How Does a Dielectric Influence Charge Induction on a Conducting Shell?

[Karl86]

Homework Statement

Imagine having a conducting sphere with free charge $Q$ surrounded by a spherical shell filled with a dielectric and then a conducting spherical shell with no free net charge. I want to find out the charge induced on the spherical conducting shell by the sphere or by the dielectric.

Homework Equations

$D=\epsilon_0E + P$

The Attempt at a Solution

The dielectric being polarized, at its interface with the conducting spherical shell there will be a charge distribution $\sigma$ attracted by the bounded surface charge $\sigma_b$ of the dielectric and also by the charged sphere. Thus both the electric field $E$ and the electric displacement field $D$ will be discontinuous. Even though there is no free net charge in the spherical shell. Is this reasoning correct?
In other words, my doubt can be restated: is there some free charge at the inner surface of the spherical shell, even though the conducting shell has no net free charge?

 

[kuruman]

You are on the right track. Consider a spherical concentric Gaussian surface entirely inside the conducting shell.
1. What is the electric flux through the surface?
2. What does the answer to the previous question imply about the free charge enclosed by the surface?
3. What do the answers to the previous two questions imply about the free charge on the outer surface of the conducting shell?