Question

For the arrangement shown in the figure, the two positive charges, $+Q$ each, are fixed. Mark out the correct statement(s) regarding a third charged particle $-q$ placed at midpoint $P$ that can be displaced along or perpendicular to the line connecting the charges.

• A. The particle will perform S.H.M. for $x<<a$
• B. The particle will oscillate about $P$ but not harmonically for any $x$
• C. The particle will perform S.H.M. for $y<<a$
• D. The particle will oscillate about $P$ but not harmonically for $y$ comparable to $a$

Answer 1

Answer: (C), (D)

The particle will oscillate about $P$ but not harmonically for $y$ comparable to $a$

The particle will oscillate or perform SHM if equilibrium is a stable one. For negative charge, equilibrium is stable if the particles are displaced along $CD$ and unstable for displacement along $AB$.


For $y<<a$, in displaced position, net resultant force towards equilibrium position is
$F_{\text{net}}=2F\sin \theta =\frac{2\times Q}{4\pi {ϵ}_0(a^2+y^2)}\times \frac{y}{(a^2+y^2{)}^{\frac{1}{2}}}=\frac{2Qy}{4\pi {ϵ}_0{a}^3}[a>>y]$
$F_{\text{net}}\propto y$, so SHM for $y<<a$
For $y\simeq a$, there would be restoring force, but $F$ is not proportional to $y$.
So, there is oscillatory motion but not SHM.

Hence, options $\left(C\right)$ and $\left(D\right)$ are the correct answers.