# Infinite Electric Field Energy [duplicate]

If the total electric energy associated with a field is:

and the electric field of a point charge follows an inverse square law:

then $$U_{ES}$$ would be infnite if you compute the integral. Can someone shed light on what it means for "infinite total energy" to be associated with a particle?

Secondly, if charges are actually "points" and do not have a finite radius, shouldn't all matter collapse into black holes?

Could someone explain what it means for an electron or other fundamental particle to be associated with a finite radius? What is the "electron radius," and how is it measured? What is self-interaction energy?

equations from https://en.wikipedia.org/wiki/Electric_field

• Related/possible duplicates: physics.stackexchange.com/q/34437/50583, physics.stackexchange.com/q/331354/50583, physics.stackexchange.com/q/62487/50583 and their linked questions Mar 19, 2020 at 20:38
• Also, please do not post formulae as screenshots, but use MathJax instead. Mar 19, 2020 at 20:39
• I did not understand how you compute the infinite energy ? Mar 19, 2020 at 22:26
• How many questions do you have? Please ask one question per post. I agree that they are valid questions. Mar 19, 2020 at 23:07
• @Reign You write the volume integral in spherical coordinates and notice that it diverges. Mar 19, 2020 at 23:50

If you consider two (or more) point charges and compute their electrostatic field energy, you’ll get an infinite term for each particle, but the “interaction terms” like $$\mathbf{E}_1\cdot\mathbf{E}_2$$ integrate to produce finite results which are precisely the potential energy
$$U=\frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{|\mathbf{r}_1-\mathbf{r}_2|}$$