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:(adsbygoogle = window.adsbygoogle || []).push({});

[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{$\scriptscriptstyle\rightharpoonup$}}

\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{$\scriptscriptstyle\rightharpoonup$}}

\over x} - \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}

\over x} '|^3 }} \approx \frac{1}{{|\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}

\over x} |^3 }} + \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}

\over x} ' \cdot [\nabla '(\frac{1}{{|\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}

\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

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# Jefimenko B-field approximation

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