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As I understand it, the concept of potential energy arises from analytical mechanics. Yet I often see the concept of electric potential $\phi$ introduced without mention of analytical mechanics. For example, Electricity and Magnetism by Purcell doesn't discuss analytical mechanics. I have always found this confusing because I don't get why it's not called electric potential energy.

Why is this? Am I just reading the wrong books? Am I mistaken that electric potential is a form of potential energy and therefore related to analytical mechanics? What's going on here?

Note: I understand electric potential is the analog of gravitational potential. That fact however hasn't helped me answer the above.

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    $\begingroup$ Potential is potential energy per unit charge ie different units. $\endgroup$ – Rob Jeffries Dec 24 '14 at 19:51
  • $\begingroup$ @RobJeffries And why is that useful as opposed to just potential energy itself? Like why not just use potential energy without adding the per unit charge qualification? $\endgroup$ – Stan Shunpike Dec 24 '14 at 19:53
  • $\begingroup$ Because "potential" is more general than "potential energy". Nor is this idea unique to electrostatics: gravitational potential is gravitational potential energy per unit mass. $\endgroup$ – dmckee Dec 24 '14 at 20:28
  • $\begingroup$ @dmckee in what sense is it more general? $\endgroup$ – Stan Shunpike Dec 24 '14 at 20:33
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    $\begingroup$ Because it allows you to do computations without knowing the charge (or mass in the gravitation case) of the body subject to the potential. $\endgroup$ – dmckee Dec 24 '14 at 20:35
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Suppose I have an electric field $E$ pointing to the right. Now I take a positive charge $q$ and push it from position $0$ to position $-l$. The force required for the push was $F=qE$, so the work I did was

$$W = Eql \,.$$

Note that the work is proportional to the charge $q$. This work is now stored as electrostatic potential energy. We define the electric "potential" as the potential energy per charge. In other words,

$$\phi = El\,.$$

That's all there is to it. Whenever potential energy is proportional to something intrinsic to the body storing the energy, we define "potential" as the energy divided by that instrinsic quantity. This is useful because it characterizes the system in a way that doesn't depend on properties of your test particle. Note that electric potential does not carry any more or less information than the electric field. In fact, electric potential and electric field are related via

$$\vec{E} = -\nabla \phi \,.$$

Another example is gravitational potential on Earth's surface, $gh$, which is the potential energy of a test particle of mass $m$, $mgh$, divided by the mass.

Note that you can remember the dimensions of electric potential to be "energy per charge".

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Electrical potential is different from potential energy because it does not represent the amount of energy stored in a battery/capacitor/etc.

Electrical potential energy can be calculated from it given the appropriate parameters.

It exists as a unit separate from electrical potential because it's useful for describing things like the electrical current, where elec. potential energy would not be enough to calculate current.

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