About

Subject Area	Electricity & Magnetism
Level	Beyond the First Year
Available Implementation	IPython/Jupyter Notebook
Learning Objectives	Students who complete this set of exercises will develop their understanding of how charged particles respond to magnetic fields (Exercises 1, 2, and 3); be able to describe in pseudo-code how to calculate the trajectory of a charged particle in a magnetic field (Exercise 1); be able to use numerical methods for ordinary differential equations to calculate the particle’s trajectory (Exercises 1, 2, and 3); be able to interpret and describe the computed trajectories (Exercises 1, 2, and 3); and be able to validate numerical solutions against analytical solutions for appropriate test cases (Exercise 1).
Time to Complete	120 min

EXERCISE 1: MOTION OF A CHARGED PARTICLE IN A UNIFORM MAGNETIC FIELD

Consider a uniform magnetic field, of strength 10−4 T, in the z-direction. An α-particle enters the magnetic field at initial position (0.0,0.0,0.0), with an initial velocity 31000ms in the x-direction.

• What do you expect the α-particle’s trajectory to be shaped like?
• Write the equations of motion for the α-particle in the uniform magnetic field. Solve the equations analytically.
• Describe in words (or pseudocode) a procedure to numerically solve these equations of motion for the trajectory of the α-particle.
• Now, use your numerical method or a differential equation solver to find a numerical solution to the equations of motion you wrote down. The special case of a uniform magnetic field has an analytical solution, but many cases do not. Validate your code: Does the shape of the α-particle’s trajectory match your expectation and your analytical calculation?
• How would your results be different for an H− ion that enters the magnetic field? Confirm by running the code with parameters for the H− ion with the same initial velocity that the α-particle had. Plot the α-particle’s trajectory and the negative ion’s trajectory on the same plot.

%matplotlib inline

import numpy

import matplotlib.pyplot as mpl

from mpl_toolkits.mplot3d import Axes3D

# Parameters for plot attributes

mpl.rc("xtick", labelsize="large")

mpl.rc("ytick", labelsize="large")

mpl.rc("axes", labelsize="xx-large")

mpl.rc("axes", titlesize="xx-large")

mpl.rc("figure", figsize=(8,8))

Exercise 1. Uniform Magnetic Field

# define key constants

m_p = 1.67E-27 # mass of proton: kg

qe = 1.602E-19 # charge of proton: C

# now, setting up an alpha particle

m = 4.0*m_p

q = 2.0*qe

QoverM = q/m

dt = 1.0E-8

t = numpy.arange(0.0, 0.002, dt)

rp = numpy.zeros((len(t), 3))

vp = numpy.zeros((len(t), 3))

v0 = numpy.sqrt(2.0*QoverM*10.0)

# negative ion

mn = 1.0 * m_p

qn = -1.0*qe

QoverMn = qn/mn

rn = numpy.zeros((len(t), 3))

vn = numpy.zeros((len(t), 3))

# Strength of Magnetic Field

B0 = 1.0E-4

expected_R = m*v0/(q*B0)

expected_T = 2.0*numpy.pi*expected_R / v0

print("v0 = ", v0)

print("Expected trajectory radius = ", expected_R)

print("Expected cyclotron period = ", expected_T)

# initial condition

rp[0,:] = numpy.array([0.0, 0.0, 0.0])

vp[0,:] = numpy.array([v0, 0.0, 0.0])

# initial condition for negative ion

rn[0,:] = numpy.array([0.0, 0.0, 0.0])

vn[0,:] = numpy.array([v0, 0.0, 0.0])

# Euler time steps

for it in range(0, len(t)-1):

Bp = numpy.array([0.0, 0.0, B0])

Ap = QoverM * numpy.cross(vp[it,:], Bp)

vp[it+1] = vp[it] + dt*Ap

rp[it+1] = rp[it] + dt*vp[it]

An = QoverMn * numpy.cross(vn[it,:], Bp)

vn[it+1] = vn[it] + dt*An

rn[it+1] = rn[it] + dt*vn[it]

# Plot the particle's trajectory, in the xy-plane

mpl.plot(rp[:,0], rp[:,1], label="Alpha particle")

mpl.plot(rn[:,0], rn[:,1], linestyle = "--", label="H\(^-\) ion")

mpl.xlabel("\(x\)")

mpl.ylabel("\(y\)")

mpl.title("Particle Trajectories in Uniform Magnetic Field")

mpl.legend(fontsize="x-large")

mpl.xlim(-13,13)

mpl.ylim(-13,13)

 

Translations

Code	Language		Translator	Run

Credits

Fremont Teng; Loo Kang Wee

 Version:

1. https://www.compadre.org/PICUP/exercises/exercise.cfm?I=111&A=ParticleInMagField
2. http://weelookang.blogspot.com/2018/06/motion-of-charged-particle-in-uniform.html

Other Resources

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