Question

Two identical beads, each have a mass m and charge $q$. When placed in a hemispherical bowl of radius $R$ with frictionless, non-conducting walls, the beads move and at equilibrium they are at a distance $R$ apart (as shown in figure). Determine the charge on each bead :

• A. ${\left(\frac{4\pi {\epsilon }_{0}mg{R}^2}{\sqrt{3}}\right)}^{1/2}$
• B. ${\left(\frac{2\pi {\epsilon }_{0}mg{R}^2}{\sqrt{3}}\right)}^{1/2}$
• C. ${\left(\frac{4\pi {\epsilon }_{0}mg{R}^2}{\sqrt{6}}\right)}^{1/2}$
• D. ${\left(\frac{\pi {\epsilon }_{0}mg{R}^2}{\sqrt{3}}\right)}^{1/2}$

Answer 1

The correct option is A ${\left(\frac{4\pi {\epsilon }_{0}mg{R}^2}{\sqrt{3}}\right)}^{1/2}$

For the charges/masses to balance at the position,

the force of electric repulsion along the tangent should balance the force of gravitation along the tangent.

$\frac{1}{4\pi ϵ}\frac{q^2}{(R{)}^2}cos\left({30}^{\circ }\right)=mgcos{60}^{\circ }$

$\frac{1}{4\pi ϵ}\frac{q^2}{(R{)}^2}\frac{\sqrt{3}}{2}=mg\frac{1}{2}$

$q=(\frac{4\pi ϵR^{2}mg}{\sqrt{3}}{)}^{1/2}$