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The diagram above demonstrates the Lorentz force, Yet I wonder... must there be two magnets/electromagnets? If the two magnets form a magnetic field of $1$ $Tesla$ acting on the wire, why not use 1 magnet and make the wire really really close to it. So that the gap would be 0. Will $B$ still be valid based on one magnet?

Let us take a bar magnet and place a current carrying wire very close to it as shown below

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The magnetic field in that region of space( very close to the wire and thus the magnet ) can be treated as a uniform field( approximately ).

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Now, the charges within the current carrying wire experience a Lorentz force given by

$$\vec{F} = q(\vec{v}\times\vec{B}) + q\vec{E}$$

Let us see how

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Consider an electron moving within the wire. This electron would experience an electric force due to the presence of an electric field inside the wire and a magnetic force due to the magnetic field created by the magnet outside the wire. So, you need not have two magnets to produce a Lorentz force. However, having two magnets would enhance the strength of the magnetic field and thus the magnetic force. Also, having two magnets would better ensure the presence of a uniform magnetic field unlike in the case of a single magnet. I conclude by saying that Lorentz force will act on the charged particle even in the presence of a single magnet because all we need is an electric field and a magnetic field( which is created by a single magnet ).

Also, you need not place the wire close to the magnet. You can place it at a considerable distance from the single magnet. In this case, you will have a non-uniform magnetic field in space. Nevertheless, a Lorentz force will continue to act on the charged particles within the current carrying wire( the magnetic force will vary from point to point ).

If you use single magnet and remove the other magnet in your experimental set up shown in figure of your question, the magnetic field B would not remain in the expected direction. In your case, using two magnets, you can have the direction of B perpendicular (approx) to the straight current carrying conductor (see the picture).
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You can also use single magnet and keep it close to north or south pole so that magnetic field remains perpendicular to the straight conductor, as explained in the other answer, but there you are going change the placement of the magnet, thus area of the magnet facing the conductor decreases.

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