The energy of a point charge
Our new relation, Eq. (8.35), says that even a single point charge $q$ will have some electrostatic energy. In this case, the electric field is given by \begin{equation*} E=\frac{q}{4\pi\epsO r^2}. \end{equation*}  So the energy density at the distance $r$ from the charge is \begin{equation*} \frac{\epsO E^2}{2}=\frac{q^2}{32\pi^2\epsO r^4}. \end{equation*}  We can take for an element of volume a spherical shell of thickness $dr$ and area $4\pi r^2$. The total energy is \begin{equation} \label{Eq:II:8:36} U=\int\limits_{r=0}^\infty\frac{q^2}{8\pi\epsO r^2}\,dr= \left.-\frac{q^2}{8\pi\epsO}\,\frac{1}{r}\right|_{r=0}^{r=\infty}. \end{equation}  Now the limit at $r=\infty$ gives no difficulty. But for a point charge we are supposed to integrate down to $r=0$, which gives an infinite integral. Equation (8.35) says that there is an infinite amount of energy in the field of a point charge, although we began with the idea that there was energy only between point charges.  In our original energy formula for a collection of point charges (Eq. 8.3), we did not include any interaction energy of a charge with itself. What has happened is that when we went over to a continuous distribution of charge in Eq. (8.27), we counted the energy of interaction of every infinitesimal charge with all other infinitesimal charges. The same account is included in Eq. (8.35), so when we apply it to a finite point charge, we are including the energy it would take to assemble that charge from infinitesimal parts. You will notice, in fact, that we would also get the result in Eq. (8.36) if we used our expression (8.11) for the energy of a charged sphere and let the radius tend toward zero. We must conclude that the idea of locating the energy in the field is inconsistent with the assumption of the existence of point charges. One way out of the difficulty would be to say that elementary charges, such as an electron, are not points but are really small distributions of charge. Alternatively, we could say that there is something wrong in our theory of electricity at very small distances, or with the idea of the local conservation of energy. There are difficulties with either point of view. These difficulties have never been overcome; they exist to this day. Sometime later, when we have discussed some additional ideas, such as the momentum in an electromagnetic field, we will give a more complete account of these fundamental difficulties in our understanding of nature.