# Magnetic Field of High Speed Ions

Staff Emeritus
2018 Award

## Main Question or Discussion Point

Hey all. Just a quick question. Would high speed positive ions in a vacuum create a magnetic field that would attract other positively charged particles? If so, why is that? I understand that moving charges creat magnetic fields, but I thought that in the absence of a conductor or opposite charges the field would be...different. I was thinking it had something to do with the explanation of magnetism by relativity and how the electric field is warped at high speeds. As I see it no matter what you do you would only increase the repulsion by warping the electric field. Am I misunderstanding anything?

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the magnetic attraction will never exceed the electric repulsion because there is always a frame in which they are stationary with respect to each other and in that frame they will repel

Staff Emeritus
2018 Award
the magnetic attraction will never exceed the electric repulsion because there is always a frame in which they are stationary with respect to each other and in that frame they will repel
What do you mean? What frame could there be where both the high speed ions and the low speed ions are stationary?

i mean 2 ions moving in the same direction and at the same speed

The situation is no different to having two wires in parrallel proximity

Remember from the text books

Electric current can be related to the number density of electrons in a conductor and the
speed with which they move by:

I = nqvdA

where n is the number density of charge carriers (electrons, usually), q is the value of their
charge, vd is the drift velocity, the speed with which the carriers actually move in the wire
(on average) and A is the cross-sectional area of the wire.

The attractive and repelling forces work the same whether the charge is in wire or partciles in the vacuum of space.

So going the same way they will attract going opposite ways they will repel for the same reasons as wires transfer of charge do.

Staff Emeritus
2018 Award
How could they attract if there are no particles of the opposite charge? From my understanding, in parallel wires from the frame of the electrons it is the positive charges that are moving and therefor the ones that are length contracted, resulting in a positive force felt by each wire. How can that be if you have ions in empty space with no negative charges? In their frame they shouldn't feel any attractive force I thought.

The problem was taught as a classical physics question so I don't know how to see it in terms of moving frames.

The question was usually posed thus

Q. Two electrons separated by distance "r" move side by side along straight paths with equal velocities. Find the electric and magnetic forces each exerts on the other.

The classic analysis goes:

(Force Electric) Fe = Keq1q1)/(r*r)

Magnetic field from q1 as seen at q2:

Bq2 = Km(-v1q1)/(r*r)

Force of magentic attraction from q2 towards q1

Fm = q2v2Bq2

Subsititute and simplify

Fm = -Kmq1q2v1v2/(r*r)

We Know Km = Ke/(c*c) so substituting that in

Fm = -Keq1q2v1v2/(r*r*c*c)

So there are your two force Fe and Fm.

I do see a problem with this however the charges are moving together how can the charge be considered to be moving through the others magnetic field. Maybe the classic physics answers we see in the text books is wrong I sort of went to the classical answer without even thinking!

Very interesting question now you bring it up.

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Interesting question.
Suppose we have a positive line charge $\lambda$ moving through space at speed v . So our current will be $I= \lambda v$
And our electric field will be $E=\frac{\lambda}{2\pi\epsilon_0r}$
from Gauss's law . And if we have a test charge moving at a speed next to the line charge. In this frame the moving proton will see the line charge coming at him with more speed and also an increased E field from the length contraction, and the line charge will go up by a factor of gamma. And we have to use the Lorentz velocity addition rule for the relative speeds between the proton and the line charge. And $E\cdot B$ will be the same in all frames. And we could pick a frame where there is just an E field and it would get repelled.
And then I would use amperes law to find the B field and it should be $B= \frac{\mu_0 \lambda v}{2\pi r}$ And Im not sure which way the current goes, if protons are going to the right would we say that the current is to the left. opposite of electrons.
And i should have been more careful but my $\lambda 's$ and v's depend on your frame.

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Staff Emeritus
2018 Award
Ok, so the proton would always be repelled from the ion beam? Looks to me like there is never a frame that either the ion beam or the proton would somehow feel an attractive force.

Ya i think so.

ZapperZ
Staff Emeritus
2018 Award
Hey all. Just a quick question. Would high speed positive ions in a vacuum create a magnetic field that would attract other positively charged particles? If so, why is that? I understand that moving charges creat magnetic fields, but I thought that in the absence of a conductor or opposite charges the field would be...different. I was thinking it had something to do with the explanation of magnetism by relativity and how the electric field is warped at high speeds. As I see it no matter what you do you would only increase the repulsion by warping the electric field. Am I misunderstanding anything?
Er... why would ions be any different than, say, electrons? After all, all we care about here is the charge.

Electrons are essentially at c after it reaches a KE of the order of MeV (all the dynamical equations treat it as being c at this energy). Yet, no such "attraction" are ever observed for bunches of electrons in particle accelerators. In fact, you have to deal with space-charge effects the larger the bunch of charge involved, which will cause the beam to "blow up" in size.

Zz.

Er... why would ions be any different than, say, electrons? After all, all we care about here is the charge.
Do I understand correctly, that this "Er..." means that in your opinion the question was dumb?

Electrons are essentially at c after it reaches a KE of the order of MeV (all the dynamical equations treat it as being c at this energy). Yet, no such "attraction" are ever observed for bunches of electrons in particle accelerators. In fact, you have to deal with space-charge effects the larger the bunch of charge involved, which will cause the beam to "blow up" in size.
If the electron bunch blows up with certain speed in rest, it will blow up slower when close to speed of light. This can be explained either by time dilation, or by magnetic forces (whose effect is attractive). Both explanations are equivalent, since the force calculations and frame change commute in special relativity.

ZapperZ
Staff Emeritus
2018 Award
Do I understand correctly, that this "Er..." means that in your opinion the question was dumb?
Er.. no. I'm just puzzled why ions are being used as an example, when electrons are more common in practice and we already know a lot about this. There are significantly more electrons accelerators in the world than there are protons or ion accelerators.

If the electron bunch blows up with certain speed in rest, it will blow up slower when close to speed of light. This can be explained either by time dilation, or by magnetic forces (whose effect is attractive). Both explanations are equivalent, since the force calculations and frame change commute in special relativity.
Yes, and your point being? At no point in the scenario does it turn around and start attracting. If they do that, FELs will have a "self-focusing" capability to reduce its emittance.

Zz.

I thought you meant that the attractive magnetic force component would not have been observed.

Dale
Mentor
Would high speed positive ions in a vacuum create a magnetic field that would attract other positively charged particles?
High speed positive ions would definitely create a magnetic field:
http://en.wikipedia.org/wiki/Liénard–Wiechert_potential

However, I don't think it is right to say that it "attracts" other positively charged particles. It does exert a force on positively charged particles, but that force depends on the velocity. I think I would use a word like "deflect" rather than "attract".

Staff Emeritus
2018 Award
Er... why would ions be any different than, say, electrons? After all, all we care about here is the charge.

Electrons are essentially at c after it reaches a KE of the order of MeV (all the dynamical equations treat it as being c at this energy). Yet, no such "attraction" are ever observed for bunches of electrons in particle accelerators. In fact, you have to deal with space-charge effects the larger the bunch of charge involved, which will cause the beam to "blow up" in size.

Zz.
I was unaware of the effect of either electrons or ions. I simply chose one.

High speed positive ions would definitely create a magnetic field:
http://en.wikipedia.org/wiki/Liénard–Wiechert_potential

However, I don't think it is right to say that it "attracts" other positively charged particles. It does exert a force on positively charged particles, but that force depends on the velocity. I think I would use a word like "deflect" rather than "attract".
Does it deflect them away from the moving ion, or towards it?

I agree assuming the classic analysis was correct and I suspect it was checked ion's will make no difference they carry charge was all the analysis used.

Two like charges moving parallel in the same direction should attract moving in opposite direction they should repel

Dale
Mentor
Does it deflect them away from the moving ion, or towards it?
Towards, away, or neither, depending on the velocity of the test charge.

Staff Emeritus
2018 Award
Towards, away, or neither, depending on the velocity of the test charge.
Hrmm...I'll have to look up more on this. I still don't see how a positive charge can attract another positive charge. Although I guess the relativistic view is probably not the full story, otherwise it doesn't look like an electromagnet should have 2 poles. At least to me.

Towards, away, or neither, depending on the velocity of the test charge.
I don't believe that one point charge could exert an attracting total force to an other point charge with the same charge sign, unless you can show an explicit construction of such geometric configuration. By "total force" I mean the electric and magnetic forces summed up.

Dale
Mentor
I don't believe that one point charge could exert an attracting total force to an other point charge with the same charge sign, unless you can show an explicit construction of such geometric configuration. By "total force" I mean the electric and magnetic forces summed up.
I was only talking about the magnetic force.

Dale
Mentor
Hrmm...I'll have to look up more on this. I still don't see how a positive charge can attract another positive charge.
If the source charge is moving in the +x direction at the origin then a test charge located on the +z axis will experience a magnetic field in the -y direction. The magnetic force can therefore be in any direction in the x-z plane, depending on the velocity of the test charge.

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