Consider a system of three charges , and placed at points , and , respectively, as shown in the figure. Take to be the centre of the circle of radius and angle . [2008]

Question

Consider a system of three charges $q / 3$ , $q / 3$ and $- 2 q / 3$ 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 $C A B = 60 °$ . [2008]

Smart Achievers · Accepted Answer

(3)

(1) The electric field due to charge $\frac{q}{3}$ at A and charge $\frac{q}{3}$ at B at O will get cancelled. The electric field at O due to charge $(\frac{- 2 q}{3})$ is

$E = \frac{1}{4 π ε_{0}} \frac{2 q / 3}{R^{2}} = \frac{q}{6 π ε_{0} R^{2}} .$

In $∆ A B C$ , $\frac{A C}{A B} = \sin 30 °$

$\Rightarrow A C = \frac{A B}{2} = R$

Also, $\frac{B C}{A B} = \sin 60 °$ $\Rightarrow B C = \frac{\sqrt{3} A B}{2} = \sqrt{3} R$

(2) Potential energy of the system $K [\frac{(q / 3) (q / 3)}{2 R}] + K [\frac{(q / 3) (- 2 q / 3)}{R}] + K [\frac{(q / 3) (- 2 q / 3)}{\sqrt{3} R}]$

$= \frac{k q^{2}}{9 R} [\frac{1}{2} - 2 - \frac{2}{\sqrt{3}}] \neq 0$

(3) Magnitude of force between B and C $F = \frac{1}{4 π ε_{0}} \frac{(2 q / 3) (q / 3)}{{(\sqrt{3} R)}^{2}} = \frac{q^{2}}{54 π ε_{0} R^{2}}$

(4) Potential $V_{O} = K [\frac{+ q / 3}{R}] + K [\frac{+ q / 3}{R}] + K [\frac{- 2 q / 3}{R}] = 0$

Solution

Q.
Consider a system of three charges $q / 3$ , $q / 3$ and $- 2 q / 3$ 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 $C A B = 60 °$ . [2008]

1 The electric field at point $O$ is $\frac{q}{8 π ε_{0} R^{2}}$ 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 $\frac{q^{2}}{54 π ε_{0} R^{2}}$

4 The potential at point $O$ is $\frac{q}{12 π ε_{0} R}$

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Solution

Q. Consider a system of three charges q/3, q/3 and -2q/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 CAB=60°. [2008]

1 The electric field at point O is q8πε0R2 directed along the negative x-axis.