The electirc potential between a proton and an electron is given by `V = V_0 ln (r /r_0)` , where r_0 is a constant. Assuming Bhor model to be applicable, write variation of `r_n` with n, being the principal quantum number. (a) `r_n prop n` (b) `r_n prop (1)/(n)` (c ) `r_n^2` (d)`r_n prop (1)/(n^2)`

Options

A. `r_(n) prop n`
B. `r_(n) prop 1/n`
C. `r_(n) prop n^(2)`
D. `r_(n) prop (1)/(n^(2))`

Answer

A

Explanation

`U=eV=eV_(0) ln (r/r_(0))implies |F|=|-(dU)/(dr)|=(eV_(0))/r`
this force will provide the necessary centripetal force.
Hence, `(mv^(2))/r=(eV_(0))/r implies v=sqrt((eV_(0))/r)` …(i)
Moreover, `mvr=(nh)/(2oi)` …(ii)
Dividing equation (ii) by (i).
We have : `mr=((nh)/(2pi)) sqrt(m/(eV_(0)))sqrt(m/(eV_(0)))implies r_(n) prop n`