About Lorentz force and Lenz' law

[The_Logos]

Hello Physics forum, I have come here seeking for some experienced help about a doubt that to some extent is not letting me advance on my studies of electromagnetism.
Basically is about the magnetic (B) field, and more specifically, about the force at each point of the vector field (The lorentz force), the thing is that I find no explanation for the equation or at least, not about the cross product that is used for the magnetic part of the force and it makes me very uneasy to continue studying all the other laws (For example, Faraday's law. ) that require such a basic definition of the magnetic field, without undestanding the why of such mathematical definition. to this point I have been able to understand most of the logic of most of the variables and functions of the system, except for this one, i don't know if anybody knows of an explanation of why the magnetic force is equal to vxB? Does it requires more advanced knowledge? or maybe it is just an empirical law without an actual mathematical proof or process to be obtained?

This last possibility leads me to my other question, which is a little less problematic. Does the minus sign of Lenz's Law has some kind of mathematical proof, or it is an experimental after thought?

Thanks beforehand.

 

[hjelmgart]

The way I understand the cross product is simple. You have vectors as you work in 3d space. The direction of the magnetic field may influence the direction of the particle, and thus you need the cross product to find the direction it moves after being affected by the magnetic field with respect to its initial direction.

The negative sign in Lenz law is to my interpretation just to illustrate, that the EMF is opposing the initial direction of the magnetic field. So I am guessing it is based on observations, simply put.

 

[athosanian]

a moving charge forms current and the current will have interaction with the magnetic field. consider a long straight cable with length l in a magnetic field B, the force on the cable will be F=I　lXＢ．where X means cross product, it indicates the force on the wire is perpendicular to the wire length and magnetic field.
if we consider only one charge in the cable, we derive the Lorentz force on the charge.

 

[The_Logos]

I understand that a charged particle moving at certain velocity is going to experiment changes on his velocity vector because of the influence of both the magnetic and the electric field on the surface.

What I don't really find logical by itself is that such influence is not going to be in the direction of the (magnetic) force at a certain point of the field, but perpendicular to both, the velocity vector of the particle and the magnetic force at the point (speaking about the magnetic part of the Lorentz force, the electrical part of the resulting force, qE seems quite straight forward). I however haven't managed to find a mathematical explanation for the vXB nor have I managed to understand the logic of it by myself, ¿Why would the influence be perpendicular, and not on the expected force direction on that point of the vector field, or for that matter, on any other direction?

 

[Nugatory]

I understand that a charged particle moving at certain velocity is going to experiment changes on his velocity vector because of the influence of both the magnetic and the electric field on the surface.

What I don't really find logical by itself is that such influence is not going to be in the direction of the (magnetic) force at a certain point of the field, but perpendicular to both

If you google around, you will find Edward Purcell's explanation of magnetism as the result of relativistic length contraction. This is, as far as I know, the only intuitive explanation for why the Lorentz force acts perpendicularly to the direction of motion.

 

[hjelmgart]

What I don't really find logical by itself is that such influence is not going to be in the direction of the (magnetic) force at a certain point of the field, but perpendicular to both, the velocity vector of the particle and the magnetic force at the point (speaking about the magnetic part of the Lorentz force, the electrical part of the resulting force, qE seems quite straight forward). I however haven't managed to find a mathematical explanation for the vXB nor have I managed to understand the logic of it by myself, ¿Why would the influence be perpendicular, and not on the expected force direction on that point of the vector field, or for that matter, on any other direction?

I no longer understand your problem. You are here concerned with the lorenz force of a particle, but it seems to me like you are mixing it up with electromagnetic waves (light).

The magnetic force does not have to be perpendicular to the direction of a particle moving through a magnetic field when concerned with the Lorenz force.
That is however the case, if you are looking at electromagnetic waves, which is light- not particles - and thus moves as waves and not like a particle, which the equation is about, and thus what explains the cross product.

 

[The_Logos]

If you google around, you will find Edward Purcell's explanation of magnetism as the result of relativistic length contraction. This is, as far as I know, the only intuitive explanation for why the Lorentz force acts perpendicularly to the direction of motion.

Gonna look into that, how advanced is the math?

I no longer understand your problem. You are here concerned with the lorenz force of a particle, but it seems to me like you are mixing it up with electromagnetic waves (light).

The magnetic force does not have to be perpendicular to the direction of a particle moving through a magnetic field when concerned with the Lorenz force.

The problem, essentially, is to understand from where the cross product comes, what justifies it.
I might be wrong, but given the variables of the lorentz force (charge in coulumbs, and velocity) it would seem that we are talking about particles (As I undertand the B-field is measured as the force that is exerted on the particles that travels on it, besides that, the faraday's law for example is a surface integration of such field when charged particles goes trought it, generating the known effect.) and by the very definition of the cross product, in this case the resulting force vector of (vxB) should be a perpendicular one to both the B-field at that point and the velocity vector.

That is however the case, if you are looking at electromagnetic waves, which is light- not particles - and thus moves as waves and not like a particle, which the equation is about, and thus what explains the cross product.

Would you mind giving the explanation (or reference) for the cross product in such a case? that might be the key to the situation in any case (given the reference that Nugatory gave me, it seems to be related to light, at least, a possible explanation)

 

[Nugatory]

Gonna look into that, how advanced is the math?

Purcell's text is suitable for the end of the first year of an undergraduate physics course.
For an intuitive understanding of where the perpendicular action comes from, all you'll need is a basic understanding of special relativity and some googling around. If you want to actually work problems or understand anything beyond the easy cases of constant motion in a straight line you'll also want at least a nodding acquaintance with vector and multi-variable calculus.

 

[hjelmgart]

I was brain farting, lol. You are only concerned about the cross product and not the contribution from the electric field, ok. So it's not really the Lorenz force, that is the problem, it is the magnetic force. I made it through my electromagnetism course without the relativistic understandings, so you can do it too!

The way I se it, is that the magnetic field consist of a lines between the poles. These lines are in fact everywhere, and whenever a particle pass through one, the line exerts a force on the particle. If the particle go along the lines it doesn't pass through any, and thus no force is exerted on it. Thus a cross product can be used to describe it, as the force must always point away from the magnetic line with respect to the movement of the particle.

Regarding electromagnetic waves, you will get to this later in your course, as you need all of Maxwell's laws to understand it fully (IMO).

 

[The_Logos]

Purcell's text is suitable for the end of the first year of an undergraduate physics course.
For an intuitive understanding of where the perpendicular action comes from, all you'll need is a basic understanding of special relativity and some googling around. If you want to actually work problems or understand anything beyond the easy cases of constant motion in a straight line you'll also want at least a nodding acquaintance with vector and multi-variable calculus.

Well if the only math needed is up to vector calculus there should be no much problem, but if they were to start using tensors it might take a little longer to get out if this part of my studying process, hahaha. though, for the moment i would prefer no to deviate more than necessary of classic electromagnetism.

I was brain farting, lol. You are only concerned about the cross product and not the contribution from the electric field, ok. So it's not really the Lorenz force, that is the problem, it is the magnetic force. I made it through my electromagnetism course without the relativistic understandings, so you can do it too!
[...]
Regarding electromagnetic waves, you will get to this later in your course, as you need all of Maxwell's laws to understand it fully (IMO).

Yes, i actually assumed that there would be not much problem continuing studying without really understanding the magnetic part of the Lorentz force, however, I prefer to not leave a single stone unturned on my way of understanding, to the point that it makes me uncomfortable to use knowledge that I can't say i understand (Thought, given the nearly infinite chain of questioning that one can make about anything, I would doubt about saying that i understand anything hahaha.)

Actually, I am not in any course, just learning with the help of various books, that's why I have the liberty to wander a lot around any idea.

Now, given that the best possible explanation for this phenomenon lies within relativistic physics, it makes me wonder, how classical electromagnetism explained it, conceived it, and applied it so freely? Did they just perceived it empirically and took it as an unexplained axiom and ran with that?

 

[Nugatory]

Now, given that the best possible explanation for this phenomenon lies within relativistic physics, it makes me wonder, how classical electromagnetism explained it, conceived it, and applied it so freely? Did they just perceived it empirically and took it as an unexplained axiom and ran with that?

Pretty much, yes.

That's usually how science works, as the answer to any "Why?" question will invite another "Why? question and the only way to end the infinite regression is to accept that some empirical fact just is what it is.

We often don't even notice the paucity of "Why?" answers out there. For example, generations of high school students have happily learned that Newton's law of gravitation "explains gravity"; it does no such thing, it just describes it with simple, useful, and powerful mathematical language. $F=Gm_1m_2/r^2$ has worked so well for so many centuries and solved so many problems for us that we tend to overlook that we "perceived it empirically and took it as an unexplained axiom and ran with that" as well.

 

[The_Logos]

Pretty much, yes.

That's usually how science works, as the answer to any "Why?" question will invite another "Why? question and the only way to end the infinite regression is to accept that some empirical fact just is what it is.

We often don't even notice the paucity of "Why?" answers out there. For example, generations of high school students have happily learned that Newton's law of gravitation "explains gravity"; it does no such thing, it just describes it with simple, useful, and powerful mathematical language. $F=Gm_1m_2/r^2$ has worked so well for so many centuries and solved so many problems for us that we tend to overlook that we "perceived it empirically and took it as an unexplained axiom and ran with that" as well.

You are undoubtly right, to progress on practical terms, at some point is necessary to establish axioms, even if they are quite arbitrary, however at the same time, the opposite manages to be true, it becomes neccesary at some point to critic this axioms to generate new hipotesis and (sometimes) better systems, another interesting double nature, hahaha.

Thought, I still believe that Maxwell and friends had at their time an hipotesis for this phenomenon, even if a more fundamental critic of their peception will show us a new perspective, thanks to Plack, creating our modern physics. I mean, we human as a whole at a time can be very meticolous with how much we justify about our epistemologies, Aristotele and Platon when to the length of inventing the occidental god to justify rational behavior of nature, idea that even after copernicus was still used until who knows what critic finally brought down the need of that god. Damn, I bet that Newton had his explanation (probably involving god at a fundamental level) for gravity, sadly I only found on the web a latin version of his "principia" and of course, could not read it.
Maybe anyone has read or has access to something directly wirtten by Maxwell or whoever invented the principles of his systems?