,

$-\frac{\partial \rho_v}{\partial t}=\frac{\sigma}{\epsilon} \rho_v$

which means that

$\rho_v=\rho_{v0}e^{-\frac{\sigma}{\epsilon}t}$. (3)

Since we are talking about general cases of Maxwell's equations, (3) looks like an unreal restriction on free charge in a conductor and does not make sense at all.

What went wrong in the above derivations? Thanks in advance!

 

[wzy75]

Would really appreciate it if someone can help me out. Thanks a lot!

 

[Barloud]

Hi,
You are mixing up equations describing conductors and dielectrics. The dielectric displacement makes no sense in a conductor.

 

[vanhees71]

The equations are correct. It describes the disappearance of charges inside a conductor. In the long-time limit you approach a stationary solution (electrostatics). In this situation there cannot be free charges inside a conductor, but those all move to its surface, leading to a distribution such that the interior of the conductor has 0 current and 0 electric field.

 

[wzy75]

Thank you so much vanhees71 for the clarification!

Is it true that in the most general case, even if the unpaired free charge will disappear inside a conductor, there still might be electric field $\textbf{E}$ and current (e.g. eddy currents induced by changing magnetic fields)?

 

[vanhees71]

For time-dependent fields/sources, there can be a field/current inside the conductor. However, also here the current flows more close to the surface. This is known as the skin effect, which is due to eddy currents due to induced electric fields from the time-varying magnetic field in the interior of the conductor counteracting the current due to the driving field:

http://en.wikipedia.org/wiki/Skin_effect

 

[Jano L.]

Is it true that in the most general case, even if the unpaired free charge will disappear inside a conductor, there still might be electric field E and current (e.g. eddy currents induced by changing magnetic fields)?

The electric field in metal is non-zero whenever the current flows through it - Ohm's law states that the current is proportional to the electric field. In the case the current is due to a battery, the electric field is due to charge distribution on the surface of the conductor and battery - this need not vanish.

If the current does not vary too fast, it flows roughly uniformly through the whole cross-section of the conductor, not just on the surface. As the frequency of the alternating current is increased, the distribution of current moves to the surface of the conductor.