Question
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 $$\frac{{{q_3}}}{{4\pi { \in _0}}}k,$$ where $$k$$ is
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 $$\frac{{{q_3}}}{{4\pi { \in _0}}}k,$$ where $$k$$ is
A.
$$8{q_1}$$
B.
$$6{q_1}$$
C.
$$8{q_2}$$
D.
$$6{q_2}$$
Answer :
$$8{q_2}$$
Solution :
We know that potential energy of discrete system of charges is given by
$$U = \frac{1}{{4\pi { \in _0}}}\left( {\frac{{{q_1}{q_2}}}{{{r_{12}}}} + \frac{{{q_2}{q_3}}}{{{r_{23}}}} + \frac{{{q_3}{q_1}}}{{{r_{31}}}}} \right)$$
According to question,
$$\eqalign{ & {U_{{\text{initial}}}} = \frac{1}{{4\pi { \in _0}}}\left( {\frac{{{q_1}{q_2}}}{{0.3}} + \frac{{{q_2}{q_3}}}{{0.5}} + \frac{{{q_3}{q_1}}}{{0.4}}} \right) \cr & {U_{{\text{final}}}} = \frac{1}{{4\pi { \in _0}}}\left( {\frac{{{q_1}{q_2}}}{{0.3}} + \frac{{{q_2}{q_3}}}{{0.1}} + \frac{{{q_3}{q_1}}}{{0.4}}} \right) \cr & {U_{{\text{final}}}} - {U_{{\text{initial}}}} = \frac{1}{{4\pi { \in _0}}}\left( {\frac{{{q_1}{q_2}}}{{0.1}} - \frac{{{q_2}{q_3}}}{{0.5}}} \right) \cr & = \frac{1}{{4\pi { \in _0}}}\left[ {10{q_2}{q_3} - 2{q_2}{q_3}} \right] = \frac{{{q_3}}}{{4\pi { \in _0}}}\left( {8{q_2}} \right) \cr} $$
We know that potential energy of discrete system of charges is given by
$$U = \frac{1}{{4\pi { \in _0}}}\left( {\frac{{{q_1}{q_2}}}{{{r_{12}}}} + \frac{{{q_2}{q_3}}}{{{r_{23}}}} + \frac{{{q_3}{q_1}}}{{{r_{31}}}}} \right)$$
According to question,
$$\eqalign{ & {U_{{\text{initial}}}} = \frac{1}{{4\pi { \in _0}}}\left( {\frac{{{q_1}{q_2}}}{{0.3}} + \frac{{{q_2}{q_3}}}{{0.5}} + \frac{{{q_3}{q_1}}}{{0.4}}} \right) \cr & {U_{{\text{final}}}} = \frac{1}{{4\pi { \in _0}}}\left( {\frac{{{q_1}{q_2}}}{{0.3}} + \frac{{{q_2}{q_3}}}{{0.1}} + \frac{{{q_3}{q_1}}}{{0.4}}} \right) \cr & {U_{{\text{final}}}} - {U_{{\text{initial}}}} = \frac{1}{{4\pi { \in _0}}}\left( {\frac{{{q_1}{q_2}}}{{0.1}} - \frac{{{q_2}{q_3}}}{{0.5}}} \right) \cr & = \frac{1}{{4\pi { \in _0}}}\left[ {10{q_2}{q_3} - 2{q_2}{q_3}} \right] = \frac{{{q_3}}}{{4\pi { \in _0}}}\left( {8{q_2}} \right) \cr} $$