Conducting Sphere Covered By Spherical Dielectric

[csnsc14320]

Homework Statement

An isolated metal sphere of radius a has a free charge Q on its surface. The sphere is covered with a dielectric layer with inner radius a and outer radius b

Calculate the polarization charge density on the inside and outside of the dielectric.

Homework Equations

The Attempt at a Solution

So I know that the electric field outside of the conducting sphere w/o dielectric is:

$$E = \frac{k Q}{r^2}$$

This field however, should have the electric field due to the dielectric subtracted from it.

If we can regard the dielectric as a capacitor since it has an equal amount of polarized charge on its inner and outer surfaces, the electric field for a < r < b should be:

$$E = \frac{k q'}{r^2}$$ where q'=charge on inner surface of dielectric

the charge on the inner surface should just be

$$q' = \sigma_{inner} 4 \pi a^2$$


now I know I want $$\sigma_{inner}$$, and then it would be simple to find $$\sigma_{outer}$$, but I am not really sure how to solve for it

 

[csnsc14320]

or is this totally wrong?