Question

The electirc potential between a proton and an electron is given by `V = V_0 in (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)`

Answer 1

`:. U = eV = eV_0 in ((r )/(r_0))`
`|F| = |-(dU)/(dr)| = (eV_0)(r )`
This force will provide the necessary centripetal force. Hence, `(mv^2)/(r ) = (eV_0)/(r )`
or `v= sqrt(eV_0)/(m)
.....(i) Moreover, ` mur = (nh)/(2pi)
....(ii) Dividing Eq. (ii) by Eq. (i) we have
`mr = ((nh)/(2pi)) sqrt(m)/(eV_0) or r_n prop n`