Question

Two equal negative charges are fixed at the points $(0,\pm a)$ on the y-axis. A positive charge $Q$ is released from rest at the points $(2a,0)$ on the x-axis. The charge $Q$ will :

• A. execute simple harmonic motion about the origin
• B. move to the origin ad remain at rest
• C. move to infinity
• D. execute oscillatory but not simple harmonic motion

Answer 1

The correct option is D execute oscillatory but not simple harmonic motion

By symmetry of the 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 $CO$. Under the action of this force, charge $Q$ will move towards $0$. If at any time charge $Q$ is at a distance $X$ from $O$.
Net force on charge $Q$,

$F_{net}=2F\cos \theta$

$=2\times \frac{1}{4\pi {\epsilon }_0}.\frac{-qQ}{(a^2+x^2)}\times \frac{x}{(a^2+x^2{)}^{1/2}}$

$F_{net}=\frac{-1}{4\pi {\epsilon }_0}.\frac{2qQx}{(a^2+x^2{)}^{3/2}}$

As restoring force $F_{net}$ is not linear, motion will be oscillatory (with amplitude $2a$) but not simple harmonic.