Induction in an inflating loop in constant B?

[Lojzek]

A loop is made from a conductive wire. The wire moves, so the area inside the loop is time dependent: S=S(t)
There is a constant homogeneus magnetic field B directed perpendicular to the wire and we are supposed to calculate induced voltage.

In my opinion there is no electric field and no voltage, since there field B is
constant. However there is a magnetic force experienced by charges moving in magnetic field:

Method 1:

F=e*B*v

If this force is integrated over the loop to gain work on a charge e after 1 circle, we get:

A=-e*B*dS/dt


The proposed solution used Faraday's law of induction:

Method 2:

U=-dfi/dt=-d(B*S)/dt=-B*dS/dt

I think that this is a misuse of the law, since corresponding Maxwell's equation can be
used only for fixed loop, but changing magnetic field. However the work gained by a charge completing one circle is exactly the same as with previous method:

A=e*U=-e*B*dS/dt

My question is:

Is the method 2 really incorrect? If yes, why is the work the same? If no, how do we prove that Faraday's law can be used in case of constant B and changing loop area?

 

[Meir Achuz]

Faraday's works for any change in the flux.
For constant B and changing area, it is often derived for a rectangle in elementary texts.
It is derived for an arbitrary change in the area in more advanced texts.

 

[Lojzek]

I found some related texts and I hope I understand the problem now: it seems that
in case of the moving loop the "induced voltage" is not an integral of a real electric field, but an integral of a fictional electric field, that would do the same work on a circling charge.