we use faraday's law with flux defined across the surface of the disk which the loop encloses. This tells you emf between two adjacent points on the conducting loop, right?
EMF mentioned in the Faraday law is meant for a loop (close curve), not for two points. If the physical circuit is disconnected so the wire has two ends that are very close together, then the emf for the closed loop will be approximately equal to voltage across the two ends (exactly equal for perfect conductor), but EMF and voltage are two different concepts.
There are other situations for which I have seen this applied, such as a spinning conducting disk with a constant magnetic field applied parallel to the rotation axis, or a rod passing through a static magnetic field. In these cases, unless a specific intuition is made about choice of surface flux change seems to be 0.
In these cases, the traditional formulation of the Faraday law indeed seems inapplicable, because there is no obvious loop that emf would be defined for. But if some loop in space is chosen nevertheless (it does not have to be inside conducting circuit at all), then the law is applicable. There are two distinct cases.
1) if the loop chosen is not moving or changing in any way, then the Faraday law is valid always without further qualification; EMF for the loop is proportional to rate of change of magnetic flux through that loop;
2) if the loop in space is changing as time passes, then the Faraday law is not always applicable. It is applicable, however, in the special case where the moving part of the loop is fixed to material conductor (for example, a rectangle loop whose one side is always inside the moving rod). If the part of the loop inside conductor was moving with respect to the conductor, the corresponding magnetic force on test charges fixed to the loop would not be in any simple relation to the actual electromotive force acting on the real conducting charges.
So, to apply the Faraday law to loops that move or change in time, one usually chooses loop where the parts in non-conductive medium are stationary with respect to the observer, but the parts in conductor are stationary with respect to the conductor.
The simplest application of this is for a coil of wire that rotates in external magnetic field (as used in simple DC motors): the loop is chosen so that it always passes through the rotating coil, which means the loop rotates in space.
If we wanted to apply this special case of Faraday law to rotating disk, we have to choose a path that is stationary where it is in air and moving along with the conductor where it is in the conductor. One way to define such a loop is this. Imagine a rectangle whose one corner is in the disk center and another corner is on its rim. Then imagine all sides of the rectangle are fixed except the side defined by these two points; this side rotates around the disk center, along with the disk.
This kind of loop is obviously changing its shape in space, and the magnetic flux through this loop will change as well. Now, we apply the Faraday law to this loop.
