# Divergence of current density and electric field within a wire

In the following exercise:  I concern myself with the validity of my interpretations of (b).

Here I am more confident slightly. The divergence of the current density is merely $$- d \rho / dt$$, so as there is charge flowing through the wire the magnitude of the current density ought to only be non-zero within the wire.

The divergence of the electric field is only non-zero at point charges. Given that it seems weird for points in the wire to be sources of divergence in the electric field if they're constantly moving, I reckon the only charges I can use is the electrons of the wire (bound charges?) not shot through the wire as a current which should be at the surface, but this seems a bit weird to me as if its a bound charge it seems weird for me if it is at the surface as that tends to be a thing free charge does as it has the freedom to do so while bound charge doesn't (as free charges can move themselves however they must to balance out the field while the bound charges more or less stays where it is but becomes polarized).

Where am I going wrong in my interpretations for (b)?

## 1 Answer

i) Nowhere, as non-zero divergence of current density ($$\nabla \cdot \mathbf j$$) would mean charge density is changing in time, which would contradict the assumption of stationary flow.

ii) Non-zero divergence of electric field ($$\nabla \cdot \mathbf E$$) means non-zero density of electric charge (does not need to be point-like though). In metal, the only place charge can be in static situation or during stationary flow assumed is on the surface of the metal object. In fact charge density must be distribution that is not everywhere zero, as the excess charge contributed with its Coulomb field to maintain the electric field and current inside.

The charges on the surface are from the metal itself, and also from those that came from the other parts of the circuit. Their origin does not matter, they can get to the surface so some of them will to in the course of establishing dynamic equilibrium in the metal. These charges are bound only in the sense they cannot escape the metal body, but other than that they are free to relocate anywhere in the body.