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Question

Three particles, each of mass m and carrying a charge q are suspended from a common point by insulating massless strings, each of length L. If the particles are in equilibrium and are located at the corners of an equilateral triangle of side a, then calculate the charge q on each particle.
[Assume L>>a].

A
[2πϵ0a3mg3L]12
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B
None
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C
[4πϵ0a3mg3L]13
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D
[4πϵ0a3mg3L]12
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Solution

The correct option is D [4πϵ0a3mg3L]12
From Fig. (a), for equilibrium of a particle along a vertical line, we get

T cos θ=mg (1)

While for equilibrium in the plane of equilateral triangle, we get

T sin θ=2F cos 30 (2)

So, from eqns. (1) and (2), we have

tan θ=3Fmg (3)


Here, F=14πϵ0q2a2 and tan θ=OAOP=OAL2OA2

Also, from Fig. (c), we get

OA=23AD=23a sin 60=a3

So,

tan θ=(a3)L2(a23)=a(3)L (as L>>a)

On substituting the above values of F and tan θ in Eq. (3), we get

a(3)L=3mgq24πϵ0a2

q=[4πϵ0a3mg3L]1/2

Hence, option (b) is the correct answer.

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