Lorentz force acting upon an electron moving in a circle

[Ale_Rodo]

So as the summary suggests, I am studying Electromagnetism, magnetic properties of matter and Magnetization vector in particular.

As a first example and to introduce the Magnetization vector (M), my textbook shows a ferromagnetic substance in a uniform magnetic field (B).
Then, every atom of the substance is oversimplified as a single electron moving in a circle, having its own magnetic moment (m) macroscopically being zero because of thermic agitation when no B is applied. When we apply said B, all those ms will point averagely in one direction, creating a macroscopic magnetic moment m_tot≠0.

My question is:
when no B is applied, a single atom of the ferromagnetic substance will still move in a circle (which I know is a simplification) and will have a centripetal acceleration (a_c) with magnitude a_c = v²/R, with R being the radius of such a circumference and v the speed of the electron.
If we now apply B properly, the Lorentz force should act on the electron in such a way that a_c increases, and I assume that for this reason also v increases.

But if v increases then it varies with time in such a way that causes the Lorentz force to increase, leading to an indefinite loop that causes the speed to diverge to infinity.

Can someone please explain what is wrong with this reason? Thank you in advance.

 

[Dale]

Summary:: Studying Electromagnetism I happened to be blocked by a doubt: if a magnetic field is applied to an already circulating electron, shouldn't it's speed increase to infinity?

If we now apply B properly, the Lorentz force should act on the electron in such a way that ac increases,

No. The centripetal acceleration (insofar as it exists at all) is a basic property of the atom itself. It does not change in response to any external condition.

 

[Ale_Rodo]

Thank for the answer but I think I didn't understand. To me it seems illogical that an increase in centripetal acceleration which magnitude, no matter which centripetal force is acting, should be a_c=v²/R doesn't follow an increase in that same "v", which is the electron's speed.

I understand that a force which is always perpendicular to the electron's trajectory can't increase its speed, but at the same time I can't find the analytical flaw in my reasoning.

Of course, all my claims are intended when R is constant.

 

[Ale_Rodo]

I'll try to translate to maths what I'm trying to say:

Say we have a B perpendicular to the circular trajectory of an electron which before being drenched in said magnetic field, had a centripetal acceleration a_c1=v_e²R.

The electron is flowing in a single circular wire (coil) which is ideal and non-deformable. An undefined centripetal force keeps the electron moving through the wire.

The effect of the B sees the Lorentz force acting in such a way that the total centripetal acceleration is now bigger than it was before. The Lorentz force's contribute is the quantity a_c2.

The total acceleration is now a_c,tot=a_c1+a_c2 and because it is still a centripetal force, to me it makes sense that all this expression must equal v_new²/R.

This is it. Isn't the only option for the initial speed v here to have increased up to v_{e_new} to justify the increment in centripetal acceleration? Also this increment in speed shouldn't cause an increment in the Lorentz force, which as we know depends on the electron's speed vector (this loop thing is what makes less sense)?

 

[Dale]

To me it seems illogical that an increase in centripetal acceleration which magnitude, no matter which centripetal force is acting, should be ac=v2/R doesn't follow an increase in that same "v", which is the electron's speed.

The thing is that there isn’t really any centripetal acceleration at all. The electron is in a stationary state which has charge and angular momentum and therefore has a magnetic moment. The magnetic moment exists and is essentially fixed in magnitude. The equivalent centripetal acceleration can be calculated from the magnetic moment, but it is a fiction.

You can tell that the centripetal acceleration is a fiction because if it were real then the charge would radiate. Since the charge is not radiating you know that it is not really accelerating.

The thing that is actually present is the magnetic moment. That is a consequence of the charge and the angular momentum, both of which are intrinsic quantized properties of the particle. As such it is fundamentally fixed and cannot vary.

So the acceleration that you calculate is not going to behave normally since it is a fiction. That is what I meant above by insofar as it exists it is fixed. It is a fiction and it is based off the magnetic moment which is fixed. So the quantity you calculate and call the centripetal acceleration is also fixed.

 

[Ale_Rodo]

I understand up to the concept but being an undergrad I can't really imagine the math behind this.
Except for the magnetic moment being the real thing and the centripetal acceleration being an auxiliary tool to grasp this physics without going through quantum mechanics.

Can I say, then, that mathematically my doubt is correct but to solve this "paradox" we need quantum mechanics?

Also do I have to avoid using the concept of centripetal force only when talking about particles or does it have problems also macroscopically?

Thanks for your patience.

 

[Dale]

Can I say, then, that mathematically my doubt is correct but to solve this "paradox" we need quantum mechanics?

I would say that if the electron were a classical point particle then your doubt would be correct. It is indeed resolved by recognizing that the electron is not classical and behaves according to the rules of QM.

 

[Ale_Rodo]

Ok, I can live with that. I would have been more worried if nobody ever issued this problem and I casually found a hole in Classical Physics, which is just more than improbable. Thank you very much!

Just for my curiosity, where and when in Quantum Mechanics is this issued? I will probably not understand, but still...

 

[Dale]

This would be in the wavefunction for ferromagnetic elements like iron

 

[Ale_Rodo]

Ok thanks!

 

[DrStupid]

My question is:
when no B is applied, a single atom of the ferromagnetic substance will still move in a circle (which I know is a simplification) and will have a centripetal acceleration (a_c) with magnitude a_c = v²/R, with R being the radius of such a circumference and v the speed of the electron.
If we now apply B properly, the Lorentz force should act on the electron in such a way that a_c increases, and I assume that for this reason also v increases.

Let me see if I understand correctly. In order to avoid quantum effects, let's say you have an elecron orbiting a positive charge on a circular path with radius 1 m and velocity 1 m/s. Now you apply a magnetic field in addition to the electrostatic field. Does that fit to your scenario?

But if v increases then it varies with time in such a way that causes the Lorentz force to increase, leading to an indefinite loop that causes the speed to diverge to infinity.

Can someone please explain what is wrong with this reason?

You seem to assume that the electron would remain in a circular path. But the orbit would become elliptical with some kind of apsidal precession. The speed of the electron does not depend on the magnetic field but on the electrostatic field only (with constant total energy).

 

[A.T.]

You seem to assume that the electron would remain in a circular path. But the orbit would become elliptical with some kind of apsidal precession. The speed of the electron does not depend on the magnetic field but on the electrostatic field only (with constant total energy).

Also, even if the electron remains in some circular path (no central charge, just magnetic fields of different strengths), that doesn't mean that v has to change with a, but that R changes.

 

[Ale_Rodo]

Let me see if I understand correctly. In order to avoid quantum effects, let's say you have an elecron orbiting a positive charge on a circular path with radius 1 m and velocity 1 m/s. Now you apply a magnetic field in addition to the electrostatic field. Does that fit to your scenario?

The scenario is quite right, although I imagined one where the electron is just spinning for some unknown reason, not really around an atom. Actually imagine there's a single coil, no batteries connected, the wire is ideal and so on. You have somehow an electron moving into the coil because of a generic centripetal force and therefore moving with uniform circular motion (thus it has an initial centripetal acceleration proportional to the square of the velocity, v_initial, over the coil's radius R).

If now you light a magnetic field perpendicular to the surface of said coil, a Lorentz force should be added to the previous centripetal force (which is unknown and moved the electron in the coil for unspecified reasons).

The total centripetal force, then, must have increased (radius, being the coil's one, is constant and can't therefore vary in any way).
We know that this total centripetal force still is a_c=v_final²/R. So, although it is indeed impossible because said force is perpendicular to the velocity, the magnitude of the velocity must be higher than it was before, leading to this kind of paradox.

This morning, by the way, I happened to come to a conclusion that could solve this. It has been said that my doubt is correct but quantum mechanics would fix it, and I could live with that. But I also said: "the coil is a solid conductor, so maybe the Lorentz force is balanced by the coil's 'internal wall' " and this would mean no increasing centripetal acceleration => no increase in velocity's magnitude => no infinite loop that generates infinite current => no Nobel Prize for me :) .

Could this be it?

 

[Ale_Rodo]

Also, even if the electron remains in some circular path (no central charge, just magnetic fields of different strengths), that doesn't mean that v has to change with a, but that R changes.

Yeah, perhaps I didn't really said it all in the question. I meant to get to a specific case (which required an electron flowing in a coil, so R would be fixed) but then I got caught from the answers all of you kindly provided and probably strayed away from my point. I should have clarified everything (and proposed a solution to my own doubt) in a reply to "DrStupid".

 

[A.T.]

I meant to get to a specific case (which required an electron flowing in a coil, so R would be fixed)

In a coil wire there are other charges that interact with the electron. Its acceleration is not just the product of the external fields.