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Correct the loop-source prefactor and regenerate its figures - #21

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Dev_Chen
Oct 6, 2026
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ARS183 merged 2 commits into
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Dev_Chen

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@ARS183 ARS183 commented Oct 6, 2026 •

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Two corrections to the loop-source section of 3_diffusion_fields/em_diffusion.md, both found by checking the chapter against the MATLAB scripts that produced its figures.

The prefactor of the loop-source electric field. eq:Exloop and eq:Eyloop carried $\mu/(8\sqrt{\pi},t)$; the correct value is $\mu/(8\pi t)$, and the pre-integration form was missing a $4\pi$ inside its square root.

The two loop figures. The $(x,z)$ panels plotted $-E_y$: the script computed the field with the two Gaussians in the reverse order from eq:Eyloop, while the $(x,y)$ block of the same file used the order the equation has.

The $(x,y)$ panels plotted $E_x^2+E_y^2$ under a colour axis labelled V/m and a caption naming $E_y$. They are regenerated as $|\boldsymbol E|$, which is what the body text describes, and the caption now names the magnitude.

That last point had a consequence in the prose. The text read that the field "has decreased by six orders of magnitude, which seems to suggest that the decay is faster in the $(x,y)$-plane for fixed depth than in the $(x,z)$-plane", and explained the difference by the horizontal distances. The six orders were three orders squared. With the panels showing $|\boldsymbol E|$, both pairs drop by the same three orders, and the two sentences are replaced by the reason the rates agree: the field falls off as $t^{-5/2}$ at every position once the diffusion distance exceeds the loop size. Measured $\mathrm{d}\log|\boldsymbol E|/\mathrm{d}\log t$ is $-2.435$ at $z=0.5$ m and $-2.434$ at $z=10$ m.

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ARS183 merged commit ec6db3b into main Oct 6, 2026
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