Electric Charge and Coulomb's Law

AI Thread Summary
The discussion centers on a physics problem involving two fixed positive charges and a negative charge placed between them, which is displaced and analyzed for simple harmonic motion. The participant calculated the resultant force and attempted to derive the period of motion but encountered discrepancies in their results compared to the expected formula. They expressed confusion over the coefficients in their calculations and sought clarification on proving the motion is simple harmonic. The participant also referenced a similar problem in another forum for additional context. The thread highlights the complexities of applying Coulomb's Law to dynamic systems and the nuances of deriving motion equations.
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Coulomb's Law Problem - Please Help

Homework Statement


Two positive charges +Q are held fixed a distance d apart. A particle of negative charge -q and mass m is placed midway between them, then is given a small displacement perpendicular to the line joining them and released. Show that the particle describes simple harmonic motion of period sqrt((epsilon not)m((pi)^3)(d^3))/(qQ).


Homework Equations





The Attempt at a Solution



SO I calculated the resultant force on the Q charge at any point and found it to be [((sqrt(2))qQ)]/[4pi(epsilon not)(r^2)). I then saw that F=-kz and T = 2pi*sqrt(m/k).
After plugging everything in and seeing that z was neglible when compared to d/2, I came up with several different answers all the same as sqrt((epsilon not)m((pi)^3)(d^3))/(qQ) only I had coefficients in the numerator and denominator. I have no idea what I am doing wrong? Also, how do you prove it is simple harmonic motion?

Here's what I did:

I said that r is approx. equal to d/2. and z is rsin(theta). After I pug this all into T = 2p*sqrt(m/k) I get T = sqrt([16(pi^3)(epsilon not)m(r^3)sin(theta)]/[sqrt(2)qQ]).
I don't get what I am doing wrong. Plz help...thx.
 
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