Electric Field in the matter - D the displacement field

[pbp]

Hi,

I try to understand the concept of the displacement field and the Gauss Law for it (the total flux equals to Q free )
1. Homework Statement
for example I try to find the electric field produced by a uniformly polarized sphere of radius.

3. The Attempt at a Solution
I try to solve this problem in two ways:
the first one is to calculate the bounded charge (equals to Pcos(teta)) and from this to calculate the potential and hence the field.
My second approach (whic probably mistaken approach) is to do the following: Since we know that the D=epsilon_0 * E +P and also according to Gauss law we know that the flux thorugh a Gaussian surface equals to the Q_free_in I built a gaussian surface inside the sphere, the total charge equals to zero and therefore I conclude that D equals to zero(the field inside the sphere is uniform) . If so, I can conclude that E= -p / (epsilon_0).

In each way I got 2 different answers...
where is the problem ?

 

[gabbagabbahey]

You seem to be saying that for a concentric spherical Gaussian surface $\mathcal{S}$ of radius $r$, you would get:

$$\int_{\mathcal{S}} \vec{D}\cdot\vec{da}=4\pi r^2 |\vec{D}|$$

...is that really true?...Don't there have to be a couple of restrictions (symmetries) on $\vec{D}$?