How Do Charge Distributions Affect a Spherical Dielectric Shell?

[fkf]

Homework Statement

Consider a spherical dielectric shell so that ε = ε_0ε_r for a < r < b and ε = ε_0 for 0 < r < a. If a charge Q is placed at the center of the shell, find

a) P for a < r < b
b) ρ_pv for a < r b
c) ρ_ps at r = a and r = b

Homework Equations

ρ_pv = -div(P)
ρ_ps = P \cdota_n

The Attempt at a Solution

I've managed to solve a) with the answer

Q/(4*pi*r^2) * (ε_r-1)/ε_r​

which is correct (checked the answers).

I've also solved b) which is ρ_pv = 0 since ρ_pv = -div(P).

I have trouble to find ρ_ps at r = a and r = b. The answer states that

-Q/(4*pi*a^2) * (ε_r-1)/ε_r and -Q/(4*pi*b^2) * (ε_r-1)/ε_r​

respectively. Where is that negative sign coming from?

 

[fkf]

On the r = a I would guess that we get a negative sign since we're having a unit vector a_r which goes outward from the origin. And since the normal to the surface when r = a is the -a_r I would guess that the negative sign when r = a is correct. But when r = b, the unit vector from P and a_r are booth positive, so why negative there?