1.A solid plastic sphere of radius a and a conducting spherical shell of inner radius b and outer radius c are shownin the figure above. The shell has an unknown charge. The solid plastic sphere has a charge per unit volume given by ρ(r)=βr, where β is a positive constant and r is the distance from the center of the sphere. Express your answers to parts (a), (b), and (c) in terms of β, r, a, and physical constants, as appropriate.
(a)Consider a Gaussian sphere of radius r concentric with the plastic sphere. Derive an expression for thecharge enclosed by the Gaussian sphere for the following regions.
i.r < a
(b)Use Gauss’s law to derive an expression for the magnitude of the electric field in the following regions.
i.r < a
(c)At any point outside of the conducting shell, it is observed that the magnitude of the electric field is zero.
i.Determine the charge on the inner surface of the conducting shell.
Justify your answer.
ii.Determine the charge on the outer surface of the conducting shell.
(d)i.On the axes below, sketch the electric field E as a function of distance r from the center of the sphere. Sketch the graph for the range r=0 at the center of the sphere to r = c at the outside of the conducting shell.
ii.The figure below shows the sphere and shell with four points labeled W, X, Y, and Z. Point W is at thecenter of the sphere, point X is on the surface of the sphere, and points Y and Z are on the inner andouter surface of the shell, respectively. Rank the points according to the electric potential at that point,with 1 indicating the largest electric potential. If two points have the same electric potential, give themthe same numerical ranking.
____W ____X ____Y ____Z