Question

Three point charges, $1\text{C}$, $2\text{C}$ and $3\text{C}$ are placed at the corners of an equilateral triangle of side $100\text{cm}$. Find the work done to move these charges to corners of similar triangle of side $50\text{cm}$.

• A. ${10}^{12}\text{J}$
• B. ${10}^{11}\text{J}$
• C. ${10}^{13}\text{J}$
• D. ${10}^{10}\text{J}$

Answer 1

The correct option is B ${10}^{11}\text{J}$

Given that,
Charge, $q_1=1\text{C}$
Charge, $q_2=2\text{C}$
Charge, $q_3=3\text{C}$
Distance between them, $r=100\text{cm}=1\text{m}$

Case I: When charges are placed on the corner of triangle having side $1\text{m}$.

The potential energy of the system will be,
$U_1=\frac{k}{r}\{q_1{q}_2+q_2{q}_3+q_1{q}_3\}$
$\Rightarrow U_1=k\{2+6+3\}$
$\Rightarrow U_1=11k=11\times 9\times {10}^9$
$\Rightarrow U_1=9.9\times {10}^{10}\text{J}\approx {10}^{11}\text{J}\dots \left(i\right)$

Case II: When charges are moved to corner of triangle having side $r^{\prime }=0.5\text{m}$

The potential energy of the system will be,
$U_2=\frac{k}{r^{{}^{\prime }}}\{q_1{q}_2+q_2{q}_3+q_1{q}_3\}$
$\Rightarrow U_2=\frac{k}{r^{{}^{\prime }}}\{2+6+3\}$
$\Rightarrow U_2=\frac{k}{0.5}\left\{11\right\}$
$\Rightarrow U_2=2k\times 11=2\times 11\times 9\times {10}^9$
$\Rightarrow U_2=19.8\times {10}^{10}\text{J}\approx 2\times {10}^{11}\text{J}\dots \left(ii\right)$

The work done to move the charges will be,
$W=U_2-U_1$
$\Rightarrow W=(2-1)\times {10}^{11}$$\text{J}$
$\Rightarrow W={10}^{11}$$\text{J}$

Hence, option (a) is correct.