# Polarized sphere field

## 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?