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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 whenr lt Rr =R?Use Gauss's theorem |
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Answer» Solution :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 ` ` (##U_LIK_SP_PHY_XII_C01_E10_029_S01.png" width="80%"> |
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