Electric field intensity due to charge non-conducting solid sphere by gauss theorem

“Electric field intensity due to charge non-conducting solid sphere by gauss theorem”:

Case(1): outside of sphere
 

Consider a non conducting sphere of centre 0 and radius R. On charging this sphere, the charge is uniformly distributed in whole volume of the sphere. we are to find electric field intensity due to this charge of sphere at a distance r from the center at point p. for this we draw  an imaginary Concentric sphere of radius r, this behaves like a Gaussian surface. The electric field is radial for this Gaussian surface S.

According to Gaussian surface
         φε = ∫∫ E ds cosθ
Here E and ds are in same direction thus θ = 0
        φε = ∫∫E ds cos0
        φε = ∫∫ E ds
        φε = E ∫∫ ds
        φε = E ×S
∵ surface area of sphere S = 4πr²

∴  φε = E×4πr²--------------(1)

By gauss theorem
    φε = q/εo
putting the value of φε  in enq (1)
    q/εo = E ·4πr²
      E = q/ 4πr²εo
     [ E = 1/4πεo × q/r²] N/C--------(2)

Hence, the volume charge density 
             ρ = charge/volume 
             p = q ÷ 4/3πR³

Hence,  q = p × 4/3πR³

This value of q is put in eqn(2)

     E = P×4/3πR³ ÷ 4πr²εo

    [ E = pR³/3r²εo ]  N/C


Case(2): At surface

For this we can put r = R in above eqn
     
          E' = p R³/ 3R²εo
           
       [ E' = p R/ 3εo ] N/C


Case (3): Inside the surface 

For this we put R = r in above eqn
            
         [ E" = p.r/ 3εo ] N/C



Note: At centre of sphere r = 0, then.   E = 0


Graph b/w E. And. r