An early model for an atom considered it to have a positively charged point nucleus of charge Ze, surrounded by a uniform density of negative charge upto a radius R. The atom as a whole is neutral. The electric field at a distance r from the nucleus is (r < R)

Options

A. `(Ze)/(4piepsilon_(0))[1/r^(2)-r/R^(3)]`
B. `(Ze)/(4piepsilon_(0))[1/r^(3)-r/R^(2)]`
C. `(Ze)/(4piepsilon_(0))[r/R^(3)-1/r^(2)]`
D. `(Ze)/(4piepsilon_(0))[r/R^(3)+1/r^(2)]`

Answer

A

Explanation

Charge on nucleus is = + Ze
Total negative charge = -Ze
(`:.` atoms is electrical neutral )
Negative charge density , `rho=("charge")/("volume")=(-Ze)/((4)/(3)piR^(3))`
i.e., `rho=-(3)/(4)(Ze)/(piR^(3))` ……….(i)
Consider a Gussian surface with radius r.
By Gauss's theorem ,
`phi=E(r) xx 4 pir^(2)=(q)/(epsi_(0))`............(ii)
Charge enclosed by Gaussian surface
`q'=Ze +(4pir^(3))/(3) rho=Ze-Ze(r^(3))/(R^(3))" "` (Using (i))
From (ii),
` E(r)=(q')/(4 pi epsi_(0)r^(2))=(Ze-Ze""(r^(3))/(R^(3)))/(4 pi epsi_(0)r^(2))=(Ze)/(4pi epsi_(0))[(1)/(r^(2))-(r)/(R^(3))]`