The calculation of extended charges moving in time, requires finite difference calculations that only a computer can perform. This is one of the principle reasons why 20th century Physicists chose to use the point charge model. Quantum Mechanics (QM) was invented, as a probability theory, to incorporate the point property and attempt to explain the wave like phenomena.
In QM, the probability density of the charged entity is of course the square magnitude of the wave function, much like the energy density of the electromagnetic (EM) field is the squared magnitude of the field densities. No matter how small the region is, there is a chance of finding the charge entity, which implies the charged entity is a point charge. As any physics student knows, this implies infinite self energy, and no physical explanation for angular momentum, magnetic spin or radiation (energy exchange).
Dirac "solved" this by equating the QM wave function to the EM fields or the QM probability density to the EM energy density. He also used the bimorphic mapping of the 2x2 group of complex unitary transformations to the proper Lorentz group, which the EM field equations are invariant to. The Lorentz transform and Poincare space were invented, in order to accommodate the illusion of a point charge, because spatially extended charge entities were just too complicated to deal with at the time.
What is most amusing is Special Relativity (SR) assumes as a hypothesis that Space has no properties (no Aether), and General Relativity (GR) states that Space is curved, yet the second order GR uses the first order SR as its foundation. As everyone knows, QM and GR are completely incompatible, so they decided to change the point charge to a Planck scale String. This String theory already has millions of Calabi-Yau manifolds that could potentially be used as the "fundamental" String, no field equations, no experimental results (Planck scale), but it has become the alter of Modern Physics.
There are powerful computers today, that can deal with spatially extended charges evolving in time, so as to truly understand Physics. We live in a 3-dim(sensional) world, so using a model of 0-dim entities as the constituents of this 3-dim Universe we live in is rather absurd. There are those who are using the Parson ring model of the electron and proton, that have avoided most all contradictions, have explained away the weak and strong forces, have described a physical mechanism for inertia, have calculated gravity as a fourth order effect of EM (anti gravity Space travel), and it is all based upon classical Physics.
If you wish to study Modern Physics, then please visit www.commonsensescience.org.