Question

A spherical charge distribution of radius R has a charge Q distributed uniformly over its volume. Find the magnitude of the electric field E(r) and the electric potential V (r) for all r.

Answer 1

Consider a point p at a distance r from the center of the sphere, draw a spherecal Guasian surface concetric with the center of the sphere and the surface passing through.

charge density of the sphere = 3Q/(4R³ )

Electric field E(r) every where on the surface is normal and uniform by symetry

Total charge enclosed inside the sphere Q' = 4r³/3

                                             =Qr³/R³

by Guass law we have

the left hand sdie of the integral = E*4r²

   E*4r²   = Qr³/R³₀

E = Qr/4R³₀   for all r<R

at the center E=0

on the surface E = Q/4R²₀

for any r>R the total charge enclosed inside the Guasian surface = Q

E = Q/4r²₀