Three point charges of $3 \mu \mathrm{C}, 4 \mu \mathrm{C}$, and $5 \mu \mathrm{C}$ are arranged at the three corners of a right angled triangle $A B C$ as shown in the figure. The work done in moving the charges at $A$ and $C$, so that the three charges are located at the three corners of an equilateral triangle of side $3 \mathrm{~cm}$ is

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In a right angle triangle, in $\triangle A B C$ $ \begin{aligned} A C^2 & =A B^2+B C^2 \\ \Rightarrow \quad A C & =\sqrt{A B^2+B C^2}=\sqrt{\left(4 \times 10^{-2}\right)^2+\left(3 \times 10^{-2}\right)^2} \\ \Rightarrow \quad A C & =5 \times 10^{-2} \mathrm{~m} \end{aligned} $ Initial electric potential energy of three charges, $ \begin{aligned} U & =\frac{k q_1 q_2}{A B}+\frac{k q_1 q_3}{A C}+\frac{k q_2 q_3}{B C} \\ & =\frac{k\left(4 \times 3 \times 10^{-12}\right)}{4 \times 10^{-2}}+\frac{k\left(4 \times 5 \times 10^{-12}\right)}{5 \times 10^{-2}}+\frac{k\left(3 \times 5 \times 10^{-12}\right)}{3 \times 10^{-2}} \\ & =k\left[3 \times 10^{-10}+4 \times 10^{-10}+5 \times 10^{-10}\right] \\ & =9 \times 10^9 \times 12 \times 10^{-10} \\ & =108 \times 10^{-1}=10.8 \mathrm{~J} \end{aligned} $

When three charges located of an equilateral triangle of side $3 \mathrm{~cm}$, the final potential energy of three charges system, $ \begin{gathered} =\frac{k q_1 q_2}{A B}+\frac{k q_2 q_3}{B C}+\frac{k q_1 q_3}{A C} \\ =\frac{k\left(4 \times 3 \times 10^{-12}\right)}{3 \times 10^{-2}}+\frac{k\left(3 \times 5 \times 10^{-12}\right)}{3 \times 10^{-2}}+\frac{k\left(4 \times 5 \times 10^{-12}\right)}{3 \times 10^{-2}} \\ =\frac{9 \times 10^9}{3}\left[12 \times 10^{-10}+15 \times 10^{-10}+20 \times 10^{-10}\right]=14.1 \mathrm{~J} \end{gathered} $ Hence, the work done in moving charges at points $A$ and $C, W=(14.1-10.8)=3.3 \mathrm{~J}$

Three point charges of $3 \mu \mathrm{C}, 4 \mu \mathrm{C}$, and $5 \mu \mathrm{C}$ are arranged at the…

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