Electric field of a point charge

[willDavidson]

TL;DR Summary

I am trying to understand what a point charge is.

I am trying to understand what a point charge is. Is it just an electron? Or is it just an idea?

 

A point charge $q$ at position $\vec{r}_0$ is described by a charge density consisting of a Dirac delta function, $\rho(\vec{r}) = q\delta^{3}(\vec{r} - \vec{r}_0)$.

Edit: N.B. you can also check that this recovers the Coulomb law, i.e. from Maxwell I,$$\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}$$Perform a volume integral over a region $\Omega$ on both sides, and then use the divergence theorem on the LHS,$$\int_{\partial \Omega} \vec{E} \cdot d\vec{S} = \frac{1}{\varepsilon_0} \int_{\Omega} q\delta^3(\vec{r} - \vec{r}_0) dV = \frac{q}{\varepsilon_0}$$That's Gauss' law in integral form, which you can convert to Coulomb's law by choosing $\partial \Omega$ to be a spherical surface concentric with the point charge, of radius $R$, $$4\pi R^2 E_r = \frac{q}{\varepsilon_0} \implies E_r = \frac{q}{4\pi \varepsilon_0 R^2}$$

 

[Kyouran]

It's just a theoretical concept. It's infinitely small. There's not much more than that to it. And yes, if you want to describe it as a charge distribution you'd use a three dimensional dirac delta function for it.