Divergence in spherical polar coordinates

Idoubt

hmm ok ok so thats a little advanced for me, but my question was about the divergence of the [itex]\frac{1}{r^2}[/itex][itex]\hat{r}[/itex] function. In simpler terms can you tell me the reason for non-zero divergence in the electric field of a point charge?

vanhees71

No, it's a distribution, the socalled Dirac [itex]\delta[/itex] distribution. It's a very useful concepts. It's even so useful that the mathematicians built a whole new subfield of mathematics to define this concept rigorously, functional analysis!

I like Serena

I suspect you're referring to Maxwell's law that says:
[tex]\textrm{div }\vec E = \frac {\rho} {\epsilon_0}[/tex]

The divergence of a spherically symmetric charge is zero everywhere except at the place where the charge actually is.
A point charge is a special case, since the charge density would be infinite (actually a Dirac delta-function).
In practice the charge would take in a certain volume.
The divergence of the electric field is not zero at the place where the charge density is not zero.

Idoubt

So can I go one more step and state that, at the points in space where the charge 'is' the electric field can no longer be defined by the function [itex]\frac{1}{r^2}[/itex][itex]\hat{r}[/itex] ?

I like Serena

Correct.
In the case of a solid sphere with constant charge density within, the electric field is proportional to [itex]r \hat {\boldsymbol r}[/itex] (inside the sphere).

Idoubt

thank you, this helped a lot