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If we establish a changing magnetic field in a region of space, circular electric field lines are induced. This electric field extends even after we exit the region of the magnetic field. If a closed path lies outside the magnetic field region, it experiences zero emf ( $\int \vec E \cdot d \vec s = -\frac{d \phi}{dt}=0$). (According to Halliday Resnick Krane vol 2)

My question is, why does the integral cancel out to zero if loop is outside the magnetic field but is non-zero inside where also the electric field lines are circular (geometry is same)? The only difference is the radial dependence of induced electric field ($\vec E \propto r$ inside and $E \propto 1/r$ outside). Is there an intuitive explanation for this?

image of induced electric field from changing magnetic field. Reference image for above text

Note: kindly use integral equations instead of differential ones in the explanation. Both would be valid, but as a high school student I don't understand them well.

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    $\begingroup$ I'm not entirely clear on where this loop is when you say it's outside the region of the magnetic field, are you saying the material of the loop is in empty space but the magnetic field is still in the loop or that the loop is physically outside the region and the magnetic field does not go through the loop? $\endgroup$
    – Triatticus
    Dec 28, 2021 at 21:26
  • $\begingroup$ not only that, the real puzzle is that you can have induced electric field even where $B=\nabla\cdot A=0$ but $ \dot A \ne 0$ $\endgroup$
    – hyportnex
    Dec 28, 2021 at 22:18
  • $\begingroup$ @Triatticus I meant physically outside the definite region where where the magnetic field acts. So, let's say from an arbitrary point there is a magnetic field till a distance of R radially. Then neither the loop nor its corresponding area will be within that region. That's why I said $d \phi / dt = 0$. $\endgroup$
    – Dodo
    Dec 29, 2021 at 10:36
  • $\begingroup$ @hyportnex I apologise but I couldn't find the formula $B=\nabla\cdot A$ on the internet. Could you share what this about? (though I did find this: en.wikipedia.org/wiki/Magnetic_vector_potential) $\endgroup$
    – Dodo
    Dec 29, 2021 at 10:51
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    $\begingroup$ apologies, not $\cdot$ but $\times$ as in $\nabla \times A$; ( $A$ is the so-called vector potential whose curl is $B$) $\endgroup$
    – hyportnex
    Dec 29, 2021 at 15:32

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I believe you are describing the Aharonov–Bohm effect total wrong in reverse.

$E_{ind}$ in your illustration is actually the $A$ magnetic vector potential:

Aharonov-Bohm effect

image source (modified): https://www.youtube.com/watch?v=1P68eba7zEs

With $B=0$ outside you still get a non-zero induced electric field $E_{in}$ (i.e. Induced Electric Vectors in your illustration are actually the magnetic vector potentials $A$).

It is an quantum interference effect. No one knows the physical explanation of this? Electric charges can somehow "smell" a totally confined and isolated changing magnetic field at a distance.

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  • $\begingroup$ Thanks for the concise answer. So what you are saying is magnetic vector potential is a more fundamental way of describing induced electric fields? $\endgroup$
    – Dodo
    Feb 15 at 17:37
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    $\begingroup$ Yes, induced by changing magnetic fields even if they are out of reach. Magnetic vector potentials are also a convenient mathematical way to describe magnetic fields in general. $\endgroup$
    – Markoul11
    Feb 15 at 17:40
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    $\begingroup$ Potentials in general are regarded as non-physical and are man made mathematical concepts to aid our analysis of physical phenomena. However, the Aharonov–Bohm effect teach us that behind every potential mathematical quantity there is an actual hidden physical phenomenon taking place causing it. $\endgroup$
    – Markoul11
    Feb 15 at 17:51
  • $\begingroup$ youtube.com/watch?v=YJGOhl8iK3o by Yakir Aharonov himself. $\endgroup$
    – Markoul11
    Feb 15 at 19:09

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