Potential difference across a wire along all paths

Question

I encountered a interesting problem. It first invokes the fact that between two points, potential difference is independent of the path between them. Then, it asks you to apply it for a wire carrying constant current I. Path 1: Along the wire. Path 2: Going in a curved path outside the wire. The thing is, a non zero field is needed outside the wire to evaluate (E . dr). It hints of a surface charge density on the wire. But, I am not able to correlate it with our day to day observation or find its distribution mathematically.