A system of three charges \( -q \), \( +Q \) and \( -q \) are placed at three points at equal distances on a straight line in the same order. If the potential energy of the system is found to be zero, find the ratio \( Q/q \).
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{1}{4} \)
The potential energy of a system of charges is the sum of \( k q_i q_j / r_{ij} \) over every pair. With three charges there are three pairs: two neighbouring pairs at distance \( d \) and the two outer charges at \( 2d \).
\( \displaystyle U = k\left[\frac{(-q)(Q)}{d} + \frac{(Q)(-q)}{d} + \frac{(-q)(-q)}{2d}\right] \)
\( \displaystyle U = \frac{k}{d}\left[-2qQ + \frac{q^{2}}{2}\right] \)
Putting \( U = 0 \):
\( \displaystyle 2qQ = \frac{q^{2}}{2} \Rightarrow \frac{Q}{q} = \frac{1}{4} \)