Let's say you have a square plate with a voltage applied across two opposite corners, connected by the hypotenuse. Presumably, the electric current density distribution throughout the plate would be such that it would follow the path of minimal resistance, i.e. through the hypotenuse. If you had a way of measuring the current density at a point in the plate (e.g. IR thermography) would the readings show that the current density has a spatial distribution throughout the plate and indeed even a change through the thickness? What factors determine how the charge carriers are distributed in a conductive medium?