Why do electrons and positrons exhibit opposite helical motion in a magnetic field?

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Solution 1

But what we never seem to see is why the electron and positron move the way that they do. Saying "they move like they do because of the force on them" doesn't explain anything at all. It's a non-answer.

The equation of motion for charge particle (electron,positron) in magnetic field is

$$ m\frac{d}{dt}\left(\frac{\mathbf v}{\sqrt{1-\frac{v^2}{c^2}}}\right) = q\mathbf v \times \mathbf B(\mathbf r,t) $$

where $\mathbf r$ is position of the particle, $\mathbf B(\mathbf r, t)$ is magnetic induction of external field at this position and time, $q,m$ are charge and mass and $\mathbf v$ is velocity of the particle.

For uniform $\mathbf B$, this equation has solutions that describe helical motion, in agreement with observations. This is an explanation of the helical trajectories; circular motion is a special case of this helical motion.

Can anybody explain why there's this rotational force, and why it rotates the electron path one way and the positron path the other?

Electron has electric charge $q=-1.6\times 10^{-19}$ C (by convention, electron is ascribed negative charge). Positron is ascribed $q=1.6\times 10^{-19}$ C. It is this difference in sign which leads to opposite directions of magnetic force. Imagine electron and positron far from each other, having the same velocity in the same uniform field. Since the magnetic forces acting on the two particles have the same magnitude but opposite directions, the particles will deflect with same rate but to opposite directions. Thus the helices they follow are left-handed and right-handed.

Solution 2

There are two factors at play here.

  1. The Lorentz force which causes the paths to bend with a radius proportional to the particles velocity and with a sense that dependent on both the particles charge and the direction of the particles velocity. In high energy (compared to $m_e$ events) such as the one pictured, the particles are nearly co-linear at the start. Note that there is nothing special about leptons (electrons and positrons) this way, other charged particles also experience this force and obey the same set of rules.

  2. The energy loss suffered by the particles in the detector medium (PDF-link, I'm afraid) which causes their momentum to fall steadily. This explains the spiral rather circular nature of the observed tracks.

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Updated on October 30, 2020

Comments

  • John Duffield
    John Duffield about 3 years

    When you throw an electron through a solenoid, it moves helically around the field lines, as per this schoolphysics illustration:

    enter image description here © Keith Gibbs 2013

    Then if we were to throw a positron through the solenoid, it would also move helically, but "the other way". One could liken their paths to left-handed and right-handed screw threads. We can see these paths in bubble-chamber pictures like this one from the BC website which anna referred to in a previous answer:

    enter image description here

    Now, we can read about this sort of thing in various textbooks, such as this. And I'm sure we all know that the force acting on an electron is perpendicular to the magnetic field lines, and we can read about the Lorentz force and the right hand rule. But what we never seem to see is why the electron and positron move the way that they do. Saying "they move like they do because of the force on them" doesn't explain anything at all. It's a non-answer. Can anybody explain why there's this rotational force, and why it rotates the electron path one way and the positron path the other? Why do electrons and positrons exhibit opposite helical motion in a magnetic field?