# 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?

• Is Coulomb's law defined for $r=0$? What direction would the electric field have at $r=0$? Apr 26 at 18:59
• Idk that's what I'm asking. What happens to coulombs law when distance between them becomes zero? Apr 26 at 19:10
• Is your question why doesn't a hydrogen atom collapse and release an infinite amount of energy? Apr 26 at 19:12
• Are you asking about electrons or classical point charges? Apr 26 at 22:06
• @sandejo in my question, I defined my electron as a classical point charge, ignoring the quantum mechanical aspect of it, which I'm now realising should come into effect in such small scales Apr 27 at 7:01

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,

• Thanks for this answer, but currently this is beyond my scope of understanding, even though it gave me a flavour of what you were trying to say Apr 27 at 7:03

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 :

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

2. 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).

• Thank you, this is a great classical analogy Apr 27 at 7:01