Question

A non-conducting spherical shell of inner radius R1 and outer radius R2 contains a uniform volume charge density p through the shell. Use Gauss's Law to derive an equation for the magnitude of the electric field at the following radial distances r from the center of the sphere. Your answer should be in terms of p,R1,R2,r Eo, and pi.

a). r < R1

b.) R1 <r<2

c.) r>R2

Answer 1

Consider Q as the total charge contained in the shell.
Q = x [4/3 (R2³ - R1³]

a)
Consider a Guassian surface of radius r., r < R1
Using Gauss law,
E1 x 4r² = 1/o [Charge enclosed]
Where E1 is the electric field
Charge enclosed = 0
E1 x 4r² = 1/o x 0
E1 = 0

b)
Consider a Guassian surface of radius r., R1 < r < R2
Using Gauss law,
E2 x 4r² = 1/o [Charge enclosed]
Charge enclosed = x (4/3 (r³ - R1³)
E2 x 4r² = 1/o x [ x (4/3 (r³ - R1³)]
E2 = {1/o x [ x (4/3 (r³ - R1³)]} / ( 4r²)
= (r³ - R1³) / (3or²)

c)
Consider a Guassian surface of radius r., r > R2
Using Gauss law,
E3 x 4r² = 1/o [Charge enclosed]
Charge enclosed = x (4/3 (R2³ - R1³)
E3 x 4r² = 1/o x [ x (4/3 (R2³ - R1³)]
E3 = {1/o x [ x (4/3 (R2³ - R1³)]} / ( 4r²)
= (R2³ - R1³) / (3or²)