Dielectric Sphere with Free Volume Charge

[jegues]

Homework Statement

See figure attached.

Homework Equations

The Attempt at a Solution

I'm confused as to how they got their D(r)

Using Gauss' Law,

For r < or = a

$$\oint_{S} \vec{D} \cdot \hat{n} dS = \int_{v} \rho dv$$

$$D(4 \pi r^{2}) = \frac{\rho 4 \pi r^{3}}{3}$$

$$\Rightarrow D = \frac{P_{0}r^{2}}{3a}$$

What am I doing wrong?

 

[rude man]

You have ρ = constant. It's not.