Find the electric field from polarization

[goohu]

Homework Statement

See picture.

Relevant Equations

Gauss law for polarizaton : $-Q = \int P dA$ , dA = element of area

Attempt at solution:
a) Since I need help with b) this section can be skipped. Results :
$ρ_{psa} = -Pa$
$ρ_{psb} = Pb$
$ρ_{p} = \frac {-1}{R^2} \frac {∂(R^2PR)}{∂R} = -3P$

b) This is where I am unsure (first time using gauss law for P) so I need some confirmation here:
$\int E ⋅ ds = \frac {Q} {ε_0} = \frac {1} {ε_0} \int -P dA$

This will become:
$E(R) 4πR^2 = -\frac {1} {ε_0} PR4πR^2$

For 0<R<a : E(R) = 0
For a<R<b : $E(R) = -\frac {1} {ε_0} PR$
For b<R : E(R) = 0

The problem here is the solution show : For a<R<b : $E(R) = -\frac {1} {ε_0} P$
Did I go wrong somewhere or is this a typo in the solutions?

 

[Delta2]

I agree with your result but depends what is that P that appears in the solution key. Is it the vector $\mathbf{P}=PR\mathbf{a_R}$ or the constant $P$. I think the solution key means the vector $\mathbf{P}$.