Particle Traveling in a Uniform Magnetic Field (SR?)

[RoyalCat]

Let a particle mass [tex]m[/tex] charge [tex]q[/tex] be placed in a uniform magnetic field [tex]\vec B = B\hat k[/tex] with an initial velocity [tex]v_0\hat i[/tex]

Clearly, it will enter a circular path of radius so and so with angular frequency so and so. But, it will not move in a straight line, not at all. Its momentum will constantly change.

Now, moving to a reference frame moving along the positive [tex]\hat i[/tex] direction with a constant velocity [tex]u[/tex] we find that the force on the particle, given by the Lorentz Force [tex]\vec F_b = q \vec v \times \vec B[/tex] has changed! That means that the change in (Classical) momentum, is not the same in both systems, which is what we would expect!

Focusing on the special case where [tex]u=v_0[/tex] we find that in the primed reference frame, there is no change in momentum at all!

I've done some reading and found that this apparent contradiction is reconciled by considering that the EM field is transformed relativistically and produces a new EM field that provides adequate results.

I could, however, only find quantitative analysis of the phenomenon (The actual transforms themselves) and qualitative analysis for a very specific situation (Current carrying wire) where the source of the new EM field was explained by length contraction and the formation of a net charge density. How this applies to the charge free region of space my question refers to eludes me.

A link or referral to a source describing the rationale behind the transforms would be much appreciated. (I only have the most basic understanding of SR and a moderate understanding of EM, so something appropriate would be wonderful)

Another thing that's been bothering me, is that this point to reflect on was given in our first lesson on Magnetism at school (The introduction to the Lorentz Force). Though it was unclear if we're supposed to be able and answer it (No one in my class has any knowledge of SR).

With thanks in advance, Anatoli. :)

 

[berkeman]

I don't know if it helps, but here's an interesting thread where the Lorentz force and reference frames were discissed:

https://www.physicsforums.com/showthread.php?t=359413

.

 

[kerimek]

I can provide a response to this content by explaining the phenomenon of a particle traveling in a uniform magnetic field in the context of special relativity (SR).

Firstly, it is important to understand that the Lorentz Force, given by the equation \vec F_b = q \vec v \times \vec B, is a classical equation and does not take into account the effects of special relativity. In other words, it only applies in non-relativistic situations where velocities are much smaller than the speed of light.

When a particle with mass m and charge q is placed in a uniform magnetic field \vec B = B\hat k, it will experience a force perpendicular to its velocity, causing it to move in a circular path with a radius determined by the magnitude of its velocity and the strength of the magnetic field. This is a well-known phenomenon in classical physics.

However, when we consider the same situation in a reference frame that is moving at a constant velocity u along the positive \hat i direction, we find that the force on the particle changes. This is due to the fact that the magnetic field is transformed under the principles of special relativity. In this new reference frame, the particle's velocity and the magnetic field are no longer perpendicular, resulting in a different direction and magnitude of the Lorentz Force.

In the special case where the velocity of the particle, v_0, is equal to the velocity of the reference frame, u, the force on the particle becomes zero. This is because the particle is now at rest in the new reference frame and there is no longer any relative motion between the particle and the magnetic field.

To reconcile this apparent contradiction, we must consider the transformation of the electromagnetic (EM) field under special relativity. This is a complex topic and requires a good understanding of both SR and EM. The mathematical equations that describe these transformations are beyond the scope of this response, but they can be found in many textbooks and online resources.

In essence, the EM field is transformed in a way that accounts for the relative motion between the particle and the reference frame. This results in a new EM field that produces the appropriate Lorentz Force in the new reference frame, thus providing consistent results.

To address your question about the source of the new EM field in the charge-free region of space, it is important to note that the EM field is not solely dependent on the presence of charges. It is also affected by