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Four point charges are arranged on the y-axis: -q at y = -2d, +q at y = -d, +q at y = +d, and -q at y = +2d. For a point on the x-axis at x = D with D >> d, how does the magnitude of the electric field E vary with D?

1. E proportional to 1/D
2. E proportional to 1/D³
3. E proportional to 1/D²
4. E proportional to 1/D⁴

Correct answer: E proportional to 1/D⁴

Solution

The total charge is -q+q+q-q = 0 (no monopole term, so no 1/D²). The system is symmetric: the pair (+q at -d, -q at -2d) and (+q at +d, -q at +2d) form two dipoles pointing in opposite directions, so the net dipole moment is also zero (no 1/D³). The leading nonzero term is the quadrupole, whose field falls off as 1/D⁴.

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