Consider a system of three charges $\frac{q}{3}, \frac{q}{3}$ and $-\frac{2 q}{3}$ placed at points $A, B$…

Consider a system of three charges $\frac{q}{3}, \frac{q}{3}$ and $-\frac{2 q}{3}$ placed at points $A, B$ and $C$, respectively, as shown in the figure. Take $O$ to be the centre of the circle of radius $R$ and angle $C A B=60^{\circ}$.
  1. The electric field at point $O$ is $\frac{q}{8 \pi \varepsilon_0 R^2}$ directed along the negative $x$-axis
  2. The potential energy of the system is zero
  3. The magnitude of the force between the charges at $C$ and $B$ is $\frac{q^2}{54 \pi \varepsilon_0 R^2}$
  4. The potential at point $O$ is $\frac{q}{12 \pi \varepsilon_0 R}$

Solution

Distance $B C=A B \sin 60^{\circ}=(2 R) \frac{\sqrt{3}}{2}=\sqrt{3} R$ $\therefore \quad\left|F_{B C}\right|=\frac{1}{4 \pi \varepsilon_0} \frac{(q / 3)(2 q / 3)}{(\sqrt{3} R)^2}=\frac{q^2}{54 \pi \varepsilon_0 R^2}$ $\therefore$ correct option is (c).

Asked in: JEE Advanced 2008 (Paper 2)

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