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
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
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