Finding Motion of Charged Particle in Changing Magnetic Field (numerically)

[Emustrangler]

I want to find the motion of an electron in a spatially varying magnetic field using finite element analysis, and have some way of estimating the error (especially in the path length). I imagine somewhere someone has written up an algorithim that does this, but I haven't had much luck googling. Anyone want to point me in the right direction?

 

[Bob S]

I have done numerical integration along the electron path (actually I did protons) using the relation:

Bρ = (βγ/c) m_oc² Tesla meters, or

ρ = (βγ/c) m_oc²/B meters

where B is the transverse magnetic field (in x and y), and ρ is the radius of curvature of particle path.

Bob S

 

[Born2bwire]

I want to find the motion of an electron in a spatially varying magnetic field using finite element analysis, and have some way of estimating the error (especially in the path length). I imagine somewhere someone has written up an algorithim that does this, but I haven't had much luck googling. Anyone want to point me in the right direction?

I'm not sure how finite element analysis would be applied, at least not the forms of FEM that I have used. I would take a look into how molecular dynamics simulations are done as they have to deal with the motion of particles in spatially/time varying potentials. In essence, you have to use an integrator, like the Euler or Verlet method, to estimate the integrations of the accelerations and velocities over time to estimate the trajectories. In your case then, you would use the forces generated by the fields via the Lorentz force as your force input but everything else should be the same I believe. These methods are designed for the simulation of large numbers of particles so part of the design of the integrator is to be numerically efficient, if you are just concerned about a single electron then I wonder if they would be ways to use the low computational costs of the problem to increase the accuracy of the integrator. The easiest would be of course to decrease the time steps used but maybe a higher order integrator may be a better way.

 

[Bob S]

In my case (see post #2), I stepped through about 10 meters, 1 mm at a time. I was tracking constant momentum particles, so β and γ were constant. I had a lookup table for the transverse dc spacially-varying magnetic fields B_x(z) and B_y(z) (actually a 4 x 4 transfer matrix) for every mm, and I tracked the particle's z coordinate and trajectory x, dx/dz, y, and dy/dz for each 1-mm step (x=0 and y=0 were the central orbit, and dx/dz and dy/dz were the divergences). The Lorentz v x B force was used at each step to update the four quantities above.

Bob S

 

[sardar]

I am excited to hear about your interest in numerically finding the motion of a charged particle in a changing magnetic field. This is a complex problem that requires a thorough understanding of electromagnetism and computational methods.

One approach to solving this problem is through finite element analysis, which involves dividing the space into small elements and using numerical techniques to solve for the particle's trajectory in each element. This method is commonly used in engineering and physics to solve complex problems that cannot be solved analytically.

In terms of estimating the error, there are various techniques that can be used depending on the specific problem and the level of accuracy required. One method is to compare the results obtained from different element sizes and refine the mesh until a desired level of accuracy is achieved. Another approach is to use error estimation techniques, such as Richardson extrapolation or error propagation, to quantify the error in the final solution.

I understand that finding an existing algorithm for this problem may be challenging, but there are resources available that can help. One option is to consult textbooks or research papers on computational electromagnetics, which often include numerical methods for solving similar problems. Additionally, reaching out to experts in the field or joining online forums and communities can also be helpful in finding the right resources and guidance.

I wish you the best of luck in your research and hope that you are able to find a suitable approach for your specific problem. Keep in mind that this is a complex and challenging task, but with perseverance and the right resources, I am confident that you will be able to successfully find the motion of the charged particle in a changing magnetic field.