Physics problem

Question

An insulating hollow sphere has inner radius a and outer radius b. Within the insulating material the volume charge density is given by ρ (r) = α/r, where α is a positive constant. (a) In terms of α and a, what is the magnitude of the electric field at a distance r from the center of the shell, where a < r < b? (b) A point charge q is placed at the center of the hollow space, at r = 0. In terms of a and a, what value must q have (sign and magnitude) in order for the electric field to be constant in the region a < r < b, and what then is the value of the constant field in this region?

Answer 1

Note that the volume of a sphere is 4/3piR³ thus dV=4piR²dR. In the region a<R<b, dQ=4piaRdR given the charge density = a/R. Performing the integral from a to r yields Q = 2pia(r²-a²) which is the amount of charge enclosed at r=R. Recall that E= kq/r² from Coulomb's law and combining that from the amount of charge found enclosed at R yields E=2akpi(1-a²/R²). (k=coulomb's constant) part a

If a point charge q is added to the center of the spherical shell, the E field contribution will be kq/R². The resultant total field will be the sum of the field in part a plus kq/R². For the total to be a constant q=2a³kpi and the constant field will be 2akpi. (use Algebra)