Induced EMF in a small volume of a conductor?

[PhiowPhi]

Assume we have the following copper slab, with volume $V$, and introduced a magnetic field covering a small area of it like so:

There is change in magnetic flux($\phi$) due to the magnetic field increasing/decreasing, and via Faraday & Len'z law of electromagnetism, there is an induced EMF($- \epsilon$) to oppose that change, and with the conductor's resistance($r$) an induced current($I$).

One thing that I'm struggling with, is knowing/understating where the induced EMF is bound to? Is the induced EMF all around the conductor like so:

Where if I placed a volt-meter anywhere on the top part (A) and the bottom part, the voltage would be the same any point there?
Or , is the induced EMF limited to the area the magnetic field interact's with like so:

The blue rectangle represents the magnetic field, and r, the resistance of the whole slab.
Likewise, will induced current be focused to that area the magnetic field interacts with, or the entire slab?

One thing I'd like to add, if we introduced and exterior Voltage source to the slab, with the induced EMF, would it be in series?

The total voltage if they we're in a circuit, and they are connected to one another the net-voltage in the circuit would be:

$V_n$ = $V$ - $\epsilon$ ?

 

[Hesch]

Assume we have the following copper slab, with volume VV, and introduced a magnetic field covering a small area of it like so:

If I understand you correct, that some magnetic changing flux, repesented by XXXXXXX, is induced perpendicular through the slab, you will get a circulating emf around the flux:

∫_circulationemf ⋅ ds = dΨ/dt , Ψ is the flux , s is a surrounding path ( the edge of the blue rectangle ).

You will get Eddy-voltage and Eddy-current.

 

[PhiowPhi]

You will get Eddy-voltage and Eddy-current.

If connected to a circuit it would be Eddy voltage & induced voltage(one via circuit and one via the surface of the conductor)?
Also, if there is current running through the conductor from a source(V+) it would be in series and oppose the applied voltage as diagramed in the last image?