Relativistic particle in non-uniform magnetic field (math)

[msailer1234]

Hi everyone,
I am currently working on a project in Physics analyzing the possibility of magnetic fields protecting against radiation particles such as fully ionized nuclei or electrons. I wrote a code in Matlab that simulates a 3-D magnetic field based on a combination of wires and wire loops. The code produces a 3D vector field of 11x11x11. So I have Bx,By, and Bz for every point in the 11x11x11 box. I am now attempting to create a code which sends a virtual particle into the magnetic field so I can see what happens.

What I am doing is treating each point as a box with a uniform magnetic field which the particle travels through. I want to find an equation that can take the point and velocity the particle enters into a box and calculates the point and velocity it leaves the box. I am tripping over the calculations. I have attached my hand written calculations (I hope they are clearly written enough), but I don't know if I am incorporating relativity correctly. I also don't think I am approaching it correctly because the equation gives me Radius_x, Radius_y, and Radius_z which doesn't make sense to me. If anyone has any corrections or ideas, please let me know! Thanks!

 

[kuruman]

Do you already have code for a virtual non-relativistic particle entering your box? If not, I would think that you need to write that first to gain the experience. Then you can attempt the more complicated relativistic case and check it by making sure that it reduces to the non-relativistic results for small speeds. I should also point out that you got the components of v wrong, if that's what you have under "given". You write the x-component as $\frac{v_x}{\sqrt{1-v^2/c^2}}.$ It should be just $v_x.$ There are other problems with your equations even in the non-relativistic limit. The equation $r=mv^2/(q\vec{v}\times \vec B )$ is nonsense.

 

[msailer1234]

Do you already have code for a virtual non-relativistic particle entering your box? If not, I would think that you need to write that first to gain the experience. Then you can attempt the more complicated relativistic case and check it by making sure that it reduces to the non-relativistic results for small speeds. I should also point out that you got the components of v wrong, if that's what you have under "given". You write the x-component as $\frac{v_x}{\sqrt{1-v^2/c^2}}.$ It should be just $v_x.$ There are other problems with your equations even in the non-relativistic limit. The equation $r=mv^2/(q\vec{v}\times \vec B )$ is nonsense.

Thank you for responding! Okay, so that looks like one of my problems. I got that equation by combining the force equations F=mv^2/r and F=qvxB. So am I correct in saying that that reasoning only works if used with the velocity being perpendicular to the B field making the equation r = mv/qB? Also, if this is a relativistic velocity, shouldn't it be relativistic in each spatial dimension?

 

[kuruman]

To begin with the symbol for v in relativistic equation stands for the relative velocity between two frames, O and O'. If Observer O sees observer O' move with velocity $\vec{v}$, observer O' will see observer O move with velocity $-\vec{v}$. That's all there is to that. You need the solve the following differential equation in 3D:
$$\frac{d\vec{v}}{dt}=\vec{\omega}\times \vec{v}$$ where $$\vec{\omega}_B=\frac{q\vec{B}}{\gamma m c};~~~\left( \gamma =\frac{1}{\sqrt{1-v^2/c^2}} \right)$$
Note that $\omega_B$ is velocity-dependent. Do you see why it is easier to do the non-relativistic calculation with $\gamma = 1$?

 

[mfb]

A radius with x,y,z coordinates is a perfectly valid thing. In addition to the curvature radius it also gives you the direction of the curvature.
You'll have to find a way to determine where your curved track (or circle) intersects the boundary of the cubes unless the deflection is very small.

 

[ChrisVer]

Since you are working with Cartesian coordinates, why are you trying to move to "r" (spherical?) ... it's only going to make things more complicated... I think you have 11x11x11 boxes (aka regions with a given Bx,By,Bz)...
if you assume that the particle in each box (i) it transcends gets a kick to its velocity in the x,y,z-direction (given by the relevant force), I think it's pretty simple.
for example if in box 0 it has a velocity (v_x, v_y, v_z), and use that to determine the box1 (next step), in box1 it would have (v_x+ F_x/m , v_y + F_y/m , v_z+ F_z/m)... ok that was for classical mechanics... for relativistic, the acceleration kick (F/m's) should be changed appropriately (see eg. https://en.wikipedia.org/wiki/Relativistic_mechanics#Force)

 

[mfb]

Do you have to work with boxes? Following the track of particles could be easier.
Knowing some key parameters of the system would help to see which approximations work how well.

 

[ChrisVer]

Do you have to work with boxes? Following the track of particles could be easier.

Well by saying we have cubic spaces of known magnetic field, I think that's the best way to deal with the problem ?
The track of the particles is supposed to be emulated based on how the particle interacts with the field [at each point the velocity should be updated to get the next point the particle will be in].
Or maybe I haven't understood the problem.

 

[mfb]

The approach so far looks more like OP wants to model the trajectory inside the cubes as circle and then let that intersect the cube boundaries to do the same again in the next cube.

 

[msailer1234]

The approach so far looks more like OP wants to model the trajectory inside the cubes as circle and then let that intersect the cube boundaries to do the same again in the next cube.

From what my code gives me, I do think having uniform magnetic fields in several boxes is the best approach for me to model this. Thank you everyone for the replies. I am still looking into it, and I am consulting some professors at my college.

 

[msailer1234]

Hi everyone,
I thought I would update you on how the project went. I apologize for taking so long to respond, but the box method did work great for an approximation. In hindsight, calculating the new magnetic field at each time step based on the particle's position would have been more efficient and practical, but it was a great learning experience. Thank you for your help!