Electric multipole expansion

For a localized charge distribution, the far electric potential is an inverse-radius expansion whose first terms are the total charge monopole, the electric dipole moment, and the electric quadrupole moment.

The trace-free electric quadrupole moment is a symmetric tensor measuring the second-order term in an electric multipole expansion. For a localized charge distribution its contribution to the far-field electrostatic potential is $Q_{ij}{\hat{r}}_i{\hat{r}}_j/(8\pi {ϵ}_0{r}^3)$, with summation over indices. The moment may be nonzero even when the total electric charge or electric dipole moment is nonzero; it should not be confused with an electric quadrupole whose lower moments vanish. Changing the origin generally changes the tensor when lower moments are nonzero.

An electric quadrupole is a charge distribution whose total charge and electric dipole moment vanish while its second charge moment does not. Its potential decays as $r^{-3}$.

This electric potential is harmonic and independent of $z$. Its electric field is linear in transverse position, with a saddle at the origin. For $A>0$, field lines approach along the $x$ axis and leave along the $y$ axis; off-axis field lines satisfy $xy=\mathrm{constant}$. Equipotentials satisfy $x^2-y^2=\mathrm{constant}$. This is a local planar electric quadrupole geometry, not the decaying far field of an isolated three-dimensional quadrupole.

For charges $-Q,NQ,-MQ$ at $z=d,0,-d$, respectively,The monopole vanishes when $N=M+1$, the dipole vanishes when $M=1$, and both vanish when $(N,M)=(2,1)$.

Subject to $N=M+1$, keeping $Qd$ fixed as $d\to 0$ gives a finite point-dipole limit when $M\ne 1$. Keeping $Q{d}^2$ fixed gives a finite pure-quadrupole limit only when $M=1$; otherwise the dipole diverges.

For charges $-Q,2Q,-Q$ at $z=d,0,-d$, the exact force on a charge $-Q$ at $(x,0,0)$ isIt points toward the origin for every nonzero $x$.