# Electrostatics of balls on a string

1. Feb 5, 2008

### Yuravian

1. The problem statement, all variables and given/known data
In Fig. 21-42 (I attached an MSPaint rendition of it), two tiny conducting balls of identical mass m and Identical charge q hang from non-conducting threads of length L. Assume $$\theta$$ is so small that $$tan \theta$$ can be replaced by its approximate equal, $$sin \theta$$. (a) Show that $$x=(\frac{q^2L}{2\pi\epsilon_{0}mg})^{1/3}$$ gives the equilibrium separation of the balls. (b) If L=120cm, m=10 g, and x=5.0 cm, what is |q|?

2. Relevant equations
Well, I think Coulomb's law is clearly involved because of the $$\frac{1}{2\pi\epsilon_{0}}$$ bit, which is equal to 2*k.
So then, restating Coulomb's law: $$F=\frac{1}{4\pi\epsilon_{0}}*\frac{\left|q_1\right|\left|q_2\right|}{r^2}$$

3. The attempt at a solution
The problem here is I don't know how to start the problem. It appears simple enough but I can't seem to get an answer out of it. Here is what I was thinking: since $$q_1$$ and $$q_2$$ are both the same, the top bit of coulomb's law becomes $$q^2$$ since any number squared is positive (no need for abs. value). splitting out coulomb's law from the full equasion within parenthesis gives $$x=(\frac{1}{4\pi\epsilon_{0}}*\frac{q^2}{r^2}*\frac{2Lr^2}{mg})^{1/3}$$. The problem with this is that I had to work in $$\frac{r^2}{r^2}$$, when I'm pretty sure there's an equivalency in there somewhere.

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• ###### fig2142.bmp
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Last edited: Feb 5, 2008
2. Feb 5, 2008

### Staff: Mentor

Pick one of the balls (the left one, say) and analyze the forces acting on it. The Coulomb force is just one of the forces involved. Hint: Consider vertical and horizontal components separately.