About
Developed by E. Behringer
This set of exercises guides the students to compute and analyze the behavior of a charged particle in a spatial region with mutually perpendicular electric and magnetic fields. It requires the student to determine the Cartesian components of hte forces acting on the particle and to obtain the corresponding equations of motion. The solutions to these equations are obtained through numerical integation, and the capstone exercise is the simulation of the $\overrightarrow{E}\times \overrightarrow{B}$ (Wien) filter.
Subject Area  Electricity & Magnetism 

Levels  First Year and Beyond the First Year 
Available Implementation  Python 
Learning Objectives 
Students who complete this set of exercises will be able to:

Time to Complete  120 min 
EXERCISE 3: THE $\overrightarrow{E}\times \overrightarrow{B}$ (WIEN) FILTER, PART 1
Regions of mutually perpendicular electric and magnetic fields can be used to filter a collection of moving charged particles according to their velocity. If we assume that a particle of velocity ${v}_{pass}={E}_{y}/{B}_{x}$ enters the field region traveling exactly along the $z$axis, the particle will experience zero net force and therefore zero acceleration and zero deflection from the $z$axis. If a small, circular aperture of radius $R$ is placed on the $z$axis at $z=L$, then this particle will be transmitted through the aperture.
(a) Use your program from Exercise 2 to determine the maximum value of ${v}_{z,max}={v}_{pass}+\mathrm{\Delta}v$ for which an aperture of radius $R=1.0$ mm will transmit the Li${}^{+}$ ion (now assuming that ${v}_{x}={v}_{y}=0$). What is the value of $\mathrm{\Delta}v/{v}_{pass}$?
(b) Repeat (a) for an aperture of radius $R=2.0$ mm. What is the value of $\mathrm{\Delta}v/{v}_{pass}$?
#
# ExB_Filter_Exercise_3.py
#
# This file is used to numerically integrate
# the second order linear differential equations
# that describe the trajectory of a charged particle through
# an E x B velocity filter.
#
# Here, it is assumed that the axis of the filter
# is aligned with the zaxis, that the magnetic field
# is along the +xdirection, and that the electric field
# is along the ydirection.
#
# The numerical integration is done using the builtin
# routine odeint
#
# The specific goal of this code is to identify the
# maximumm value of v_z that permits transmission of the
# particles through the velocity filter with
# a specified exit aperture.
#
# By:
# Ernest R. Behringer
# Department of Physics and Astronomy
# Eastern Michigan University
# Ypsilanti, MI 48197
# (734) 4878799 (Office)
# This email address is being protected from spambots. You need JavaScript enabled to view it.
#
# Last updated:
#
# 20160624 ERB
#
from pylab import figure,plot,xlim,xlabel,ylim,ylabel,grid,title,show,legend
from numpy import sqrt,array,arange,linspace,zeros,absolute
from scipy.integrate import odeint
#
# Initialize parameter values
#
q = 1.60e19 # particle charge [C]
m = 7.0*1.67e27 # particle mass [kg]
KE_eV = 100.0 # particle kinetic energy [eV]
Ex = 0.0 # Ex = electric field in the +x direction [N/C]
Ey = 105.0 # Ey = electric field in the +y direction [N/C]
Ez = 0.0 # Ez = electric field in the +z direction [N/C]
Bx = 0.002 # Bx = magnetic field in the +x direction [T]
By = 0.0 # By = magnetic field in the +x direction [T]
Bz = 0.0 # Bz = magnetic field in the +x direction [T]
R_mm = 1.0 # R = radius of the exit aperture [mm]
L = 0.25 # L = length of the crossed field region [mm]
u = [1.0,1.0,100.0]/sqrt(10002.0) # direction of the velocity vector
Ntraj = 1000 # number of trajectories
# Derived quantities
qoverm = q/m # charge to mass ratio [C/kg]
KE = KE_eV*1.602e19 # particle kinetic energy [J]
R = 0.001*R_mm # radius of the exit aperture [m]
vmag = sqrt(2.0*KE/m) # particle velocity magnitude [m/s]
vzpass = Ey/Bx # zvelocity for zero deflection [m/s]
# Set up the array of zvelocities to try
vz = vzpass + linspace(0.25*vzpass,0.25*vzpass,Ntraj+1) # the set of initial zvelocities
particle_pass = zeros(Ntraj+1)
#
# Over what time interval do we integrate?
#
tmax = L/vzpass;
#
# Specify the time steps at which to report the numerical solution
#
t1 = 0.0 # initial time
t2 = tmax # final scaled time
N = 1000 # number of time steps
h = (t2t1)/N # time step size
# The array of time values at which to store the solution
tpoints = arange(t1,t2,h)
#
# Here are the derivatives of position and velocity
def derivs(r,t):
# derivatives of position components
xp = r[1]
yp = r[3]
zp = r[5]
dx = xp
dy = yp
dz = zp
# derivatives of velocity components
ddx = qoverm*(Ex + yp*Bz  zp*By)
ddy = qoverm*(Ey + zp*Bx  xp*Bz)
ddz = qoverm*(Ez + xp*By  yp*Bx)
return array([dx,ddx,dy,ddy,dz,ddz],float)
# Specify initial conditions that don't change
x0 = 0.0 # initial xcoordinate of the charged particle [m]
dxdt0 = 0.0 # initial xvelocity of the charged particle [m/s]
y0 = 0.0 # initial ycoordinate of the charged particle [m]
dydt0 = 0.0 # initial yvelocity of the charged particle [m/s]
z0 = 0.0 # initial zcoordinate of the charged particle [m]
# Start the loop over the initial velocities
for i in range (0,Ntraj):
# Specify initial conditions
dzdt0 = vz[i] # initial zvelocity of the charged particle [m/s]
r0 = array([x0,dxdt0,y0,dydt0,z0,dzdt0],float)
# Calculate the numerical solution using odeint
r = odeint(derivs,r0,tpoints)
# Extract the 1D matrices of position values
position_x = r[:,0]
position_y = r[:,2]
# Check if the particle made it through the aperture
if absolute(position_x[N1]) < R:
if absolute(position_y[N1]) < sqrt(R*R  position_x[N1]*position_x[N1]):
particle_pass[i] = 1.0
else:
particle_pass[i] = 0.0
else:
particle_pass[i] = 0.0
# Look for the specific value of
for i in range (int(Ntraj/2),int(Ntraj)):#Frem:Added int
if absolute(particle_pass[i]particle_pass[i1]) > 0.5:
print("i = %d"%(i1),"vz[i] = %.3e"%vz[i1]," m/s.")#Frem:Added brackets
Deltav = vz[i1]  vzpass
print("Delta v = %.3e"%Deltav," m/s.")#Frem:Added brackets
Deltavovervzpass = Deltav/vzpass
print("Delta v/vzpass = %.3e"%Deltavovervzpass)#Frem:Added brackets
# start a new figure
figure()
# Plot the particle pass function versus zvelocity
plot(vz,particle_pass,"b",label='\(R = \)%.2f mm'%R_mm)
xlim(min(vz),max(vz))
ylim(0.0,1.2)
xlabel("\(v_z\) [m/s]",fontsize=16)
ylabel("Transmitted truth value",fontsize=16)
grid(True)
title('Wien filter: \(v = \)%.2e m, length \(L = \)%.2f m'%(vmag,L))
legend(loc=1)
show()
# start a new figure
figure()
# Plot the particle pass function versus scaled zvelocity
plot(vz/vzpass,particle_pass,"b",label='\(R = \)%.2f mm'%R_mm)
xlim(min(vz/vzpass),max(vz/vzpass))
ylim(0.0,1.2)
xlabel("\(v_z/v_{z,pass}\)",fontsize=16)
ylabel("Transmitted truth value",fontsize=16)
grid(True)
legend(loc=1)
title('Wien filter: \(v_{z,pass} = \)%.2e m, length \(L = \)%.2f m'%(vzpass,L))
show()
Translations
Code  Language  Translator  Run  

Software Requirements
Android  iOS  Windows  MacOS  
with best with  Chrome  Chrome  Chrome  Chrome 
support fullscreen?  Yes. Chrome/Opera No. Firefox/ Samsung Internet  Not yet  Yes  Yes 
cannot work on  some mobile browser that don't understand JavaScript such as.....  cannot work on Internet Explorer 9 and below 
Credits
Fremont Teng; Loo Kang Wee; based on codes by E. Behringer
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Sample Learning Goals
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For Teachers
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Research
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Video
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Version:
 https://www.compadre.org/PICUP/exercises/Exercise.cfm?A=ExB_Filter&S=6
 http://weelookang.blogspot.com/2018/06/wienexbfilterexercise123and4.html
Other Resources
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end faq
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