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On Wikipedia, voltage is defined to be the electric potential difference. However, I am still not certain as to whether voltage is the electric potential ($PE/q$) or electric potential change ($\Delta PE/q$).

My confusion partially comes from talking about circuits and defining the amount of volts a point in a circuit has.

So, is voltage electric potential or its change? If voltage is the change of electric potential, then how can one point in a circuit be measured in volts?

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Related: physics.stackexchange.com/q/15402/2451 –  Qmechanic Feb 26 '13 at 8:53
By other questions you have posted, you seem to be using Matter & Interactions by Chabay and Sherwood (Wiley, 2010), I suspect the third edition. Having been involved with the authors to a large extent, I can tell you in no uncertain terms that their discussion of potential difference and potential energy is without peer. See my extended answer below. –  user11266 Mar 10 '13 at 23:31
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6 Answers

you can never measure the absolute electric potential at any point. its just not possible. that is why we talk about difference in potential. similar concept goes to concept of internal energy of a gas. the difference is measurable but not the absolute value. now if you want to strictly define a "volt" then it would go like this: the amount of work done to bring a unit positive charge from infinity to the desired point. so that amount of work done equals the potential at that point OR the potential difference between that point and a point in infinity(that is any point outside the electric field).

so bottom line is that when we talk about potential difference we mean we have two specified points. and when its just potential at a point, then it is actually potential difference between that specified point and a point in infinity(outside the field).

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The electric potential, which I will denote by $\Phi$, is originally defined by the following relation to the electric field (if the math is unfamiliar, don't worry I'm just including it for completeness) $$ \mathbf E = -\nabla \Phi $$ One consequence of this is that

The electric potential is only defined up to an additive constant

This means, in particular, that one has the freedom to pick any point in space, usually called a reference point, at which the potential is zero. Once you pick this point, then the value of the potential $\Phi$ at any other point is completely determined by the definition above.

However, if you don't choose such a point, then additive ambiguity in the definition of potential makes it so only calculating differences in potential makes sense. In this case, it wouldn't make sense to say that "so and so point in the circuit has such and such value."

Punchline. The electric potential is defined in such a way that only differences in potential make sense unless one picks a reference point at which the value of the potential is specified.

Additionally, voltage is usually used as a term for differences in electric potential between two points, so it does not suffer from the same ambiguity as the term electric potential. So in standard parlance, it would be appropriate to say "potential at a point A" (provided a reference point has been chosen) but it would not be appropriate to say "voltage at point A."

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This is potentially confusing, and an excellent example of sloppy terminology perpetuated by careless textbook authors. One should never substitute a quantity's unit for the name of that quantity. We never ask people about their "yearage" when we mean age or "footage" when we mean height. We frequently, however, use "mileage" as a substitute for fuel efficiency, which is also incorrect. Add to this that too many authors use the same symbol for both electric potential and electric potential difference.

The physical quantity here is electric potential, which is a comparison of electric potential energy and electric charge and is represented by the symbol $V$. This quantity constitutes a scalar field around charged particles. Each point in space has assigned to it a value for electric potential. The difference in electric potential between any two points is called an electric potential difference and is represented by the symbol $\Delta V$. Too many authors use $V$ to represent both quantities, but this is erroneous because they are different quantities and are not interchangeable despite having the same unit, volt. To make matters even more confusing, the volt is denoted by the symbol $\mathrm{V}$, which is too similar to the symbol used for electric potential. Electric potential difference is directly measurable with a multimeter but electric potential is not.

The bottom line is that one should never substitute a quantity's unit for that quantity. Call the quantity what it is, electric potential or electric potential difference, and use distinct and unambiguous symbols.

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potentially confusing :) –  askewchan Feb 26 '13 at 17:59
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For a layman answer... Voltage is measured between two or more points. So, it is the difference in potential. If you have a volt battery and a 6 volts battery connected by their negative terminals each would read the proper voltage if measured from ground, but if you were to measure the voltage across their positive terminals the volt meter would read 3 volts because it is measuring the difference in potential between those two points.

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The term "voltage" can initially be quite confusing, I was pretty confused when I learned it first.

It refers to (as you said) the electric potential difference between two points on a circuit. If you have a piece of wire, you would define the voltage across the wire as the potential difference between the two ends of the wire.

Now to apply this to circuits, you can use a handy tool called Khirchoff's circuit laws which explain how to calculate voltages and currents. Essentially, in a circuit one cannot define a "potential" unless you have a source of a potential difference (like a battery). The voltage is defined for every voltage source (battery, charged capacitor) and then added according to the circuit laws for the element (whether in series, parallel etc). Then the total potential difference across the circuit (the voltage) is the sum of these individual voltages.

What all this boils down to is - If you connect a plain wire between two terminals of a battery $^{[1]}$ the voltage at any point on the wire is the voltage of the battery. That's how you define the voltage at one point.

$[1]$ - Please never try and actually do this. That'll cause a short circuit in the battery and bad things will happen.

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Voltage is always a difference. HOWEVER, you usually have some designated point * in a circuit diagram, with respect to which you measure voltages, i.e the voltage at point P in a circuit diagram is given by the difference $$V_P-V_*$$ Usually this point is (any) ground in your circuit (these are assumed to be connected to each other thus are at the same potential).

For instance look at http://www.hoodcomputing.com/blog/HeadlightCircuit.gif , the weird black parallel lines at the bottom left corner denote the ground, so all voltages are measured with respect to this point. Sometimes they say "we set the voltage at point X equal to zero" which amounts to measuring voltages with respect to $V_X$ since point X clearly has voltage 0 with respect to itself.

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