Consider an electrical conductor in the shape of a parabola (or any other curve) connected to the ends of a dc power supply such as a battery. The electric field inside the wire at any point is supposed to be tangential to the conductor at the point. What is the mechanism which generates a curved field inside the conductor? I believe that the arrangement of the lattice ions have a role to play in this.
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Welcome to physics.SE! A very simple gedankenexperiment: Consider the wire as a chain of single atoms separated by a small space d and all in line. The power supply will "steal" one electron from the leftmost atom and give it to the rightmost atom. There will then be a field between two adjacent atoms of different charge . This will basically "propagate" from atom to atom, no matter how you arrange them. So the electric field will always be of the shape of your wire.
