Lorentz Force and Apparent Conservation of Momentum Violation Useful for Unidirectional Force?

Question

My understanding is that the apparent violation of Newton's Third Law by the Lorentz Force necessitates a description of the system that describes the "missing" momentum as being absorbed/carried by the magnetic field itself. What is not clear to me, however, is why this apparent violation cannot yield an apparent unidirectional force to the system.

For example: In a system with 2 magnets aligned across from one another and a current carrying rod placed parallel to the stage between them (perpendicular to the $B$ field lines), application of current to the rod can result in the Lorentz force vector pointing upwards acting on the rod. If the rod were to be affixed to the magnets themselves by a non-conducting support, preventing the rod from moving upward out of the $B$ field between the magnets, would the upward force acting on the rod not be transmitted to the magnets themselves? This seems to imply that the entire system (rod, electrons within the rod, and the magnets supplying the $B$ field) would experience an upward force.

This can't possibly be the case because that'd basically produce an anti-gravity device, but the math seems to suggest it, meaning I'm missing something fundamental somewhere. If the back-reaction from the Lorentz Force is not being transmitted to the magnets (pushing them "down" to counter the the upward vector force applied to the rod) but instead is being absorbed by the $B$ field, what prevents the system in this setup from experiencing a net upward force?

Answer 1

The moment you rigidly connect the magnet (or two magnets in your example) to the rod, they become one body.

Therefore a force between them should be treated as an internal force, which means that it cannot move the body as a whole, due to conservation of momentum.

Answer 2

If someone told you that magnets don't feel a force from magnetic fields then you may have been misled. But if you want to use the Lorentz Force Law then consider using all wires or (more realistically) all resistors.

So you could have a super large parallel plate capacitor (make it planet sized) and have it charged up a lot and have two small thin resistors (with large resistance) near the center connecting the two plate. The resistors can run parallel to each other about a meter apart. And next to one of the long thin resistors you could have a series of batteries. So imagine a series of shorter resistors all connected in series with equally long wires and next to each battery you have a battery that isn't connected.

With the batteries disconnected you would have two resistors, each with a steady current, each producing a magnetic field and each feeling a force from the field of the other.

If you then hooked up all the batteries to the resistors it is next to by moving the battery to where the wire was then the current in that series of resistors changes right away and it starts feeling a stronger force right away.

But a meter away over at the other resistor, the field there is still due to the old current (new current doesn't change the magnetic field far away fight away) so it still feels a smaller force.

This is becasue wires don't exert forces on each other. Wires exchange momentum with the fields right where they are, and changes in the fields propagate at the speed of light and only when a new field value gets to a distant location does that distant location start exchanging momentum with the fields next to it at a different rate.

As for whether a wire moves up, it can only get momentum from the field and it only gets the opposite of what it can give. So wires move a certain way by giving an equal and opposite momentum to the fields where it is located.

Now, when the mobile charges feel a force they get deflected outwards from the wire, which creates a charge imblance in the wire that pulls it oppositely and pulls the nonmobile charges in the wire the direction the mobile charges were pulled. That's how the whole wire moved even though only the mobile charges felt a magnetic force. But that's also how work gets done, from the electric force.

So you do have limits to how much can be done. And the batteries are getting drained as well. If you track the energy and the momentum flow you'll see them coming in through the sides of the resistors.

I don't see a clear explanation why the force exerted on the electrons in the wire could not be transmitted to the B field source

A wire feeling a force from a magnetic field exchanges momentum with the field. A wire getting energy (which doesn't happen from magnetic forces) takes energy from the field. So energy and momentum went into the field long ago. Energy and momentum moved through space in the fields. Energy and momentum gets exchanged with the wire based in the current and charge in the wire and the fields at the wire.

The same arguments could be made for electric forces. The energy and momentum went into the field long ago, the energy and momentum moved around in space through the fields. And the charges get energy and momentum from the fields where they are located by exchanging energy and momentum with the fields right there.

Answer 3

No, in this case Lorentz force is reciprocal: current in rod creates it's own magnetic force and acts on electron current loops in magnetic materials... The same stands for current attracting iron rod and etc... So no violation of Newton's Laws...

Answer 4

The reason why the Lorentz Force $\vec F = q(\vec v\times\vec B)$ (or, in this case, $\vec F = I(\vec L\times\vec B)$) moves a wire (let's assume it is horizontal) is because charges build up inside the wire, and then subsequent flowing charges are forced upward. This continuous bombardment of charges is what causes a current-carrying wire to deflect vertically in the presence of a magnetic field.

Because the wire continuously moves away from incident flowing charges as they make contact, this is considered an inelastic collision; so these particles deflect, lose kinetic energy, and are swept up into the applied current.

If this wire is held fixed, these collisions are then considered elastic. These charges are now bouncing off the top, and then redirecting with enough energy to bombard the wire on the other side too. This suggests that any particle deflects off of opposite sides of the wire an equal number of times, effectively attenuating any unidirectional forces one might expect to observe.

Answer 5

The observation is correct that the Lorentz force does not conserve momentum and equivalently does not obey Newton's third law. This is a paradox as momentum is actually conserved in electromagnetic theory. The solution is found by inspecting the force law:

$\begin{align} d_t P_k &= q \epsilon_{ijk} \vec v_i B_j \\ &= - q \epsilon_{ijk} \vec v_i \epsilon_{jmn} d_m A^n \\ &= q \left( \delta_{im}\delta_{kn} - \delta_{in}\delta_{km} \right) \vec v_i d_m A_n \\ &= q v_i d_i A_k - q v_i d_k A_i = q d_t A_k - q v_i d_k A_i ~~. \end{align}$

From this it follows that $d_t (P-qA)_k = -q v_i d_k A_i ~~$. The total momentum of a charge is the sum of kinetic momentum $m \vec v$ and potential momentum $-q \vec A$, just like its total energy is the sum of kinetic and potential energy. This momentum is conserved and, equivalently, the force $q v_i \vec \nabla A_i$ obeys Newton's third law.

Answer 6

I think the missing link for you is that you have concluded that after the momentum is absorbed by the magnetic field it vanishes. Instead, the momentum gained by the field is transferred back to what generated the field.

So to answer your question, the magnets do feel an equal and opposite exchange of momentum with the wire. The change in momentum is just not transferred directly, it is transferred though the magnetic field. This means that your conclusion that you can't make an anti gravity device is correct.

It could help to think that a magnet "wants" to stay in the center of it's own field. If you give the field momentum, it would start floating away from the magnet and the magnet's resistance to the field floating away transfers the momentum from the field to the magnet.

If you want a proper explanation with math there is a nice description here and here.