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Let the charge q is displaced slightly by
perpendicular to the line joining the two fixed charges. The net force on the charge q will be towards O.The motion of charge -q to be simple harmonic if the force on charge q must be proportional to its distance from the centre O and is directed towards O.
The net force on the charge
Here
And
Hence,![F_{net}=2 \left [ \frac{1}{4\pi \varepsilon _0}\frac{q^2}{\left ( d^2+x^2 \right )} \right ]\left [ \frac{x}{\sqrt{x^2+d^2}} \right ]](https://entrancecorner.codecogs.com/gif.latex?F_%7Bnet%7D%3D2%20%5Cleft%20%5B%20%5Cfrac%7B1%7D%7B4%5Cpi%20%5Cvarepsilon%20_0%7D%5Cfrac%7Bq%5E2%7D%7B%5Cleft%20%28%20d%5E2+x%5E2%20%5Cright%20%29%7D%20%5Cright%20%5D%5Cleft%20%5B%20%5Cfrac%7Bx%7D%7B%5Csqrt%7Bx%5E2+d%5E2%7D%7D%20%5Cright%20%5D)
As
i.e. force on charge q is proportional to its displacement from the centre O and it is directed towards O.
Hence, the motion of charge q would be simple harmonic, where