Suppose potential energy between electron and proton at separation `r` is given by `U = klog r`, where `k` is a constant. For such a hypothetical hydrogen atom , calculate the radius of nth Bohr and its energy level

Options

A. `(k)/(2) [1 + log (n^(2) h^(2))/(4 pi^(2)mk)]`
B. `2k [2 + log (n^(2) h^(2))/(4 pi^(2)mk)]`
C. `k [2 + log (n^(2) h^(2))/(4 pi^(2)mk)]`
D. `(k)/(2) [1 + log (n^(2) h^(2))/(2 pi^(2)mk)]`

Answer

A

Explanation

For a conservative force field.
`(dU)/(dr) = F`
Since `U = k log r`
`(dU)/(dr) = - (k)/( r) = F`
This force `F = -k//r` provides the centripetal force for the circular motion of electron
`(mv^(2))/( r) = (k)/( r)`
Applying Bohr's quantization rule,
`mvr = (nh)/(2 pi)`
From Eqs. (i) and (ii), we get
`r = (nh)/(2 pi sqrt(mk))`
From Eq. (i),
`KE` of electron `= (1)/(2) k`
Total energy of electron `= KE + PE`
`PE` of electron `= k log r`
`= (1)/(2) k + k log r`
`E = (K)/(2) [1 + log (n^(2) h^(2))/(4 pi^(2)mk)]`