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Homework Statement
Two positive charges +Q are held fixed a distance apart. A particle of negative charge -q and mass m is placed midway between them, then is given a small displacement x perpendicular to the line joining them and released. Show that the particle describes simple harmonic motion of period:
##\sqrt{\frac{\epsilon_0 m \pi^3 d^3}{Qq}}##
Homework Equations
##T = 2\pi\sqrt{m/k}##
##F = \frac{1}{4\pi\epsilon_0} \frac{Qq}{r^2} = -kx##
The Attempt at a Solution
I wasn't getting it to work initially by working forwards so I worked backwards from the equation:
##2\pi\sqrt{m/k} = \sqrt{\frac{\epsilon_0 m \pi^3 d^3}{Qq}}##
I knew that k was effectively F/x. so doing that I tried to get F by itself and got:
##F = \frac{2Q(-q)x}{4\pi\epsilon_0(\frac{d}{2})^3}##
with some really creative algebra.
I don't know how to get this from the original equation, but I know it's correct courtesy a hint from a friend. I know the significance of the 2 I got up top is because when you sum the forces in the x direction you get double one of the forces. I also know that at least 2 of the d/2 has to do with the distance between charges approaching that if the displacement it small. But I don't know where the x or the third d/2 come into the picture. I figure since the displacement is small we use some approximation method, but the only ones I know of are the small angle approximations for trig functions and Taylor series. Thoughts?