Calculating Point Charge Potential at (0,0,0)

[Yegor]

Hallo!

Homework Statement

Consider electrostatic Potential of point charge at point (0,0,0)
$$\phi = 1/r$$
I'm trying to calculate $$\Delta\phi$$

Homework Equations

The Attempt at a Solution

Actually it's not a difficult problem outside (0,0,0):

$$\nabla\phi = -\frac{\vec r}{r^3}$$
$$\Delta\phi = 0$$

But i also know, that i should become Dirac-Delta function (charge density for point particle). What is the problem in point (0,0,0)?
I understand that Potential goes there to infinity, but how can i work it out mathematically?

 

[Meir Achuz]

Using the integral definition of div, it can be shown that
div(r(vec)/r^3)=4pi delta(r).
This can also be shown by appying the div theorem and taking the limit as r-->0.
What book are you using? Graduate texts do this.