Faraday's law of electromagnetic induction reveals that the induction electromotive force generated in a metal coil is proportional to the change rate of the magnetic flux passing through the coil. Lorentz's magnetic field force theorem reveals that an electric charge moving in a magnetic field is affected by the Lorentz magnetic field force. Lorentz's magnetic field force theorem is the microscopic physical essence of the induction electromotive force. An induction electromotive force will be generated between the two ends of a metal wire moving in a magnetic field. In this study, calculation formulas of the electromotive force of metal wires were separately derived based on Faraday's law of electromagnetic induction and Lorentz's magnetic field force theorem. When a metal wire moves at a uniform speed in a magnetic field, the calculation formulas derived from both of them are the same. When a metal wire moves back and forth sinusoidally in a magnetic field, the electromotive forces of the wire derived from Faraday's law of electromagnetic induction and Lorentz's magnetic field force theorem are different. Lorentz's magnetic force theorem is a universal fundamental electromagnetic theorem. Therefore, Faraday's law of electromagnetic induction is an engineering approximation formula. This study proposes the electron motion resistance force theorem: When an electron moves in the metal wire, it will be affected by the motion resistance force, and the electron motion resistance force is proportional to the speed of the electron. An electric charge moving in a uniform magnetic field is affected by the Lorentz magnetic field force, which is the microscopic physical essence of the motional electromotive force. An electric charge at rest in a changing magnetic field wave is also affected by the Lorentz magnetic field force, which is the microscopic physical essence of the induced electromotive force. The electromotive force in metal wires and coils is essentially the result of the counter-potential movement of electric charges under the action of the Lorentz magnetic field force. This study reveals that Faraday's law of electromagnetic induction is an engineering approximation formula, which is a great challenge for Maxwell's equations and the fundamental electromagnetic theorems.