The classical concept of atomic structure is that negative charges are uniformly distributed inside a sphere of radius R keeping the nucleus of positive charge Ze at the centre of that sphere. An atom as a whole is electrically neutral. find out the electric field at a distance r from the nucleus, according do this atomic model.

Explanation

Charge of the nucleus = `+ Ze `
` therefore ` Amount of charge outside the nucleus = ` - Ze`, as an atom is electrically neutral.
Volume density of negative charge inside the sphere,
`rho = ( -Ze )/( (4 ) /( 3 ) pi R ^ 3 ) = - ( 3 Ze ) /( 4 pi R^ 3 ) `
Electric field at external points ` ( r gt R ) ` : Let A be any such point outside the sphere . A concentric sphere of radius r, passing through A, is a convenient Gaussaian surface.
The change enclosed by this Gaussian surface,
`q` = charge of the nucleus + charge distributed inside the sphere of radius R `
` = + Ze+ ( - Ze ) = 0`
` therefore ` From Gauss' theorem, ` oint _ S vecE * dvecS = (q ) /( in _ 0 ) = 0 `
` therefore E = 0`, at all external points.
Electric field at internal points (` r lt R ` ) : Let B be any such point inside the sphere. A concentric sphere of radius r, passing through B , is a convenient Gaussian surface . It is to be noted that this surace encloses the entire positive charge `+ Ze ` of the nucleus, but a part of the negative charge ` - Ze ` remains outside it.
The amount of negative charge inside this Gaussian surface,

` q ' ` = volume of this enclosed sphere ` xx ` volume density ` rho ` of negative charge
` = ( 4 )/(3) pi r ^ 3 ( - (3 Ze )/( 4pi R ^ 3 ) ) = - Ze ( r^ 3 )/( R ^ 3 ) `
` therefore ` Net charge enclosed by the sphere
` q = + Ze + q' = Ze - Ze ( r ^ 3 ) /( R ^ 3 ) = Ze ( 1 - ( r ^ 3 )/( R ^ 3 )) `
For any surface element ` d vecS ` on the Gaussian surface, the electric field ` vec E ` and the area vector ` d vec S ` are parallel to each other
` therefore vec E * d vecS = Ed S cos 0^ @ = EdS `
Again, from symmetry, E is uniform in magniturde over the Gaussian surface. then, from Gauss' theorem,
` oint _ S vec E * d vecS = ( q ) /( in _ 0 ) `
or, ` oint_S EdS = (q )/( in _ 0 ) ` or, ` E oint _ S dS = ( q )/( in _ 0 ) ` or , ` ES = (q ) / ( in _ 0 ) `
or, ` E * 4pi r^ 2 = (1 ) /( in _ 0 ) Ze ( 1 - (r ^ 3 )/( R ^ 3 )) `
or, ` E = ( Ze )/( 4pi in _ 0) * ( 1 ) /( r ^ 2 ) ( 1 - (r^ 3 )/( R ^ 3 ) ) = ( Ze ) /( 4pi in _ 0) (( 1 )/( r^ 2 ) - ( r) /( R ^ 3 )) `