# Simple Harmonic Motion by Charges

After a brief mention of simple harmonic motion by charges (and a strong hint that we should all get blogging) in class today, I've been inspired to look into how charges oscillate. I set up a hypothetical problem as follows:

A charge of +1 (charge A) is located a distance x from a stationary +1 charge (, which in turn is located a distance of 2 units from another stationary charge of -2 ©

I then continued- with complete and utter disregard for units- to solve for the force of charge B on charge A and C on A:

F_{B on A} = k(1)(1)/x^{2 } F_{C on A} = k(-2)(1)/(2+x)^{2}

I then surmised that at the point where the two opposing forces equal one another good things will happen (I'm pretty sure this would be the equilibrium position for oscillation).

k/x^{2} = -2k/(x^{2}+4x+4) → -2x^{2} = x^{2}+4x+4 → 3x^{2}+4x+4 = 0

Unfortunately, this quadratic is unsolvable and I'm left at a dead end. In class Mr. Fullerton mentioned how at some distance from stationary charges B and C, A would only feel the combined attractive force of the +1 and -2 charges. However, I'm stumped for the moment. Ideas?

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