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Question

Consider a small positive charge q and mass m placed as shown, between two positive charges Q (fixed) each. For a small push given to q as shown, find the time period of simple harmonic oscillations.


A
T=2πma2π0Qq
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B
T=2πma2π0Qq
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C
T=2πma3π03Qq
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D
T=2π3ma3π0Qq
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Solution

The correct option is A T=2πma2π0Qq
Net force will be,
F=[kQq(a+x)2kQq(ax)2]
F=KQq[1(a+x)21(ax)2]
F=4kQqax(a2x2)2

This is not SHM, but if x<<a
F=4kQqaxa4=4kQqxa3
Fx
Now this is SHM, where 4kQqa3=mω2
i.e ω=4kQqma3
Thus, time period T=2πω
T=2πma34kQq
T=2πma3π0Qq

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