A particle of mass m and charge `+q` is located midway between two fixed charged particles each having a charge `+q` and at a distance 2L apart. Assuming that the middle charge moves along the line joining the fixed charges, calculate the frequency of oscillation when it is displaced slightly.

Options

A. `2sqrt((pi^3Mepsilon_0d)/(Qq))`
B. `2sqrt((pi^2Mepsilon_0d^3)/(Qq))`
C. `2sqrt((pi^3Mepsilon_0d^3)/(Qq))`
D. `2sqrt((pi^3Mepsilon_0)/(Qqd^3))`

Answer

C

Explanation

Force on displacement x
`F=-K[(Qq)/((d-x)^2)-(Qq)/((d+x)^2)]`
`implies F =- (4KQqx)/d^2 ` if `d gt gt x`
( -ve sign is due to restoring force)
For SHM `a= -omega^2 x " " ....(i)`
`a=F/M=(-4KQqx)/(Md^3)" "...(ii)`
By (i) & (ii)
`omega^2= (4KQq)/(Md^3) implies omega= sqrt((4Qq)/(4pi epsilon_0d^3M))`
`T=2pi sqrt((pi epsilon_0d^3M)/(Qq))=2sqrt((pi^3 Md^3epsilon_0)/(Qq))`