A conducting spherical shell with inner radius aa and outer radius bb has a positive point charge Q located at its center. The total charge on the shell is -3Q, and it is insulated from its surroundings.  Derive the expression for the electric field magnitude in terms of the distance r from the center for the region r<a. Express your answer in terms of some or all of the variables Q, a, b, and appropriate constants. Derive the expression for the electric field magnitude in terms of the distance rr from the center for the region a<r<b. Derive the expression for the electric field magnitude in terms of the distance rr from the center for the region r>b. What is the surface charge density on the inner surface of the conducting shell? What is the surface charge density on the outer surface of the conducting shell?

A small sphere with a mass of 3.00×10−3 g and carrying a charge of 4.80×10−8 C hangs from a thread near a very large, charged insulating sheet, as shown in the figure (Figure 1). The charge density on the sheet is −2.20×10−9 C/m2 . Find the angle of the thread.

A small conducting spherical shell with inner radius aa and outer radius bb is concentric with a larger conducting spherical shell with inner radius c and outer radius d (Figure 1). The inner shell has total charge +2q, and the outer shell has charge −2q. Calculate the magnitude of the electric field in terms of q and the distance rr from the common center of the two shells for r<a. Calculate the magnitude of the electric field for a<r<b. Calculate the magnitude of the electric field for b<r<c.