What feels the force that is equal and opposite to the Laplace force?

Question

The Laplace force is the force applied to a current-carrying conductor in a uniform magnetic field:

$$ \vec F = I \vec l \times \vec B $$

It has been described as a macroscopic view of the Lorentz force.

I'm assuming there must be an equal and opposite force. So to what is that force applied? To the magnetic field? The magnet itself? Does it manifest as a voltage opposing current in the wire?

I would guess the opposing force manifests as a voltage across the length of the wire, because this instance of Faraday's Paradox implies that work can be extracted from a homopolar generator when the magnet rotates in sync with the conductor. Evidently there is no energy exchange between the magnet and the conductor (as is confirmed by other answers that show mathematically that the magnetic field supplies no net work). From that I'm deducing that the opposing force (through which work can be extracted) is conveyed through the only thing left... the conducting circuit.

But if the opposing force arises as a voltage across the length of wire, how is that force "opposite" since it is perpendicular to the Laplace force? I thought equal and opposite meant equal in magnitude and in the exactly opposite direction?

As you can tell, my background isn't in math or physics so I'm easily confused by these sorts of things. I appreciate any help in clearing up my confusion!

Answer 1

(a) "I'm assuming there must be an equal and opposite force. So to what is that force applied? To the magnetic field? The magnet itself?" Yes, to the magnet. This is easily verified by placing a U-shaped magnet upright on a weighing machine and passing a current through a horizontal conductor between the poles of the magnet and at right angles to the magnet's field. We measure a force on the magnet that accords with $\vec F=-I \vec l \times \vec B$. The force is transmitted by the magnetic field.

(b) "Does it manifest as a voltage opposing current in the wire?" No; an induced voltage arises in the wire only if it is moving with at an angle to itself and to the field. The charge carriers in the wire will then have a velocity component at right angles to the wire and will experience a Lorentz force component along the wire, and this is responsible for an induced voltage (emf) in the wire.

(c) "The Laplace force is the force applied to a current-carrying conductor in a uniform magnetic field:

𝐹⃗=𝐼𝑙⃗×𝐵⃗

It has been described as a macroscopic view of the Lorentz force."

This is not always the whole truth. As discussed in (b), if the wire is moving at right angles to itself, the Lorentz force will give rise to an emf in the wire.