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I'm having a hard time determining the relationship/differences between interaction energy and forces. Say we have a system of two charged particles. Each particle will exert a force on each other (from Coulomb's Law), but also defined is an interaction energy between the two particles given by Coulomb's Law multiplied by the distance between them (reducing the $r^2$ term in Coulomb's Law to just $r$).

How are these concepts related? What is the correct way to interpret the interaction energy?

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The force is just the gradient of the potential energy. So it's not true that the energy is just force multiplied by $r$. It is the integral of the force w.r.t. $r$, which gives you the $1/r$ dependence.

Edit: here "potential energy" and "interaction energy" are used interchangeably.

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