Calculate electric field due to a charge uniformly distributed in a spherical volume

Explanation

Consider a sphere of radius R having total charge q uniformly distributed in its entire volume. Then , the charge density `p = (q)/( (4pi)/(3)R^(3))` `( :' ` volume of sphere ` = ( -4pi)/(3)R^(3))`
We have to find the electric field at a point at a distance `r=OP` from the centre O of the sphere . Due to spherical symmetry the field is radial , and hence a sphere of radius r concentric with the sphere of charge is taken as Gaussian surface. Suppose , E be the field at P.
Case-I `:`
When P lies outside the sphere of charge (i.e., `r ge R )`

In this case , the total charge inside the Gaussian surface is q and total flux `bar(E). bar(DeltaS)= E Delta S cos 0^(@)` summed over the entire Gaussian surface is `E4 pi r^(2)` . So ,as per Gauss's law.
`E( 4 pi r^(2)) = ( q)/( epsilon_(0))`
or `E= ( q)/(4pi epsilon_(0)r^(2)) ( ` for `r ge R)`
i.e., total charge appears as concentrated at centre O for points outside the charged sphere.
Case-II `:`
When P lies inside the sphere of charge (i.e., ` r lt R )`

In this case the charge inside the Gaussian surface is `q=p(4pir^(3))/(3)`
So, from Gauss'slaw `E ( 4pir^(2)) = ( q')/( epsilon_(0))=( 4pi)/(3)(r^(2)p)/(epsilon_(0))`
`:. E = (pr)/( 3 epsilon_(0))`
or , `E = ( qr)/( 4piepsilon_(0)R^(3)) ` `( :' p = ( q)/((4pi )/(3) R^(3)))`