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The Wien (E x B) Filter

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 Estyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">×style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">Bstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> (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:

  • generate equations predicting the Cartesian components of force acting on the charged particle and generate the equations of motion for the particle (Exercise 1);
  • calculate particle trajectories by solving the equations of motion (Exercise 2);
  • produce two-dimensional and three-dimensional plots of the trajectories (Exercise 2); and
  • simulate the operation of an Estyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">×style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">Bstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> (Wien) filter (Exercise 3) and determine the range of particle velocities transmitted by the filter, and how these are affected by the geometry of the filter (Exercise 4).
Time to Complete 120 min
Exercise 2

EXERCISE 2: COMPUTING THE TRAJECTORY

Solve the equations of motion to obtain the trajectory of the Listyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">+style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> ion from Exercise 1 while it traverses the field region from zstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">=style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">0style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> to zstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">=style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">Lstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">=style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">0.25style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; 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(a) On separate graphs, plot xstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">, ystyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">, and zstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> versus time.

(b) Plot the trajectory in space. What does the trajectory of the ion look like? What did you expect (Exercise 1)? What happens if you reduce the initial kinetic energy of the ion by a factor of 100? A factor of 10,000?

(c) What is the kinetic energy of the ion at the end of its trajectory? How does it compare to its initial energy?

ExB_Filter_Exercise_2.py with Bug Fix

#

# ExB_Filter_Exercise_2.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 z-axis, that the magnetic field

# is along the +x-direction, and that the electric field

# is along the -y-direction.

#

# The numerical integration is done using the built-in

# routine odeint.

#

# By:

# Ernest R. Behringer

# Department of Physics and Astronomy

# Eastern Michigan University

# Ypsilanti, MI 48197

# (734) 487-8799 (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

from numpy import sqrt,array,arange

from scipy.integrate import odeint

#

# Initialize parameter values

#

q = 1.60e-19 # particle charge [C]

m = 7.0*1.67e-27 # 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]

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

# Derived quantities

qoverm = q/m # charge to mass ratio [C/kg]

KE = KE_eV*1.602e-19 # particle kinetic energy [J]

vmag = sqrt(2.0*KE/m) # particle velocity magnitude [m/s]

v1x = vmag*u[0] # v1x = x-component of the initial velocity [m/s]

v1y = vmag*u[1] # v1y = y-component of the initial velocity [m/s]

v1z = vmag*u[2] # v1z = z-component of the initial velocity [m/s]

vzpass = -Ey/Bx # vzpass is the z-velocity required for no deflection [m/s]

#

# Over what time interval do we integrate?

#

tmax = L/v1z;

#

# 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

x0 = 0.0 # initial x-coordinate of the charged particle [m]

dxdt0 = v1x # initial x-velocity of the charged particle [m/s]

y0 = 0.0 # initial y-coordinate of the charged particle [m]

dydt0 = v1y # initial y-velocity of the charged particle [m/s]

z0 = 0.0 # initial z-coordinate of the charged particle [m]

dzdt0 = v1z # initial z-velocity of the charged particle [m/s]

r0 = array([x0,dxdt0,y0,dydt0,z0,dzdt0],float)

# Set up the time interval

t1 = 0.0 # initial time

t2 = tmax # final scaled time

N = 1000 # number of time steps

h = (t2-t1)/N # time step size

# The array of time values at which to store the solution

tpoints = arange(t1,t2,h)

# Calculate the solution using odeint

r = odeint(derivs,r0,tpoints)

#

# Extract the 1D matrices of position values

#

position_x = r[:,0]

xmin = min(position_x)

xmax = max(position_x)

position_y = r[:,2]

ymin = min(position_y)

ymax = max(position_y)

position_z = r[:,4]

zmin = min(position_z)

zmax = max(position_z)

# Calculate the final velocity

vx = r[:,1]

vxf = vx[N-1]

vy = r[:,3]

vyf = vy[N-1]

vz = r[:,5]

vzf = vz[N-1]

vf = sqrt(vxf*vxf+vyf+vyf+vzf*vzf)

KEf_eV = 0.5*m*vf*vf/1.60e-19

print("The initial x-velocity = %.3e"%v1x," m/s.") ##Frem: Added brackets

print("The initial x-velocity = %.3e"%vx[0]," m/s.")##Added brackets

print("The pass velocity = %.3e"%vzpass," m/s.")##Added brackets

print("The magnitude of the initial velocity = %.3e"%vmag," m/s.")##Added brackets

print("The magnitude of the final velocity = %.3e"%vf," m/s.")##Added brackets

print("The initial kinetic energy = %.3e"%KE_eV," eV.")##Added brackets

print("The final kinetic energy = %.3e"%KEf_eV," eV.")##Added brackets

# start a new figure

figure()

# Plot the x-position versus time

plot(tpoints,position_x,"b-")

xlim(t1,t2)

ylim(xmin,xmax)

xlabel("Time \(t\) [s]",fontsize=16)

ylabel("\(x\) [m]",fontsize=16)

grid(True)

title('Wien filter: \(v = \)%.2e m, length \(L = \)%.2f m'%(vmag,L))

show()

# start a new figure

figure()

# Plot the y-position versus time

plot(tpoints,position_y,"b-")

xlim(t1,t2)

ylim(ymin,ymax)

xlabel("Time \(t\) [s]",fontsize=16)

ylabel("\(y\) [m]",fontsize=16)

grid(True)

title('Wien filter: \(v = \)%.2e m/s, length \(L = \)%.2f m'%(vmag,L))

show()

# start a new figure

figure()

# Plot the z-position versus time

plot(tpoints,position_z,"b-")

xlim(t1,t2)

ylim(zmin,zmax)

xlabel("Time \(t\) [s]",fontsize=16)

ylabel("\(z\) [m]",fontsize=16)

grid(True)

title('Wien filter: \(v = \)%.2e m/s, length \(L = \)%.2f m'%(vmag,L))

show()

# start a new figure

plot_trajectory = figure()

# Plot the trajectory in 3D

ax = plot_trajectory.gca(projection='3d')

ax.plot(position_x,position_y,position_z,"b-")

ax.set_xlabel("\(x\) [m]")

ax.set_ylabel("\(y\) [m]")

ax.set_zlabel("\(z\) [m]")

ax.set_title("Wien filter: \(v = \)%.2e m/s, length \(L\) = %s"%(vmag,L))

grid(True)

show()

 

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Android iOS Windows MacOS
with best with Chrome Chrome Chrome Chrome
support fullscreen? Yes. Chrome/Opera No. Firefox/ Sumsung 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:

  1. https://www.compadre.org/PICUP/exercises/Exercise.cfm?A=ExB_Filter&S=6
  2. http://weelookang.blogspot.com/2018/06/wien-e-x-b-filter-exercise-123-and-4.html

Other Resources

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