- #1

- 1,170

- 3

## Homework Statement

A sphere with radius R carries a uniform polarization in the radial direction P = k/r

^{2}

Find the bound charges and calculate the field they produce outside the sphere.

## Homework Equations

ρ = -∇[itex]\bullet[/itex]P

σ = P[itex]\bullet[/itex][itex]\hat{r}[/itex]

## The Attempt at a Solution

We find σ = k/r

^{2}

ρ = 0

So the field they produce is E = kR

^{2}/(ε

_{0}r

^{2})

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 = ε

_{0}E

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 ∇[itex]\bullet[/itex]k/r

^{2}= 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?