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With generators and motors, is there a form of magnetic attraction or repulsion between the conductive element(rotor/solenoid) and the magnet's magnetic fields? If so, what formula would be best to calculate the magnetic forces given each element's magnetic field and distances?

I'm somewhat confused, because when a motor's conductor experiences a Lorentz force(ILxB) simultanously the conductor is being attracted/repelled to the electromagnet due to the magnetic fields? Likewise with a generator,aside from the Lorentz force opposing the input mechanical energy, there is a magnetic force as well?

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The force that causes motion is due to the interaction of two magnetic fields, at least one of which is created by a current flowing through a conductor. Force is given by F=BIL, which is, essentially, the Lorentz force.

In a generator, a voltage (and consequent current if there's a circuit) is created by moving a conductor in a magnetic field. Mechanical energy is thereby converted to electrical energy (if a current flows). If there is no circuit, there will be zero current, and zero force is required to move the conductor. The induced voltage (actually, an EMF) is E=BLv, where v is the velocity of the conductor wrt the magnetic field.

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