Question

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

Answer 1

An axial electric quadrupole, made of four inline charges (+q,−q,−q,+q)(+q,−q,−q,+q) with opposite charges a distance aaapart, and the two −q−q charges adjacent, has an electric field at a remote point PP a distance r≫ar≫a.

Eq=14πϵ03Qr4,where Q=2a2q(the quadrupole moment).Eq=14πϵ03Qr4,where Q=2a2q(the quadrupole moment).

This can be found by (vectorially) adding the fields of each individual charge with

E=14πϵ0qr2.E=14πϵ0qr2.

Individual charges distances are (r−a),r,r,(r+a)(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:

Ed=12πϵ0pr3,where p=2aq(the dipole moment),Ed=12πϵ0pr3,where p=2aq(the dipole moment),

the field strength at point PP will come out twice as large:

Eq=12πϵ03Qr4

Dipole distance is not aa but (r−0.5a)(r−0.5a) and (r+0.5a)(r+0.5a).

Answer 2

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