We use the result of Problem 3. The force on a particle with charge $-q$ that is on the axis of symmetry is
$\begin{align*}(-q)\frac{-dV}{dx} = \frac{- Q qx }{4 \pi \epsilon_0 (R^2 + x^2)^{3/2}}\approx \frac{- Q qx }{4 \pi \epsilon_0 ...$
This follows from the fact that
$\begin{align*}\frac{-dV(x,y,z)}{dx}\end{align*}$
evaluated at $(x_0,0,0)$ equals
$\begin{align*}\frac{-dV(x,0,0)}{dx}\end{align*}$
evaluated at $x_0$ .
The angular frequency of oscillations in the case where we have a spring with spring constant k is $\sqrt{k/m}$ , so by analogy, the angular frequency of oscillations here is:

$\begin{align*}\boxed{\sqrt{\frac{Q q }{4 \pi \epsilon_0 R^3 m}}}\end{align*}$
Therefore, answer (A) is correct.

Solution to 1996 Problem 4