Can Faraday's law be deduced from Lorentz force? Consider a rectangular circuit with one side being a rod of length $L$, which is free to move, and a constant magnetic field perpendicular to it. The rod moves at a constant velocity $v$, expanding the area of the circuit (this system). Let there be some resistor $R$ in the circuit.

One can use Faraday's law to calculate the current $I$ from the E.M.F induced by the changing magnetic flux. But the same result follows from Lorentz force considerations, ignoring Faraday:

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*In a stationary configuration (assuming constant current), the total force on the charge carriers in the rod, in the direction along its axis, must vanish. This gives $vB=E$.


*We deduce that the electric field is constant along the rod, and so there is a voltage $\mathcal E=EL=BLv$. We can calculate the current $I$ and get the same result as we would have gotten using Faraday's law.



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*Does this mean that there is some way to deduce Faraday's law from the other Maxwell equations plus the Lorentz force? If so, how can we show this generally?


*Or maybe this can be done only when there are no time-varying magnetic fields?
 A: I've done the math, and will answer my own question.
It is important to distinguish between Faraday's law of induction, $\mathcal E=- \frac{d\phi_B}{dt}$ and the Maxwell–Faraday equation, $\nabla \times E=-\frac{\partial B}{\partial t}$. The latter is not a consequence of the Lorentz force, while the former is, in some cases.
The change in magnetic flux can be due to a time-varying magnetic field, or due to a time-varying area (as in the example cited in the question).
In the latter case (e.g the moving rod), the Maxwell–Faraday equation is irrelevant, as $\frac{\partial B}{\partial t}=0$. We can, however, use Lorentz-force considerations to deduce Faraday's law of induction for these cases. However, we cannot derive Faraday's law of induction as a general result using only Lorentz-force considerations. If we consider a static loop of wire, in the background of a time-varying magnetic field, Lorentz-force considerations do not imply a current. The Maxwell–Faraday equation is a fundamentally new result.
For more information on the mathematical derivation, see here.
