Question

Two identical positive charges $Q$ each are fixed at a distance of ' $2 a$ ' apart from each other. Another point charge $q _0$ with mass ' $m$ ' is placed at midpoint between two fixed charges. For a small displacement along the line joining the fixed charges, the charge $q_0$ executes SHM. The time period of oscillation of charge $q _0$ will be :

• A. $\sqrt{\frac{4 \pi^3 \varepsilon_0 m a^3}{q_0 Q}}$
• B. $\sqrt{\frac{q_0 Q}{4 \pi^3 \varepsilon_0 m a^3}}$
• C. $\sqrt{\frac{2 \pi^2 \varepsilon_0 m a^3}{q_0 Q}}$
• D. $\sqrt{\frac{8 \pi^3 \varepsilon_0 m a^3}{q_0 Q}}$

Answer 1

Answer: (A)

$ F = macc { }^{ n }=\frac{ KQq _0}{( a - x )^2}-\frac{ KQq _0}{( a + x )^2}$
$m acc ^{ n }=\frac{ KQq _0[2 a ][2 x ]}{\left( a ^2- x ^2\right)^2}$
$\Rightarrow \operatorname{acc}^{ n } \approx\left(\frac{4 kQq_{0 } }{ ma ^3}\right) x$
$ T =2 \pi \sqrt{\frac{\pi \varepsilon_0 ma ^3}{ Qq _0}} $
$ T =\sqrt{\frac{4 \pi^3 \varepsilon_0 ma ^3}{ Qq _0}} $