(a) Show that the quadrupole term in the multipole expansion can be written as

${\mathit{V}}_{\mathbf{q}\mathbf{u}\mathbf{a}\mathbf{d}}\mathbf{\left(}\overrightarrow{\mathbf{r}}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}\mathbf{\pi }{\mathbf{\epsilon }}_{\mathbf{0}}}\frac{\mathbf{1}}{{\mathbf{r}}^{\mathbf{3}}}\mathbf{\sum }_{\mathbf{i}\mathbf{,}\mathbf{j}\mathbf{-}\mathbf{1}}^{\mathbf{3}}\widehat{{\mathbf{r}}_{\mathbf{i}}}\widehat{{\mathbf{r}}_{\mathbf{j}}}{\mathit{Q}}_{\mathbf{i}\mathbf{j}}$ ............(1)

(in the notation of Eq. 1.31) where

${\mathit{Q}}_{\mathbf{ij}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\int }\mathbf{[}\mathbf{3}\mathbf{r}{\mathbf{\text{'}}}_{\mathbf{j}}\mathbf{r}{\mathbf{\text{'}}}_{\mathbf{j}}\mathbf{-}{\mathbf{(}\mathbf{r}\mathbf{\text{'}}\mathbf{)}}^{\mathbf{2}}{\mathbf{\delta }}_{\mathbf{ij}}\mathbf{]}\mathit{\rho }\mathbf{\left(}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{r}\mathbf{\text{'}}}\mathbf{\right)}\mathit{d}\mathit{\tau }\mathbf{\text{'}}$ ..........(2)

Here

${\mathit{\delta }}_{\mathbf{i}\mathbf{j}}\mathbf{=}\{\begin{array}{l}1\\ 0\end{array}\begin{array}{cccc}\mathbf{i}\mathbf{f}& \mathbf{i}& \mathbf{=}& \mathbf{j}\\ \mathbf{i}\mathbf{f}& \mathbf{i}& \mathbf{\ne }& \mathbf{j}\end{array}$ ..........(3)

is the Kronecker Delta and $Q_{ij}$is the quadrupole moment of the charge distribution. Notice the hierarchy

${\mathit{V}}_{\mathbf{mon}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}{\mathbf{\pi \epsilon }}_{\mathbf{0}}}\frac{\mathbf{Q}}{\mathbf{r}}\mathbf{}\mathbf{;}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}{\mathit{V}}_{\mathbf{dip}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}{\mathbf{\pi \epsilon }}_{\mathbf{0}}}\mathbf{\sum }_{\mathbf{}}\frac{\widehat{{\mathbf{r}}_{\mathbf{j}}{\mathbf{p}}_{\mathbf{j}}}}{{\mathbf{r}}^{\mathbf{2}}}\mathbf{}\mathbf{;}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}{\mathit{V}}_{\mathbf{quad}}\mathbf{\left(}\hat{\mathbf{r}}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}{\mathbf{\pi \epsilon }}_{\mathbf{0}}}\frac{\mathbf{1}}{\mathbf{r}{\mathbf{3}}^{}}\mathbf{\sum }_{\mathbf{i}\mathbf{}\mathbf{j}\mathbf{-}\mathbf{1}}^{\mathbf{3}}\hat{{\mathbf{r}}_{\mathbf{i}}\widehat{{\mathbf{r}}_{\mathbf{j}}}}\mathit{Q}\mathit{i}\mathit{j}\mathbf{}\mathbf{}\mathbf{}\mathbf{;}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}$

The monopole moment (Q) is a scalar, the dipole moment $\left(\stackrel{\rightharpoonup }{p}\right)$ is a vector, the quadrupole moment $\left(Q_{ij}\right)$ is a second rank tensor, and so on.

(b) Find all nine components$Q_{ij}$ of for the configuration given in Fig. 3.30 (assume the square has side and lies in the x-y plane, centered at the origin).

(c) Show that the quadrupole moment is independent of origin if the monopole and

dipole moments both vanish. (This works all the way up the hierarchy-the

lowest nonzero multipole moment is always independent of origin.)

(d) How would you define the octopole moment? Express the octopole term in the multipole expansion in terms of the octopole moment.

Step by step solution

01

Given data

The quadrupole moment can be written as

.

The Kronecker Delta function can be defined as

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!