Calculating the Divergence of an Electrostatic Field at the Origin

[rado5]

Homework Statement

The electrostatic field of a point charge q is E=\frac{q}{4 \pi \epsilon r^3} r. Calculate the divergence of E. What happens at the origin?

Homework Equations

The Attempt at a Solution

Well the solution is: \nabla.E= \partialE_x/\partialx + \partialE_y/\partialy + \partialE_z/\partialz

E_x= \frac{qx}{4 \pi \epsilon r^3} and E_y=\frac{qy}{4 \pi \epsilon r^3} and E_z=\frac{qz}{4 \pi \epsilon r^3} and r= \sqrt{x^2 + y^2 + z^2}

After calculation I found the result \nabla.E= \partialE_x/\partialx + \partialE_y/\partialy + \partialE_z/\partialz= 0
Is it correct? I think it is wrong! Then why it is wrong?
Somewhere else I saw that the result was \nabla.E= \frac{\rho}{\epsilon} !

 

[clamtrox]

Somewhere else I saw that the result was \nabla.E= \frac{\rho}{\epsilon} !

What is \rho in this case (a point particle)? It's definitely zero at most places, but where is it not?

 

[rado5]

What is \rho in this case (a point particle)? It's definitely zero at most places, but where is it not?

I really don't know what \rho is here! But do you mean that I calculated it in the correct way?

 

[clamtrox]

All the charge of a pointlike particle is located at a single point (that's why it's called pointlike). At what coordinates is the charge located in this case? What values does the electric field get there?

 

[rado5]

All the charge of a pointlike particle is located at a single point (that's why it's called pointlike). At what coordinates is the charge located in this case? What values does the electric field get there?

Oh thank you for telling me the definition of a pointlike particle, because I didn't know that all the charge is located at a single point! The coordinate is cartesian. I have to translate the main problem, well I will try.
The main problem is this:

If E is the electrostatic field and \Phi is the potential of the electrostatic field prove that: \int\rho\Phidv= \epsilon\int E²dv by using this theorem
\nabla.(\PhiE)=\Phi(\nabla.E)+(\nabla\Phi).E

It is actually an example and in the solution it says that we know \nabla.E=\frac{\rho}{\epsilon} and also E=-\nabla\Phi and also it says that \int\nabla.(\PhiE)dv= 0
Do you think it's a correct problem? I mean is it right or wrong, the solution!

 

[clamtrox]

Oh thank you for telling me the definition of a pointlike particle, because I didn't know that all the charge is located at a single point! The coordinate is cartesian.

So do you now know what the charge distribution is?

You also misunderstood my question. At what location is the particle in? In terms of x,y,z? And what is the value of the electric field there?

If E is the electrostatic field and \Phi is the potential of the electrostatic field prove that: \int\rho\Phidv= \epsilon\int E²dv by using this theorem
\nabla.(\PhiE)=\Phi(\nabla.E)+(\nabla\Phi).E

It is actually an example and in the solution it says that we know \nabla.E=\frac{\rho}{\epsilon} and also E=-\nabla\Phi and also it says that \int\nabla.(\PhiE)dv= 0
Do you think it's a correct problem? I mean is it right or wrong, the solution!

This problem has nothing to do with the question you asked earlier :) To solve it, you only have to use the formulas that you are given.