= Coulomb potential energy
{c}
{title2=$U(r)=kq_1q_2/r$}
= Coulomb potential
{c}
{synonym}
The <Coulomb potential energy> is the interaction <potential energy> of two <point charges> in <vacuum>. With separation $r>0$, <electric charges> $q_1,q_2$, $k=1/(4\pi\varepsilon_0)$ and zero <potential energy> at infinite separation,
$$
\boxed{U(r)=\frac{kq_1q_2}{r}.}
$$
This is mutual interaction <energy>; it does not include the divergent self-energies of ideal <point charges>. Unlike <electric potential>, which has units of <volts>, $U$ has <SI> units of joules. With charge 1 fixed and positive radial direction away from it, the <conservative force> in <Coulomb's law> gives
$$
U(r)-U(\infty)=-\int_\infty^r\frac{kq_1q_2}{s^2}\,ds=\frac{kq_1q_2}{r},\qquad
F_r=-\frac{dU}{dr}=\frac{kq_1q_2}{r^2}.
$$
Thus the $1/r$ <potential energy> and the <inverse-square law> for <force> are two descriptions of the same interaction. https://openstax.org/books/university-physics-volume-2/pages/7-1-electric-potential-energy[OpenStax, University Physics II, §7.1: electric potential energy]{ref}.
For $q_1=+2\,\mathrm{nC}$, $q_2=-3\,\mathrm{nC}$ and $r=0.20\,\mathrm m$, using the stipulated approximation $k=8.99\times10^9\,\mathrm{N\,m^2/C^2}$,
$$
U=\frac{(8.99\times10^9)(2\times10^{-9})(-3\times10^{-9})}{0.20}\,\mathrm J=-2.697\times10^{-7}\,\mathrm J.
$$
At $r=0.40\,\mathrm m$, $U=-1.3485\times10^{-7}\,\mathrm J$. Separating the pair from $0.20$ to $0.40\,\mathrm m$ therefore requires external <work> $\Delta U=+1.3485\times10^{-7}\,\mathrm J$ if the initial and final <kinetic energies> agree; the <electric force> does $-1.3485\times10^{-7}\,\mathrm J$ of <work>. Separating them all the way to infinity requires $+2.697\times10^{-7}\,\mathrm J$. The negative initial <potential energy> means that the pair lies below the chosen separated reference; it does not mean a negative <kinetic energy>.
Like-signed <electric charges> have positive $U$, which decreases as they separate, so their <electric force> is repulsive. Opposite-signed <electric charges> have negative $U$, which decreases as they approach, so their <electric force> is attractive. This attractive interaction supplies the <Coulomb potential> used in the <hydrogen atom>; determining its stationary quantum states also requires <quantum mechanics>. https://www.feynmanlectures.caltech.edu/III_19.html#Ch19-S1[Feynman lectures III.19, §19–1: the Schrödinger equation for hydrogen]{ref}. Dividing $U$ by a nonzero test <electric charge> $q_2$ gives the source's <electric potential> $\Phi_1=kq_1/r$. More generally, the <electrostatic potential energy of point charges> sums each unordered charge pair once. This <energy> description is useful because <work> between two separations follows from endpoint values, without integrating a complicated trajectory.
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