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Two equal negative charge $-q$ are fixed at the fixed points $(0,\,a)$ and $(0,\, - a)$ on the $Y$-axis. A positive charge $Q$ is released from rest at the point $(2a,\,0)$ on the $X$-axis. The charge $Q$ will
Execute simple harmonic motion about the origin
Move to the origin and remain at rest
Move to infinity
Execute oscillatory but not simple harmonic motion
Solution
(d) By symmetry of problem the components of force on $Q$ due to charges at $A$ and $B$ along $y$-axis will cancel each other while along $x$-axis will add up and will be along $C_O$. Under the action of this force charge $Q$ will move towards $O$. If at any time charge $Q$ is at a distance $x$ from $O$. Net force on charge $Q$
${F_{net}} \Rightarrow 2F\cos \theta $ $ = 2\frac{1}{{4\pi {\varepsilon _0}}}\frac{{ – qQ}}{{({a^2} + {x^2})}}$ $ \times \frac{{ x}}{{({a^2} + {x^2})^{1/2}}}$
i.e., ${F_{net}} = – \frac{1}{{4\pi {\varepsilon _0}}}.\frac{{2qQx}}{{{{({a^2} + {x^2})}^{3/2}}}}$
As the restoring force Fnet is not linear, motion will be oscillatory (with amplitude $2a$) but not simple harmonic.
