Electric field of an electron when r is zero I know this is a stupid question but according to the coloumb's formula for electric field, if I have a charge right next to another similar charge, such that the distance between them is zero, the electric field should be infinite. Now we obviously don't see infinite electric fields in such cases then what's the problem?
 A: In mainstream physics electrons are point elementary particles in the standard model of particle physics, which is a quantum field theoretical model.
They do have a charge, but their behavior at short distances from each other are governed by quantum mechanical equations, and not by the equations of classical electrodynamics, as is Coulomb's law. When distances and energies are such that h, Plancks constant, can be effectively zero, then the classical equations apply for electrons too, as can be seen in the tracks an electron leaves in a bubble chamber turning in a perpendicular magnetic field, it is just a charged particle.
For the quantum mechanical short range interaction quantum field theory can calculate the probability of interaction of two electrons, where the 1/r Coulomb potential is taken into account in the  solution. Here is the Feynman diagram that allows the calculation to first order of electron electron scattering,

A: Classical analogy:
Assuming the charged items (electrons, protons) have a radius $a$ (small but finite) and the density of charge is constant in the items (if we use a classical analogy), then :

*

*The distance is never zero (the minimum is $2 . a$), hence the field in never infinite.


*inside the sphere with radius a (assuming, the objects are spherical)
, the charge varies linearly with the radius $r, (r < a)$ (Gauss theorem):
$E=\frac{q \ r}{4πε_0a^{3}}$
Within the framework of this classical model, the field decreases linearly inside the object, and therefore is not infinite at the center (in fact, it has a null value at the center of the sphere).
