Question

A system of three charges are placed as shown in the figure :

If \(D \gg d\), the potential energy of the system is best given by :

• A. \(\frac{1}{4\pi\varepsilon_0}\left[\frac{q^2}{d} + \frac{qQd}{D^2}\right]\)
• B. \(\frac{1}{4\pi\varepsilon_0}\left[-\frac{q^2}{d} - \frac{qQd}{2D^2}\right]\)
• C. \(\frac{1}{4\pi\varepsilon_0}\left[-\frac{q^2}{d} - \frac{qQd}{D^2}\right]\)
• D. \(\frac{1}{4\pi\varepsilon_0}\left[-\frac{q^2}{d} + \frac{2qQd}{D^2}\right]\)

Hint:

Add the pair energies: the \(+q, -q\) pair, plus the interaction of \(Q\) with each of the two charges, then expand for \(D \gg d\).

Answer 1

The Correct Option is C

Figure: charges \(+q\) and \(-q\) are separated by \(d\); \(Q\) is at distance \(D\) from the midpoint of \(+q\) and \(-q\), on the side of \(-q\).
Distances: \(Q\) to \(-q\) is \(D - \frac{d}{2}\); \(Q\) to \(+q\) is \(D + \frac{d}{2}\).
\(U = \frac{1}{4\pi\varepsilon_0}\left[-\frac{q^2}{d} + \frac{qQ}{D + d/2} - \frac{qQ}{D - d/2}\right]\).
\(\frac{1}{D + d/2} - \frac{1}{D - d/2} = \frac{-d}{D^2 - d^2/4} \approx -\frac{d}{D^2}\) for \(D \gg d\).
\(U \approx \frac{1}{4\pi\varepsilon_0}\left[-\frac{q^2}{d} - \frac{qQd}{D^2}\right]\).
Answer: (3) \(\frac{1}{4\pi\varepsilon_0}\left[-\frac{q^2}{d} - \frac{qQd}{D^2}\right]\)