# Subarea within a changing magnetic flux?

If I were to introduce a boundary area $$\tau$$:

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).

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:

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

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?