Component of the quadrupole Q_ij

[ellocomateo]

Hello,

I do not manage to visualize the link between the component of the quadrupole Q_ij and the spatial distribution of the electric quadrupole field.

I was told to imagine the Q_ij as an ellipsoid, which I understand (the ellipsoid "radius" in a given direction being the strength of the quadrupole along this direction). Yet what is the link between the Q_ij and the usual representation in Slide 12 of this file?:
cems.uvm.edu/~oughstun/LectureNotes141/Topic_09%20%28ElectrostaticMultipoles%29.pdf

In particular, I want to find out when does the gradient \nabla_k Q_ij equal zero? When i,j =! k ?

Pleeeease, help!

 

[Bob S]

The azimuthal gradient of V(r,θ,\varphi) in slide 11 is proportional to \frac{d}{d\theta}\left(3\cos^2\theta-1 \right)=6\sin\theta\cos\theta=3\sin\left(2\theta \right)

 

[ellocomateo]

Thank you for commenting, but how does this translate to the i and j ?

 

[Bob S]

Thank you for commenting, but how does this translate to the i and j ?

Perhaps you and I are looking at different slides and/or files. I am looking at slide 12 of the file
www.cems.uvm.edu/~oughstun/LectureNotes141/Topic_09%20%28ElectrostaticMultipoles%29.pdf
which is a plot of the equipotential lines of V(r,θ,φ) of a linear electric quadrupole.

 

[ellocomateo]

We are looking on the same graph, but I still do not see how I should label the axes.
There is no information whatsoever on this point. Or at least I do not see it.

 

[Bob S]

The plot is a combination of the gradient and the equipotential lines of V(r,θ,φ) using r and z as axes. The four-fold symmetry indicates it is a quadrupole field.