Question

Problem 2 Consider a solid sphere of radius R with a uniform charge density with total charge Q distributed throughout its volume. a) Show that the electric field for this system is given by for r> R Ameoks for TSR b) What is the potential V (r) for the regions () r > R and r S R2 Use lim and remember that V (r) is continuous at r R r)0 as your reference point HINT: You might be able to just write down what the potential is for r > R. Be sure to explain your reasoning if you just write it down. c) What is the total electrostatic energy of this system?

Answer 1

first we use gauss law to calculate electric field inside and outside the sphere. then we use potential gradient to calculate potential . in part c we use direct formula of electrostatic energy of sphere.
Radius of the sphere is R total charge of the sphere is Q charge is uniformly distributed so volume charge density of the sphere is S = Q/4/3 pi R^3 = 3Q/4 pi R^3 S = 3Q/4 pi R^3 (a) First we calculate electric field in the region r > R We draw a gaussian surface of radius r concentric with the sphere. \r\n total charge inclosed by the gaussian surface is q_in = Q from Gauss's law contourintegral E vector middot A vector = q_in/epsilon_0 E(r) middot 4 pi r^2 = Q/epsilon_0 E(r) = Q/4 pi epsilon_0 r^2 direction of this field will radially outward so this field in vector form is E vector (r) = Q/4 pi epsilon_0 r^2 r hat r > R Now we draw a gaussian surface of radius r lessthanorequalto R \r\n total charge by the gaussian surface is q_in = rho times volume of gaussian surface q_in = 3Q/4 pi R^3 times 4/3 pi r^3 q_in = Qr^3/R^3 \r\n