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induced emf = - d(flux)/dt

If this is applied to a loop where induced emf causes currents, and thus flux itself, do we have to consider that flux (of course we don't if it's constant)?

If the external flux has a nonzero second derivative, then the induced emf is changing with time, thus the induced flux has a nonzero first derivative. Will this varying induced flux need to be considered when applying Faraday's law?