# The electric field generated by a point charge becomes infinite when distance tends to zero. What physical meaning does this have? [duplicate]

What follows is from Professor Barton Zwiebach of MIT:

The only problem with such a large field is an infinite self-energy of the point particle limit. This problem is not really there in QED, as any infinite self energy is renormalized away. There is actually an effective version of Maxwellâ€™s equations, called nonlinear electrodynamics in which point charges have finite self energy, and electric fields cannot exceed a limit (just like the speed of objects is $< c$). This is also true in classical string theory. It is explained in Chapter 20 of my book on string theory."

What does it mean that the energy is renormalized?

## 1 Answer

Without a great understanding of QED, I will not attempt to go into renormalization.

The short answer, which I hope is satisfactory, is that this problem arises from us humans using a simplified model to make sense of reality.

It is simple for us to use $\vec{E} = \frac{q}{\epsilon_0r^2}\hat{r}$ to describe the field from a point charge. And it works for most cases, but you notice there's a big problem when you try to evaluate E at $r = 0$! Since you don't really need to do that for practical purposes, you stick with the point charge model.

Maxwell's Electrodynamics ignores the $r=0$ problem, and does just fine. QED zooms in on $r=0$ to try and figure out what's going on.

• As the quote correctly points out, the only problem with such a large field is an infinite self-energy of the point particle limit. This answer does not consider that self-energy at all. Maybe that's enough to sketch just how far from complete it is? – Emilio Pisanty Jan 2 '18 at 18:15