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A particle of mass M and charge q is at ...

A particle of mass M and charge q is at rest at the midpoint between two other fixed similar charges each of magnitude Q placed a distance 2d apart. The system is collinear as shown in the figure. The particle is now displaced by a small amount `x(x lt lt d)` along the line joining the two charges and is left to itself. It will now oscillate about the mean position with a time period ( `epsilon_0` = permittivity of free space)

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))`

Text Solution

Verified by Experts

The correct Answer is:
C
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))`
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