Magnetic fields, Newton's Second Law, and GR

[Jetro]

I have a question regarding the interactions of electromagnetic fields.

Say you have two superconducting electromagnets A and B. The properties of the magnets are known such that you can tell precisely how quickly it takes each one to produce it's full strength magnetic field and how long it takes for that field to no longer be present at the magnet when it is turned off. They are connected by a non-conductive rod at opposite ends at a distance equal to (time it takes for the field to no longer be present)x(speed of light). A________B

A magnetic field is induced in magnet A. A))))))))B Using a very precise computer, you shut off magnet A's field.

For a very brief period of time there should be no field present directly at magnet A, but the field is still propagating through space. A__))))))B

Now if I turn on magnet B at that moment, producing a field with opposite polarity, magnet B should be repelled by that field. A__))))((B --->

My question is, if magnet B is pushing off of the field magnet A produced. But that field is no longer present in magnet A itself. What prevents the rod they are attached too from moving in a single direction, as there wouldn't be an opposite force acting on magnet A to cancel it out?

 

[sophiecentaur]

There is a much simpler scenario that we could discuss. Take two parallel wires, each carrying current. You could ask is the force between them due to the two fields interacting or the fields from each wire interacting with the current flowing in the other. Alternatively, there is a good SR explanation which does without the Magnetic Field entirely and explains the force as being due to the different speeds of the electrons and protons in the wires and the resulting apparent density of charges - so the force and be just an Electric force.
There would be a delay in the force, due to a finite c.

 

[Jetro]

There is a much simpler scenario that we could discuss. Take two parallel wires, each carrying current. You could ask is the force between them due to the two fields interacting or the fields from each wire interacting with the current flowing in the other. Alternatively, there is a good SR explanation which does without the Magnetic Field entirely and explains the force as being due to the different speeds of the electrons and protons in the wires and the resulting apparent density of charges - so the force and be just an Electric force.
There would be a delay in the force, due to a finite c.

Right, but in this case, only one wire is carrying a current at a time. As I understand it, an electromagnetic field has momentum and the fields carry a charge. So if, like in the example, wire A was not carrying an opposite charge of wire B, wire A would not be repelled by the opposite charges from wire B. But since wire A did previously create a electromagnetic field with an opposite charge and has momentum, the field that wire A previously produced seems like it would still repel wire B.

 

[sophiecentaur]

Right, but in this case, only one wire is carrying a current at a time.

In the short period of time that it takes for the EM wave to traverse the distance, the Force will still be there.
There is the other issue of deciding that one magnet just switches off. In fact this will take a significant time to happen and the changing magnetic field will be inducing an emf in the other coil / wire.
It looks to me that any apparent paradox could be due to ignoring all such factors.

 

[Jetro]

In the short period of time that it takes for the EM wave to traverse the distance, the Force will still be there.
There is the other issue of deciding that one magnet just switches off. In fact this will take a significant time to happen and the changing magnetic field will be inducing an emf in the other coil / wire.
It looks to me that any apparent paradox could be due to ignoring all such factors.

Ok, that does make sense. But if you positioned the two wires, such that they were arbitrarily far enough away for the force in wire A to be no longer present, or greatly reduced by the time the field from wire B arrived, would that interaction violate conservation of momentum?(I do understand that there are limits to materials that might prevent that interaction from having a meaningful force at the arbitrary distance)

As far as I understand it, the field that is generated has it own volume, energy density, and would extend outside of the system as described. Assuming you had some perfect materials that could change their magnetic fields quickly enough for the force to be meaningful at an arbitrary distance, would that interaction be considered an open system where conservation of momentum is concerned?

 

[Orodruin]

You cannot simply neglect the momentum carried by the EM field itself when you consider situations like this.

 

[Nugatory]

would that interaction violate conservation of momentum?

No. You're failing to take into account the momentum exchanged between the wires and the fields, and the momentum carried by the changing fields.

You may be confusing yourself by thinking that there is a force between the wires (mainly because you can get away with thinking that way in static situations) but in fact the wires are interacting with the fields at the location of the wire, not one another.

 

[sophiecentaur]

Assuming you had some perfect materials that could change their magnetic fields quickly enough

Whichever way you produce the field, currents are involved and the energy stored in the field takes time to dissipate. An Inductance must govern the time constant of the decay. Changing the decay time would be altering the radiated Power.

 

[Jetro]

You cannot simply neglect the momentum carried by the EM field itself when you consider situations like this.

I'm not neglecting that. Any change in wire A's field propagate at the speed of light. So if wire A's field is shut off the change in the field would take time to reach wire B. So there would be a period of time where the field produced by wire A is present in space occupied by wire B's, but no longer present at wire A itself. If wire A's field carries momentum, and you produced a field with opposite polarity at wire B's location then the fields should still interact and repel wire B. But if there is no field present at wire A any longer then it can't be repelled when wire B's field occupies wire A's space.

No. You're failing to take into account the momentum exchanged between the wires and the fields, and the momentum carried by the changing fields.

You may be confusing yourself by thinking that there is a force between the wires (mainly because you can get away with thinking that way in static situations) but in fact the wires are interacting with the fields at the location of the wire, not one another.

But in the moment I described, is possible to interact with wire A's field independently of wire A in the brief period between were there is no field being produced by wire A, and those changes in that field reaching wire B? The field carries it own momentum and energy density that extends beyond wire A itself. How would it be any different that interacting with a massive electromagnetic field that was a light-minute away? The object producing that field wouldn't be repelled by an opposing field for one minute.

 

[tech99]

It seems that when we switch on wire A, we accelerate charges and create an EM wave. Photons travel away from A. As consequence of their momentum, they create mechanical reaction forces on A which act radially and equally in all directions, summing to zero. If photons then pass close to wire B, they create radiation pressure on it acting away from A. Let's assume wire B has electrical resistance, so it absorbs energy.
The force on B occurs after a time delay from switching on A, caused by the velocity of propagation, c.
If the two objects are mechanically linked by a rod, they will move together. Radiation pressure on B will cause the combination to move in that direction. When the current is switched off in A, it is interesting to notice that another EM wave will be radiated and will give B another impulse in the same direction. This is because it is the flow of energy from A to B which is causing the effect and it is insensitive to electrical phase.
Although the movement of A and B as a unit looks impossible, it is analogous to a rocket, where the momentum of the ejected gas is directional. In our case, photons are absorbed in one direction - we have a shadow of B.
If B has zero resistance and so re-radiates the energy, it is still possible that the radiation from A and B combined will be directional due to the relative phases of these two sources. In this case there will still be a resultant propelling force.
After the EM wave passes we are left with a static magnetic field, which does not have momentum nor transfer energy.
Regarding the inductance of A, it is a store for magnetic energy. As it fills and empties, the acceleration of the charges creates radiation, the frequency being determined by the RLC constants of the circuit.
The static magnetic and electric fields cannot vary in strength without charges being accelerated, and so creating EM waves. For example, the building of a static magnetic field after switch-on is preceded by an EM wave. It is like the tsunami which preceeds the flood.
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