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

Options

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

Answer

A

Explanation

As atom is neutral hence total negative charge `= -Ze`
Negative charge density, `rho("charge")/("volume")=(-Ze)/(4/3piR^(3))`
i.e., `rho= -(3)/(4)(Ze)/(piR^(3))` ….(i)
Consider a Gaussian surface with radius `r`,
By Gauss's theorem, `oint vec(E).vec(d)s =(q_("encl."))/(epsilon_(0))`
`E(r ) xx 4pi r^(2) = (q)/(epsilon_(0))` ...(ii)
Charge enclosed by Gaussian surface,
`q_(enc)= Ze+(4pir^(3))/(3)rho=(Ze-Ze(r^(3))/(R^(3)))` (from (i)) From (ii),
`E(r )=(q_(enc))/(4piepsilon_(0)r^(2))=((Ze-Ze(r^(3))/(R^(3))))/(4piepsilon_(0)r^(2))=(Ze)/(4piepsilon_(0))[(1)/(r^(2))-(r )/(R^(3))]`