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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.
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Solution
For the particle to execute simple harmonic motion:
F' = mw2x
\[\Rightarrow \frac{4}{\pi \epsilon_0}\frac{qQx}{d^3} = m \left( \frac{2\pi}{T} \right)^2 x\]
\[ \Rightarrow T^2 = \frac{m \pi^3 \epsilon_0 d^3}{Qq}\]
\[\Rightarrow T= \left[ \frac{m \pi^3 \epsilon_0 d^3}{Qq} \right]^{1/2} \]
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