How can I calculate the magnetic field of a relativistic muon?

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Discussion Overview

The discussion revolves around calculating the magnetic field generated by a relativistic muon, particularly focusing on the mathematical approach and theoretical considerations involved in such a calculation. Participants explore various aspects of the problem, including the effects of relativistic speeds and the detection of muons through their magnetic fields.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses difficulty with the mathematics involved in calculating the magnetic field produced by a muon moving at near light speed, seeking assistance.
  • Another participant suggests that the calculation is not straightforward and recommends consulting a good text on the subject.
  • A participant proposes that the magnetic field due to a negative muon current can be treated similarly to that of an electron moving at the same speed, outlining a step-by-step approach to the calculation.
  • Another participant expresses interest in whether a stream of energetic muons could be detected solely by the magnetic field they produce, speculating on the expected flux density.
  • A different viewpoint suggests that the problem may be overly complicated and proposes a simpler method of calculating the magnetic field using the current element of charge and velocity.
  • One participant recommends considering the time-dependent field of a single relativistic muon by first analyzing the muon at rest and then applying Lorentz transformations to understand the resulting magnetic field in a relativistic frame.
  • A note is added regarding the width of the magnetic field pulse produced by a muon at a certain distance, indicating a specific scenario with a gamma factor.

Areas of Agreement / Disagreement

Participants express differing views on the complexity of the problem and the methods to approach it. There is no consensus on a single method or solution, and multiple competing perspectives on how to calculate the magnetic field remain present.

Contextual Notes

Some participants highlight the need for a proper text or reference to clarify the calculations, indicating that assumptions and definitions may vary. The discussion also reflects uncertainty regarding the expected magnetic field strength and the conditions under which it can be detected.

magphys
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I know this is basic stuff but my maths is truly terrible. I hope someone can help.

Assuming you have a muon moving at near light speed, it will generate a magnetic field due to its movement. I'm assuming there is no external magnetuc field present. How can I calculate the field produced please?
 
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look up a good text .. it is not as straightforward as it seems
 
nirax said:
look up a good text .. it is not as straightforward as it seems

Probably explains why I can't find it on the web. I don't have a text with me. I was hoping someone might know ...
 
magphys said:
IAssuming you have a muon moving at near light speed, it will generate a magnetic field due to its movement. I'm assuming there is no external magnetuc field present. How can I calculate the field produced please?
First, the magnetic field due to the negative muon current is the same as an electron at the same speed.
Second, calculate the field due to 1 Coulomb of electrons per second.
Third, multiply this result by 1.6 x 10-19 Coulombs per electron.
Fourth (this is the hard part), considering relativistic contraction of the EM field of a relativistic charged particle, what is the observed magnetic field of a single charged particle as a function of time?
 
That will be fun!

What I am really interested is knowing whether you could detect a stream of energetic muons purely by the magnetic field they produce and, if so, what kind of flux density to expect. I'm guessing picoTesla (wild guess) or less?

Anyone guess what sort of magnetic field a cosmic ray muon might produce, for instance?
 
This is a classic lesson in making a problem too hard. You have a current element of qv, where q is the charge of a muon and v is the velocity. Calculate the magnetic field from that and you're done.
 
In order to completely understand the time-dependent field of a single relativistic muon, one way is first to consider a muon at rest, and then transform it to a relativistic reference frame using the Lorentz EM transformations. See the last four lines in:
http://pdg.lbl.gov/2009/reviews/rpp2009-rev-electromag-relations.pdf
The last four equations deal with the relativistic transformation of electric and magnetic fields. At rest, the muon has no external magnetic field. So only the last equation can transform an electric field of the muon at rest to a magnetic field of a relativistic muon.

[Added note] If you are 1 cm away from a muon with gamma = 10, the magnetic field pulse is picoseconds wide.
 
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