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
evaluated at (x_0,0,0) equals
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.

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