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I would like to prove mathematically that the electric field produced by a punctual charge is isotropic and radial, i.e.

$$\vec{E}(r,\phi,\theta)=E(r)\vec{e}_r\tag{1}$$

I think that this statement is visually understandable by invoking the idea that a sphere would produce a spherical electric field, but it seems hard to show it mathematically.

I know that the the charge distribution of a punctual charge located in $\vec{x}=\vec{x_0}$ is expressed as $\rho(\vec{x})=Q\delta(\vec{x}-\vec{x}_0)$. And from Gauss law, one would get:

$$\iint_{\mathcal{S}}\left(\vec{E}\cdot\vec{n}\right)\,dS=\frac{Q}{\varepsilon_{0}}\tag{2}$$

where $\mathcal{S}$ is a closed surface which interior contains $\vec{x}_0$. I don't know how to proceed after this step. Maybe different kinds of surfaces $\mathcal{S}$ should be tried. What would you do to prove the statement?

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    $\begingroup$ I'm not sure, but I think that it might be impossible to use the integral form of Gauss' law to show this without invoking the kind of symmetry argument that you are trying to avoid. $\endgroup$ – garyp Mar 18 at 21:15
  • $\begingroup$ You seem to be forgetting that Coulomb's law is fundamentally experimental. $\endgroup$ – FGSUZ Mar 18 at 21:19
  • $\begingroup$ I would just invoke symmetry and be done with it. There is no other feature but the distance from the charge in this scenario. $\endgroup$ – Jasper Mar 18 at 21:24
  • $\begingroup$ How would you prove the validity of Gauss law if not equipped with the field of single particle already? And if your answer is that Gauss law can be experimentally proven, well, so can the field of single particle. $\endgroup$ – gented Mar 18 at 22:07
  • $\begingroup$ Your question makes sense. I took the Maxwell laws as granted, and I learn't that they can't be proved mathematically. But because no one told me, in the past, the same about the electric field of an electric charge, I made this question here. As I can see now by the comments, there's no way to prove the statement mathematically. I just didn't like the way I learnt some things at the university. I'm not a computer, and I can't simply apply formulas without questioning them. I would do this question again, because for me it's important to know when something can or cannot be derived. $\endgroup$ – Élio Pereira Mar 18 at 23:03
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Field of single point charge isn't necessarily radial. There is infinity of different fields consistent with such source. If the field is assumed purely retarded and the charge was at rest forever, then the field will be radial everywhere.

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