Problem 1

Two small spheres of equal mass

m

hang from strings of length

l

from the same point, initially touching. Each is given an equal charge

q

; they repel and reach equilibrium at a small angle

\theta

from the vertical. Find

q

in terms of

m

l

\theta

. Here

g=9.8\,\mathrm{m\,s^{-2}}

and

k=1/(4\pi\epsilon_0)

is Coulomb’s constant; neglect sphere size relative to the center separation, so the repulsive force is horizontal and the two strings make symmetric angles

\theta

with the vertical. (a) Use force balance to find the electric force in terms of

m,g,\theta

. (b) Express the center separation using

l,\theta

and obtain the exact charge in terms of the given quantities. (c) For

\\theta\ll1

rad, derive the small-angle formula and check its dimensions.

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

k\frac{q^2}{r^2} = mg\tan\theta \approx mg\theta \ \Rightarrow\ q \approx \sqrt{\frac{4mgl^2\theta^3}{k}}

problems.proof.analysis

Substituting $F_C = kq^2/r^2$ and $r\approx 2l\theta$ into $F_C = mg\tan\theta \approx mg\theta$ (small angle) and solving for $q$ shows $q \propto \theta^{3/2}$ — a non-trivial relationship, testable experimentally by measuring $\theta$ for different charges.

problems.proof.pitfall. Use radians for

\sin\theta\approx\theta

and keep the center separation equal to

2l\sin\theta