Question

A particle of mass $m$ and charge $q$ is located midway between two fixed charged particles each having a charge $q$ and a distance $2\ell$ apart. If it is displaced along the line connecting them and released. Also find its time period for SHM.

• A. $4\pi \sqrt{\frac{m\pi {ϵ}_0{\ell }^3}{q^2}}$
• B. $\pi \sqrt{\frac{m\pi {ϵ}_0{\ell }^3}{2q^2}}$
• C. $2\pi \sqrt{\frac{m\pi {ϵ}_0{\ell }^3}{q^2}}$
• D. $\pi \sqrt{\frac{2m\pi {ϵ}_0{\ell }^3}{q^2}}$

Answer 1

Answer: (C)

When the charge is displaced by an amount $x$. Force on it
$F=\frac{K{q}^2}{(\ell +x{)}^2}-\frac{K{q}^2}{(\ell -x{)}^2}$
$=\frac{K{q}^2}{{\left[{\ell }^2-x^2\right]}^2}(-4\ell x)$
or $F=-\frac{4K\ell x{q}^2}{{\left({\ell }^2-x^2\right)}^2}$
$=-\frac{\ell x{q}^2}{\pi {ϵ}_0{\left({\ell }^2-x^2\right)}^2}$
When $x$ is very small
$F=-\frac{q^2}{\pi {ϵ}_0{\ell }^3}x$
or $a=-\frac{q^2}{\pi {ϵ}_{0}m{\ell }^3}x$
${\omega }^2=\frac{q^2}{\pi {ϵ}_{0}m{\ell }^3}\&T=\frac{2\pi }{\omega }$
$\omega =\frac{q}{\sqrt{\pi {\epsilon }_{0}m{\ell }^3}}$
$T=2\pi \sqrt{\frac{\pi {\epsilon }_{0}m{\ell }^3}{q^2}}$