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Is there a magnetic attraction between two parallel electron beams?

Two electron beams exert different forces on each other depending on frame of reference?Applying $nablatimesmathbfB = mu_0mathbfJ$ in the presence of magnetic shieldingTwo electron beams exert different forces on each other depending on frame of reference?Relativistic explanation of attraction between two parallel currentsMagnetic force between moving chargesWhy do magnetic coils consist of many thin wires?Can relativity explain the magnetic attraction between two parallel electrons or electron beams comoving in a vacuum? (No wires)Can free electrons in a cathode ray pair up?Why is there no B-field parallel force on an electron orbiting a magnetic field line?

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$begingroup$

I am refering to Ampere's force law, and to the beams accelerated after the cathode, so the deflection is not due to their respective cathode. In other words, do two electrons accelerating parallel to each other converge because of magnetic attraction?
Does it apply to wires only or to a beam of charge carriers too?
If coulomb interaction is stronger and make them diverge, is there at least a small attraction limiting it?

electromagnetism electrons

share|cite|improve this question

asked 8 hours ago

ExocytosisExocytosis

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$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  See physics.stackexchange.com/questions/71378/… and many others
  $endgroup$
  – Rob Jeffries
  2 hours ago
  

  
  
  
  

add a comment
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3

$begingroup$

I am refering to Ampere's force law, and to the beams accelerated after the cathode, so the deflection is not due to their respective cathode. In other words, do two electrons accelerating parallel to each other converge because of magnetic attraction?
Does it apply to wires only or to a beam of charge carriers too?
If coulomb interaction is stronger and make them diverge, is there at least a small attraction limiting it?

electromagnetism electrons

share|cite|improve this question

asked 8 hours ago

ExocytosisExocytosis

65511 silver badge1515 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  See physics.stackexchange.com/questions/71378/… and many others
  $endgroup$
  – Rob Jeffries
  2 hours ago
  

  
  
  
  

add a comment
 | 

$begingroup$

I am refering to Ampere's force law, and to the beams accelerated after the cathode, so the deflection is not due to their respective cathode. In other words, do two electrons accelerating parallel to each other converge because of magnetic attraction?
Does it apply to wires only or to a beam of charge carriers too?
If coulomb interaction is stronger and make them diverge, is there at least a small attraction limiting it?

electromagnetism electrons

share|cite|improve this question

asked 8 hours ago

ExocytosisExocytosis

65511 silver badge1515 bronze badges

$endgroup$

share|cite|improve this question

asked 8 hours ago

ExocytosisExocytosis

65511 silver badge1515 bronze badges

share|cite|improve this question

asked 8 hours ago

ExocytosisExocytosis

65511 silver badge1515 bronze badges

asked 8 hours ago

ExocytosisExocytosis

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asked 8 hours ago

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1 Answer
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8

$begingroup$

In the lab frame there is a magnetic attraction, but it will never overpower the Coulomb repulsion between the two beams.

This is easiest to see in a frame of reference which moves with the electrons themselves: there, the electrons are stationary, and the only force between them is the repulsive Coulomb force. That said, if the electrons are moving fast enough (and, since the problem is scale-free, any velocity is "fast enough"), special relativity will require some minor tweaks to how that repulsion is observed from the lab frame, because of effects coming from length contraction and time dilation.

In the lab frame, those relativistic corrections to the Coulomb repulsion can be interpreted as an additional force which is proportional to the velocities and to the charge of the electrons. This is what we know as the magnetic interaction between the two beams.

If you want to see this line of understanding in all its glory, I recommend the relativity-and-magnetism chapter ('The fields of moving charges') in Ed Purcell's Electricity and Magnetism.

share|cite|improve this answer

edited 3 hours ago

answered 7 hours ago

Emilio PisantyEmilio Pisanty

91.4k2323 gold badges230230 silver badges475475 bronze badges

$endgroup$

• 
  
  
  

  
  5
  

  
  
  
  
  
  

  
  $begingroup$
  A fun computation is to find the speed at which the magnetic attraction would exactly match the electrical repulsion between the two electron beams. If you're smarter than I am, you can argue from relativity that this critical speed must be $c$ or faster, because otherwise different observers would disagree about whether the beams were attracted or repelled by each other; I had to do all the Biot-Savart stuff to figure out that there wasn't any paradox.
  $endgroup$
  – rob♦
  7 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Emilio: very informative thanks, I will look for the book you refer to. Just so I get it right: the magnetic interaction due to relativistic corrections is the total magnetic component, or is it to be added to the classical one you seem to refer to in your first paragraph?
  $endgroup$
  – Exocytosis
  6 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The relativistic correction and the magnetic force are one and the same thing. All magnetic interactions (with the possible exception of intrinsic magnetic moments of elementary particles) are relativistic shifts of electrostatic forces.
  $endgroup$
  – Emilio Pisanty
  6 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Emilio: I downloaded the book, which is going to be useful in its own regard, browsing the contents. Special relativity although mentioned more than 600 times only makes an appendix but no such chapter, so I guess you are refering to another edition, which is it? (mine 3rd, 2013)
  $endgroup$
  – Exocytosis
  6 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It should be chapter 5 on all editions - 'The fields of moving charges'.
  $endgroup$
  – Emilio Pisanty
  6 hours ago
  

  
  
  
  

add a comment
 | 

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8

$begingroup$

In the lab frame there is a magnetic attraction, but it will never overpower the Coulomb repulsion between the two beams.

This is easiest to see in a frame of reference which moves with the electrons themselves: there, the electrons are stationary, and the only force between them is the repulsive Coulomb force. That said, if the electrons are moving fast enough (and, since the problem is scale-free, any velocity is "fast enough"), special relativity will require some minor tweaks to how that repulsion is observed from the lab frame, because of effects coming from length contraction and time dilation.

In the lab frame, those relativistic corrections to the Coulomb repulsion can be interpreted as an additional force which is proportional to the velocities and to the charge of the electrons. This is what we know as the magnetic interaction between the two beams.

If you want to see this line of understanding in all its glory, I recommend the relativity-and-magnetism chapter ('The fields of moving charges') in Ed Purcell's Electricity and Magnetism.

share|cite|improve this answer

edited 3 hours ago

answered 7 hours ago

Emilio PisantyEmilio Pisanty

91.4k2323 gold badges230230 silver badges475475 bronze badges

$endgroup$

• 
  
  
  

  
  5
  

  
  
  
  
  
  

  
  $begingroup$
  A fun computation is to find the speed at which the magnetic attraction would exactly match the electrical repulsion between the two electron beams. If you're smarter than I am, you can argue from relativity that this critical speed must be $c$ or faster, because otherwise different observers would disagree about whether the beams were attracted or repelled by each other; I had to do all the Biot-Savart stuff to figure out that there wasn't any paradox.
  $endgroup$
  – rob♦
  7 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Emilio: very informative thanks, I will look for the book you refer to. Just so I get it right: the magnetic interaction due to relativistic corrections is the total magnetic component, or is it to be added to the classical one you seem to refer to in your first paragraph?
  $endgroup$
  – Exocytosis
  6 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The relativistic correction and the magnetic force are one and the same thing. All magnetic interactions (with the possible exception of intrinsic magnetic moments of elementary particles) are relativistic shifts of electrostatic forces.
  $endgroup$
  – Emilio Pisanty
  6 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Emilio: I downloaded the book, which is going to be useful in its own regard, browsing the contents. Special relativity although mentioned more than 600 times only makes an appendix but no such chapter, so I guess you are refering to another edition, which is it? (mine 3rd, 2013)
  $endgroup$
  – Exocytosis
  6 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It should be chapter 5 on all editions - 'The fields of moving charges'.
  $endgroup$
  – Emilio Pisanty
  6 hours ago
  

  
  
  
  

add a comment
 | 

8

$begingroup$

In the lab frame there is a magnetic attraction, but it will never overpower the Coulomb repulsion between the two beams.

This is easiest to see in a frame of reference which moves with the electrons themselves: there, the electrons are stationary, and the only force between them is the repulsive Coulomb force. That said, if the electrons are moving fast enough (and, since the problem is scale-free, any velocity is "fast enough"), special relativity will require some minor tweaks to how that repulsion is observed from the lab frame, because of effects coming from length contraction and time dilation.

In the lab frame, those relativistic corrections to the Coulomb repulsion can be interpreted as an additional force which is proportional to the velocities and to the charge of the electrons. This is what we know as the magnetic interaction between the two beams.

If you want to see this line of understanding in all its glory, I recommend the relativity-and-magnetism chapter ('The fields of moving charges') in Ed Purcell's Electricity and Magnetism.

share|cite|improve this answer

edited 3 hours ago

answered 7 hours ago

Emilio PisantyEmilio Pisanty

91.4k2323 gold badges230230 silver badges475475 bronze badges

$endgroup$

• 
  
  
  

  
  5
  

  
  
  
  
  
  

  
  $begingroup$
  A fun computation is to find the speed at which the magnetic attraction would exactly match the electrical repulsion between the two electron beams. If you're smarter than I am, you can argue from relativity that this critical speed must be $c$ or faster, because otherwise different observers would disagree about whether the beams were attracted or repelled by each other; I had to do all the Biot-Savart stuff to figure out that there wasn't any paradox.
  $endgroup$
  – rob♦
  7 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Emilio: very informative thanks, I will look for the book you refer to. Just so I get it right: the magnetic interaction due to relativistic corrections is the total magnetic component, or is it to be added to the classical one you seem to refer to in your first paragraph?
  $endgroup$
  – Exocytosis
  6 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The relativistic correction and the magnetic force are one and the same thing. All magnetic interactions (with the possible exception of intrinsic magnetic moments of elementary particles) are relativistic shifts of electrostatic forces.
  $endgroup$
  – Emilio Pisanty
  6 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Emilio: I downloaded the book, which is going to be useful in its own regard, browsing the contents. Special relativity although mentioned more than 600 times only makes an appendix but no such chapter, so I guess you are refering to another edition, which is it? (mine 3rd, 2013)
  $endgroup$
  – Exocytosis
  6 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It should be chapter 5 on all editions - 'The fields of moving charges'.
  $endgroup$
  – Emilio Pisanty
  6 hours ago
  

  
  
  
  

add a comment
 | 

$begingroup$

In the lab frame there is a magnetic attraction, but it will never overpower the Coulomb repulsion between the two beams.

This is easiest to see in a frame of reference which moves with the electrons themselves: there, the electrons are stationary, and the only force between them is the repulsive Coulomb force. That said, if the electrons are moving fast enough (and, since the problem is scale-free, any velocity is "fast enough"), special relativity will require some minor tweaks to how that repulsion is observed from the lab frame, because of effects coming from length contraction and time dilation.

In the lab frame, those relativistic corrections to the Coulomb repulsion can be interpreted as an additional force which is proportional to the velocities and to the charge of the electrons. This is what we know as the magnetic interaction between the two beams.

If you want to see this line of understanding in all its glory, I recommend the relativity-and-magnetism chapter ('The fields of moving charges') in Ed Purcell's Electricity and Magnetism.

share|cite|improve this answer

edited 3 hours ago

answered 7 hours ago

Emilio PisantyEmilio Pisanty

91.4k2323 gold badges230230 silver badges475475 bronze badges

$endgroup$

share|cite|improve this answer

edited 3 hours ago

answered 7 hours ago

Emilio PisantyEmilio Pisanty

91.4k2323 gold badges230230 silver badges475475 bronze badges

answered 7 hours ago

Emilio PisantyEmilio Pisanty

91.4k2323 gold badges230230 silver badges475475 bronze badges

answered 7 hours ago

Emilio PisantyEmilio Pisanty

91.4k2323 gold badges230230 silver badges475475 bronze badges