SHM in charges

 

\text{Geometry:} \[ r = \sqrt{x^2 + L^2} \] \text{Force on charge:} \[ F = \frac{kQq}{r^2} \] \text{Resolving components:} \[ F_{\text{restoring}} = 2F \sin\theta \] \[ \sin\theta = \frac{x}{\sqrt{x^2 + L^2}} \] \textbf{Net restoring force:} \[ F = 2 \cdot \frac{kQq}{(x^2 + L^2)} \cdot \frac{x}{\sqrt{x^2 + L^2}} \] \[ F = \frac{2kQq \, x}{(x^2 + L^2)^{3/2}} \] \textbf{Equation of motion:} \[ ma = - \frac{2kQq \, x}{(x^2 + L^2)^{3/2}} \] \text{For small displacement } (x \ll L): \[ (x^2 + L^2)^{3/2} \approx L^3 \] \[ ma = - \frac{2kQq}{L^3} x \] \textbf{Comparing with SHM:} \[ a = -\omega^2 x \] \[ \omega^2 = \frac{2kQq}{mL^3} \] \[ \omega = \sqrt{\frac{2kQq}{mL^3}} \] \textbf{Time period:} \[ T = \frac{2\pi}{\omega} \] \[ T = 2\pi \sqrt{\frac{mL^3}{2kQq}} \]