How Do Surface Charge Densities Calculate on a Conducting Spherical Shell?

[Trentonx]

Homework Statement

A point charge of strength q1 = -4 µC is located at the center of a thick, conducting spherical shell of inner radius a = 2 cm and outer radius b = 3 cm. The conducting shell has a net charge of q2 = 5 µC
(a) Calculate the surface charge densities on the inner (sa) and outer (sb) surfaces of the spherical shell.

Homework Equations

Gauss's law
Surface area of sphere - 4*pi*r^2

The Attempt at a Solution

So I thought that on a conductor, the charge was all on the outside, so there would no charge on the inner surface, but that was wrong, so I'm at a lose on how to approach this. Gauss's law give me the electric field through an area, right? And then relates that to the charge enclosed, which I know? I might know the equation, but the concepts and how to apply them escape me.

 

[rl.bhat]

In the absence of the charge q1 at the center , all the charge q2 on the sphere would have been on the outer surface of the sphere. But due to the presence of q1, same amount of charge will be attracted towards inner surface of the sphere. The remaining charge will be on the outer surface of the sphere.

 

[Trentonx]

Alright, so I need to find the "spread" of charges, how much remained on the outside surface, and how much went towards the inside. How do I apply Gauss's law then? Can I assume that 4 µC of the sphere's charge went to the inner surface then to compensate for the charge inside?

 

[Trentonx]

So I found the charge densities, and it makes sense, as 4 µC were attracted to the inside of the sphere, since -4 µC were enclosed and left 1µC on the outer surface. Now, the follow up questions are asking about the electric field at various radii.
(b) Calculate the net radial electric field component at the following radii:
At r = 1cm?
At r = 6cm?

It seems this would be an application of Coulomb's Law or that E=(k)(Q)/(r^2), but these aren't the point charges that we have worked with. Gauss's Law tells me the net electric flux, right, so it doesn't seem to be useful in finding the electric field.

 

[rl.bhat]

According to Gauss's law charged sphere acts like a point charge.