Magnetic Force Paradox: How Does Einstein's Theory of Relativity Solve It?

[kkiddu]

Hi

We've just begun studying magnetic force in school. Our teacher told us to explore the possibility of a paradoxical situation as

F=qBv

From different inertial frames, the apparent force will be different, won't it ? Our teacher told us that Einstein's theory of relativity solved the paradox, but it's not in our syllabus.

So can you guys tell me how theory of relativity solves the paradox by including the actual details of the theory as little as possible ?

And while you're at it, can you tell me why does a magnetic bottle cause reversal of the direction of helical motion ?

Thanks a lot.

 

[GRDixon]

Hi

F=qBv

From different inertial frames, the apparent force will be different, won't it ? Our teacher told us that Einstein's theory of relativity solved the paradox, but it's not in our syllabus.

Before relativity, it was believed that F' equals F, as measured in inertial reference frames K and K'. (The pre-relativity transformations, from frame K to frame K', are generally known as the Galilean transformations.) However, in relativity theory F transforms into a different F'. Amazingly, the new transformation (called a Lorentz transformation) is entirely consistent with the way a magnetic force may exist in K' but not in K, etc. In your researching, look for the words "Lorentz transformation." There is a Lorentz transformation for most (if not all) of the quantities in classical physics. The transformations for the components of F are one and the same as the transformations of d(mv)/dt, as per Newton's second law.

 

[kkiddu]

Could you please explain it again in cruder and simpler terms ? Knowing that I don't even understand spacetime or Lorentz Transformation ?

I tried googling for stuff, but all of it went way over my head. I'm getting a feeling that I won't be able to understand it unless I take a book and read from the very first page.

 

[GRDixon]

Could you please explain it again in cruder and simpler terms ? Knowing that I don't even understand spacetime or Lorentz Transformation ?

I tried googling for stuff, but all of it went way over my head. I'm getting a feeling that I won't be able to understand it unless I take a book and read from the very first page.

Special Relativity is neither crude nor simple. It isn't obvious to me how to explain things more clearly. Actually, reading a book is a good idea. The three phenomena at the root of the Lorentz transformation of space and time coordinates are: (1) moving objects are measured to be shorter than when at rest; (2) moving clocks are measured to run more slowly than when at rest; and (3) the clocks distributed in one inertial reference frame are not measured to be syinchronized using the grid/clocks of a second reference frame. Good luck!

 

[kkiddu]

Right. Thanks. I understand. And could anyone answer the second part of my question ?

"Can you tell me why does a magnetic bottle cause reversal of the direction of helical motion ?"

 

[netheril96]

Could you please explain it again in cruder and simpler terms ? Knowing that I don't even understand spacetime or Lorentz Transformation ?

I tried googling for stuff, but all of it went way over my head. I'm getting a feeling that I won't be able to understand it unless I take a book and read from the very first page.

In fact,the transform of spacetime isn't the primary issue here.Even at a very low speed (relative to c),when the time dilation and length contraction aren't obvious,the transform of electromagnetic field is significant.

The key is that when in one frame only magnetic field exists,in another frame there are both magnetic and electric field.

I'm not able to answer your second question.In our textbook it is proved by introducing an adiabatic invariant,which we will learn next semester in Analytic Mechanics.

 

[Born2bwire]

In fact,the transform of spacetime isn't the primary issue here.Even at a very low speed (relative to c),when the time dilation and length contraction aren't obvious,the transform of electromagnetic field is significant.

The key is that when in one frame only magnetic field exists,in another frame there are both magnetic and electric field.

I'm not able to answer your second question.In our textbook it is proved by introducing an adiabatic invariant,which we will learn next semester in Analytic Mechanics.

This is an important fact to note. When we transform between reference frames in special relativity, we experience both transformations to the coordinate axes and time and the strength of these transformations are dependent upon the relative velocity of the two frames with respect to the speed of light. This would imply, and this is correct for most situations, that very low speed differences between frames result in negligible differences between the classical Galilean transformation (simple additive transformations that you have used in basic mechanics) and the Lorentz transformations (transformations under Special Relativity).

This is not the case for electromagnetics because in addition to the length contraction and time dilation you also have transformation of the electromagnetic fields. The Lorentz transformation for the electromagnetic fields can convert some of the electric fields into magnetic fields and vice-versa. The transformation of the fields in addition to the length contraction and time dilation are important regardless of the relative velocity. The easiest (well, relatively easiest) example is with two current carrying wires.

If we have two wires that have a current on them, then in the lab frame we will see only a magnetic field produced by the currents and we have a force of attraction/repulsion between the wires. If we shifted ourselves to the frame of the moving carriers of one of the currents so that we are moving along with the electrons, then in that frame the charges of the wire do not have a velocity. Thus, if we observe only a magnetic field in the new frame just like we did in the lab frame then there would not be any forces acting on the charges like we see in the lab frame. The answer comes from the Lorentz transformation that allows us to first convert part of the magnetic fields from the lab frame into an electric field and second to apply length contraction on the charges that result in a net charge on the wire in the frame. The latter part here is a bit tricky to imagine but if we think of positive carriers stationary or moving in the opposite direction of the electrons in the lab frame, then in the frame of the electrons we will see that positive carriers still moving. Since they are moving in the opposite direction as the electrons were in the lab frame, they experience a different length contraction and so the charge per unit length of the positive charges is now different than the negative charges which gives rise to a net charge in the moving frame. The net charge (via length contraction) coupled with the electric field (via the Lorentz transformation of the fields) allows us to see the same force in the electron's frame as we do in the lab frame.

This analysis is valid for any non-zero speeds of the electrons and serves to demonstrate that Lorentz transformations for electromagnetic fields can be necessary in situations where we normally would not expect them. In addition, it also serves to demonstrate how the Lorentz force would be frame dependent using a classical viewpoint but can be reconciled via Special Relativity. Most textbooks on electrodynamics discuss this problem as an example in, say, Griffiths and Purcell.

 

[kkiddu]

I'm not able to answer your second question.In our textbook it is proved by introducing an adiabatic invariant,which we will learn next semester in Analytic Mechanics.

That means the stuff, or the math in it, is a bit advanced for us to do it at this point, so I can just accept it as a fact and understand it when I go to college. Thanks. :)

This is an important fact to note. When we transform between reference frames...

...electrodynamics discuss this problem as an example in, say, Griffiths and Purcell.

So, in short, from the lab frame, we see two current carrying wires producing magnetic fields and attracting each other. When we are in a moving frame, something that's related with length contraction happens, that gives the rise to electric field of such a value that the total Lorentz force in the lab frame and the moving frame is the same.

Am I right ?

 

[sagardip]

In a magnetic bottle as you go from one end to the other then you will find that the direction of force (determined here by Flemings left hand rule) on the moving charge slowly changes from the previous to the opposite direction.So the moving charged particle experiences decceleration and then finally repeats its original helical path in the opposite direction till it reaches the other end and viceversa.
That is probably a simple explanation which you can remember now

 

[sagardip]

Did you get it?

 

[Born2bwire]

That means the stuff, or the math in it, is a bit advanced for us to do it at this point, so I can just accept it as a fact and understand it when I go to college. Thanks. :)



So, in short, from the lab frame, we see two current carrying wires producing magnetic fields and attracting each other. When we are in a moving frame, something that's related with length contraction happens, that gives the rise to electric field of such a value that the total Lorentz force in the lab frame and the moving frame is the same.

Am I right ?

Hmmm... yes and no. First, the magnetic field is transformed into a magnetic field. In solving the problem, this is treated as being independent of the length contraction as we are working in the absence of sources. The next problem though is that without the length contraction, the wire is still electrically neutral. However, the electrons are moving with respect to the stationary ions. In the lab frame the charge density of the electrons is equal and opposite to that of the ions and thus the wire is neutral. But with length contraction, in the frame of the electrons the ions undergo a different length contraction (because their relative velocities are different) which means the the ions per unit length is now different from the electrons per unit length.

Thus, length contraction allows us to see a net charge on the wire which is then acted upon by the electric field that arises from the transformations of the magnetic field. And yes, the resultant Lorentz force comes out to be the same as we find in the lab frame.

However, I do think we should expect that the newly transformed fields can be related back to the effective sources in the new frame. The sources, the other wire, will experience a length contraction as well and we would expect that we would see a similar net charge that would produce an electric field and similarly there are still moving charges to create a current that generates a magnetic field. This would have to be true in my mind since the same physics operates in all inertial frames and thus we must have an associated set of sources for our transformed fields. Still, the transformations of the fields directly is applied even when we are considering a source free problem. Though I guess to satisfy causality one must assume that there was a source somewhere that would undergo transformations.

So I think that you are correct in saying that the length contraction produces net charges which creates our electric field. However, the contraction that produces this field is on the other wire. For simplicity we do not consider the problem to such fine detail as we can work with the fields directly.

 

[kkiddu]

Did you get it?

Yes, what remains is to find how exactly does the force change direction. Let this be a journey I undertake alone. :)

Hmmm... yes and no. First, the magnetic field is transformed into a magnetic field. In solving the problem, this is treated as being independent of the length contraction as we are working in the absence of sources...

...So I think that you are correct in saying that the length contraction produces net charges which creates our electric field. However, the contraction that produces this field is on the other wire. For simplicity we do not consider the problem to such fine detail as we can work with the fields directly.

Thanks born2bwire. I think I understand as much as I can without going into the actual theory. I get a basic idea at least. You guys rock. :)

 

[oldman]

I've heard it said, and believe it is often taught, that magnetism is a purely relativistic phenomenon (see for example Purcell's 1963 book and Feynman's published lecture notes). From a moving observer's viewpoint electric and magnetic fields transform into one another, as pointed out here. Electric charge, though, is conserved from one observer to another. I'm quite happy to accept this, and to accept also that while the electric potential is a scalar, its magnetic counterpart is a vector. Perhaps one should expect that when relative velocities are involved vectors crop up somewhere in the mathematical description. It's all a consequence of Maxwells equations, which fully describe electromagnetism, being Lorentz invariant. Well and good.

But nobody has yet told me whether other field theories that are Lorentz invariant behave in a similar way. For instance do particles which feel the strong force (quarks)do so in ways that depend on how observers move relative to the source of colour charges? Perhaps they do, although this is impossible to observe and may therefore be of little interest.

But what about gravity? Here there seems to be no gravitational equivalent to a magnetic field, and no vector potential. Is this because the "source charge" is mass, which is not invariant for all moving observers? Or is there some other deep reason why with gravity, described like its long-range cousin, electromagnetism, by a tensor field (the metric 4-tensor instead of the field 4-tensor), there is no "gravimagnetic" field?

 

[GRDixon]

I've heard it said, and believe it is often taught, that magnetism is a purely relativistic phenomenon (see for example Purcell's 1963 book and Feynman's published lecture notes). From a moving observer's viewpoint electric and magnetic fields transform into one another, as pointed out here. Electric charge, though, is conserved from one observer to another. I'm quite happy to accept this, and to accept also that while the electric potential is a scalar, its magnetic counterpart is a vector. Perhaps one should expect that when relative velocities are involved vectors crop up somewhere in the mathematical description. It's all a consequence of Maxwells equations, which fully describe electromagnetism, being Lorentz invariant. Well and good.

But nobody has yet told me whether other field theories that are Lorentz invariant behave in a similar way. For instance do particles which feel the strong force (quarks)do so in ways that depend on how observers move relative to the source of colour charges? Perhaps they do, although this is impossible to observe and may therefore be of little interest.

But what about gravity? Here there seems to be no gravitational equivalent to a magnetic field, and no vector potential. Is this because the "source charge" is mass, which is not invariant for all moving observers? Or is there some other deep reason why with gravity, described like its long-range cousin, electromagnetism, by a tensor field (the metric 4-tensor instead of the field 4-tensor), there is no "gravimagnetic" field?

The same thoughts have often puzzled me. I would say that, given any situation where Newton's 2nd law applies (F = d(mv)/dt ), a force law (such as the Lorentz force law) must transform identically to the transformation of d(mv)/dt. To the extent gravity is a force, and given the identical forms of Coulomb's law and Newton's law of universal gravitation, mass/mass interactions in general seem to beg for some analogue of Maxwell's magnetic field. If I know how to do this, you might find the link <a href="www.maxwellsociety.net/PhysicsCorner/Miscellaneous Topics/equilibrium.htm">Link Maxwell and Gravity</a> amusing. Thanks for your well written comment.

 

[netheril96]

I've heard it said, and believe it is often taught, that magnetism is a purely relativistic phenomenon (see for example Purcell's 1963 book and Feynman's published lecture notes). From a moving observer's viewpoint electric and magnetic fields transform into one another, as pointed out here. Electric charge, though, is conserved from one observer to another. I'm quite happy to accept this, and to accept also that while the electric potential is a scalar, its magnetic counterpart is a vector. Perhaps one should expect that when relative velocities are involved vectors crop up somewhere in the mathematical description. It's all a consequence of Maxwells equations, which fully describe electromagnetism, being Lorentz invariant. Well and good.

But nobody has yet told me whether other field theories that are Lorentz invariant behave in a similar way. For instance do particles which feel the strong force (quarks)do so in ways that depend on how observers move relative to the source of colour charges? Perhaps they do, although this is impossible to observe and may therefore be of little interest.

But what about gravity? Here there seems to be no gravitational equivalent to a magnetic field, and no vector potential. Is this because the "source charge" is mass, which is not invariant for all moving observers? Or is there some other deep reason why with gravity, described like its long-range cousin, electromagnetism, by a tensor field (the metric 4-tensor instead of the field 4-tensor), there is no "gravimagnetic" field?

1.You can never find a electromagnetic field when in one frame it is purely electric and in another purely magnetic.In other words,you can never transform a pure magnetic field into a pure electric field just by changing the frame of reference.So the "magnetism is a purely relativistic phenomenon" isn't correct at all.

2.Electric field having only scalar potential is not its nature.In fact,one can designate a vector potential for vortex electric fields just as for magnetic field.We don't do it because this vector potential can be written as the derivative of magnetic vector potential,not because electric field doesn't have one.

3.I know little about strong force.But "gravimagnetic field" does exist.It is a phenomenon of general relativity,and you cannot find it in Newton's gravitation theory.(Note that general relativity does not treat gravitation as a force,but the result is quite similar to a "gravimagnetic field".)

 

[oldman]

...you might find (this) link amusing...

Thanks, I've followed it and I'll explore your website further, young man (I'm much older than you!). It looks interesting.

1.You can never find a electromagnetic field when in one frame it is purely electric and in another purely magnetic.In other words,you can never transform a pure magnetic field into a pure electric field just by changing the frame of reference.So the "magnetism is a purely relativistic phenomenon" isn't correct at all.

Thanks for these comments. Nevertheless, there are knowledgeable writers and teachers who claim that regarding magnetism as a relativistic phenomenon is a valid perspective.

2.Electric field having only scalar potential is not its nature.In fact,one can designate a vector potential for vortex electric fields just as for magnetic field.We don't do it because this vector potential can be written as the derivative of magnetic vector potential,not because electric field doesn't have one.

I'm not familiar with 'vortex electric fields' --- could you point me to an example of one?

3. ... "gravimagnetic field" does exist.It is a phenomenon of general relativity,and you cannot find it in Newton's gravitation theory.(Note that general relativity does not treat gravitation as a force,but the result is quite similar to a "gravimagnetic field".)

I'm afraid that I didn't know that others have written about what I thought would be just an invented fiction of mine, namely a gravitational analogue of magnetism. Now I see that there's even a Wikipedia article about this. I fear this idea, proposed by Oliver Heaviside, is a can of worms better not opend.

I just wondered if the reason why moving electric charges have magnetic fields, wheras moving masses don't interact with other masses in a similar velocity-dependent way, is this: we choose, in our relativistic descriptions of nature, to designate electric charge as a conserved observer-independent quantity and, in contrast, to designate mass as an observer's-relative-velocity-dependent quantity. I'm not suggesting that our descriptions are wrong.

 

[kkiddu]

Thanks for these comments. Nevertheless, there are knowledgeable writers and teachers who claim that regarding magnetism as a relativistic phenomenon is a valid perspective.

It does seem logical. I mean, I don't know even 5% of the physics that exists in the world, but I don't know how to explain it to you, electric force just felt right. It was, as if, there was a reason for it to exist, a property called charges. I was completely befuddled about the very nature of magnetic field itself when we began studying it. There was no different 'source' of magnetic field. Since when I came to know about the relativistic relation between electricity and magnetism, it all has started to make some kind of sense.

Please don't set much store by what I said, these are the words of an amateur, but I did feel like I had to say it.

And about 'gravomagnetic effect', could this analogy that magnetic fields and forces are far weaker than electric fields and forces (I'm talking by interpreting the constants 9 X 10^9 vs 10^-7), and since gravitation is already a weak force, it's magnetic analogue would be way too weak ?

 

[Dale]

Thanks for these comments. Nevertheless, there are knowledgeable writers and teachers who claim that regarding magnetism as a relativistic phenomenon is a valid perspective.

You are probably referring to something like this:
http://physics.weber.edu/schroeder/mrr/MRRtalk.html

1.You can never find a electromagnetic field when in one frame it is purely electric and in another purely magnetic.In other words,you can never transform a pure magnetic field into a pure electric field just by changing the frame of reference.So the "magnetism is a purely relativistic phenomenon" isn't correct at all.

This comment is also correct:
http://www.fourmilab.ch/etexts/einstein/specrel/www/

You are both right.