# EMF generated by moving square

I have been studying electromagnetic induction and I have trouble understanding some things. Say we have a square wire going through a magnetic field like in the picture (height is "h" and total length in a magnetic field is "x"). If we moved that whole conductor to the right because of the magnetic field going into the screen (j) the left-most part of the wire would experience an upwards force and therefore generate movement of electrons in that direction. That's easy enough to calculate F = q * v X B. And generated EMF is now ε = vBh.

Now if we wanted to get the same answer using flux which is Φ = ∫B * da:

Φ = B * h * x so a change in flux is Φ = Bh * dx/dt = Bh * (-v) which makes ε = Bhv, precisely what we got the first time.

My questions is: If we bring in a new magnetic field that is constant everywhere wouldn't the change in flux be 0 which would mean induced EMF is 0 but if we used the first method to calculate EMF nothing would change and EMF would still be ε = vBh.