Question

Why is the energy of the electric field created by point charge infinite?

Answer 1

The diverging self energy of a point charge is indeed still a very open problem in physics today. The problem arises when we consider the formula for the energy stored in an electromagetic field: W = ϵ o 2 ∫ a l l s p a c e E 2 d τ W=ϵo2∫allspaceE2dτ For a point charge, such as an electron, this reduces to: W = q 2 8 π ϵ o ∫ ∞ 0 1 r 2 d r W=q28πϵo∫0∞1r2dr This integral diverges, leading to the infinite value for the energy contained in the field of a point charge. From a qualitative perspective: The formula derived above is derived by adding up all the energy needed to assemble a charge distribution. Well, to assemble a point charge, we would need to take a finite amount of charge and "cram" it into a point, a space infinitely small, having no dimension. Considering this, it makes sense that to "assemble" a point charge will require an infinite amount of energy, since we would need to create a distribution with infinite an charge density. Thus, the problem is that classical electromagnetism predicts that the energy required to "create" a point charge distribution is infinite, which makes no sense. This is a great example of a "singularity" appearing in a physical theory. At this singularity, the laws of physics don't work and predict ridiculous infinite answers. This problem is not just present in classical electrodynamics but is also present in the quantum theory as well. Fixing or explaining this problem is an open area of theoretical research.

Reference https://www.physicsforums.com/threads/the-energy-of-a-point-charge.232921/