# Electric Force

• Jan 19th 2009, 10:37 AM
Aryth
Electric Force
Two tiny conducting balls of identical mass m and identical charge q hang from non-conducting threads of length L. Assume that $\displaystyle \theta$ is so small that $\displaystyle \tan{\theta}$ can be replaced by its approximate equal, $\displaystyle \sin{\theta}$.

(a) Show that

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

(b) If L = 120cm, m = 10g, and x =5.0, what is |q|?
• Jan 19th 2009, 10:52 AM
skeeter
forces acting on a single ball ...

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

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

equilibrium ...

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

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

$\displaystyle T\sin{\theta} = F$

$\displaystyle mg\tan{\theta} = F$

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

$\displaystyle mg\sin{\theta} = F$

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

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

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

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