Electric dipole moment of a neutral system For a system of point charges Qi located at position vectors ri, the electric dipole moment is p = Σ (Qi * ri). What must be true of the charges Qi for the system to be electrically neutral?

Introduction / Context:
Dipole moment is a first-order measure of charge separation. For arbitrary charge distributions, neutrality (zero net charge) is a separate condition from the presence of a dipole moment. This question clarifies the neutrality condition for a set of discrete point charges when discussing multipole expansions in electromagnetics.

Given Data / Assumptions:

• Point charges Qi at positions ri.
• Dipole moment defined as p = Σ Qi ri.
• Neutral system means net charge is zero.

Concept / Approach:

Net charge Q_total = ΣQi. Electrical neutrality requires Q_total = 0, independent of the actual value of the dipole moment p. A neutral system can still possess a nonzero dipole (e.g., +q and −q separated by vector d), and conversely a non-neutral system can have any p but is not neutral. Therefore, the necessary and sufficient condition for neutrality is ΣQi = 0.

Step-by-Step Solution:

Verification / Alternative check:

Example: charges +q at r = +d/2 and −q at r = −d/2 give ΣQi = 0 and p = q d (nonzero), illustrating neutrality with finite dipole moment.

Why Other Options Are Wrong:

(a) or (b) alone would imply nonzero net charge unless complemented by opposite charges; (d) also yields non-neutral systems. Neutrality depends on the sum, not on individual signs being all the same.

Common Pitfalls:

Assuming that zero dipole moment is required for neutrality; that condition relates to charge symmetry, not to net charge.

Final Answer:

The algebraic sum ΣQi must be zero