Charged particle movement in arbitrary electromagnetic field

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SUMMARY

The discussion focuses on developing a simulation code for charged particles in an arbitrary electromagnetic field, specifically addressing the equations of motion for particles under a magnetic field vector B. The participant presents equations for the velocities ux, uy, and uz, which incorporate constants that determine the direction of B. The need for initial conditions to calculate these constants is emphasized, and the conversation suggests utilizing standard aligned equations along with a rotation matrix to handle arbitrary orientations of the magnetic field.

PREREQUISITES
  • Understanding of classical mechanics, particularly the motion of charged particles in magnetic fields.
  • Familiarity with electromagnetic theory, specifically the Lorentz force law.
  • Knowledge of matrix operations, particularly rotation matrices.
  • Basic programming skills for simulation development.
NEXT STEPS
  • Research the Lorentz force law and its application to charged particle dynamics.
  • Learn about rotation matrices and their use in transforming coordinate systems.
  • Explore numerical methods for solving differential equations related to particle motion.
  • Investigate simulation frameworks suitable for modeling electromagnetic fields, such as MATLAB or Python with SciPy.
USEFUL FOR

Engineers, physicists, and simulation developers interested in modeling the behavior of charged particles in complex electromagnetic environments.

FlatronL1917
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Hello there! This is my first post here, hopefully I am not posting in the wrong place.
Also, I am an engineer and have not used this stuff for years, so please be patient with me, I am pretty sure that my question is stupid :-)

I would like to develop a simulation code for charged particles moving in a electromagnetic field.
My thought is that, we may not always have the luxury to align the coordinate system to our magnetic field.
Assuming an arbitrary magnetic field vector B in every cell and ignoring the electric field for the moment, I tried solving the equations of motion to see if I can avoid discretization and got the following:

ux = c1x + (c2x+c3x)*cos(qB/m * t) + (c2x-c3x)*sin(qB/m * t)
uy = c1y + (c2y+c3y)*cos(qB/m * t) + (c2y-c3y)*sin(qB/m * t)
uz = c1z + (c2z+c3z)*cos(qB/m * t) + (c2z-c3z)*sin(qB/m * t)

It seems that the cij constants determine the direction of B. I only have the initial conditions for ux, uy, uz, obviously so I can not calculate them. Where do I go from this point? Is my solution wrong? Any help/pointer would be greatly appreciated.
 
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It looks like you plane on linear electric and magnetic fields, rather than "arbitrary" ones ... just with axis arbitrarily rotated.

In which case, all you need is the standard aligned equations and the rotation matrix.
 

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