electromagnetism - Why the induced field is ignored in Faraday's law?

Suppose we have a conducting ring in a constant magnetic field $\vec{B}$. Suppose that the ring is being deformed. We know from Faraday's law that such an action will induce a current in the loop. Because of this currect a magnetic field $\vec{B}_{\text{induced}}$ would appear, which will weaken the net magnetic field (Lenz's law). So in total, the magnetic field would change ($\vec{B}+\vec{B}_{\text{induced}}$). However in every texbook there is an assumption that $B$ is always constant, that is: $\mathcal{E}=-\mathrm{d} \Phi/\mathrm{d} t=B(\mathrm{d} A/\mathrm{d} t)$, even though there are induced fields which change the magnetic field. Why do we ignore the induced fields in Faraday's law?