Obtain the expression of electric field by thir spherical shell with uniform charge distribution at a point outside it.

Explanation

Consider a uniformly charged sphere of radius R having the charge density `rho`.
Gauss.s law is `intvecE.vecda =q/epsilon_(0)`
Here we imagine one spherical Gaussian surface of radius r which is greater than R. `(r gt R)`
Total flux passing through the surface = `4pir^(2)E(r)`
Total charge enclosed `q=(4pi)/3R^(3)rho/epsilon_(0)`
`therefore E(r) = (R^(3)rho)/(3rho^(2)epsilon_(0))(r gt R)`
This equation gives the value of the electric field for the points lying in the region outside the sphere at a distance r from the centre of the sphere `(r gt R)`
Total charge of the sphere `Q = 4/3 pi r^(3)rho`
`therefore rho = (3Q)/(4piR^(3))`
Using this in above equation:
`therefore E(r) = R^(3)/(3r^(2)epsilon_(0)).(3Q)/(4piR^(3))`
`therefore E(r) = Q/(4pi epsilon_(0)).1/r^(2)`
Thus, for a point outside the sphere the entire charge of the sphere can be treated as concentrated at its centre.