How Does Polarization Affect Electric Fields in Spherical Objects?

[KeyToMyFire]

ρ

Homework Statement

A sphere of radius a has a radial polarization P = krⁿ$\hat{r}$ where k and n are constants and n $\geq$ 0.

a.) Find the volume and surface charge densities of bound charge.

b.) Find E outside and inside the sphere. Verify that you results for E satisfy the appropiate boundary conditions.

c.) Find V (potential) outside and inside the sphere.

d.) Sketch the results.

Homework Equations

σ_b = P $\cdot$ $\hat{n}$

ρ_b = -$\nabla$ $\cdot$ P

The Attempt at a Solution

I got

σ_b = krⁿ

and

ρ_b = -(n+2)kr^n-1

which I'm pretty sure is right, but then for E I get

E_inside = -krⁿ$\hat{r}$/ε₀

which seems okay and then

E_outside = 0

which doesn't seem right.Can anybody tell if this is right or not? And if not how to do it?

 

[TSny]

I think everything is correct, except you need to specify a specific value of r in your expression for σ_b.

To see if it's reasonable that E = 0 outside the sphere, calculate the net charge of the sphere due to both the bound volume charge density and the bound surface charge density.