Two equal point charges each of charge 'q' are placed at a distance of \( 2a \) from each other and a third charge \( -2q \) is placed at the mid point between them. The potential energy of the system is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{-7q^{2}}{8\pi\varepsilon_0 a} \)
The potential energy of a system of point charges is the sum, over every pair, of \( \frac{1}{4\pi\varepsilon_0}\frac{q_iq_j}{r_{ij}} \). Three charges make three pairs: the two charges q (distance 2a apart), and each q with the central charge \( -2q \) (distance a).
\( \displaystyle U = \frac{1}{4\pi\varepsilon_0}\left[\frac{q\cdot q}{2a} + \frac{q(-2q)}{a} + \frac{q(-2q)}{a}\right] \)
\( \displaystyle U = \frac{q^{2}}{4\pi\varepsilon_0 a}\left[\frac{1}{2} - 2 - 2\right] = \frac{q^{2}}{4\pi\varepsilon_0 a}\times\left(-\frac{7}{2}\right) \)
\( \displaystyle U = \frac{-7q^{2}}{8\pi\varepsilon_0 a} \)