2007E2. In the figure above, a nonconducting solid sphere of radius a with charge +Q uniformly distributed throughout its volume is concentric with a nonconducting spherical shell of inner radius 2a and outer radius 3a that has a charge −Q uniformly distributed throughout its volume. Express all answers in terms of the given quantities and fundamental constants.
(a) Using Gauss's law, derive expressions for the magnitude of the electric field as a function of radius r in the following regions.
i. Within the solid sphere ( r<a )
ii. Between the solid sphere and the spherical shell ( a<r<2a )
iii. Within the spherical shell ( 2a<r<3a )
iv. Outside the spherical shell ( r>3a )
(b) What is the electric potential at the outer surface of the spherical shell ( r=3a )? Explain your reasoning.
(c) Derive an expression for the electric potential difference Vx−Vy between points X and Y shown in the figure.