Struggling to understand what Faraday's law says

[Johnls]

Faraday's law talks about the change in the flux through a loop, but I never see how people take into account the change in the flux due to Faraday's law itself and I am wondering if that leads to a contradiction.

For example when calculating the current through a physical loop with area A and resistance R in a uniform field that changes with time I see people do this:

flux = B*A (lets assume that the area vetor is parallel to the field) emf = A* d B dt
I = emf / R = (A/R) * d B dt

but this current immediately (at least in our context) produces a magnetic field, so the flux is not B*A, and because of this we should get a different current in the first place.

So what is going on here?

 

[albertrichardf]

It is actually true that the induced current produces its own magnetic field and thus changes the magnetic flux. In considerations of Faraday's law, however, the current is usually assumed to be small enough so as to neglect the magnetic field it produces, so any change in flux that the current causes is also ignored.
In reality, an induced current is not stable from the get-go. The current is high at first but it gradually decreases due to its own magnetic field, and eventually a balance is reached and the current is stable. This is a phenomenon known as self-inductance.

 

[Delta2]

Just to add a general advice, when studying physics we often neglect the minor effects (though what is minor and what is major might not be so clear at some times), in order to simplify the study of the major effects. In your case minor effect is the magnetic field produced by the induced current while major effect is the time varying flux of the initial field. In your case if we want to take into account the magnetic field from the induced current, that is the self induction effect, we should add a term $L\frac{dI}{dt}$ to the equation you wrote so that it will become

$L\frac{dI}{dt}+IR=A\frac{dB}{dt}$ where L is the self induction coefficient of the loop. (L is going to be pretty small number for 1 single loop, that's the reason we can neglect it without affecting much our study).

Of course this makes the study a bit more complex since we now have to study a differential equation instead of the simpler algebraic equation you initially wrote in your post.