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I have a spherical shell with inner radius [itex]R_1[/itex] and outer radius [itex]R_2[/itex] and a point charge [itex]Q[/itex] in its center. It is NOT a conducting sphere. In the region [itex]R_1 < r < R_2[/itex] there is another constant charge density [itex]\rho_0[/itex]. So total charge density could be expressed as:

[itex]\rho(\vec{r}) = Q \delta(\vec{r}) + \rho_0 \Theta(r-R_1) \Theta(R_2-r)[/itex]

Gauss's law:

[itex]\int_{\partial V} \! \vec E \, d\vec{S} = \frac{1}{\epsilon_0} \int_V \! \rho(\vec{r}) \, d^3r[/itex]

The right hand side is what interests me.

I have to look at 3 different areas obviously.

[itex]r < R_1[/itex]: In this are total charge is simply Q.

The next part is where I'm insecure though.

[itex]R_1 < r < R_2[/itex]:

Is it [itex]\frac{4 \pi \rho_0 (r^3-R_1^3)}{3}[/itex] or is it [itex]Q + \frac{4 \pi \rho_0 (r^3-R_1^3)}{3}[/itex]?

Does the point charge in the center add up or not for the total charge?