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Jefimenko Bfield approximation 
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#1
Feb2610, 01:42 AM

PF Gold
P: 7,120

I'm looking to evaluate the magnetic field using Jefimenko's equations. There is two parts to it but I'm just looking at the first. The approximation is r>>r' where r' is localized about the origin. The Jefimenko's equation for the magnetic field (the first term that I'm having trouble with) has:
[tex]B(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$ }} \over x} ,t) = \frac{{\mu _0 }}{{4\pi }}\int\limits_v {d^3 x'\{ [J(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$ }} \over x}' ,t')]_{ret} \times \frac{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpo onup$}} \over R} }}{{R^3 }}} \} [/tex] [tex]R = \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$} } \over x}  \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over x} '[/tex] using the taylor expansion: [tex]\frac{1}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\righth arpoonup$}} \over x}  \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over x} '^3 }} \approx \frac{1}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\righth arpoonup$}} \over x} ^3 }} + \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over x} ' \cdot [\nabla '(\frac{1}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\righ tharpoonup$}} \over x}  \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over x} '^3 }})]_{\vec x' = 0} [/tex] Now, 4 terms are generated. One will go away, one is easily solved, but what I'm having trouble with are terms like this: [tex]\frac{{  \mu _0 }}{{4\pi }}\int\limits_v {(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$ }} \over x} ' \cdot [\nabla '(\frac{1}{{R^3 }})]_{\vec x' = 0} )\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$} } \over x} ' \times [J(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$ }} \over x} ,t')]_{ret} d^3 x} [/tex] I want to pull the derivative operator off the 1/R^3 through an integration by parts but since it's evaluated at r'=0, I'm not exactly confident on how to do that. I want the derivative operator on things such as [tex]\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over x} ' \times [J(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$ }} \over x} ',t')]_{ret} [/tex] which are somewhat like the magnetic moment once integrated but wonder if I still evaluate the original part at r'=0 or does the evaluation switch over to the part I'm now using the derivative operator on? Or does it go onto both now? Signed, Confused in Antarctica 


#2
Feb2610, 10:37 AM

HW Helper
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#3
Feb2610, 10:45 AM

HW Helper
P: 5,003

Also, you seem to be missing a term involving [tex]\mathbf{\dot{J}}(\textbf{x}',t_r)[/tex] in your original equation.



#4
Feb2610, 02:58 PM

PF Gold
P: 7,120

Jefimenko Bfield approximation
Yes, I left out that term becuase I haven't looked at it yet, I was just wanting to post what part I was having problem with. After talking to a few people we came to the same conclusion however, doing the evaulation makes it impossible to pull off the gradiant I guess. I found out the hopefuly correct way of doing it and I shall give it a shot.



#5
Feb2810, 11:36 PM

PF Gold
P: 7,120

So now I'm stuck with doing these 3dimensional integration by parts. I need to work with this integral. I'm trying to do it by integration by parts but since it's in 3 dimensions, I'm not sure how it's done. I'm quite amazed that I don't think I've ever run across a 3dimensional IVP that didnt use Gauss' law or anything. The steps so far are:
[tex]\begin{array}{l} \int_V {(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$ }} \over x} \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over x} ')(} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over x} ' \times \left. {[J(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$ }} \over x} ',t')]} \right_{ret} )d^3 x' \\ u = (\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$} } \over x} \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over x} ') \\ dw = \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over x} \times \left. {[J(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$ }} \over x} ',t')]} \right_{ret} )d^3 x' \\ w = 2\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$} } \over m} \\ du = ? \\ \end{array}[/tex] Now I figure I can't simply naively say that [tex]du = \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over x} d^3 x'[/tex] because then I have a vector multiplying a vector. Doing the actual computation seems to give me: [tex]du = xdx' + ydy' + zdz'[/tex] and now I have just a sum of 3 integrations. Is this all correct mathematically? It's worrisome because the integral over prime coordinates diverge since the integration is over an arbitrary volume. 


#6
Mar110, 11:40 AM

HW Helper
P: 5,003

In one dimesion, integration by parts is derived from the product rule:
[tex](fg)'=f'g+fg'\implies \int f'g dx=\int(fg)'dx\int fg' dx= fg\int fg' dx[/tex] In 3 dimensions, things are not so simple; there are actually 8 product rules and each leads to a different variation of integration by parts. So, in order to use IBP in vector calculus, you need to select an appropriate product rule. For example, if I wanted to calculate [itex]\int f(\textbf{r})(\mathbf{\nabla}g(\textbf{r}))\cdot d\textbf{r}[/itex] over some curve, I might find it useful to use the product rule [itex]\mathbf{\nabla}(fg)=(\mathbf{\nabla}f)g+f(\mathbf{\nabla}g)[/itex] to transfer the derivative to [itex]g[/itex] instead. 


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