Question: In my syllabus about electromagnetism, they state: 
"This surface charge density will not always be present, e.g. when considering two non-conducting dielectrics such surface charge density remains absent. However, at a perfect conductor, a surface charge density will be present. If one of the media (or both) carry a conduction current, a surface charge can also be present (explain why!)"

I don't really see though, how this can be explained. They seem to imply that there is a relation between the conductivity and the possibility of a surface charge, but I can't figure how they are related.

They also state somewhere earlier that there can't be a surface current between two lossy dielectrics. This seems to be analogous to my previous question. I thought that this was due to the fact that dielectrics don't conduct current, but I wonder if someone knows a better explanation here too (that perhaps explains why they explicitly mention that the dielectrics are lossy).

Answer: 
The following relation describes the total electric flux density in a medium:

![enter image description here](https://i.stack.imgur.com/RcF57.png)

In the previous equation, D is the total electric flux density, epsilon is the free space permittivity . E is the external applied electric field, and P is the polarization vector. The polarization vector represents the reaction of the medium to the externally applied electric field. In general, one can write:

![enter image description here](https://i.stack.imgur.com/1YQTM.png)

The previous equation shows the “induced” or “effective” volume charge density in a medium as a response to the external electric field. Now let us consider what happens in the case of perfect dielectric (non-lossy) and perfect conductor.

In case of perfect dielectric, the induced volume charges can’t move. This means they are going to stay where they were induced. The charges stay in the volume. In the case of perfect conductor, the induced charges move freely, such that instead of being distributed in the volume they all go and accumulate at the surface from which the electric field is applied. The number of charges or more accurately the surface charge density is just enough to cancel the external electric field within the medium. This basically means there won’t be any other charges induced in the volume.

A lossy dielectric is a dielectric that has finite conductivity, which means induced charges can move but not as freely as they would in a perfect conductor. If two lossy dielectrics are in contact, there will be no surface charge between them because that surface charge would move to the boundary from which the electric field is applied.

To make it clearer, have a look at the attached figure. Figure a shows the dielectric case, where charges are induced in pairs but they can’t move. That is why they are distributed everywhere. Figure b shows the perfect conductor case, where all negative charges moved to surface creating surface charge density. The positive charges left behind were neutralized by negative charges coming from ground. That is not a concern in your question. Figure c shows the two lossy dielectrics, if surface charge accumulated on the surface separating between the dielectric, it will eventually move to the boundary at which the external field is applied.

![enter image description here](https://i.stack.imgur.com/zXul5.png)

The difference between a perfect conductor and a lossy dielectric is the time scale at which the induction of charge in the volume and the motion of the negative charges happen. In perfect conductors the whole thing is immediate, while in lossy dielectrics it takes some time described by what is called relaxation time. That is defined as:

![enter image description here](https://i.stack.imgur.com/wYEYh.png)

Epsilon is the permittivty of material, sigma is the conductivity of the material. For perfect conductor sigma is infinite and the relaxation time is zero, which is why it is instantaneous.

I recommend you to have a look at chapter 5 in Sadiku’s book “Elements of Electromganetics”. It describes more details on the same topic

Hope that was useful
A simple line is as follows: 

1. **Conductor is an equipotential volume.** If there were potential difference between any two points, free charges would flow to compensate for this difference, hence produce currents. If there is a current, it produces heat. However, due to energy conservation the heat cannot be produced forever. Hence over time all the currents have to stop. And this is possible only when there is no potential difference between any two points in the conductor.
2. Hence $\vec E=-\nabla \phi=\vec 0$ inside the conductor.
3. Hence $\nabla\cdot\vec E=4\pi \rho=0$ everywhere throughout the volume.

One cannot write $\nabla \cdot \vec E$ on the surface, because $\vec E$ is not continuous there. However, surface charge can make $\vec E$ be non-zero outside the boundary and zero inside the volume.

The reasoning applies to conductors and conducting materials, but does not apply to dielectrics. However, this does not mean, that dielectrics cannot have surface charges.