Question

Two balls of same mass and carrying equal charge are hung from a fixed support of length $l$. At electrostatic equilibrium, assuming that angles made by each thread is small, the separation, $x$ between the balls is proportional to :

• A. $l$
• B. $l^2$
• C. $l^{2/3}$
• D. $l^{1/3}$

Answer 1

The Correct Option is D

In equilibrium, $F_{e} = T \,sin\, \theta$ $mg = T\,cos\,\theta$ $tan\,\theta = \frac{F_{e}}{mg} = \frac{q^{2}}{4\pi\,\in_{0}\,x^{2}\times mg}$ also $tan\, \theta\,\approx\, sin = \frac{x/2}{\ell}$ Hence, $ \frac{x}{2\ell } = \frac{q^{2}}{4\pi \,\in _{0}\,x^{2}\times mg}$ $\Rightarrow x^{3} = \frac{2q^{2}\ell}{4\pi \,\in_{0}\,\times mg}$ $\therefore x = \left(\frac{q^{2}\ell }{2\pi \,\in _{0}\,\times mg}\right)^{1/3}$ Therefore $x \,\propto \,\ell^{1/3}$