Three charges $q, q, a n d - 2 q$ are fixed on the vertices of an equilateral triangular plate of edge length a. This plate is in equilibrium between two very large plates having surface charge density $σ_{1} a n d σ_{2},$ respectively. Find the time period of small angular oscillation about an axis passing through its centroid and perpendicular to the plane. Moment of inertia of the system about this axis is $l .$

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$2 π \sqrt{\frac{ε_{0} l}{2 q a | σ_{1} - σ_{2} |}}$

$2 π \sqrt{\frac{2 ε_{0} l}{\sqrt{3} q a | σ_{1} - σ_{2} |}}$

$2 π \sqrt{\frac{2 ε_{0} l}{q a | σ_{1} - σ_{2} |}}$

$2 π \sqrt{\frac{ε_{0} l}{q a | σ_{1} - σ_{2} |}}$

answer is C.

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Detailed Solution

Net dipole moment $= \sqrt{3} q a$

$τ = - \sqrt{3} q a \frac{| σ_{1} - σ_{2} | θ}{2 ε_{0}} = l α$

$\frac{\sqrt{3} q a | σ_{1} - σ_{2} |}{2 ε_{0}} = l {(\frac{2 π}{T})}^{2}$

$T = 2 π \sqrt{\frac{2 ε_{0} l}{\sqrt{3} q a | σ_{1} - σ_{2} |}}$

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