Harmonic Motion of a Charged Particle

[FS98]

Homework Statement

Two positive point charges Q are located at points (±l, 0). A particle with positive charge q and mass m is initially located midway between them and is then given a tiny kick. If it is constrained to move along the line joining the two charges Q, show that it undergoes simple harmonic motion (for small oscillations), and find the frequency.

Homework Equations

F = (kqQ)/r^2

The Attempt at a Solution

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I believe the force on the center particle should be F = (kqQ)/(l-x)^2 - (kqQ)/(l+x)^2

I think I need to find that the force is equal to some constant multiplied by x to show that there is simple harmonic motion, but I’m not sure how to do it.

 

[TSny]

When $y \ll 1$,

$\large \frac{1}{(1+y)^n}$ $\approx 1 - ny$ .

This is an example of the binomial approximation https://en.wikipedia.org/wiki/Binomial_approximation