Is Coulomb potential valid at extremely short distances? As we know, the Coulomb potential is in the form
\begin{equation}
V(\mathbf{r}) = Q/\mathbf{r}
\end{equation}
Mathematically, as $\mathbf{r} \rightarrow 0$, $V(\mathbf{r}) \rightarrow \infty$. But is this [as $\mathbf{r} \rightarrow 0$, $V(\mathbf{r}) \rightarrow \infty$] physically true? I can’t imagine at an extremely short distance, say, on a sub-nuclear scale, we would encounter an infinite electric potential (indeed, quarks carry electric charges and produce electromagnetic fields). If this is not the case, how should we modify the Coulomb potential and describe electromagnetic interactions at extremely short distances? (Sure, I know that there are other interactions such as strong and weak interactions at this scale, but they are not what I am concerned here.)
 A: The Coulomb potential formula with fixed values of electric charges isn't verified to be valid "all the way down" and it seems unlikely, because at some small distance electric potential energy of two point charges becomes so large a negative number, it was never observed. For example, if a negative electron got too close to a positive electron (positron), the formula would predict very large negative energy of bound state. But the largest energy we know we can extract is $2m_ec^2$ where the two particles disappear and energy turns into radiation. This excludes validity of the formula for distances smaller than $\frac{Ke^2}{2m_ec^2}\approx $ 1e-15 m (around the size of proton).
A: The Coulomb potential stays the same at shorter distances but what happens is that the value of the electron charge gets renormalized. The standard story behind this is that at normal energies, there is an "electron cloud" which screens the "true" charge of the electron, which leads to an observable effective value that we are familiar with from every day life. The more technical story is that you need to compute vacuum polarization diagrams, which contribute to the potential.
It turns out though that the perturbative expansion in QED is not always valid, since the value of the charge seems to grow at shorter length scales, so beyond a certain energy scale (or equivalently under a certain length scale), the theory breaks down. This signals the need for a more fundamental theory that can explain what exactly happens at these short scales.
A: $V(r) = Q/r$ is the Coulomb potential for a pointlike charge. Yes,
it is valid for any distance $r$ and it goes
to infinity as the distance from the charge goes to zero. If this is physically true depends on what you mean with that, which is not trivial.
When you consider subatomic particles, however, you have to take into account the nuclear forces, and in particular the strong nuclear force, which would prevail.
A: Coulomb potential is accurate even at small distances. In fact it is using this very form that the Schrödinger equation for a hydrogen atom is solved to surprising accuracy. This is enough to model the electron in the atom to good accuracy.
Further corrections and for nuclear behaviour we need to incorporate nuclear forces.
A: For electrons no derivation from 1/r behaviour has been found and the electron radius is below the detection limit. Protons experimentally have a radius of about 0.8 fm below which the 1/r potential no longer holds.
