Quadrupole Potential: Help Derive and Decipher This Beast

[UAR]

[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]

 

[tiny-tim]

Welcome to PF!

Hi UAR! Welcome to PF!

(use \cdot instead of \bullet )

[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 )

 

[UAR]

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] ?

 

[tiny-tim]

why is:

[tex](\textbf{q}\cdot\nabla)(\textbf{p}\cdot\textbf{r})=\textbf{p}\cdot\textbf{q}[/tex] ?

Because p.r = xp_x + yp_y + zp_z,

so (q·∇)(p.r) = … ?

 

[UAR]

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!

 

[tiny-tim]

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

No idea … it could be anything, and the equation would still work

 

[UAR]

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!