For a hypothetical hydrogen like atom, the potential energy of the system is given by `U(r) =(-ke^2)/(r^4)`, where ris the distance between the two particles. If Bohr's model of quantisation of angular momentum is applicable, then the velocity of the particle is given by

Options

A. `(nh)/(16kepi^2 m^(3//2))`
B. `(n^2h^2)/(8k^2e^2pi^4 m^3)`
C. `(n^3h^3)/(2k^2e^3pi^3 m^4)`
D. `(n^2h^2)/(8sqrt(kepi^2 m^(3//2)))`

Answer

D

Explanation

`(d[U(r) ])/(dr) =(4ke^2)/(r^5) =` force
`(4ke^2)/(r^5) =(mv^2)/r and mvr = (nh)/(2pi)`
or `r= (nh)/(2pi mv) implies 1/r =(2pimv)/(nh)`
`4ke^2 xx 1/r^5 = (mv^2)/r`
`4ke^2 xx 1/r^4 =mv^2`
`4ke^2 xx (16pi^4m^4v^4)/(n^4h^4)=mv^2`
`v^2 = (n^4h^4)/(64ke^2pi^4m^3)`
`v=(n^2h^2)/(8sqrt(kepi^2 m^(3//2)))`