An axial electric quadrupole, made of four inline charges $(+q, -q, -q, +q)$ with opposite charges a distance $a$ apart, and the two $-q$ charges adjacent, has an electric field at a remote point $P$ a distance $r \gg a$. $$E_q = {1 \over 4\pi\epsilon_0}{3Q \over r^4}, \qquad \hbox{where } Q = 2a^2q \quad \hbox{(the quadrupole moment)}.$$ This can be found by (vectorially) adding the fields of each individual charge with $$E = {1 \over 4\pi\epsilon_0}{q \over r^2}.$$ Individual charges distances are $(r-a), r, r, (r+a)$.

Although the axial quadrupole is physically identical to two adjacent axial dipoles, if the quadrupole is treated as two dipoles, and the axial dipole field equation is applied: $$E_d = {1 \over 2\pi\epsilon_0}{p \over r^3}, \qquad \hbox{where }p = 2aq \quad \hbox{(the dipole moment)},$$ the field strength at point $P$ will come out twice as large: $$E_q = {1 \over 2\pi\epsilon_0}{3Q \over r^4}.$$ Dipole distance is not $a$ but $(r-0.5 a)$ and $(r + 0.5 a)$.

Question: Why? Our mathematical analysis should be independent of the physics of the situation, and the answer should be the same.

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    $\begingroup$ You need vectors or else specify the direction from the origin to your point $\vec r$. $\endgroup$ – ZeroTheHero Jun 21 '18 at 1:39
  • $\begingroup$ Of course you need vectors, but this is an axial quadrupole, so there are no other components except those on the axis. The origin can just as well be the point $P$. $\endgroup$ – Falsoon Jun 22 '18 at 14:20
  • $\begingroup$ If a two-dipole approach and the individual four-charge approach give two different answers to the same physical situation, then it seems that the analytical approach is incorrect--but which one to use? $\endgroup$ – Falsoon Jul 6 '18 at 12:39
  • $\begingroup$ "There are no other components except those on the axis" ─ this is false. There's a nonzero off-axis electric field component at all points except on the axis, and your post does not specify whether $P$ lies on or off the axis. $\endgroup$ – Emilio Pisanty Jul 6 '18 at 12:48
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    $\begingroup$ The magnitude of the dipole moment of two equal and opposite charges $\pm q$ a distance $a$ apart is $p = qa$, not $p = 2qa$. $\endgroup$ – Michael Seifert Jul 6 '18 at 13:27

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