An early model for an atom considered it to have a positivelly charged point nucleus of charge +Ze surrounded by a uniform density of negative charge up to a radius R. The atom as whole is neutral. For this model , what is the electric field at a distance r from the nucleus when
`r lt R `

` r =R ` ?Use Gauss's theorem

Explanation

Let as shown in positive charge at the centre of the atom be +Ze and let density of negative charge be `rho `, such that total negative charge ` (4)/(3) pi R^(3) rho =-Ze`
`rArr " " rho = ( 3Ze)/( 4 pi R^(3)) `
For any point P situated at a distance r `(rlt R ) ` from the nucleus considering a sphere of radius r as the Gasussian surface , we have
Charged enclosed
` " " q= ( +Ze) + ` negative charge enclosed within the sphere of radius r
`rArr " " q= Ze +(4)/(3) pi R^(3)rho =Ze -( Zer^(3))/( R^(3)) =Ze [1-( r^(3))/( R^(3)) ]`
` therefore ` Flux on Gaussian surface `phi_in =E 4 pir^(2) =(1)/( in_0) q = (Ze)/( in_0) [1-(r^(3))/( R^(3)) ]`
` rArr " "E= " " (Ze)/( 4 pi in _0) [(1)/( r^(2)) -(r)/( R^(3)) ]`
(b) For any point P situated on or outside the atom (i.e., ` ge R ) ` total charge enclosed
` " " q= ( +Ze) +(-Ze) =0 `
Hence ` " " phi_in =E. 4 pi r^(2) =(1)/( in_0) (0) =0 `
` rArr " " E=0 `
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