Physics - Current Electricity Question with Solution | TestHub

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 $y\ll a$, in displaced position, net resultant force toward equilibrium position is

$$F_{res}=2F\sin \theta =\frac{2\times Q}{4\pi {\epsilon }_0\left(a^2+y^2\right)}\times \frac{y}{{\left(a^2+y^2\right)}^{1/2}}=\frac{2Qy}{4\pi {\epsilon }_0{a}^3}[a\gg y]$$

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