Quadrupole Potential: Help Derive and Decipher This Beast!

  • Thread starter UAR
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  • #1
UAR
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Main Question or Discussion Point

[tex](\textbf{q}\bullet\nabla)(\textbf{p}\bullet\nabla)\frac{1}{r} = -(\textbf{p}\bullet\textbf{q})\frac{1}{r^{3}} + 3(\textbf{p}\bullet\textbf{r})(\textbf{q}\bullet\textbf{r})\frac{1}{r^{5}}[/tex]

r is distance between field point and dipole source, [tex]\textbf{p}[/tex] is dipole moment, and I believe [tex]\textbf{q}[/tex] may be quadrupole moment tensor (what is that anyways?),

How is the above equation derived ? and exactly how is it related to the more physically and mathematically lucid dipole potential below:

[tex](\textbf{p}\bullet\nabla)\frac{1}{r} = -(\textbf{p}\bullet\textbf{r})\frac{1}{r^{3}}[/tex]
 
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Answers and Replies

  • #2
tiny-tim
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Welcome to PF!

Hi UAR! Welcome to PF! :smile:

(use \cdot instead of \bullet :wink:)
[tex](\textbf{q}\bullet\nabla)(\textbf{p}\bullet\nabla)\frac{1}{r} = -(\textbf{p}\bullet\textbf{q})\frac{1}{r^{3}} + 3(\textbf{p}\bullet\textbf{r})(\textbf{q}\bullet\textbf{r})\frac{1}{r^{5}}[/tex]

r is distance between field point and dipole source, [tex]\textbf{p}[/tex] is dipole moment, and I believe [tex]\textbf{q}[/tex] may be quadrupole moment tensor (what is that anyways?),

How is the above equation derived ? and exactly how is it related to the more physically and mathematically lucid dipole potential below:

[tex](\textbf{p}\bullet\nabla)\frac{1}{r} = -(\textbf{p}\bullet\textbf{r})\frac{1}{r^{3}}[/tex]
It's derived from that equation simply by using it twice, first with p· and then with q·

(and because (q·)(p·r) = p·q :wink:)
 
  • #3
UAR
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Thanks Tiny-tim!

However, while you are still online, excuse my slowness: why is:

[tex](\textbf{q}\cdot\nabla)(\textbf{p}\cdot\textbf{r})=\textbf{p}\cdot\textbf{q}[/tex] ?
 
  • #4
tiny-tim
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why is:

[tex](\textbf{q}\cdot\nabla)(\textbf{p}\cdot\textbf{r})=\textbf{p}\cdot\textbf{q}[/tex] ?
Because p.r = xpx + ypy + zpz,

so (q·∇)(p.r) = … ? :smile:
 
  • #5
UAR
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Aaah! Thanks! I was hesitant to do that due to a (poor notation)-induced irrational fear that [tex]p_{x'}[/tex] was a function of [tex]x[/tex]. But now I see it is not, since [tex]\nabla[/tex] is w.r.t field point [tex]\textbf{r}[/tex], while [tex]\textbf{p}[/tex] depends only on source pts [tex]\textbf{r'}[/tex].

One more question: What is [tex]\textbf{q}[/tex] ?

By the way, you are truly a good mentor. Thanks for your help and keep up the great work!!!
 
  • #6
tiny-tim
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One more question: What is [tex]\textbf{q}[/tex] ?
No idea :rolleyes: … it could be anything, and the equation would still work :wink:
 
  • #7
UAR
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Hi Tiny Tim,

Thanks for your help and for your honesty.

Anyone else care to help physically and mathematically elucidate [tex]\textbf{q}[/tex] in the quadrupole potential equation above. I read somewhere that it is called (or is related to ?) the "quadrupole moment tensor" (what exactly is that by the way?).

Thanks!!!
 

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