How Does a Charged Particle Move in a Magnetic Field?

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A charged particle in a magnetic field moves along the field direction while also drifting in yz space due to its initial velocities. The discussion highlights that although the particle has x and z velocities, its average position over the cyclotron period shows minimal change in the xz plane. The displacement along the y direction depends on the particle's velocity in that direction. This understanding clarifies the relationship between the particle's motion and the magnetic field. Overall, the dynamics of charged particles in magnetic fields involve complex interactions between their velocities and the field direction.
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Here is an excerpt from a book: Notice also how the charged particle is captured by the magnetic field, just drifting along the field direction. In my book, the field direction is in the y-direction, but the in addition to the particle having an initial y velocity, it had an initial z velocity as well. So it seems to me that the particle is drifting not in the direction of the field, but traveling in yz space. Am I understanding this right? Thank you.
 
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What the author is talking about is probably the fact that *averaging* the particles position over its cyclotron period (m/qB), it really has not changed it's xz position even though it has xz velocity. However, it may have been displaced along the y direction (field line), depending on its velocity in the y direction.
 
I understand now, thank you
 
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