Question

Repeat the previous problem if the particle C is displaced through a distance x along the line AB. 

Answer 1

Net force

\[= \frac{1}{4\pi \in_0}\left[ \frac{Qq}{\left( \frac{d}{2} - x \right)^2} - \frac{Qq}{\left( \frac{d}{2} + x \right)^2} \right]\] 

\[ = \frac{Qq}{4\pi \in_0}\frac{\left[ \left( \frac{d}{2} \right)^2 + x^2 + xd - \left( \frac{d}{2} \right)^2 - x^2 + xd \right]}{\left[ \left( \frac{d}{2} \right)^2 - x^2 \right]^2}\]

when, 
x << d
So, net force = \[\frac{qQ}{4\pi \in_0}\frac{\left( 2xd \right)}{d^4}\] 

\[ = \frac{qQ}{4\pi \in_0}\frac{2x}{d^3}\] 

\[\text{ Or  m }\left( \frac{2\pi}{T} \right)^2 x = \frac{2xqQ}{4\pi \in_0 d^3}\] 

\[T =  \left[ \frac{\pi^3 \in_0 m d^3}{2Qq} \right]^{1/2}\]