Three point charges $1 \mathrm{C}, 2 \mathrm{C}$ and $3 \mathrm{C}$ are placed at the comers of an…
Three point charges $1 \mathrm{C}, 2 \mathrm{C}$ and $3 \mathrm{C}$ are placed at the comers of an equilateral triangle of side one metre. The work done to move these charges to the corners of another equilateral triangle of side $0.5 \mathrm{~m}$ is
$199 \times 10^9 \mathrm{~J}$
$19 \times 10^9 \mathrm{~J}$
$99 \times 10^9 \mathrm{~J}$
$29 \times 10^9 \mathrm{~J}$
Solution
Initial potential energy of system of charges
$\mathrm{U}_{\mathrm{i}}=\mathrm{k}\left[\frac{\mathrm{q}_1 \mathrm{q}_2}{\mathrm{r}}+\frac{\mathrm{q}_2 \mathrm{q}_3}{\mathrm{r}}+\frac{\mathrm{q}_3 \mathrm{q}_1}{\mathrm{r}}\right]$
Substitute the necessary values,
$\begin{aligned} & \mathrm{U}_{\mathrm{i}}=\mathrm{k}\left[\frac{1 \times 2}{1}+\frac{2 \times 3}{1}+\frac{3 \times 1}{1}\right] \\ & \mathrm{U}_{\mathrm{i}}=11 \mathrm{k}\end{aligned}$
Potential energy of the system of charges whom they move across the corner of equilateral triangle of side $0.5 \mathrm{~m}$.
$\mathrm{U}_{\mathrm{f}}=\mathrm{k}\left[\frac{\mathrm{q}_1 \mathrm{q}_2}{\mathrm{r}}+\frac{\mathrm{q}_2 \mathrm{q}_3}{\mathrm{r}}+\frac{\mathrm{q}_3 \mathrm{q}_1}{\mathrm{r}}\right]$
Substitute necessary values:
$\begin{aligned} & \mathrm{U}_{\mathrm{f}}=\mathrm{k}\left[\frac{1 \times 2}{0.5}+\frac{2 \times 3}{0.5}+\frac{3 \times 1}{0.5}\right] \\ & =22 \mathrm{k}\end{aligned}$
Work done is given by:
$\begin{aligned} & W=U_f-U_i \\ & =(22-11) \mathrm{k}=11 \mathrm{k} \\ & =99 \times 9 \times 10^9 \\ & W=99 \times 10^9 \mathrm{~J}\end{aligned}$