EM Phenomenon: Magnetism from a Cathode Ray?

[Chris Frisella]

I have recently studied about relativity being the reason for magnetism. The example given was about a magnetic field generated from current down a long wire. When two such wires with current in the same direction were parallel to each other there was an attractive force between them because from the reference frame of the electrons, the protons in the opposite wire were condensed, therefore a net attractive force.

Now my question: Would a cathode ray in a vacuum create a magnetic field like the wire does? I wonder this because in the cathode ray beam there are no corresponding protons like there are in the wire so there would be nothing to attract to...or what?

Thanks,
Chris

 

[Simon Bridge]

Welcome to PF;

Would a cathode ray in a vacuum create a magnetic field like the wire does?

Yes.

It's a good question because the usual description showing how special relativity gives rise to a magnetic force in a wire is restricted to that kind of example (with positive and negative charge present) only. Probably because the more general case is kinda hard to follow. In a nutshell, relativity gives moving charges a magnetic field because of the finite time for changes in the electric field to propagate.
Try it just for a single charge to start with.

http://farside.ph.utexas.edu/teaching/em/lectures/node125.html

 

[sophiecentaur]

The density of charges in the cathode Rays is much less but the speed is much greater. Relative to the cathode Rays, there is still a relativistic difference in + and - charge density in any conductor (electromagnet / nearby wire). As for how it would interact with a permanent magnet field, the 'simple' SR explanation for two wires would still need to be modified to give Flemming's rule.

 

[ZapperZ]

Another addition to this.

In particle accelerators, we can make a non-destructive measurement of the charge of the particle beam, especially when they are in bunches. We can use what is called an Integrating Charge Transformer (ICT), which is nothing more than a toroid around the beam pipe (see pg. 19 and 20 of https://cas.web.cern.ch/cas/France-2008/Lectures/Denard.pdf, which is from a particle accelerator school). The charges passing through is the current going through the coil, inducing a magnetic field and generating an induced current in a secondary winding.

So heck yes, moving electrons can generate magnetic fields, without needing any positive background.

Zz.

 

[Merlin3189]

I'm only looking at this thread by accident, but I recently saw another thread discussing the drift velocity of electrons in a wire. So I was a bit puzzled by the explanation of the magnetic attraction as a relativity effect. The velocity quoted for electrons in a wire was of the order 10^-5 m/sec. This is not the sort of speed I associate with relativistic effects ( more like 10⁸ m/sec)

I don't doubt the explanation for a moment. I think I have even heard of it before. I just wondered if anyone had any simple comment to reconcile these ideas in my mind?
The only thing I can think of so far, is that the relativistic effect is indeed miniscule, of the order of (10^-5)²/ (10⁸)², but that when you take account of the number of electrons in the wire, I suppose around 10²⁰ per metre, it starts to enter the realm of normality again.

 

[Charles Link]

I'm only looking at this thread by accident, but I recently saw another thread discussing the drift velocity of electrons in a wire. So I was a bit puzzled by the explanation of the magnetic attraction as a relativity effect. The velocity quoted for electrons in a wire was of the order 10^-5 m/sec. This is not the sort of speed I associate with relativistic effects ( more like 10⁸ m/sec)

I don't doubt the explanation for a moment. I think I have even heard of it before. I just wondered if anyone had any simple comment to reconcile these ideas in my mind?
The only thing I can think of so far, is that the relativistic effect is indeed miniscule, of the order of (10^-5)²/ (10⁸)², but that when you take account of the number of electrons in the wire, I suppose around 10²⁰ per metre, it starts to enter the realm of normality again.

In the case of two charges moving parallel to each other, (or the currents in two wires), if you go to the rest frame of the electrical charges there is no magnetic field or magnetic force. (All you observe is electrostatic repulsion.) This would seem to indicate that the magnetic fields and forces are a relativistic effect even for the slower velocities.

 

[sophiecentaur]

"So heck yes, moving electrons can generate magnetic fields, without needing any positive background. "
The 'pinch effect' In plasma beams would (?) need the + and - charges to be traveling at different speeds for a 'simple' SR explanation, I think.

 

[Chris Frisella]

Thank you Simon and all; that was helpful! I would be interesting in seeing an experiment of this. I have searched all of youtube and haven't found any attempt to have multiple electron beams interact with each other in different ways (only with electromagnets). Perhaps I'll have to build an apparatus...

 

[Simon Bridge]

You are thinking of an electron beam experiment equivalent to the one where two wires deflect or attract each other?
The trouble here is that two charged beams will repel each other just from the electric charge being the same ... drowning out the magnetic effect.
ie you'd need quite sensitive measurements to see the difference.

Wires don't have that problem since they are just about electrically neutral.

You are, in fact, demonstrating the magnetic properties of the beam when you deflect it in a magnetic field ... just like you deflect a wire carrying a current.
It's the same effect.

 

[sophiecentaur]

A wire has + and - charges and the SR explanation works fine even for just one passing free electron. But where is there an SR model for two beams of electrons? (Is my point)

 

[Chris Frisella]

You are thinking of an electron beam experiment equivalent to the one where two wires deflect or attract each other?

I think I know what would happen if the beams were parallel. Now I am just wondering why an incoming beam perpendicular to the wire would receive a force which is parallel to the wire. I made a little graphic to illustrate what I think would happen. My question is why does that force exist?

 

[sophiecentaur]

That's yet another instance where an SR explanation seems to be lacking. Some well informed PF member should be able to put us out of our misery. Hellooooo out there?

 

[Vanadium 50]

That's yet another instance where an SR explanation seems to be lacking. Some well informed PF member should be able to put us out of our misery. Hellooooo out there?

Maybe you want to rethink what you wrote. First, the PF membership is not some cadre of trained seals at your beck and call. Second, this is a holiday weekend. And finally, what you ask for is simply unreasonable.

SR has no problem in calculating the force on a charge from an arbitrary collections of charges and currents. It can even do so starting from electrostatics, superposition and Lorentz transformations. What it cannot do is do this at a "B" level.

 

[Chris Frisella]

Vanadium, do you have a basic cause>effect explanation to my question? Not interested in math proofs, just a logical explanation.

 

[Vanadium 50]

A moving charge creates a magnetic field.

 

[jtbell]

In general, the electromagnetic force that a point charge A exerts on a point charge B depends on the positions of A and B, and on their velocities (magnitude and direction). For historical reasons, we call the position dependent part of the force "electric" and the velocity-dependent part "magnetic."

 

[Charles Link]

To the OP: (Referring to the diagram=post # 11) The magnetic force that an electron in the beam will experience is given by $F=e(v \times B )$ where the $v \times B$ is a vector cross product. Once you compute the magnetic field $B$ from the current in the wire, you should be able to compute the force if you can estimate the electron velocity. From the vector cross product you can compute the direction of the force, with or without an estimate on the magnitude of the electron's velocity.

 

[Chris Frisella]

Vandium, granted. Referring to my illustration, why would there be a force on the beam parallel to the current carrying wire? How does this force arise along that vector? The common SR explanation doesn't really connect to my example well because I don't see there being any charge directly perpendicular to the electron beam, you know what I'm saying?

 

[Charles Link]

Vandium, granted. Referring to my illustration, why would there be a force on the beam parallel to the current carrying wire? How does this force arise along that vector? The common SR explanation doesn't really connect to my example well because I don't see there being any charge directly perpendicular to the electron beam, you know what I'm saying?

The magnetic field $B$ from the wire is in a circular direction around it. (Comes from Ampere's law and/or Biot-Savart's law). (The $B$ is perpendicular to the moving current in the wire so that a $v \times B$ can be parallel to the current.( $v$ is electron beam velocity.))

 

[Chris Frisella]

Understood, JT, thank you. Charles, thank you, perhaps I need to clear up "vector cross product".

 

[Charles Link]

Understood, JT, thank you. Charles, thank you, perhaps I need to clear up "vector cross product".

In a vector cross product, the result is always perpendicular to the two vectors of the product. Another item is, if the two vectors are parallel, the vector cross product is zero.

 

[Chris Frisella]

Charles, got you.

 

[sophiecentaur]

Vanadium, do you have a basic cause>effect explanation to my question? Not interested in math proofs, just a logical explanation.

A moving charge creates a magnetic field.

That isn't an 'SR explanation'. The Magnetic Field explanation (Flemming's Rules) is well known and it's the standard one we start with. I think what we're after is something along the same lines as the SR explanation for the force between two current carrying wires. That doesn't need an explicit Magnetic Field and it is unlikely that there is not a suitable Non-Magnetic Field answer for moving electron situations.
P.S. I was not 'demanding' and answer in my previous post; I was simply saying that there "must" be such an explanation. "Trained Seals" we are not.

 

[Delta2]

I 've to say I don't know much about relativity but are we dealing here with cases where classical physics provide us with a simpler explanation but relativity give us a much more complex explanation (yet might explain it just a bit better).

For example using classical physics the moving electrons of the beam constitute a current density , hence they create a magnetic field due to Ampere's Law and this magnetic field just exerts a force on other nearby moving charges regardless if they are electrons or protons..

 

[Chris Frisella]

Welcome to PF;
In a nutshell, relativity gives moving charges a magnetic field because of the finite time for changes in the electric field to propagate.

http://farside.ph.utexas.edu/teaching/em/lectures/node125.html

Simon,

Understanding how an electric field force converts to a magnetic field force once the particle starts moving would be a big help! Can you explain it?

 

[Simon Bridge]

Did you try the link?
I can give it a go - but if you are hoping for explaining relativity phenomena in terms you find familiar already I don't think I can. The reason we need relativity is because there are things the more familiar does not explain.

The principle assumption of relativity is that nothing travels faster than light in a vacuum (formally: the speed of light is the same for all observers).
Consequences of that are consequences of special relativity.

naively you'd think that the electric field of a charge moving with velocity v in the z direction would be: $\vec E = (kq/|\vec r - vt\hat k|^3)(\vec r - vt\hat k)$ ... basically just the regular point charge field lines whose center moves along the z axis. But this violates the principle assumption above: do you see why?

 

[Simon Bridge]

You may like to play around with some simulations:
http://www.cco.caltech.edu/~phys1/java/phys1/MovingCharge/MovingCharge.html
... I'm sure there's a better one than that tho.

 

[Charles Link]

Besides the delayed effect of the fields that Simon has mentioned that makes for special relativity to enter the picture in treating some of these scenarios, I think magnetic forces even in a steady state and/or for v/c <<1 case (c=speed of light) are by their very nature relativistic because both the magnetic field that a particle creates depends on its velocity and the magnetic force another particle experiences in that field also depends on its (own) velocity. (In contrast, electrostatic forces are velocity independent. The electric field E around a charged particle (for v/c<<1) is velocity independent, and the force another charge experiences in that electric field is also velocity independent.) As was even the subject of a discussion on PF about two months ago, when there are only two moving particles present, the magnetic force that one moving particle experiences isn't necessarily equal and opposite to the magnetic force the other particle experiences. Suggestion for the OP would be to read about Biot-Savart's law for the magnetic field $B$ that a moving charge generates (and this is for speeds v where v/c<<1, so you don't need to consider the propagation of the field), and also study the equation for the magnetic force a moving charged particle experiences: $F=q v \times B$ where this is a vector cross product. The total force is given by $F_t=q(E+v \times B)$ where E is any electric field that may be present as well. The case that Simon has presented where the fields propagate is quite interesting as well, but the calculations are much more difficult.

 

[arydberg]

The idea of electrons traveling at slow speeds in a wire seems not to happen. ( yes it is predicted) It comes from the numbers of electrons in a wire carrying a fixed current.

It seems to me that we should be able to calculate the apparent speed of the electrons in a wire by using the force between two parallel wires and applying SR, finding the length contraction and thus the change in charge density to explain the expected force.

 

[sophiecentaur]

The idea of electrons traveling at slow speeds in a wire seems not to happen. ( yes it is predicted) It comes from the numbers of electrons in a wire carrying a fixed current.

It seems to me that we should be able to calculate the apparent speed of the electrons in a wire by using the force between two parallel wires and applying SR, finding the length contraction and thus the change in charge density to explain the expected force.

It sounds ridiculous but the actual free charge that's sitting inside the wire is massive. It's just that a tiny fraction of it is entering and leaving at the ends in the form of the Current you measure with a meter. I have never done the sums myself but I take 'their' word for it.

 

[Charles Link]

It sounds ridiculous but the actual free charge that's sitting inside the wire is massive. It's just that a tiny fraction of it is entering and leaving at the ends in the form of the Current you measure with a meter. I have never done the sums myself but I take 'their' word for it.

@sophiecentaur The calculation you mentioned is readily performed. Copper has atomic weight 63.5 and atomic number 29, so that in one gram of Copper, the electrical charge of the protons is Q=+(29/64.5)(6.02 E+23)(1.602 E-19)= 4.5 E+4 Coulombs and an equal and opposite amount for the electrons. (If a current of 1 Ampere flows for 1 second, you get 1 Coulomb). (The 6.02 E+23 is Avagadro's number=the number of particles per mole. Number of Moles=Mass(in grams)/Atomic Wt.) The 1.602 E-19 is the proton or electron charge).

 

[sophiecentaur]

. . . . . and then there's the Lorenz sums to do. Let's face it, I am just idle.

 

[nasu]

@sophiecentaur The calculation you mentioned is readily performed. Copper has atomic weight 63.5 and atomic number 29, so that in one gram of Copper, the electrical charge of the protons is Q=+(29/64.5)(6.02 E+23)(1.602 E-19)= 4.5 E+4 Coulombs and an equal and opposite amount for the electrons. (If a current of 1 Ampere flows for 1 second, you get 1 Coulomb). (The 6.02 E+23 is Avagadro's number=the number of particles per mole. Number of Moles=Mass(in grams)/Atomic Wt.) The 1.602 E-19 is the proton or electron charge).

Actually in copper there is only 1 "free" electron per atom. So the result will be 29 times less.

 

[Charles Link]

Actually in copper there is only 1 "free" electron per atom. So the result will be 29 times less.

I computed the electrical charge "sitting in the wire". Yes, the charge involved in any current flow is 29 times less.

 

[sophiecentaur]

29 times less

I always get twitchy when I read a phrase like that. That way of putting things is very dodgy. Just perhaps, I / we know what you mean but why not use the Mathematical "one twenty ninth'? This is a relatively new phenomenon all goes back to the use of percentage reductions of prices in the Bank Holiday Sales etc., but that always refers to a Subtraction. You would never get 'a reduction of 101%'
I am not just being picky for the sake of it. It just adds confusion and I just wish that on PF, at least, we could avoid using the phraseology of the uninformed press and commercial market when discussing matters in which accuracy is vital.