electromagnetic induction and magnetic shielding

In the figure I have a circular conducting wire. Somehow, in the middle circular region I have a magnetic field (this means the magnetic field is shielded in this region and it is possible from what I have read in the wiki page).

Now if I vary the magnetic field in this region, will that induce a current in the wire through Faraday's law? If so, how?

If I assume that the circular shaped wire has the same diameter as the galaxy. How will the electrons in the conductor know that they should move because the magnetic field has changed? EDIT:i saw a similar question in Griffiths example 7.8

According to Maxwell's equations $$\textrm{curl}\,\textbf{E} = -\frac{\partial}{\partial t}\textbf{B}$$ therefore a variation of the magnetic field in time generates a non-zero curl for the electric field, whose solution, together with the other set $$\textrm{div}\,\textbf{E} = \frac{\rho}{\epsilon_0}$$ describes the electric fields at any point $(x,y,z,t)$. Once so, electrons (or any other charge carriers) will feel the electric field and move accordingly because they have charges. Everything that has a charge feels the electromagnetic field and moves according to $$m\textbf{a} = q (\textbf{E} + \textbf{v}\times\textbf{B})$$ where the fields on the right hand side are solutions of the Maxwell equations. Notice that, given any point in space-time $(\textbf{r},t)$, the solutions of the Maxwell's equations behaves as advanced or retarded function of the form $f(|\textbf{r}| - ct)$, meaning that it takes a finite amount of time for the wave to reach a point in space since it moves at finite velocity $c$.