In the arrangement shown in figure, two positive charges, $+ Q$ each, are fixed. Mark the correct statement(s) regarding a third charged particle q placed at the midpoint P that can be displaced along or perpendicular to the line connecting the charges.

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The particle will oscillate or perform SHM if equilibrium is a stable one. For negative charge, equilibrium is stable if the particles are displaced along CD and unstable for displacement along AB .

For $y ≪ a$ , in displaced position, net resultant force toward equilibrium position is

$F_{res} = 2 F sin θ = \frac{2 \times Q}{4 π ε _{0} ( a ^{2} + y ^{2} )} \times \frac{y}{( a ^{2} + y ^{2} ) ^{1/2}} = \frac{2 Q y}{4 π ε _{0} a ^{3}} [a ≫ y]$

$F_{res} \propto y$ , so SHM for $y ≪ a$ For $y = a$ , there would be restoring force, but F is not proportional to y . So there is oscillatory motion but not SHM.

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