Question

Two charges $-q$ each are fixed separated by distance $2d.$ A third charge $q$ of mass m placed at the mid-point is displaced slightly by $x(x<<d)$ perpendicular to the line joining the two fixed charged as shown in figure. Show that $q$ will perform simple harmonic oscillation of time period.

$T={\left[\frac{8{\pi }^3{ϵ}_{o}m{d}^3}{q^2}\right]}^{\frac{1}{2}}$

Answer 1

Net force $F$ on $q$ towards the centre $O$

$F_{net}=2F\cos \theta$

Where $F=\left(\frac{1}{4\pi {ϵ}_o}\right)\frac{q^2}{r^2}$

$=2\frac{q^2}{4\pi {ϵ}_0{r}^2}\cos \theta$

$=-\frac{2q^2}{4\pi {ϵ}_0{r}^2}\cdot \frac{x}{r}$

$F=\frac{-2q^2}{4\pi {ϵ}_0}\frac{x}{(d^2+x^2{)}^{3/2}}$

As $(x<<d)$

$F\approx \frac{-2q^2}{4\pi {ϵ}_0{d}^3}x=-Kx$

Where $K=\frac{2q^2}{4\pi {ϵ}_0{d}^3}$

This is the equation of $SHM,$ and time period of $SHM$ is given by

$T=2\pi \sqrt{\frac{m}{K}}={\left[\frac{4{\pi }^{2}m}{K}\right]}^{\frac{1}{2}}$

$T={\left[\frac{8{\pi }^3{ϵ}_{o}m{d}^3}{q^2}\right]}^{\frac{1}{2}}$

Final Answer : $T={\left[\frac{8{\pi }^3{ϵ}_{o}m{d}^3}{q^2}\right]}^{\frac{1}{2}}$