MATLAB BASE CODE:

clc

clear all

close all

$\%$n $=2$; $\%$mumber of particles

n $=3$;

qe $=-1\mathrm{.}6$e$-19$; $\%$ the charge of an electron

me $=9\mathrm{.}11$e$-31$; $\%$ mass of an electron

cell $=5$e$-3$; $\%$observing a box of $1$ cm

$\%$initial positions

$\%$ Xs $=[-3$e$-3,0]$; $\%$ the list of particles x$-$cordinates

$\%$ Ys $=[0,0]$; $\%$ the list of particles y$-$cordinates

Xs $=[-3$e$-3,0,2$e$-3]$; $\%$ the list of particles x$-$cordinates

Ys $=[0,0,15$e$-9]$; $\%$ the list of particles y$-$cordinates

$\%$initail velocities

$\%$ Us $=[1$e$5,0]$; $\%$ the list of particles x$-$velocity

$\%$ Vs $=[0,0]$; $\%$ the list of particles y$-$velocity

Us $=[1$e$5,0,0]$; $\%$ the list of particles x$-$velocity

Vs $=[0,0,0]$; $\%$ the list of particles y$-$velocity

$\%$timesteps

dt $=1$e$-13$; $\%$runnig $10$ itteratins for every $1$ ps

imax $=1$e$-7/$dt; $\%$ duration of sim: $100$ns

ireport $=1$e$-9/$dt; $\%$ print once every $1$ ns

$\%$create a matrix of particle paris that makes sure we dont double count

pairs$=$ triu$($ones$($n$),1)$;

$\%$Force initialization

$\%$caculate the force as a sum of each particles from each particle pair

Fx $=$ zeros$([1,$n$])$;

Fy $=$ zeros$([1,$n$])$;

for i $=1$:n

for j $=$ i$+1$:n

if pairs$($i$,$j$)==1\%$only applies to unique particle pairs

dx $=$ Xs$($j$)-$ Xs$($i$)$; $\%$ ditance between the electrons in x dim

dy $=$ Ys$($j$)-$ Ys$($i$)$; $\%$ ditance between the electrons in y dim

r $=$ sqrt $($dx$^2+$ dy$^2)$; $\%$ the toltal distance magnitude between the two

electrons

R $=[$dx$,$ dy$]/$r; $\%$normalized vector for direction of the force

F $=$ F$\_$Coulomb$($r$,$ qe $,$ qe$)$; $\%$force between i$-$th and j$-$th

$\%$particle i

Fx$($i$)=$ Fx$($i$)+$ F$*$R$(1)$; $\%$force componets in X

Fy$($i$)=$ Fy$($i$)+$ F$*$R$(2)$; $\%$force componets in y

$\%$particle i

$\%$apply Newton's third law

Fx$($j$)=$ Fx$($j$)-$ F$*$R$(1)$; $\%$force componets in X

Fy$($j$)=$ Fy$($j$)-$ F$*$R$(2)$; $\%$force componets in y

end

end

end

$\%$main simulation using the verlt

for t $=1$:imax

$\%$update the position

Xs $=$ Xs $+$ Us$*$dt $+$ Fx$/(2*$me$)*$dt$^2$; $\%$position update in x

Ys $=$ Ys $+$ Vs$*$dt $+$ Fy$/(2*$me$)*$dt$^2$; $\%$position update in y

$\%$half update velocity

Us $=$ Us $+$ Fx$/(2*$me$)*$dt; $\%$velocity in X

Vs $=$ Vs $+$ Fy$/(2*$me$)*$dt; $\%$velocity in y

$\%$ recalculate the force

Fx $=$ zeros$([1,$n$])$;

Fy $=$ zeros$([1,$n$])$;

for i $=1$:n

for j $=$ i$+1$:n

if pairs$($i$,$j$)==1\%$only applies to unique particle pairs

dx $=$ Xs$($j$)-$ Xs$($i$)$; $\%$ ditance between the electrons in x dim

dy $=$ Ys$($j$)-$ Ys$($i$)$; $\%$ ditance between the electrons in y dim

r $=$ sqrt $($dx$^2+$ dy$^2)$; $\%$ the toltal distance magnitude between the two

electrons

R $=[$dx$,$ dy$]/$r; $\%$normalized vector for direction of the force

F $=$ F$\_$Coulomb$($r$,$ qe $,$ qe$)$; $\%$force between i$-$th and j$-$th

$\%$particle i

Fx$($i$)=$ Fx$($i$)+$ F$*$R$(1)$; $\%$force componets in X

Fy$($i$)=$ Fy$($i$)+$ F$*$R$(2)$; $\%$force componets in y

$\%$particle i

$\%$apply Newton's third law

Fx$($j$)=$ Fx$($j$)-$ F$*$R$(1)$; $\%$force componets in X

Fy