How does EM induction and the motor effect work?

How does EM induction and the motor effect work?

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(1) How is the Lorentz force law derived? (2) For the motor effect, is a fair explanation that the moving electrons experience a force downwards due to a magnetic force, and push the wire down with them due to EM repulsion between them and the metal atoms? (3) For EM induction, why is there a dB/dt term if there is not in the Lorentz force law? I am confused intuitively about it; I understand what happens, but not why above the level of rote.

induced emf was first suggested by Michel Faraday and by applying circuit for close loop using inductor and applying dc or ac voltage we can measure induced emf

lorentz force law is \[F=qE+q(v \times B)\]

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