What information does the quadrupole provide?

I've seen many definitions on the internet, but I don't understand the relation between them.

• What is the relation between knowing that the quadrupole moment is between four charges and that it is a measurement of how much the nucleus charge distribution deviates from spherical symmetry?
• Why does a dipole come from cancelling two monopole fields and a quadrupole comes from cancelling two dipole fields?
• Does a tripole field exist?

Moments measure how charge (or any other quantity of interest) is distributed in space. Instead of specifying a continuous charge density function $$\rho(x,y,z)$$, you can specify discrete numbers: its monopole moment (i.e., total charge)

$$q=\int \rho\,dV,$$

its dipole moments

$$p_i=\int \rho x_i\,dV,$$

$$Q_{ij}=\int \rho x_ix_j\,dV,$$

its octupole moments

$$O_{ijk}=\int \rho x_ix_jx_k\,dV,$$

etc. The complete set of moments contains the same amount of info, but in a form that can be more useful. In many cases, only the first few moments are important, because the effects (field strengths, forces, energies, torques, etc.) of higher-order moments drop off more quickly with distance.

Do not focus on the two-point-charges, four-point-charges, etc. cases. Moments are much more general than that.

From the definition of the quadrupole moment, you can see that spherical symmetry implies $$Q_{11}=Q_{22}=Q_{33}$$ and the off-diagonal elements are zero. Departures from this indicate a non-spherical distribution.

The moments I have shown are Cartesian moments, which are the simplest to understand. You can also define moments using spherical harmonics, and these are more elegant because you don’t get a proliferation of indices.

• How are the off-diagonal elements zero? Mar 31, 2020 at 20:57
• Express $x$, $y$, and $z$ in terms of $r$, $\theta$, and $\phi$. Assume $\rho$ is only a function of $r$, and do the angular integrals. Mar 31, 2020 at 21:25
• Ok got it thank you Mar 31, 2020 at 21:50