# Why can the induced magnetic field be ignored in this problem?

I am given the following conducting track, where on its right side there is a conducting rod with mass $$m$$ and length $$a$$ which is free to slide on it. There is a constant magnetic field throughout space $$\mathbf{B}=B_0\hat{z}$$ where $$\hat{z}$$ is perpendicular to the frame and goes outwards from the page. At time $$t=0$$ there is no charge on the capacitor and the rod is moving at a velocity $$v_0$$ to the right.

I am asked to find the induced EMF on the closed path and the force on the rod at time $$t$$ given that the velocity of the rod is $$v(t)$$ and the current through it is $$I(t)$$.

I'm having difficulty with this problem since the induced EMF that would result from the initial velocity of the rod would itself induce a magnetic field that will oppose the original magnetic field so as to oppose the change in the flux, and that induced magnetic field will, in turn, induce a correction to the EMF which will also induce a correction to the induced magnetic field and so on, seemingly ad infinitum. The magnitude of the force on the rod is given by $$F=aIB$$. My problem here is also that the induced magnetic field should affect the force acting on the rod.

Now, I checked the published answer to this problem and it seems that they completely neglected the effect of the induced magnetic field in this problem, both when finding the force acting on the rod and in finding the induced EMF. My question is, why is this valid?