# Homework Help: Polarized sphere field

1. Jun 17, 2012

### aaaa202

1. The problem statement, all variables and given/known data
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.

2. Relevant equations
ρ = -∇$\bullet$P
σ = P$\bullet$$\hat{r}$

3. 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?

2. Jun 17, 2012

### 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?