Induced Currents and Magnetic Forces

An induced EMF is fixed by $-N\,d\Phi_B/dt$ and by nothing else; the circuit then sets the current through $I = \varepsilon/R$, so an open loop carries a full EMF across its gap with zero current. The polarity of a motional EMF follows from $q\vec{v}\times\vec{B}$ acting on the carriers inside the moving conductor. The magnetic force on whatever current results always opposes the relative motion, since the $I^2R$ heat has to be paid for by the agent doing the moving, and for a bar on rails the books balance exactly: $Fv = B^2L^2v^2/R = I^2R$.

The errors cluster around the two-stage structure. Letting resistance shrink the EMF instead of the current, or declaring that a cut loop has no induction. Guessing which end of a moving rod goes positive, or running the cross product with the rod's own direction, or with $\vec{B}\times\vec{v}$, which names the wrong end every time. Predicting an induced force that assists the motion, which would build a free-energy machine. Drawing a single one-way pulse for a magnet that passes all the way through a coil, when the trace has two opposite lobes. And expecting a steady primary current to drive a steady secondary current, which is exactly what transformers cannot do.