Two charges \[{{q}_{1}}\] and \[{{q}_{2}}\] are placed 30 cm apart as shown in the figure. A third charge \[{{q}_{3}}\] is moved along the arc of a circle of radius 40 cm from C to D. The change in the potential energy of the system is \[\frac{{{q}_{3}}}{4\pi {{\varepsilon }_{0}}}\,k,\] where k is:                                               [AIPMT (S) 2005]

[a] Key Idea: The change in POTENTIAL energy of the system is \[{{U}_{D}}-{{U}_{C}}\] as discussed under.             When charge \[{{q}_{3}}\] is at C, then its potential energy is                         \[{{U}_{C}}=\frac{1}{4\pi {{\varepsilon }_{0}}}\LEFT( \frac{{{q}_{1}}\,{{q}_{3}}}{0.4}+\frac{{{q}_{2}}\,{{q}_{3}}}{0.5} \right)\]             When charge Q3 is at D, then                         \[{{U}_{D}}=\frac{1}{4\pi {{\varepsilon }_{0}}}\left( \frac{{{q}_{1}}\,{{q}_{3}}}{0.4}+\frac{{{q}_{2}}\,{{q}_{3}}}{0.1} \right)\]                Hence, change in potential energy                         \[\Delta U={{U}_{D}}-{{U}_{C}}\]                         \[=\frac{1}{4\pi {{\varepsilon }_{0}}}\left( \frac{{{q}_{2}}\,{{q}_{3}}}{0.1}-\frac{{{q}_{2}}\,{{q}_{3}}}{0.5} \right)\]             but        \[\Delta U=\frac{{{q}_{3}}}{4\pi {{\varepsilon }_{0}}}K\]             \[\therefore \]      \[\frac{{{q}_{3}}}{4\pi {{\varepsilon }_{0}}}k=\frac{1}{4\pi {{\varepsilon }_{0}}}\,\left( \frac{{{q}_{2}}\,{{q}_{3}}}{0.1}-\frac{{{q}_{2}}\,{{q}_{3}}}{0.5} \right)\]             \[\RIGHTARROW \]   \[k={{q}_{2}}\,(10-2)=8{{q}_{2}}\]