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Electric potential in a space as like as gravity is something understandable but when it comes to circuit I always get puzzled about this. I am looking for help regarding this topic.

Is electric potential of a circuit a merged concept of basic potential energy and energy lost due to the electron transition throughout the circuit?

I was wondering why wouldn't the potential of a circuit increase at the apex or any other point closer to apex in such an arrangement where a extremely long wire connected to two terminal of a source is remaining in a parabolic shape? Is it because in case of circuit we don't take the equation of potential energy (derived using coulomb's law) too seriously rather we are only concerned about the energy it is supposed to lose and conservation of energy?

Any help will be appreciated.

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Electric potential in the context of electric circuits is the same thing as in electrostatics - it is a single real number at any position in space. For a fixed point anywhere in the circuit, it is a function of charged particles' charges and positions, as given by the Coulomb potential formula.

This formula takes into account all charges: those on the voltage source, those on the wires, those on circuit elements like resistors, capacitors, etc.

Potential in a wire in a DC circuit decreases in the direction of electric field. In a metallic conductor at rest, this is also the direction of electric current. In the simplest circuit where voltage source drives current through a resistor, or just a wire, potential decreases when going from positive terminal towards negative terminal.

This is also true if the wire is shaped into a parabola with an apex. There is no reason why the apex should be the point of maximum potential. If current goes through it, then potential decreases in direction of the current.

If current is not flowing, then potential of the whole wire is the same everywhere.

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