Three point charges of $1 C, 2 C$ and $3 C$ are placed at the corners of an equilateral triangle of side $1 m$ . The work required to move these charges to the corners of a smaller equilateral triangle of side $0.5 m .$ is $x \times 10^{9} J$ , Find $x$

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Solution

Given,

$q_{1} = + 1 C, q_{2} = + 2 C, q_{3} = + 3 C$

Side of an equilateral triangle , $r = 1 m$

Workdone by the electrostatic force $W = U_{i} - U_{f}$

For three charge distribution we know that ,

$Δ U = \frac{[q_{1} q_{2} + q_{2} q_{3} + q_{3} q_{1}]}{4 π ε_{0}} (\frac{1}{r_{f}} - \frac{1}{r_{i}})$

$∴ W = ∣ ∣ ∣ \frac{[q_{1} q_{2} + q_{2} q_{3} + q_{3} q_{1}]}{4 π ε_{0}} (\frac{1}{r_{f}} - \frac{1}{r_{i}}) ∣ ∣ ∣$

Substituting the given values we get,

$W = \frac{(9 \times 10^{9}) [3 \times 2 + 3 \times 1 + 2 \times 1] (1 - 0.5)}{0.5 \times 1}$

$\Rightarrow W = 99 \times 10^{9} J$

Acceptable answers : $99, 99.0, 99.00$

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