# Math Help - Electric Force

1. ## Electric Force

Two tiny conducting balls of identical mass m and identical charge q hang from non-conducting threads of length L. Assume that $\theta$ is so small that $\tan{\theta}$ can be replaced by its approximate equal, $\sin{\theta}$.

(a) Show that

$x = \left(\frac{q^2L}{2\pi\epsilon_0 mg}\right)^{\frac{1}{3}}$
gives the equilibrium separation x of the balls.

(b) If L = 120cm, m = 10g, and x =5.0, what is |q|?

2. forces acting on a single ball ...

T = tension in the string
mg = weight
F = electrostatic force

let $\theta$ = angle the string makes with the vertical

equilibrium ...

$T\cos{\theta} = mg$

$T = \frac{mg}{\cos{\theta}}$

$T\sin{\theta} = F$

$mg\tan{\theta} = F$

since $\tan{\theta} \approx \sin{\theta}$ ...

$mg\sin{\theta} = F$

$mg\sin{\theta} = \frac{q^2}{4\pi \epsilon_0 \cdot x^2}$

since $\sin{\theta} = \frac{x}{2L}$ ...

$\frac{mgx}{2L} = \frac{q^2}{4\pi \epsilon_0 \cdot x^2}$

$x^3 = \frac{q^2 L}{2\pi \epsilon_0 mg}$