Question

Two particles A and B, each carrying a charge Q, are held fixed with a separation dbetween them. A particle C of mass m and charge q is kept at the middle point of the line AB.   Under what conditions will the particle C execute simple harmonic motion if it is released after such a small displacement? Find the time period of the oscillations if these conditions are satisfied.

Answer 1

For the particle to execute simple harmonic motion:
F' = mw²x

\[\Rightarrow \frac{4}{\pi \epsilon_0}\frac{qQx}{d^3} = m \left( \frac{2\pi}{T} \right)^2 x\] 

\[ \Rightarrow  T^2  = \frac{m \pi^3 \epsilon_0 d^3}{Qq}\] 

\[\Rightarrow T= \left[ \frac{m \pi^3 \epsilon_0 d^3}{Qq} \right]^{1/2} \]