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

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

• Lorentz says: F = q *(v x B). So, force is perp. to field and velocity. Since positrons have positive and electrons have negative charge, the one turns right and the other left. Is that the answer, or are you asking why Lorentz is valid? – sweber Aug 23 '15 at 12:34
• So your question seems to be a roundabout way of asking for more detail about Lorentz force. The specific example of electrons and positrons in a magnetic field only serves to obfuscate this. – dukwon Aug 23 '15 at 12:34
• "Because of Lorentz force" and "Because they have opposite charge" are still correct, even if they're not complete. What you're asking for is a derivation of Lorentz force from first principles. This is hidden in a question that is mostly devoted to the consequences of Lorentz force. – dukwon Aug 23 '15 at 14:11
• If you take the attitude 'Saying "they move like they do because of the force on them" doesn't explain anything at all. It's a non-answer.' then science ultimately never provides any "answers", it only provides useful predictive capabilities. Which is fine as far as it goes, but does leave you needing certain verbal gymanstics to express yourself. – dmckee --- ex-moderator kitten Aug 23 '15 at 15:09
• "The field is the same, the particles aren't, so what's the difference?" I have to say that this sentence confuses me. The difference is that the particles are not the same, and in particular that they have opposite charges. All positively charged particle bend in the sense that positrons do and all negatively charged particles bend in the sense that electrons do. – dmckee --- ex-moderator kitten Aug 24 '15 at 21:03

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.

• Thanks for responding Ján, that's worth an upvote from me. But with respect, the equation of motion merely says "it will move like this". It doesn't say why. – John Duffield Aug 23 '15 at 13:16
• It seems that you're more interested in metaphysics than physics, if you insist on answering "why"-type questions, @JohnDuffield. – Danu Aug 23 '15 at 14:13
• Any specific reason that you ascribe an electron a positive and a positron a negative charge? – Merlin1896 Aug 23 '15 at 14:42
• @JohnDuffield, the equation itself does not imply how the trajectory looks like - the initial conditions are needed as well. The equation is a result of generalization and formalization of experience with electrons. It is a physics law, like Newton's equation of motion for planet motion. The ultimate answer to the question "why this physics law" is : experience. – Ján Lalinský Aug 23 '15 at 15:46
• @Danu : this is physics. We do physics to understand the world. Not to give up on understanding the world. Ján I'm afraid that it's the law is not enough. – John Duffield Aug 24 '15 at 11:36

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.

• Re your comment above, we do physics to understand the world, science does provide answers. – John Duffield Aug 23 '15 at 15:38
• Not by the definition of "answer" that you adopted in the question. Every single piece of science is founded—after you peel back the layers of increasingly fine exposition—on "this is the way the world is observed to behave", and you rejected an observation that rests directly on that foundation as "a non-answer". – dmckee --- ex-moderator kitten Aug 23 '15 at 16:16
• Yes, I reject that. When a child asks why does my pencil fall down, we do not say it's because this is the way the world is observed to behave. Not if we're scientists. – John Duffield Aug 24 '15 at 11:39
• "why does my pencil fall down" I would be extremely surprised if child asked this. Falling down is basic human experience children and most people experience daily, they have no need of explaining it. In any case, the answer to these kinds of questions is "nobody knows". – Ján Lalinský Aug 24 '15 at 18:26
• @Ján Lalinský : I know exactly why my pencil falls down. If you'd like to know too, ask the question, and I will answer it. And what I will not say, is because that's the law. – John Duffield Aug 25 '15 at 13:01

The electron has three well known properties, its electric charge, its magnetic dipole moment and its intrinsic spin. All three are constant quantities. And to prevent contradition about the reality of this intrinsic spin, it was shown in the Einstein-de-Haas experiment, that this spin really has to do with a rotation of the electron.

It has to be stated, that the magnetic dipole moment and the intrinsic spin in the electron are aligned. This is a very important fact for the following explanation.

Being under the influence of a magnetic field, the electron's magnetic dipole moment get aligned. If the electron is not moving or if the electron moved parallel to the magnetic field that is it, nothing more happens.

But if the electron moves non parallel to the external magnetic field there came in the game the electron intrinsic spin. Due to the torque induced precession (gyroscopic effect) a rotating body tries to resistant its deflection. One can feel this by deflecting a rotating wheel from a bicycle. The magnetic field align the magnetic dipole moment and this time the intrinsic spin too. The spin resists against the deflection and emitting photons go back in the direction of his previous state. This repeats many times until the electron comes to rest. So the sketch in your question is not complete. The electron moves in tangerine slices and of course the path is a spiral and ends with the electron in rest.

The positron has the same values of the magnetic dipole moment and the intrinsic spin, but the direction of the spin in relation to the direction of the magnetic dipole moment is opposite to this relation of electrons. This is the reason, the "why", you are asking for.

• The movement described in the OP and the Lorentz force are completely classical phenomena. They have nothing to do with spin. The Lorentz force acts on particles as in Jan Lalinsky's answer regardless of their spin. – ACuriousMind Aug 23 '15 at 14:31
• @ACuriousMind There are two interpretations, one well accepted and one new. Not to tell, where is the first mistake in my causality chain, is not is a scientist not worthy. I reviewed your answers and comments to this question and there was no mentioned any mistake in my interpretation. – HolgerFiedler Aug 23 '15 at 18:31
• @ACuriousMind : please offer your own answer instead of playing the naysayer. Holger has at least tried to answer the question, and offered evidence in the form of the Einstein de Haas effect, which "demonstrates that spin angular momentum is indeed of the same nature as the angular momentum of rotating bodies as conceived in classical mechanics." – John Duffield Aug 24 '15 at 11:43
• The Einstein-de-Haas experiment show that spin is angular momentum, which is a fact you'll find in every single treatment of the quantity. It does not show that it is $\mathbf{r} \times \mathbf{p}$, and moreover the quantization of spin disagrees with that of $\mathbf{r} \times \mathbf{p}$. – dmckee --- ex-moderator kitten Aug 24 '15 at 21:05