# Why is the electric field of an axial quadrupole not the same as the electric field of two axial dipoles, at far distance?

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.

• You need vectors or else specify the direction from the origin to your point $\vec r$. – ZeroTheHero Jun 21 '18 at 1:39
• 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 $P$. – Falsoon Jun 22 '18 at 14:20
• 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? – Falsoon Jul 6 '18 at 12:39
• "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. – Emilio Pisanty Jul 6 '18 at 12:48
• The magnitude of the dipole moment of two equal and opposite charges $\pm q$ a distance $a$ apart is $p = qa$, not $p = 2qa$. – Michael Seifert Jul 6 '18 at 13:27