In the classic problem of the induced EMF in a moving conductor (as in the picture above), the calculation for the induced EMF is as follows
The derivation assumes that the magnetic field B is constant throughout the entire time the flux is changing. I am confused because it does not seem like the magnetic field is constant because while there is an induced EMF in the wire, there must also be self induction that occurs (since the induced current in the coil also produces a magnetic field) thereby changing the already present magnetic field. So in fact the magnetic field is a function of time inside the coil of wire. Is the self induction so insignificant that the calculation can assume that the magnetic field does not change?