Jackson("Classical Electrodynamics", Ch.6)(adsbygoogle = window.adsbygoogle || []).push({});

uses the theorem of curl of curl to separate current density into transverse and parallel,

[tex]\vec J = \vec{J_p}+\vec{J_t}[/tex] to say,

[tex]\begin{align*}\vec{J}(\vec{x}) &= \int\vec{J}(\vec{x'})\delta(\vec{x}-\vec{x'})d^{3}x'\\

&= -{1\over{4\pi}}\int\vec{J}(\vec{x'})\nabla^2 \left({1\over|\vec{x}-\vec{x'}|}\right)d^{3}x'

\end{align*}[/tex]

Since the del is about [tex]x[/tex] and independent of the integral variable,

[tex]\begin{align*}{}&=-{1\over{4\pi}}\nabla^2\int{\vec{J}(\vec{x'})

\over|\vec{x}-\vec{x'}|}d^{3}x'

\end{align*}[/tex]

And using the theorem

[tex]\nabla\times(\nabla\times\vec{A})=\nabla(\nabla\cdot\vec{A})-\nabla^2\vec{A}[/tex]

[tex]\begin{align*}\vec{J}(\vec{x}) &=

{1\over{4\pi}}\nabla\times\nabla\times\int{\vec{J}(\vec{x'})

\over|\vec{x}-\vec{x'}|}d^{3}x'-{1\over{4\pi}}\nabla\left(\nabla\cdot\int{\vec{J}(\vec{x'})

\over|\vec{x}-\vec{x'}|}d^{3}x'\right)\end{align*}[/tex]

But here Jackson take some hidden procedure to get from the second term of ther right side

[tex]-{1\over{4\pi}}\nabla\left(\int{\nabla'\cdot\vec{J}(\vec{x'})

\over|\vec{x}-\vec{x'}|}d^{3}x'\right)={1\over{4\pi}}\nabla\left(\int{\partial\rho(\vec{x'})/\partial t

\over|\vec{x}-\vec{x'}|}d^{3}x'\right)[/tex]

to use the continuity theorem to get the term about a time derivative of charge density at [tex]x'[/tex].

And I cannot see how is the differential about [tex]x[/tex] changed into a differential about [tex]x'[/tex] and got inside the integral, and is only applied to the current density, but not the denominator.

Can somebody explain it for me? Thank you.

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# Homework Help: Current density and theorem of curl of curl

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