Can Electromagnetism Violate Newton's Third Law?

[KamiKage]

Hi,
My physics teacher the other day proposed an interesting question while we were studying how magnetic fields are produced when an electric current is present in wires. He asked if we could figure out a configuration where Newton's Third Law of Motion is violated for an instant using electromagnetism. At first I thought maybe quantum mechanics might be the answer but he said the solution was much simpler than that. I am really perplexed by the question and thought you guys might be able to answer it.

Thanks a lot,
KK

 

[Galileo]

Think of the Lorentz-force:

$$\vec F = q(\vec v \times \vec B)$$
on a charged particle with velocity v. This is some pretty weird law as far as directions are concerned. Now you also now an electric current generates an magnetic field. So see if you can find a configuration of wires where the force on one wire due to the magnetic field of the other is not opposite to the force on the other wire due to the magnetic field of the first.

 

[KamiKage]

But I thought

But I thought this might be impossible because since the currents are opposite the forces will always be opposite.

 

[Meir Achuz]

Closed current loops obey NIII. The force between two moving charges on perpendicular paths does not. This is true either using relativity or not.
The reason is that the EM field has momentum.
It is the sum of the EM momentum and the mechanical momentum that is conserved.
Cons of mom is more fundamental than NIII.

 

[Andrew Mason]

Closed current loops obey NIII. The force between two moving charges on perpendicular paths does not. This is true either using relativity or not.
The reason is that the EM field has momentum.
It is the sum of the EM momentum and the mechanical momentum that is conserved.
Cons of mom is more fundamental than NIII.

Are you saying that Newton's Third Law is not obeyed? Why? Momentum is conserved because dp/dt = 0. If dp/dt = 0 = net force, the sum of all forces = 0 at all times. If the sum of all forces at any time is not 0, dp/dt is not 0 and momentum is not conserved.

AM

 

[shyboy]

Momentum should be conserved, but EM is essentially a relativitic theory, while Newton's mechanics is not. So paradoxes may arise if we stumble on relativistic properties of EM. For example, using straightforward thinking, no force should act on a particle in an inertial systym which moves with the particle velocity. According to the Newton the acceleration (and force) is absolute, so if we do not have it in some inertial system, we do not have in any inertial systems.

However I could not imagine how to make a paradox with the third law :(

 

[jdavel]

Newton believed that the force of gravity was transmitted intstantaneously. So he believed that when one planet exerts a force on second planet, the second one immediately exerts an equal and opposite force on the first. But gravity is not transmitted instantaneously so this is not true.

The same problem arises with EM fields and charged particles. If two charges are at rest in each other's fields a distance L apart, then if one moves, the force from the other charge immediately changes. But the force on the other charge doesn't change for a time L/c. So the force of one body on a second body isn't quite always equal and opposite to the force of the second body on the first. And that's what Newton's third law says is true. So Newton wasn't quite right.

But momentum is still conserved, because the fields carry the momentum during the time, L/c, when the momentum of the two objects is not conserved.

 

[Andrew Mason]

Newton believed that the force of gravity was transmitted intstantaneously. So he believed that when one planet exerts a force on second planet, the second one immediately exerts an equal and opposite force on the first. But gravity is not transmitted instantaneously so this is not true.

Can you give us an example of how the speed at which "gravity is transmitted" will affect the force that one body exerts on another? I am having difficulty understanding why gravity has to be 'transmitted'. If you could demonstrate this, I think you would have proven the existence of gravitons.

AM

 

[jdavel]

Well, suppose that the graviational attraction between the Earth and the sun were transmitted at a speed that took it 3 months to get from one to the other. Then as the Earth revolves through the sun's stationary field, the force on the Earth would always point toward the sun. But the force exerted on the sun would be 90 degrees behind, and so, while equal, certainly not opposite.

Of course we know it's not that slow. But even if it's c, the Earth goes about 20" of arc in the 8 mins it takes for light to get from here to there. So, again the forces wouldn't be opposite.

Isn't that right?

 

[jtbell]

The force between two moving charges on perpendicular paths does not. This is true either using relativity or not.
The reason is that the EM field has momentum.

Are you saying that Newton's Third Law is not obeyed? Why?

Note Meir's last sentence. The total momentum of a system that includes electromagnetic fields is the sum of the momenta of the particles and the momentum carried by the fields. This total is conserved, whereas the sum of the momenta of the particles alone is not conserved in general.

 

[Andrew Mason]

Well, suppose that the graviational attraction between the Earth and the sun were transmitted at a speed that took it 3 months to get from one to the other. Then as the Earth revolves through the sun's stationary field, the force on the Earth would always point toward the sun. But the force exerted on the sun would be 90 degrees behind, and so, while equal, certainly not opposite.

Of course we know it's not that slow. But even if it's c, the Earth goes about 20" of arc in the 8 mins it takes for light to get from here to there. So, again the forces wouldn't be opposite.

Isn't that right?

If we model gravity as a gravitational 'flux' transmitted from one mass into all of space, I don't think it matters what speed it travels at. Mass, hence gravity, cannot be created nor destroyed. Locally, momentum is conserved. So, the centre of mass, hence centre of gravity, of a body does not change even though its parts may move around. There is no need for any new information to be be transmitted to the distant mass.

Consider the mass of the sun. The rest mass of the sun is continually decreasing due to its release of energy. But that energy has relativistic mass equal to the loss of rest mass, so gravitational effect on the Earth is not affected until the energy actually reaches us.

If the sun were to explode, under either GR or Newton there would be no effect on the gravity felt until such time as the matter actually reached us.

So, it seems to me, whether you treat gravity has having a finite or infinite speed, you get the same result.

AM

 

[Meir Achuz]

It has nothing to do with relativity or time delays.
It is a simple P101 exercise to find the forces on two electrons,
one at the origin moving up the z axis with speed v,
and the other at the point z=10 on the z-axis, but moving with speed v in the x direction. The magnetic force on; one does not equal the magnetic force on the other.
Poor Newton did not know about electromagnetic fields, because he placed out of P101. That's why he thought conservation of momentum and his NIII were equivalent concepts.

 

[nmondal]

It's easy!

Ok, put it this way:--
Have a magnetic field and a charge running with a velocity v.
Now run parallely to the charge...you see that the charge is still bending without any force!
That is the *vilolation* of every law, unless you know from a moving-ref-frame the E field certainly has a curl!
There are more to it.
In case of eddy current dumping, the Action [The force you give to the conductor] is not the same as the conductor sends you back!

 

[nmondal]

No ...force means sum of all forces...

It has nothing to do with relativity or time delays.
It is a simple P101 exercise to find the forces on two electrons,
one at the origin moving up the z axis with speed v,
and the other at the point z=10 on the z-axis, but moving with speed v in the x direction. The magnetic force on; one does not equal the magnetic force on the other.
Poor Newton did not know about electromagnetic fields, because he placed out of P101. That's why he thought conservation of momentum and his NIII were equivalent concepts.

No this is not the case.
The net force on one electron must be balanced by the another.
Get the concepts back again.
In a system with no *external* force, the total momentum
must remain conserved. And, study more conservation laws. There is never a *real* viloation of NIII per say.



Instead take the eddy current dumping case, where it *seems* that the NIII is viloated.

Conservation of momentum is indeed the NIII.
DOT.

 

[jdavel]

nmondal,

I believe NIII refers to the forces that each of two bodies exert on the other. If by "bodies" we can assume that Newton meant objects with mass (and he did), then NIII is temporarily violated when one of two charges, separated by some distance is accelerating. The violation is equal to the momentum carried by the EM radiation produced by the accelerating charge, and it lasts for a time greater than or equal to the distance between the charges divided by the speed of light.

 

[Meir Achuz]

Didn't any of you add the forces on electron one and electron two in the case I described? The vector sum of these two forces is not zero. Nothing esoteric is needed.

 

[jdavel]

Meir Achuz,

If what you're saying were true then, since there is not acceleration (and therefore no radiation) not only would NIII be violated, so would conservation of momentum in general. This of course is not the case.

While the force from magnetic fields is not the same for the two particles, the total force on each particle is the same, as measured in any frame. I think you're forgetting to take the Lorentz transforms of the electric fields in your calculations.

 

[Meir Achuz]

I can include both fields and do it either NR or SR.
So could you.
NIII is simply violated.
Conservation of mechanical momentum does not hold, unless the changing
EM field momentum is included.
This is just a demonstration that there is momentum in the EM field.
When you use "of course", I know you are confused.
You don't need radiation for there to be momentum in the EM field.
Anyway, there is acceleration due to the forces, but the radiation is not an important part of the EM fields here.

 

[jdavel]

Meir Achuz,

But if you consider the problem in the center of mass frame, the two charges are moving at the same speed, in opposite directions. Surely the symmetry of that situation requires that the forces between them be equal and opposite. Doesn't it?

 

[Meir Achuz]

Yes, if the two particles move on the same straight line, the EM field momentum is constant so the mechanical momentum is conserved and NIII is satisfied.
But there are also situations, such as the one I suggested, where the EM field momentum is changing so mechanical momentum is not conserved and NIII is violated.
The original question "asked if we could figure out a configuration where Newton's Third Law of Motion is violated for an instant using electromagnetism."
I answered that question.

 

[jdavel]

Meir Achuz,

In your post #18 you said that you could show the forces are not equal with a non-relativistic calculation. Does that mean terms in v^2/c^2 can be ignored? If so, could you show the calculation that gives dp/dt of the EM field just matching the non-zero difference in the the forces on the two particles?

 

[Meir Achuz]

I didn't mean quite that. The NR calculation is just wrong, because relativity must be used for two moving charges. What I meant by "I can include both fields and do it either NR or SR." was that even doing the incorrect NR calculation, the two mechanical forces did not add up to zero. A problem with the NR calc is that the magnetic forces already are of order v^2/c^2 so terms of that order in the E field coming from SR cannot be ignored. A general proof that the sum of the mechanical and the EM momentum is constant is made (sometimes incorrectly) in most high level EM texts, in connection with the "Maxwell Stress Tensor".
Doing a specific calculation, even in the simplest case I mentioned, turns out to be difficult because even the simpler NR fields are too messy to integrate over all space, except numerically, which I haven't done. I believe the general theorem.

 

[jdavel]

Meir Achuz,

Well, I looked up the Maxwell stress tensor, and I'm not sure if I had forgotten it or whether we skipped that chapter. At any rate it certainly does seem to mean what you've been saying. But I don't understand it well enough to be sure that the momentum contained in the field as measured in one frame, isn't just making up for the dfference in the forces on each particle when you change frames.

Here's what I mean. Call your electron at the origin (the one moving in the z direction) q1, and call the other one q2. Then calculate the force on q2 in the frame where q1 is at rest. That's easy because the source charge is at rest, so the Lorentz force is just the coulomb force of a point charge. The only thing that changes is that r^2 gets multiplied by gamma squared. Now Lorentz transform that force back to the original frame (where both particles are moving as you originally described). Then do the same thing for the force on q1 in the frame where q2 is at rest. Transform back to your original frame and, unless I made a mistake, the two forces (in your original frame) are equal and opposite.

What do you think?