# Is it strange that there are two directions which are perpendicular to both field and current, yet the Lorentz force only points along one of them?

By "strange" I mean 'Is there a reason for this, or is it something we accept as a peculiarity of our universe?'

I see no reason why if magnetic field is in the $+x$ direction and a charge's velocity is in the $+y$ direction, that the force experienced by the charge can't be in either the $+z$ or $-z$ direction, both are perpendicular to $x$ and $y$. The equation for Lorentz force just tells us that it goes in the $+z$ direction, but it seems equally valid that the force would be in the $-z$ direction (I mean there's nothing to distinguish $+z$ from $-z$ anyway, you can turn one into the other by swapping the handedness of your coordinate system). It seems as if the universe is preferentially selecting one direction over the other.

As a practical example, if a current carrying wire is in a magnetic field and it experiences an upward force, why shouldn't it experience a downward force?

• Is it strange that there are two components of the nonzero real numbers, yet the electron's charge lies in only one of them? – WillO Oct 16 '17 at 17:08
• Are you saying that real numbers can be positive or negative either side of zero, but the electron is on the negative side, which means it has a sense of negative direction, and this sense of direction is then carried forward by the equation for Lorentz force, so that the force has a sense of direction? @WillO – Rational Function Oct 16 '17 at 17:14
• "I see no reason why if magnetic field is in the +x direction and a charge's velocity is in the +y direction, that the force experienced by the charge can't be in either the +z or −z direction" - the force on a positive (negative) charge is in the +z (-z) direction. – Alfred Centauri Oct 16 '17 at 17:24
• I see, yes that would be the extension of my first comment - if a negative charge has a sense of negative orientation and experiences a force in the negative direction, a positive charge has a sense of positive orientation and experiences a force in the positive direction. What I really meant to ask was why this relationship couldn't be flipped, and it seems that Floris' answer addresses this. @Alfred Centauri – Rational Function Oct 16 '17 at 17:32
• This brings to mind an answer I posted once about the nature of pseudovectors. – David Z Oct 16 '17 at 20:46

## 4 Answers

The universe is not preferentially selecting one direction over another. The fact that it appears that this is happening is an artifact of how we represent the magnetic field.

It is well-known that the existence of magnetic forces can be inferred from a Lorentz-invariant theory involving electric forces. For example, see this answer.

The magnetic force so derived necessarily has the property that parallel currents attract while antiparallel currents repel.

The magnetic field can be thought of as being the field that needs to be introduced into the theory in order to give a local description of this attraction between parallel currents. It is therefore necessary for the Lorentz force law to be written in such a way so that it gives the correct direction for the magnetic force between two currents. Otherwise the law would violate the observed Lorentz invariance of our universe. A law itself does not determine what actually happens; that can only be determined by experiment.

Because the direction of the magnetic field is assigned through a right-hand rule, a second application of the right-hand rule is needed in the Lorentz force law in order to get the correct direction for the actual force between the two currents. If the magnetic field direction were assigned through a left-hand rule, the Lorentz force law would also involve a left-hand rule. In neither case does the universe enforce an arbitrary choice of one over the other. We are simply describing the phenomenon in a way that requires us to put in the rule by hand in order to get the correct result.

This contrasts with the situation with weak interactions, which really do violate parity symmetry.

• I'm not sure this is completely correct: the universe does sometimes select one thing over the other and breakdown of symmetries do occur. Only the simple fact that there are two different types of charges is a manifestation thereof (and especially in electromagnetism there are many similar examples). – gented Oct 17 '17 at 7:59
• @GennaroTedesco The first sentence of this answer isn't meant to be a general statement to be taken out of context, it's meant to be an answer to this question. – JiK Oct 17 '17 at 9:01
• I think that the best (or simplest, at least) way to think about this is to visualize the magnetic current as circular. A current flows, and there is a circular magnetic field. We can think of it as the field per se not having any direction, but it has an orientation. The orientation is given by that current. A current flowing "into the page" has the magnetic field in one orientation, and current flowing "out of the page" has it in the other orientation. – Kaz Oct 17 '17 at 17:09
• ... then representing the orientation itself of the magnetic field as it having a particular direction is just a representational trick, involving an arbitrary choice of symbol: like whether we use a circle for zero and stick for one or vice versa. The orientation of the field is not separable from the current direction though; they are one and the same, so it is meaningless to ponder about one of them being flipped while the other stays. – Kaz Oct 17 '17 at 17:12
• @Kaz Do you know about exterior algebras? What you're hitting on here are the kinds of thoughts and intuitions that Élie Cartan had about 110 years ago as well as Hermann Grassmann before him. A vector is not the truest description of the magnetic field, even if you're not interested in manifestly Lorentz covariant Maxwell Equations. You might like this article ....... – WetSavannaAnimal Oct 17 '17 at 23:07

The magnetic field is not a [polar] vector, but a pseudovector. In fact, the cross-product of a vector (e.g. Velocity) and a pseudovector (Magnetic Field) is a [polar] vector (e.g. Force).

In a more abstract view, the magnetic field is better represented by a two-form or by a bivector [depending on your abstract point of view]. In 3 spatial dimensions, these abstract objects can be mapped to a pseudovector. In any case, it is this additional sense of orientation that prefers one "perpendicular" direction over the other.

You can see this distinction in the way the electric and magnetic fields are represented in the field tensor. The magnetic field components are in the entries of an antisymmetric 3-by-3 submatrix.

• Does this just arise from mathematical rules, and not have any underlying physical reason why the sense of orientation prefers one direction? @robphy – Rational Function Oct 16 '17 at 17:19
• It's not just mathematical rules... but mathematics is used to formulate the physics that is observed in nature. I'm not prepared to elaborate.. but I can offer these two references: arxiv.org/abs/physics/0005084 (A gentle introduction to the foundations of classical electrodynamics by Hehl) and scholarsarchive.byu.edu/cgi/… (Teaching electromagnetic field theory using differential forms by Warnick). – robphy Oct 16 '17 at 17:35
• Check out the links in my comments to Kaz on Brian Bi's answer. There are some lovely introductions to forms and exterior algebras around, written in the last ten years. – WetSavannaAnimal Oct 17 '17 at 23:14
• Okay, so the convention comes in when a bivector is mapped to a pseudo vector; you can choose a normal on either side of the plane of the bivector as long as you're consistent with the convention. The convention that humans chose corresponds to the right hand rule. Right? – EL_DON Oct 18 '17 at 4:32

You just discovered something pretty fundamental about the universe. It is conceivable that there would be a "mirror image universe" in which the laws of physics are exactly the other way around. But that's not the universe we live in.

There is an interesting Feynman lecture one the topic of symmetry - in particular, on the difficulty of trying to explain "clockwise" to someone without any visual aids. See this link

• I'll not sure if this is the correct answer, though maybe I'm misunderstanding. The sign of the magnetic field is intrinsically tied to the convention for the cross product. There's no universe where the electric force is the same but the magnetic force is backwards, relativity demands a certain relationship between them. – Javier Oct 16 '17 at 18:58
• @Dawood ibn Kareem: "Mirror image" is not a quality of any universe, it is a quality of the relationship between two universes. - When two universes are related in this way, which one is the "mirror image universe" depends on which one you're standing in. If you're not currently in any universe then the term remains undefined. – A. I. Breveleri Oct 16 '17 at 19:32
• I think this answer is somewhat misleading, actually, since the choice of right-handed vs left-handed orientation used in calculating magnetic fields is arbitrary, and is not something fundamental. – David Z Oct 16 '17 at 20:46
• @Kaz Of course there's an orientation. But that has nothing to do with having to choose a hand. The reason you have to choose a hand is that the framework we use to describe magnetic fields requires you to convert the inherent orientation of the magnetic field (a bivector) into an orientation of an axis (a single vector, or more precisely a pseudovector), and there are two ways to make that conversion. I'm just saying that choice isn't fundamental; it's an artifact of how we describe magnetic fields. – David Z Oct 17 '17 at 7:37
• @WetSavanna I'll be sure to tell my friend Humpty Dumpty. – Dawood ibn Kareem Oct 17 '17 at 23:19

... there are two directions which are perpendicular to both field and current, yet the Lorentz force only points along one of them

Your statement is correct for electrons and anti-protons. If one do the same experiment with positrons are protons he’ll observe that the move in the opposite direction.

I see no reason why if magnetic field is in the +x direction and a charge's velocity is in the +y direction, that the force experienced by the charge can't be in either the +z or −z direction.

The reason lays in the nature of the mentioned subatomic particles. All they obey the intrinsic property of a magnetic dipole moment and an associated intrinsic spin. The spin does not mean that the particle is rotating, the name points to the macroscopic phenomenon that is known as the gyroscopic effect.

The mechanism behind the Lorentz force is the following. If an electron is under the influence of an external magnetic field it’s magnetic dipole moment gets aligned. Nothing more happens. But if this electron is moving into the external magnetic field the alignment is accompanied by a sideway deflection like in the gyroscopic effect.

As you know any deflection is an acceleration and under acceleration a electron emits photons. This photon with its moment dis-align the magnetic dipole moment again and the cycle with alignment, gyroscopic effect, photon emission repeats until the kinetic energy of the electron is exhausted. So the electron moves in a spiral path or more precise in tangerine slices.

For the positron the process is the same except the direction of the emission of the photons. So

if a current carrying wire is in a magnetic field and it experiences an upward force,

With anti-matter

it experience a downward force.

• Tom, downvotes shows that somebody doesn't agree. So take my answer as a tail. But as long as nobody tells you a better one ... – HolgerFiedler Oct 18 '17 at 5:31

## protected by Qmechanic♦Oct 16 '17 at 21:40

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