The problem is with energy of electromagnetic field(adsbygoogle = window.adsbygoogle || []).push({});

[tex]-\frac{d}{dt}\int_V\frac{1}{2}(\vec{E}\cdot\hat{\epsilon}\vec{E}+\vec{B}\hat{\mu^{-1}}\vec{B})dV=\oint(\vec{E}\times\vec{H})\cdot d\vec{S}+\int_V\vec{j}\cdot\vec{E}dV[/tex]

I have this relation

[tex]\hat{\epsilon}, \hat{\mu}^{-1}[/tex] are symmetric tensors. Now we looktotal field. This is or finite area in which bounaries electric and magnetic field are equal to zero, or whole space with a condition that electric and magnetic field goes to zero at least as [tex]\frac{1}{r^2}[/tex] in infinity.

So electric and magnetic field must be functions of [tex]\frac{1}{r^{2+\epsilon}}[/tex] where [tex]\epsilon \geq 0[/tex]

Why this condition must be satisfied?

In first case of finite area [tex]\oint(\vec{E}\times\vec{H})\cdot d\vec{S}\equiv 0[/tex].Why is that? I don't understand?

And in second case with take sphere infinitely long away and have

[tex]lim_{S \rightarrow \infty}\oint(\vec{E}\times\vec{H})\cdot d\vec{S}=lim_{S\rightarrow \infty} [\overline{(\vec{E}\times\vec{H})}_n 4\pi r^2]=0[/tex]

because [tex]\overline{(\vec{E}\times\vec{H})}_n[/tex] goes to zero at least as [tex]\frac{1}{r^4}[/tex].

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# One hard theoretical question

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