# Derivation of the Potential Energy of an Electric Charge System

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## Main Question or Discussion Point

Hi,

I learned about how PE = U=kq1q2/r is the electrical potential energy for the system. It is found by taking the integral of electrical force and dr from infinity to the point of location we are interested in.

So that is the intregral(F*dr) from r=inf to r=ro.

My question is that do I need to know to know this work integral for other applications? I know the formula U, so is the derivation of that important for any other physics problems?

I have a exam that includes the topic of work, potential energy, and potential. I'm kinda shaky on how to utilize the work integral for finding U in other cases. On the other hand, the work integral for Potential (V) is integration of E and dr which is useful is many applications cause it will always work (like in parallel charged plates and a point charge).

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kuruman
Homework Helper
Gold Member
Hi lorx99 and welcome to PF.

$V$ is the potential energy per unit charge while is just the potential energy. The two are related by $V = U/q$. If you know $V$ everywhere in space, you can find $U$ everywhere in space. Both are useful depending on what you are being asked to do. If you know one, you can find the other. Electrostatic potential $V$ is equivalent to $gh$ and electrostatic potential energy $U$ is equivalent to $mgh$ in the following sense: You can find the gravitational potential energy of any mass $m$ in a region of space by multiplying $gh$ by the mass $m$ that you bring in that region of space. Likewise, you can find the electrostatic potential energy $U$ of any charge $q$ in a region of space by multiplying $V$ by the charge $q$ that you bring in that region of space. It's really one integral. Yes, you have to know it. As an educator, I cannot tell you that it's OK not to know something. A word of caution: the equation you quoted, $U=kq_1q_2/r$ is the potential energy of two point charges separated by distance $r$ and applies to that case only and not any other case.

hilbert2
For any conservative force field, including that of gravity, a potential energy function can be found in that way. Note that an arbitrary constant can be added to the potential energy, as you can choose freely what is its value at $r\rightarrow\infty$.