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If I were to introduce a boundary area $\tau$: enter image description here

And after sometime $t$, I introduced a constant magnetic field(let's imagine it spawned suddenly and ignored the change in flux from $t_o$ $\rightarrow$ $t$ for the sake of simplicity). enter image description here

I define the magnetic flux $\Phi_\tau$ w.r.t the boundary area. The magnetic field would change: decrease in strength, and based on Maxwell-Faraday's law I know there is an electric field that would curl within the boundary area:

$$ \nabla \times E_\tau = \frac{\partial B_\tau}{\partial t}$$

enter image description here

Can a subarea $\tau_{sub}$ exist within the boundary area $\tau$ and the same method above is applied?

Focusing on the magnetic field within that subarea, and the electric field($E_{sub}$) that curls around that region would this representation below be correct? If not, why wouldn't it be? enter image description here

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The following representation is correct, and i would be using the integral form of the maxwell law you have written to justify why is it so.

enter image description here

So what i wanted to show was that the boundary area is not really what matters for the induced electric field to be generated. The electric field is particular to the loop that you might choose to analyse.
There can be infinitely many such loops when such a time variable magnetic field is introduced over a conductor.

Under such circumstances a conductor will experience arbitrary amounts of heat generation due to the induced elecric fields which also are arbitrary on such a plane sheet.

So the smaller electric field loop inside the larger one will surely exist.

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