Question

Two small spheres, each with a mass of 20 g, are suspended from a common point by two insulating strings of length 40 cm each. The spheres are identically charged and the separation between the balls at equilibrium is found to be 4 cm. Find the charge on each sphere.

Answer 1

Tcosθ = mg ...(i)
Tsinθ = F_e ...(ii)
Here, $F_e=\frac{1}{4{\mathrm{\pi \epsilon }}_0}\frac{q^2}{r^2}$
$\tan \theta =\frac{2}{39.9}$
Dividing equation (ii) by (i), we get
$$\begin{gathered} \tan \theta =\frac{F_e}{mg} \\ \Rightarrow \tan \theta =\frac{1}{4{\mathrm{\pi \epsilon }}_0}\frac{q^2}{r^2}.\frac{1}{mg} \\ \Rightarrow q^2=\frac{\tan \theta .r^2.mg}{1/4{\mathrm{\pi \epsilon }}_0} \\ \Rightarrow q^2=17\times {10}^{-16} \\ \Rightarrow q=4.123\times {10}^{-8}\mathrm{C} \end{gathered}$$