Electric field intensity due to charged metallic sphere [solid or hollow]

“Electric field intensity due to charged metallic sphere [solid or hollow]”

consider a metallic sphere of centre O and radius R. When +q is imparted to the sphere. Thus this  charge uniformly distributed on outer surface of a sphere and having no charge inside the sphere. We are to find the electric field intensity due to the charged sphere at the distance r from its centre at point P , for this we draw an imaginary sphere of centre O and radius r. this imaginary sphere behaves like a Gaussian surface. taking smaller element of area ds  around the point P . let the electric field intensity at this point be E.

Case(1): when the observation point P lies outside the sphere:

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)

Case(2): When point p lies on the spherical surface

For this we put r = R in eqn(2)
 
   [ E' = 1/4πεo × q/R²] N/C


Case(3): when point p lies inside the sphere

We know that there is no charge inside the charged metallic sphere thus we can put  q = 0 in eqn(2)

Hence,  E" = 0

Graph b/w E and r