Question

Consider a system of 4 charged particles each of mass m placed at the vertices and centre of an equilateral triangle of side a as shown in figure. Under mutually electrostatic interaction charges at vertices of the triangle are revolving in a circle. Find speed of charges. [$k=\frac{1}{4\pi {ϵ}_0}$ and neglect gravitational effect]

• A. $q\sqrt{\frac{k}{a}(6-2\sqrt{3})}$
• B. $q\sqrt{\frac{k}{a}(2-\sqrt{3})}$
• C. $q\sqrt{\frac{k}{a}(2\sqrt{3}-1)}$
• D. $q\sqrt{\frac{3k}{a}(2\sqrt{3}-1)}$

Answer 1

The correct option is D

$q\sqrt{\frac{3k}{a}(2\sqrt{3}-1)}$

$F_{net}ofq=\frac{k\left(2q^2\right)}{{\left(\frac{a}{\sqrt{3}}\right)}^2}-\frac{2k{q}^2}{a^2}cos{30}^{\circ }\frac{m{v}^2}{\left(\frac{a}{\sqrt{3}}\right)}=\frac{k{q}^2}{a^2}(6-\sqrt{3})v^2=\frac{k{q}^2}{ma}(6\sqrt{3}-3)\Rightarrow v=q\sqrt{\frac{3k}{a}(2\sqrt{3}-1)}$