Polarized sphere with P = k/r² producing conflicting field results

[aaaa202]

Homework Statement

A sphere with radius R carries a uniform polarization in the radial direction P = k/r²
Find the bound charges and calculate the field they produce outside the sphere.

Homework Equations

ρ = -∇$\bullet$P
σ = P$\bullet$$\hat{r}$

The Attempt at a Solution

We find σ = k/r²
ρ = 0
So the field they produce is E = kR²/(ε₀r²)

But this can't be true! How can a polarization create a field outside the sphere? Something is definitely wrong. Because we also have:
D = ε₀E
And using a Gaussian surface you find:
E=0

And I am confident in the last result. There is something disappearing in the above - where have all the charges of opposite sign gone? I believe it has to do with the mathematics behind ∇$\bullet$k/r² = 0. For that the pesky delta function is involved in that I believe. If ρ=0 we also find no field inside the sphere. Can that really be true?

 

[BruceW]

Let me check a couple of things: The polarisation is k/r² in the radial direction, so how is it uniform polarisation? Also, I agree with your answer E = kR²/(ε₀r²) due to the bound charges. Why do you think it is wrong?