## Solution to 1996 Problem 4

 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.