The potential energy of a particle in a force field is:
`U = (A)/(r^(2)) - (B)/(r )`,. Where `A` and `B` are positive
constants and `r` is the distance of particle from the centre of the field. For stable equilibrium the distance of the particle is

Options

A. B/A
B. B/2A
C. 2A/B
D. A/B

Answer

A

Explanation

`U = A/(r^(2)) - B/r `
Differenetiate w.r.t r
`(dv)/(dr) = (-2A)/(r^(3)) +B/(r^(2))`
Foe maximum or minimum `(dU)/(dr) = 0 `
`(2A)/(r^(3)) = B/(r^(2)) rArr r = (2A)/B `
Again deivative
`(d^(2)U)/(dr^(2) = (6A)/(r^(4)) - (2B)/(r^(3))`
`[(d^(2)U)/(dr^(2))] _(r = (2A)/B) = (6AB^(4))/(16A^(4)) - (2BB^(3))/(8A^(3))`
` = (B^(4))/(24A^(4)) gt 0 `
U is minima at r = `(2A)/B `
When potential energy is minimum , body is in stable equilibrium
` r = (2A)/B `