Before I get roasted in the comments, let me say I recently got a BS in electrical engineering, so I’m not a newbie in these concepts. I’ve studied electric potential energy, electric potential, voltage (a.k.a. electric potential difference, electric tension, electric pressure), time-varying/non-conservative electromagnetic fields, Maxwell’s equations, etc. I’ve read textbooks on university physics, on electromagnetic theory, on circuit analysis/theory. (But I haven’t studied quantum mechanics or relativity.)
Before explaining why I think voltage and electric potential actually describe the same physical phenomena/process, first I’ll briefly recall certain facts about the two, so that you can see I have some standard understanding on these concepts.
Electric potential
Electric potential is the electric potential energy, per unit charge, at a point in space. Recall that electric potential energy is a type of potential energy (i.e. energy that a particle has by virtue of its position in space or in a field, it is energy that could be used to do work), associated with the position of a charged particle.
Electric potential is defined at a point in space, provided we’ve previously defined the zero electric potential point (usually Earth ground or a point infinitely far from the region of space under study). Since the potential is defined at any point in space (at least for conservative electric fields), and since it is a scalar value at each point, then we say the electric potential is a scalar field. It makes perfect sense. I can say “the electric potential at point $P_0$ is $\phi_0$”.
We sometimes use the electric potential (a scalar field) to calculate a static electric field (a vector field) as the negative of the gradient of the electric potential, because the former is easier to calculate than the latter.
We can plot the electric potential as a 2D or 3D scalar field.
We can write Maxwell’s equations in terms of the electric potential and the magnetic vector potential.
Etc.
Voltage
Voltage is defined as the work to be done (or energy to be transferred), per unit charge, to move a charged particle with unit charge, from one point in space to another point in space, along some path or trajectory. So voltage is a quantity between two points, while electric potential is a quantity at a single point.
In the presence of conservative electric fields only, voltage can also be calculated as the difference of the electric potential at the two points, thus the name electric potential difference.
Voltage is usually thought of as a scalar, without being a (scalar) field like electric potential. It makes perfect sense. I can say “the voltage between point (or node in the context of circuits) $a$ and point $b$ is $V_{ab}$”. But it wouldn’t make sense to say “the voltage at point $c$ is $V_c$”, because we’re not specifying with respect to which other point we’re measuring it, unless it is implicitly obvious.
Voltage (and current), not electric potential, is the quantity in which Ohm’s law, the voltage-current relationship for inductors, capacitors, and diodes, are written.
Voltage, not electric potential, is the quantity in which Kirchhoff’s voltage law is described.
In the presence of conservative electromagnetic fields only (zero currents [electrostatics] or constant currents [magnetostatics]), the work to be done between two points is independent of any of the possible paths connecting the two points, and so the work to be done in moving a charge around a closed loop is zero, and since voltage is work per charge, then voltage is also independent of path in such case. But in the presence of non-conservative electromagnetic fields (time-varying currents [electrodynamics], for example AC circuits, with non-negligible leakage electromagnetic fields outside circuit elements/devices such as inductors), the work to be done in moving a charge from one point to another does depend on the path, and so the work to be done in moving a charge around a closed loop is in general not zero, and so voltage also depends on the path.
Etc.
Why I think voltage and electric potential are the same
Okay, now to my question. I recently had an online discussion with someone, where I said voltage and electric potential were not the same, while the person said they were. I explained them in a similar manner I just did above. But after talking and me thinking, I think those two quantities are really describing the same thing. Below I’ll try to convince you, or at least explain you why to me those quantities seem the same.
Suppose I choose one point in space, which I’ll call the reference point, with respect to which I measure the voltage at all other points in space. Isn’t that, then, the same as electric potential? For example, I could choose the reference point to be the Earth ground or a point infinitely far from us, thus the voltage would seem to have the same meaning as electric potential.
And since I’ve chosen one reference point for voltage measuring, then at each point in space the voltage has a certain scalar value. Thus voltage is now a scalar field, like electric potential. So, we can also compute a conservative electric field from this voltage “field”, and we can write Maxwell’s equations in terms of this voltage “field” (and the magnetic vector potential), just like we could with the electric potential.
And this idea of choosing a reference point for voltage measuring is not uncommon. It is widely used in a method of circuit analysis known as nodal analysis; all or most circuit simulators use that method; it is also used in power systems analysis to obtain the admittance matrix that mathematically describes the electrical behavior of a power system; even using oscilloscopes in real-life electronic circuits is the same, because we attach the negative probe to one node (usually called ground in electronics) of the circuit and then only move the positive prove for measuring the voltage.
So, as you can see, if we measure the voltage at any point by choosing the same reference point that we defined as zero electric potential, then voltage and electric potential are the same.
Please note I’m suggesting that voltage (electric tension, electric pressure, electric potential difference) and electric potential are the same thing, not that any of them is the same as electric potential energy. I know the former two are not the same as the latter.
I searched if this question had been already asked, but didn’t find any. I found the following which ask different questions: