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How does EM induction and the motor effect work?

Physics
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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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Other answers:

here q= charge F=force and its a vector E=electric field and its a vector v=velocity of charge q and its a vector B= magnetic field and its vector
if in any place both electric field E and magnetic field B present then this lorentz force law will b applied
Lorentz force law can b writen as\[F=q(E+v \times B)\]
ur question 2 is not clear to me
now i wanna talk abt ur question 3
That's not the derivation, it's just a statement of the law.
ya

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