Usually a voltage is practically all used up (dropped) in driving the current through circuit elements. However a very small portion of the voltage is needed to drive the current through the connecting wiring because that does have some (small) resistance. The wiring is therefore not quite equipotential - there is a small voltage drop along it.

It's not an equipotential surface when current flows. The current produces a small voltage drop across the conductor.

What is the power source ? AC or DC ? If it is AC, then how high is the frequency ?

Remember, extremely high frequency current ( group of electrons ) only flows on the surface of a conductor rather than flows through the whole solid conductor.

If a voltage is applied to the conductor, current must flow to keep the potential equal.

The conductor is just a very small or 0 ohm resistor. I=E/R