# Conduction current in a lossy dielectric due to a nearby static point charge

I'm a bit confused about how a steady current can apparently exist in an isolated object. Note that I'm only considering the static case here.

Consider an object in free space with permittivity $$\varepsilon$$ and conductivity $$\sigma$$. Now suppose I bring an external point charge close to this object, and that point charge has value $$Q_\mathrm{ext} > 0$$.

Qualitatively, I think that the free charges within the object should redistribute themselves on the surface to try to cancel out the electric field inside the object. So all the positive free charges will crowd away from $$Q_\mathrm{ext}$$, and equally many negative charges will crowd near it.

Also, the bound charges would realign themselves to create an opposing electric field within the object.

(I am thinking of this as a generalization of the "perfect" conductor, where the free charges are successful in cancelling out the electric field inside, and there are no bound charges.)

However, in a mildly lossy material, there should still be some non-zero electric field $$\vec{E}$$ inside the object, because there are only so many free charges available. Then, if the object obeys Ohm's law, there would also be a non-zero steady current inside the object: $$\vec{J} = \sigma\vec{E}$$.

My question is, if the object is isolated and not connected to any circuit, how can there be a current $$\vec{J}$$ flowing in it? Where does it go?

I can understand that in the transient phase when I first bring the charge $$Q_\mathrm{ext}$$ close, then some current will flow as the free charges in the object redistribute themselves. But from my analysis above, it seems there will be some $$\vec{J}$$ even in the steady state. Where did I go wrong?

If the object has so few free charges that all of them migrate to the surface, and still cannot cancel out the applied electric field, then the macroscopic concept of conductivity doesn't really apply anymore, and $$\vec{J} = \sigma\vec{E}$$ is no longer the correct constitutive relation.
Another way to see this may be: if all the free charges flow to the surface, then the bulk of the object is not "conductive" anymore, and effectively its bulk $$\sigma\to0$$. So again, there would not be any DC current in the isolated object.