Question

Two charges each of magnitude $Q$ are fixed at $2a$ distance apart and a third charge $q$ of mass $m$ is placed at the midpoint of the line joining the two charges; all charges being of the same sing. If $q$ charge is slightly displaced from its position along the line and released then determine its time period.

Answer 1

If $q$ is displaced slightly released

Then fined period of oscillation.

When +q is displaced $x$. towards A

Then force experienced by q due to charge at A

−

$F_1=\frac{KQq}{(a-x{)}^2}$

& Force experienced by the $+q$ due to charge at B.

$F_2=\frac{KQq}{(a+x{)}^2}$Net force on $q=F_1-F_2$

$=\frac{KQq}{(a-x{)}^2}-\frac{KQq}{(a+x{)}^2}$

$=\frac{KQ2\left[\right(a+x{)}^2-(a-x{)}^2]}{(a-x{)}^2(a+x{)}^2}$

$=\frac{KQq\left[\right(a^2+x^2+2ax)-(a^2+x^2-2ax\left)\right]}{[a^2-x^2{]}^2}$

$=\frac{KQq\left[4ax\right]}{(a^2{)}^2}$ $\{A>>>x\}$

$F_{Net}=\frac{4KQqx}{a^3}$ $F=Kx,$ $F=m{\omega }^{2}x$

Hence $m{\omega }^{2}x=\frac{4KQq}{a^3}x$

${\omega }^2=\frac{4KQq}{m{a}^3}$

$\Rightarrow \omega =\sqrt{\frac{4KQq}{m{a}^3}}$

So time period

$T=2\pi \sqrt{\frac{m{a}^3}{4KQq}}$