Question

Consider a small positive charge $q$ and mass $m$ placed as shown, between two positive charges $Q$ (fixed) each. For a small push given to $q$ as shown, find the time period of simple harmonic oscillations.

• A. $T=2\pi \sqrt{\frac{m{a}^2\pi {\in }_0}{Qq}}$
• B. $T=2\pi \sqrt{\frac{m{a}^2\pi {\in }_0}{Qq}}$
• C. $T=2\pi \sqrt{\frac{m{a}^3\pi {\in }_0}{3Qq}}$
• D. $T=2\pi \sqrt{\frac{3m{a}^3\pi {\in }_0}{Qq}}$

Answer 1

The correct option is A $T=2\pi \sqrt{\frac{m{a}^2\pi {\in }_0}{Qq}}$

Net force will be,
$F=[\frac{kQq}{(a+x{)}^2}-\frac{kQq}{(a-x{)}^2}]$
$F=KQq[\frac{1}{(a+x{)}^2}-\frac{1}{(a-x{)}^2}]$
$F=-\frac{4kQqax}{(a^2-x^2{)}^2}$

This is not SHM, but if $x<<a$
$F=\frac{-4kQqax}{a^4}=\frac{-4kQqx}{a^3}$
$F\propto -x$
Now this is SHM, where $\frac{4kQq}{a^3}=m{\omega }^2$
i.e $\omega =\sqrt{\frac{4kQq}{m{a}^3}}$
Thus, time period $T=\frac{2\pi }{\omega }$
$T=2\pi \sqrt{\frac{m{a}^3}{4kQq}}$
$T=2\pi \sqrt{\frac{m{a}^3\pi {\in }_0}{Qq}}$