From Wikipedia

The electrostatic potential energy, UE, of one point charge q at position r in the presence of an electric field E is defined as the negative of the work W done by the electrostatic force to bring it from the reference position rref[note 1] to that position r.1[2]:§25-1[note 2]

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Since the potential energy is defined as negative of work done by the conservative force. I will proceed with the proof by calculation work done by the force in bringing the point charge q from $\infty$ to $r$.

I have used $dx$ instead of $ds$ as depicted in the diagram.

$$W=F_{Qq}\cdot dx$$
Since the direction of the force by Q on q ($F_{Qq}$) is antiparallel the dot product is resolved as
$$W=F_{Qq}\times dx \cos(180^{\circ})$$
$$W=-F_{Qq}\times dx$$

Now, subbing in the value of $F_{Qq}= \frac{k Qq}{x^2}$

$$W=-\int_{\infty}^{r} \frac{k Qq}{x^2}dx$$

Since this is the work calculated therefore potential energy is its negative:


However the correct expression for the potential energy of two point charges is


I can't find anything wrong with my proof but it is wrong, why?

  • 1
    $\begingroup$ Duplicate which explains that the direction of $d\vec x$ is entirely determined by the path taken or the limits of integration. $\endgroup$
    – Farcher
    Jan 14, 2019 at 18:39

1 Answer 1


You should work with $d\vec{r}$ instead of $d\vec{x}$. As they are opposite, that's your problem.


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