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When talking about a circuit, why does voltage stand for electrical potential difference equal to $\frac {Work}{q}$ when instead we can use electrical potential energy?

Is it because the charge of an electron is too small and plugging the charge of an electron into the equation $U_E = k\frac{q_1q_2}{r}$ gives a very small number?

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All you can use of potential energy is by definition a potential difference. You can always globally shift your potential energy by an arbitrary amount, as long as you do this consistently in all equations, because by definition the work done when changing the potential energy is the difference between the potential in the two considered states.

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To understand this, we must make the terms electric potential energy and electric potential and voltage clear:

  • Electric potential energy $U$ is the energy associated with a point in the circuit compared to some reference (similarly to when gravitational potential energy for a ball placed on a shelf is in reference to the ground): $$U=k\frac{q_1q_2}r$$This reference can be some point infinitely far away or simply the grounding level.
  • Electric potential $V$ is this electric potential energy per charge at a point: $$V=U/q$$Some would say that the reason for using this is that it is easier for comparing points when figuring out how they will interact with a single charge.
  • And finally, voltage $\Delta V$ (sometimes just symbolized $V$) is just the difference in the electric potential between two points in the circuit: $$\Delta V=V_2-V_1=\frac{U_2}q-\frac{U_1}q$$So, voltage is the actual comparison between points in the circuit.

The terms are not always used accurately, so be aware of context.

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If you know the potential difference between two points $V$ then knowing the charge which flows between these two points (= current $\times$ time = $It$) the change in electrical potential energy can be evaluated (VIt).
That change in electrical potential energy manifests itself in other forms of energy, eg heat in a resistor, light and heat in a led etc.

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We came up with the idea of potential beacuse we needed to come up with a quanity that was unique for a given system (or arrangement of charged particles) contanining electric field.

Potential at a point in an electric field is a unique number which gives us the work done by briniging a unit positive charge from infinity to that point. This makes the calculation of work very easy when the test charge had magnitude other than unity. All we have to do to get the work is to multiply the particle's charge and the potential at that point.

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  • $\begingroup$ This is not entirely correct, as we do not always (though usually) choose infinity as zero reference point for the potential and the potential is only unique with respect to the chosen reference point. There are situations where choosing infinity is not valid or depends on the direction. One example is the idealized situation of a homogeneous field, where the potential diverges at infinity. $\endgroup$ Commented Apr 24, 2018 at 11:05
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Apparently the definition of potential is defined as potential energy per unit charge ,same as electric field is defined as the force per unit charge.

Thus,

V=$\frac U{q}$ or U=qV

The work done by electrostatic force in displacing a test charge q from a to b in an electric field is defined as negative of change in potential energy between them,or

$\Delta$U=-Wa-b

$Ub - Ua$ = $-$ $Wa-b$

We can divide this equation by q

$\frac {Ub}q$-$\frac {Ua}q$=-$\frac{Wa-b}q$

or

Va-Vb=$\frac{Wa-b}q$ as V=$\frac{U}q$

Hope this helps you

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It seems that you are talking about using a potential energy per unit charge (electron and proton) to characterize electrical field.

If so, the charge would have to be unitless, i.e., expressed in a number of electrons or protons, so that the potential energy, associated with a non-unit charge, could still be measured in units of energy, say, joules.

I don't know why such system is not used and I am not sure what advantages it would have, but it, possibly, has something to do with the fact that the ampere and coulomb had been defined before the charge of an electron was measured.

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