#

# 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)

# ebehringe@emich.edu

#

# 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()