Excel Simulation help - Electron runaway with given frictional force (Bethe)

[Matt atkinson]

Homework Statement

I'm trying to create a simple excel simulation for runaway electrons using a given energy-loss per unit distance equation (Bethe formula).
I'm stepping through time (pico-seconds) and giving the the electric field some initial value at a distance of 1mm and from there basically using energy and equations from special relativity to calculate other parameters.
I'm having serious issues setting the spreadsheet up correctly I've been trying for a few weeks now and keep having problems with circular references.
I've literally got no idea how to fix it so if someone could please help, it would be greatly appreciated.

Homework Equations

The Attempt at a Solution

The equation given above is only correct for kinetic energies greater than the kinetic energy which causes the peak of the graph below, so i had to find a linear fit to approximate the beginning portion of the graph.

I'm starting to doubt if my physics is correct.

Spreadsheet is attached! please take a look if possible.

Ps:- not sure if this was the correct place to post.

 

[Matt atkinson]

I really have no idea, I've been struggling with this for over a week now.

 

[mfb]

At some point, you have to say "okay, I'll use this value for the next time step". It does not matter much where you do it, but it has to be done.

Also, your simulation won't start if you do not use an initial velocity and use the energy as main parameter instead of the velocity. No motion, no distance, no gained kinetic energy.

A picosecond looks like a long step, especially for low energy.

Why do you have a quadratic dependence of your field on distance?

Your calculation of x looks odd. Why do you take the difference between velocities?

 

[Matt atkinson]

Okay so I've played around with the spreadsheet some more, thank you for pointing out my x calculation It was a mistake on my part.
I lowered the step to $$10^{-14}s$$ and fixed the calculation of x.
The electric field is essentially a ratio of the initial field $$E_o$$ and initial distance $$x_o$$ with the field at a distance x, this is because i have to vary the initial field to investigate energy loss against distance, I believe the ratio is correct.
The only thing confusing me now, is the resolution of forces, (if i have fixed the circular reference fine) would the accelerating force be equal to; $$f_a=E$$for an electron when working in eV? or; $$f_a=eE$$.

I was also told to start the simulation at zero velocity, because I initially gave the electron some thermal velocity but I was told to change it.

Attached: Updated version of spreadsheet.

Thank you so much for your help, I've been tearing my hair out for days trying to get assistance.

 

[mfb]

I was also told to start the simulation at zero velocity

Then you should use position and velocity as main variables. All other parameters are just helping to get the next velocity and position.

Some formulas are still odd.

I think the electric field has some keV <-> eV mixture.
The total energy uses the position instead of the position difference to get updated.

I would start all over again, fixing bugs in this mess is more work.

 

[Matt atkinson]

I'm sorry, but i do not understand what do you mean the electric field has keV <-> mixture?
Okay I'll try to start again thank you!

 

[mfb]

The electric field is given in eV/m but you use it as keV/m in other columns.

 

[Matt atkinson]

Oh thank you, I did not notice that !

 

[Matt atkinson]

Oh but wait if, its an initial field coming from another point source would it still be in eV/m?

 

[mfb]

It does not matter which units you use, but you cannot mix them without conversion factors.

 

[Matt atkinson]

okay, so basically if i use the electric field in another column i need to divide it by 1000? to change it to KeV?
also for the resultant force column I am not sure whether i should be multiplying the electric field by e because right now I'm just using E+drag force=resultant force.

 

[mfb]

If you want to use the electric field given in eV/m as keV/m in another column, then sure you have to divide it by 1000.

If you have a length in kilometers in one column and want to know the length in meters in another formula, then you have to multiply it by 1000 as 1km=1000m. Same concept.

E is not the force. The force is F=qE. And make sure your units match (in eV, this is quite easy).

 

[Matt atkinson]

I know it is quite easy its just It's had me so confused for a while now.
Thanks so much for the help I think I've got it working now !