How can it be that, if like charges repel, they don't repel themselves? In other words, why don't charges break apart?
About the possible duplicate: I want to know about charges in general, not just that of an electron.
My response to Lawrence B. Crowell's answer:
Thank you very much for the extensive explanation. Unfortunately it's a bit above my level (I'm a first year electrical engineering bachelor student). This is how I understand it: If the charge is the sum of multiple separate charges, there has to be an external force that keeps this charges together.
However we most often think of electrons and protons as point charges. In other words, we don't think of them as if they are made up from different (smaller) parts.
But the idea of electrons and protons as point charges has its own problems. I didn't know that the electric field has a mass. From the formula, I understand that $m\propto \frac{1}{r}$. This would mean that $m\rightarrow \infty$ as $r \rightarrow 0$ (or $r = 0$), which of course is physically not possible.
But there is a way to calculate the radius of an electron based on the mass, the speed of light and the constant of Planck. (Unfortunately I don't know why.)
You can get around this by a technique called called renormalization, which causes the integral to converge.
Unfortunately, I don't understand the rest of the answer due to my level in physics. Nevertheless I am grateful for you answer. It would be awesome if you could confirm or correct my understanding.
Best Answer
Composite particles like protons don't break apart because of the strong interaction which holds their constituents (the quarks) together. Elementary particles like electrons don't break apart because they are point-like particles, i.e. they are not made of “parts” (if the Standard Model is correct).