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Question

Three equal negative charges, −q1 each, form the vertices of an equilateral triangle. A particle of mass m and a positive charge q2 is constrained to move along a line perpendicular to the plane of triangle and through its centre which is at a distance r from each of the negative charges as shown in the figure. The whole system is kept in gravity free space. Find the time period of vibration of the particle for small displacement from equilibrium position.

A
2π4πϵ0mr3q1q2
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B
2π3πϵ0mr34q1q2
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C
2π4πϵ0mr33q1q2
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D
2ππϵ0mr33q1q2
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Solution

The correct option is C 2π4πϵ0mr33q1q2
Let the charge q2 is moved a distance x perpendicular to the plane of the triangle.

Distance of charge q2 from one of the vertices of triangle,
R=r2+x2

Let the angle between x and R is θ.
Net force on charge q2, perpendicular to plane of triangle is 3Fcosθ

Fnet=3Fcosθ=3kq1q2R2xR=ma
here ve sign indicates that force is opposite to displacement.

a=3kq1q2xm(x2+r2)3

a=[3q1q24πϵ0mr3]x=ω2x (as for small x i.e. r>>x)

ω=3q1q24πϵ0mr3

T=2π/ω=2π4πϵ0mr33q1q2
Hence, option (a) is correct.

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