Why does the polarization point inwards for this problem?

[NucEngMajor]

Homework Statement

Griffiths 4.15: A thick spherical shell of inner radius a and outer radius b is made up of dielectric material with a frozen in polarization. P = k / r in the r-hat direction. There is no free charge (why?). Find the electric field inside and out.

Homework Equations

The Attempt at a Solution

I have the solution, but the only part that I don't understand is why the polarization is negative at r=a (pointing inwards, why not radially out like at b?). This may be trivial, but a thorough explanation would be helpful, and perhaps and generalization to other cases. Also they tell us there is no free charge, but could someone explain why physically? Thank you

 

[TSny]

I have the solution, but the only part that I don't understand is why the polarization is negative at r=a (pointing inwards, why not radially out like at b?).

See if this helps: http://www.a-levelphysicstutor.com/field-capacit-1.php#dielectrics
You can see how the polarization of the molecules creates a negative surface charge on one of the surfaces of the dielectric and positive on another surface.

Also they tell us there is no free charge, but could someone explain why physically? Thank you

This is just one of the assumptions for this problem. It is assumed that the only macroscopic charge density in the system is due to the polarization of the molecules. Free charge would be extra charge added to the system. I think Griffiths defines "free charge" a couple of pages previous to this problem.

 

[TSny]

Additional comment: The polarization vector $\vec{P}$ is radially outward everywhere inside the dielectric. But, at the inner surface (r = a) the radially outward polarization produces a negative, bound surface charge density. It's not the polarization that's negative, it's the surface charge that's negative.