Charged sphere place within a cylindrical capacitor

[ces75]

Homework Statement

The outer conductor (radius b, length l) of a long cylindrical capacitor is earthed. The inner conductor (radius a, length l) is hollow, insulated and uncharged. A sphere of radius R is charged to a potential V far from any other bodies and is inserted inside the inner conductor of the cylindrical capacitor without touching it. (The length l of the capacitor is very much greater than R so end effects can be neglected.)

Draw diagrams showing the distributions of the induced charges and the E field lines in two perpendicular planes through the centre of the sphere one parallel and one perpendicular to the axis of the cylinders.

Show that the electric field strength at a radius a<r<b is E=2RV/rl and hence find the potential of the inner cylinder. Do your answers depend on whether or not the sphere is on the axis of the cylinders?

Homework Equations

I would assume Gauss' Law, potential of a point charge and charge/charge density on a sphere.

The Attempt at a Solution

I have attempted the first part but was not really confident with my answer to start on the second.
I'm not sure what the significance of cylinder a being insulated is, and did not include any effects from it in my diagrams. The field lines will radiate symmetrically from the surface of the sphere, I think straight through a, but stop at b as this is earthed. In the diagrams I have place the centre of the sphere on the axis of the cylinder.

I'm less sure on the induced charges, or even quite what the question wants.

The charge on the sphere will be Q=V4∏ε₀ and σ_s=ε₀ this should induce an equal and opposite charge on the inner surface of a, that is over the total length of the wire, but the charge density will not be uniform down the wire, the greatest charge will be at points in the plane of the centre of the sphere.

the outer surface of a will have the opposite charge again, ie. the charge of the sphere b will zero due to the earth.

the only thought I have has so far for the second part is to use a gaussian sphere with radius a. Again I'm not sure if I am over simplifying the question to assume the field r is only being considered when r fall in the plane of the centre sphere, ie as in the axis perpendicular diagram, in which case the question might be straight forward, but otherwise I can't see how E would be uniform along the whole length of wire.

 

[rude man]

What's the charge on the sphere? So then what's the total charge distributed on the inner cylindrical shell? Then, what does Gauss's law tell you about the E field between the inner and outer cylindrical shells?