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When a magnetic field is applied perpendicular to a current carrying wire, a Lorentz force

$$\mathrm{\vec{F}=i(\vec{L}\times\vec{B})}$$

acts on the conducting wire. Of course the applied magnetic field interacts with the magnetic field generated by the current carrying wire.

My question is, does the Lorentz force affect the flow of electrons in the conducting wire? Does it slow them down? Does it affect the resistance of the wire? Does the wire heat up, even slightly?

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Let's say the wire is in the $\overrightarrow x$ direction, while the applied magnetic field is in the $\overrightarrow z$ direction.

The Lorentz force law $\overrightarrow F=I\overrightarrow v\times \overrightarrow B $ tells us that the the force will be in the $\overrightarrow y$ direction.

The electrons in the wire will incur a net force in the direction perpendicular to their motion and the applied magnetic field. So if you do further calculations, you will see that the electrons will slow down a bit. If your wire is now a conducting plane, you'll have also a voltage difference across the conductor in the same direction of the force, see Hall effect

The resistance of the wire is an intrinsic parameter of the material of which it's made, so it cannot be modified by the magnetic field. As for the temperature, I think you'll need a strong alternating magnetic field for a non-negligible increase.

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To all of your question: Yes you are right.

Lorentz force of moving charge in magnetic field is based on the electron’s magnetic dipol moment. The magnetic field align the electron's magnetic dipol moment in the direction to this field. The motion of the electron undergoes a - predictable and perpendicular to the two vectors of the velocity and the magnetic field - acceleration according to the cross product of this two vectors. This acceleration leads to the emission of photons from the electron. As photon has a pulse, this pulse reduces the velocity of the electron, and acts against the alignment of the electron’s magnetic dipol moment by the external magnetic field too. This process is repeated periodically until the kinetic energy of the electron is consumed.

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