Divergence of ##\vec{x}/\vert\vec{x}\vert^3##

Lambda96 said:

Homework Statement: Calculate the following ##\vec{\nabla}\cdot \vec{E}(\vec{r})##
Relevant Equations: none

Hi,

unfortunately, I am not sure if I have calculated the task correctly

View attachment 325180
The electric field of a point charge looks like this ##\vec{E}(\vec{r})=\frac{Q}{4 \pi \epsilon_0}\frac{\vec{r}}{|\vec{r}|^3}## I have now simply divided the electric field into its components i.e. #E_x , E-y, E_z#.

$$\vec{E}(\vec{r})=\frac{Q}{4 \pi \epsilon_0}\left( \begin{array}{rrr}
\frac{x}{(x^2+y^2+z^3)^{\frac{3}{2}}} \\
\frac{y}{(x^2+y^2+z^3)^{\frac{3}{2}}} \\
\frac{z}{(x^2+y^2+z^3)^{\frac{3}{2}}} \\
\end{array}\right)$$

Then I calculated the divergence

$$\vec{\nabla}\cdot \vec{E}(\vec{r})=\frac{Q}{4 \pi \epsilon_0} \Bigl( \frac{\partial}{\partial x}\frac{x}{(x^2+y^2+z^2)^{\frac{3}{2} }} +\frac{\partial}{\partial y}\frac{y}{(x^2+y^2+z^2)^{\frac{3}{2} }} +\frac{\partial}{\partial z}\frac{z}{(x^2+y^2+z^2)^{\frac{3}{2} }} \Bigr)=\Bigl( \frac{-2x^2+y^2+z^2}{(x^2+y^2+z^2)^{\frac{5}{2} }} +\frac{x^2-2y^2+z^2}{(x^2+y^2+z^2)^{\frac{5}{2} }} +\frac{x^2+y^2-2z^2}{(x^2+y^2+z^2)^{\frac{5}{2} }} \Bigr)=0$$

With the result of 0 I am a bit confused, in the task is not mentioned the sign of the charge, but with a positive charge I would expect as a divergence a source, so the divergence would have to be positive and with a negative charge, a sink, so a negative divergence.

Have I somehow miscalculated, or do I have a thinking error in the physical interpretation of the result?

PhysicsRock said:

Is it as simple as to say that, since you can't divide by 0, the origin isn't part of the field, and thus, there is no particular source? Or have I made a mistake in my calculations and the result isn't 0?

PhysicsRock said:

I am asked to calculate what's effectively the divergence of the vector field ##\vec{F} = \vec{x}/\vert\vec{x}\vert^3## over ##\mathbb{R}^3##.

PhysicsRock said:

Is it as simple as to say that, since you can't divide by 0, the origin isn't part of the field, and thus, there is no particular source?