PICUP Wien (E x B) Filter Exercise 1 JavaScript HTML5 Applet Simulation Model

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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 (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 (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 1

EXERCISE 1: FORCES ACTING ON A CHARGED PARTICLE AND THE EQUATIONS OF MOTION

Imagine that we have a particle of mass , charge , and velocity with entering a region of uniform magnetic field and uniform electric field . As shown below, and . The length of the field region along the -axis is .

Neglecting any other forces (e.g., gravitational forces), show that the Cartesian components of the combined electric and magnetic forces are

F x = 0 (7)

F y = q E y + q v z B x (8)

F z = - q v y B x (9)

resulting in the equations of motion

x ¨ = 0 (10)

y ¨ = q m (E y + v z B x) (11)

z ¨ = - q m v y B x (12)

where the dot accents indicate differentiation with respect to time. Note that the particle will not experience any transverse acceleration if .

(a) Assume that V/m and T, and that all other field components are zero. Calculate, by hand, the Cartesian components of the acceleration at the instant when a Li ion of mass 7 amu and kinetic energy 100 eV enters the field region traveling along the direction .

How will these acceleration components compare to those for a doubly ionized nitrogen ion (N)?

(b) Write a code to perform the calculation in part (a). Note that, as soon as the particle enters the field region, the velocity components will change, and therefore so will the forces. To calculate an accurate trajectory, it is necessary to repeatedly calculate the forces, a task for which the computer is very well suited.

ExB_Filter_Exercise_1.py with Bug Fix

# ExB_Filter_Exercise_1.py

# This file is used to calculate

# the Cartesian components of the acceleration 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.

# By:

# Ernest R. Behringer

# Department of Physics and Astronomy

# Eastern Michigan University

# Ypsilanti, MI 48197

# (734) 487-8799 (Office)

# ebehringe@emich.edu

# Last updated:

# 20110309 = March 9, 2011 ERB Initial writing in Matlab.

# 20160616 ERB conversion from Matlab.

from numpy import sqrt

import time

# 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]

D = 2.0 # D = diameter 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

# 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]

# Calculate the Cartesian components of the acceleration

a_x = qoverm*(Ex + v1y*Bz - v1z*By)

a_y = qoverm*(Ey + v1z*Bx - v1x*Bz)

a_z = qoverm*(Ez + v1x*By - v1y*Bx)

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

print ("The x-component of the acceleration is %.3e"%a_x," m/s $^2$ .") #Frem:Added brackets

print ("The y-component of the acceleration is %.3e"%a_y," m/s $^2$ .") #Frem:Added brackets

print ("The z-component of the acceleration is %.3e"%a_z," m/s $^2$ .") #Frem:Added brackets

time.sleep(10)##Frem:Added Time to prevent the program from closing too fast

Translations

Code	Language		Translator	Run

Credits

Fremont Teng; Loo Kang Wee; based on codes by E. Behringer

1. Introduction:

This briefing document summarizes the key themes and important concepts presented in the provided excerpts from the PICUP Wien filter exercise. Developed by E. Behringer, this resource serves as a guide for students to understand and analyze the behavior of charged particles within a region containing mutually perpendicular electric and magnetic fields. The ultimate goal is to simulate the operation of an E x B (Wien) filter.

2. Main Themes and Important Ideas:

Charged Particle Motion in Perpendicular E and B Fields: The core focus of the exercise is on understanding the forces acting on a charged particle moving through a space where electric and magnetic fields are perpendicular to each other. Students are expected to calculate the Cartesian components of these forces.
Equations of Motion: A significant part of the exercise involves deriving the equations of motion for the charged particle based on the forces acting upon it. The text explicitly mentions the need for students to "generate the equations of motion for the particle (Exercise 1)."
Numerical Integration for Trajectories: Due to the complexity of the motion under the influence of both electric and magnetic fields, the solutions to the equations of motion are obtained through numerical integration. This allows for the calculation of the particle's trajectory over time.
Simulation of the Wien Filter: The "capstone exercise" involves simulating the operation of the E x B (Wien) filter. This implies that students will utilize their understanding of the forces and equations of motion to model how the filter selects particles based on their velocity.
Velocity Selection: A key aspect of the Wien filter is its ability to selectively transmit particles with a specific velocity. The learning objectives include determining "the range of particle velocities transmitted by the filter, and how these are affected by the geometry of the filter (Exercise 4)."
Mathematical Formulation: The excerpt introduces a scenario with a particle of mass m, charge q, and velocity v entering a region with a uniform magnetic field B = B_x x^ and a uniform electric field E = E_y y^, where the fields are mutually perpendicular. The text then provides the Cartesian components of the Lorentz force:
"F_x = q v_y B_x (7)"
"F_y = q E_y + q v_z B_x (8)"
"F_z = -q v_y B_x (9)"
Equations of Motion in Terms of Acceleration: By applying Newton's second law (F = ma), the excerpt presents the equations of motion in terms of the Cartesian components of acceleration:
"ẍ = 0 (10)"
"ÿ = q/m (E_y + v_z B_x) (11)"
"z̈ = -q/m v_y B_x (12)" where the double dot notation signifies the second derivative with respect to time (acceleration).
Pass Velocity: The text highlights a crucial condition for a particle to pass through the filter undeflected: "Note that the particle will not experience any transverse acceleration if v_z = v_pass = -E_y / B_x." This defines the velocity at which the electric and magnetic forces in the y-direction cancel each other out.
Application Example: The excerpt provides a practical example involving a Li+ ion entering a region with specific electric and magnetic field strengths (E_y = 1.20 x 10^5 V/m, B_x = 2.00 x 10^-3 T). It asks students to calculate the Cartesian components of the acceleration for this ion with a given mass (7 amu), kinetic energy (100 eV), and initial velocity direction. It also poses a comparative question regarding a doubly ionized nitrogen ion (N++).

3. Key Quotes:

"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."
"The solutions to these equations are obtained through numerical integation, and the capstone exercise is the simulation of the E ⃗ × B ⃗ (Wien) filter."
"Students who complete this set of exercises will be able to: ... simulate the operation of an E ⃗ × B ⃗ (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 )."
"Note that the particle will not experience any transverse acceleration if v_z = v_pass = -E_y / B_x."

4. Implications and Next Steps:

This exercise provides a comprehensive approach to understanding the physics behind velocity selectors. Students will gain hands-on experience in:

Applying the Lorentz force law.
Deriving and solving equations of motion (numerically).
Visualizing particle trajectories in electromagnetic fields.
Understanding the principle of velocity selection in a Wien filter.

The final question in the excerpt regarding the comparison of acceleration components for different ions highlights the dependence of particle motion on charge-to-mass ratio, a fundamental concept in mass spectrometry and related applications of electromagnetic fields. Further exploration of Exercises 2, 3, and 4 would likely delve into the implementation of the numerical integration, the simulation of the filter, and the analysis of factors affecting the transmitted velocity range.

Wien Filter Study Guide

Key Concepts

Charged Particle Motion in Fields: Understand how electric and magnetic fields exert forces on moving charged particles.
Lorentz Force: Comprehend the vector equation describing the total electromagnetic force on a charged particle.
Cartesian Components of Force: Be able to break down the Lorentz force into its x, y, and z components.
Equations of Motion: Know how to apply Newton's second law (F=ma) to a charged particle in electromagnetic fields to derive differential equations describing its motion.
Numerical Integration: Recognize that the equations of motion can be solved numerically to determine particle trajectories.
E × B Drift: Understand the concept of the crossed-field drift where the electric and magnetic forces balance for a specific velocity.
Velocity Selector: Grasp the principle of a Wien filter as a device that selects particles based on their velocity.
Particle Trajectory: Be able to visualize and describe the path of a charged particle in combined electric and magnetic fields.
Filter Geometry: Understand how the physical dimensions and field strengths of a Wien filter affect its performance.
Charge-to-Mass Ratio: Recognize the importance of the q/m ratio in determining a particle's response to electromagnetic fields.

Quiz

What are the two types of forces that act on a charged particle moving through a region with both electric and magnetic fields? Describe each force in terms of the particle's charge, velocity, and the field vectors.
Write down the vector form of the Lorentz force law. Explain what each symbol in the equation represents, including their vector or scalar nature.
Given perpendicular electric ( $\vec{E}$ ) in the y-direction and magnetic ( $\vec{B}$ ) in the x-direction, write down the y-component of the Lorentz force acting on a particle with charge $q$ and velocity $\vec{v} = (v_x, v_y, v_z)$ .
Explain why numerical integration is often necessary to find the exact trajectory of a charged particle in a Wien filter. What information does numerical integration provide?
What condition on the velocity of a charged particle allows it to pass through a Wien filter undeflected when the electric and magnetic fields are mutually perpendicular? Express this condition mathematically.
Describe the motion of a charged particle in a Wien filter if its velocity in the z-direction is significantly larger than its velocities in the x and y directions. Assume the fields are oriented as described in the source.
According to the provided equations of motion (10, 11, and 12), if there is only a magnetic field in the x-direction and a velocity component in the y-direction, which component(s) of the acceleration will be non-zero? Explain your reasoning.
What does the term "transverse acceleration" refer to in the context of a charged particle moving primarily along the z-axis through a Wien filter with $\vec{E}$ in the y-direction and $\vec{B}$ in the x-direction?
How does the charge-to-mass ratio ( $q/m$ ) of a particle affect its acceleration in a given electric and magnetic field? Explain this relationship based on the equations of motion.
Based on the provided code snippet, if a Lithium-ion ( $Li^+$ ) and a doubly ionized Nitrogen ion ( $N^{++}$ ) enter the Wien filter with the same kinetic energy and direction, what primary property will cause their accelerations to differ?

Answer Key

The two forces are the electric force ( $\vec{F}_E = q\vec{E}$ ) and the magnetic force ( $\vec{F}_B = q\vec{v} \times \vec{B}$ ). The electric force is proportional to the charge and the electric field, while the magnetic force is proportional to the charge, the velocity of the particle, and the magnetic field, acting perpendicular to both.
The Lorentz force law is $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$ , where $\vec{F}$ is the total electromagnetic force (vector), $q$ is the charge of the particle (scalar), $\vec{E}$ is the electric field (vector), $\vec{v}$ is the velocity of the particle (vector), and $\vec{B}$ is the magnetic field (vector). The " $\times$ " denotes the vector cross product.
Given $\vec{E} = (0, E_y, 0)$ and $\vec{B} = (B_x, 0, 0)$ , the y-component of the magnetic force is $q(v_z B_x - v_x B_z) = q v_z B_x$ (since $B_z = 0$ ). Therefore, the total y-component of the Lorentz force is $F_y = q E_y + q v_z B_x$ .
The equations of motion for a charged particle in a Wien filter often involve velocity-dependent forces (due to the magnetic field), resulting in coupled differential equations that may not have simple analytical solutions. Numerical integration provides an approximate solution by breaking the motion into small time steps and iteratively calculating the particle's position and velocity.
A charged particle passes through undeflected when the net force in the transverse direction (perpendicular to the initial velocity) is zero. For $\vec{E}$ in the y-direction and $\vec{B}$ in the x-direction with a primary velocity $v_z$ , this occurs when $qE_y + qv_zB_x = 0$ , which simplifies to $v_z = -E_y / B_x$ . This specific velocity is the pass velocity ( $v_{pass}$ ).
If $v_z >> v_x, v_y$ , the particle will primarily move along the z-axis through the field region. However, the electric and magnetic fields will exert forces causing deflections in the x and y directions. The magnetic force will depend on $v_z$ , leading to a y-directed force, while the electric field directly causes a force in the y-direction.
According to the equations, if $E_y = 0$ , $B_x \neq 0$ , and $v_y \neq 0$ with all other field and initial velocity components being zero, then $\ddot{x} = 0$ , $\ddot{y} = 0$ , and $\ddot{z} = -\frac{q}{m} v_y B_x$ . Therefore, only the z-component of the acceleration will be non-zero in this specific scenario.
Transverse acceleration refers to the acceleration components perpendicular to the primary direction of motion, which is assumed to be along the z-axis in this context. Therefore, $a_x = \ddot{x}$ and $a_y = \ddot{y}$ are the transverse acceleration components.
The acceleration of a charged particle is directly proportional to its charge-to-mass ratio ( $q/m$ ). As seen in equations (11) and (12), the acceleration components $\ddot{y}$ and $\ddot{z}$ are both multiplied by the factor $q/m$ . A larger $q/m$ will result in a larger magnitude of acceleration for the same electric and magnetic fields and velocities.
The primary property that will cause their accelerations to differ is their charge-to-mass ratio ( $q/m$ ). Although they have the same kinetic energy and initial direction, Lithium ( $Li^+$ ) and doubly ionized Nitrogen ( $N^{++}$ ) have different charges and significantly different masses, leading to distinct $q/m$ values and therefore different accelerations in the same electromagnetic fields.

Essay Format Questions

Discuss the fundamental principles behind the operation of a Wien filter as a velocity selector for charged particles. Explain how the mutually perpendicular electric and magnetic fields interact with particles of different velocities and how this interaction leads to velocity selection.
Derive the Cartesian components of the Lorentz force acting on a charged particle with velocity $\vec{v} = (v_x, v_y, v_z)$ in a region with an electric field $\vec{E} = (0, E_y, 0)$ and a magnetic field $\vec{B} = (B_x, 0, 0)$ . Then, using Newton's second law, write down the corresponding equations of motion for the particle.
Explain the significance of numerical integration in analyzing the motion of charged particles in electromagnetic fields, particularly in the context of a Wien filter. What are the limitations of analytical solutions in such scenarios, and how does numerical integration provide valuable insights?
Analyze how the geometry (e.g., length of the field region) and the strengths of the electric and magnetic fields in a Wien filter affect the range of velocities of particles that can pass through undeflected. Consider the implications for the filter's resolution and transmission efficiency.
Compare and contrast the forces experienced by charged particles in purely electric fields, purely magnetic fields, and combined electric and magnetic fields as found in a Wien filter. How do these different force configurations affect the trajectory and energy of the charged particles?

Glossary of Key Terms

Charged Particle: A subatomic particle or ion that carries an electric charge.
Electric Field ( $\vec{E}$ ): A region of space around an electrically charged object in which a stationary test charge would experience an electric force. Measured in volts per meter (V/m).
Magnetic Field ( $\vec{B}$ ): A region of space around a magnet or moving electric charge in which a moving test charge would experience a magnetic force. Measured in Tesla (T).
Lorentz Force: The total force exerted on a charged particle moving with velocity $\vec{v}$ in an electric field $\vec{E}$ and a magnetic field $\vec{B}$ , given by $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$ .
Cartesian Components: The projections of a vector along the x, y, and z axes of a Cartesian coordinate system.
Equations of Motion: Mathematical equations that describe the physical movement of a system as a function of time. For a particle, these are often differential equations relating its position, velocity, and acceleration to the forces acting on it.
Numerical Integration: A set of algorithms for numerically approximating the value of a definite integral. In this context, used to solve differential equations of motion by stepping through time.
$\vec{E} \times \vec{B}$ Drift: The motion of a charged particle in crossed electric and magnetic fields where the particle drifts in a direction perpendicular to both fields when the electric and magnetic forces do not perfectly balance.
Wien Filter (Velocity Selector): A device that uses perpendicular electric and magnetic fields to select charged particles based on their velocity. Only particles with a specific velocity will pass through undeflected.
Particle Trajectory: The path followed by a moving particle through space as a function of time.
Filter Geometry: The physical dimensions and arrangement of the components of a filter, such as the length and spacing of the plates in a Wien filter.
Charge-to-Mass Ratio (q/m): The ratio of the electric charge of a particle to its mass. This ratio is a fundamental property that determines how a particle will accelerate in electromagnetic fields.

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Questions:

1. What is the purpose of the PICUP Wien (E x B) filter exercises?

The purpose of these exercises is to guide students in understanding and analyzing the motion of a charged particle within a region containing mutually perpendicular electric and magnetic fields. Students will learn to calculate the forces acting on the particle, derive the equations of motion, solve these equations numerically to determine trajectories, visualize these trajectories in 2D and 3D plots, and ultimately simulate and analyze the operation of a Wien filter.

2. What fundamental concepts from physics are required to engage with these exercises?

To engage with these exercises, students need a solid understanding of electricity and magnetism, specifically the forces exerted on a charged particle by electric and magnetic fields (Lorentz force). They should also be familiar with basic mechanics, including Newton's laws of motion, and have some mathematical background in vector algebra, differential equations (specifically, the ability to work with accelerations and relate them to velocity and position), and numerical integration techniques.

3. What are the main learning objectives for students completing these exercises?

Upon completing these exercises, students should be able to:

Generate equations that predict the Cartesian components of the force acting on a charged particle in combined electric and magnetic fields and derive the corresponding equations of motion.
Calculate the trajectory of a charged particle by numerically solving its equations of motion.
Create two-dimensional and three-dimensional plots to visualize these particle trajectories.
Simulate the functionality of a Wien (E x B) filter.
Determine the range of particle velocities that can pass through the filter and understand how the filter's geometry influences this range.

4. How are the equations of motion for a charged particle in perpendicular electric and magnetic fields obtained?

The equations of motion are obtained by first determining the net force acting on the charged particle. This force is the vector sum of the electric force (F = qE) and the magnetic force (F = q(v x B)), known as the Lorentz force. By considering mutually perpendicular electric (Ey in the y-direction) and magnetic (Bx in the x-direction) fields, and a particle with velocity components (vx, vy, vz), the Cartesian components of the force can be determined. Applying Newton's second law (F = ma) then yields the differential equations of motion in terms of the particle's acceleration components (ẍ, ÿ, ž), which are the second time derivatives of the position coordinates.

5. Why is numerical integration necessary to solve the equations of motion in this scenario?

The equations of motion for a charged particle in combined electric and magnetic fields are typically coupled and nonlinear, especially when the velocity components are not constant. This complexity often prevents finding analytical (closed-form) solutions. Numerical integration techniques, such as the Euler method or more advanced methods like Runge-Kutta, are employed to approximate the particle's trajectory by breaking the motion into small time steps and iteratively updating the velocity and position based on the forces and accelerations at each step.

6. What is a Wien filter, and how does it operate based on the principles explored in these exercises?

A Wien filter, also known as a velocity selector, is a device that uses perpendicular electric and magnetic fields to select charged particles with a specific velocity. The electric force (qE) and the magnetic force (q(v x B)) acting on a charged particle moving through the filter are in opposite directions for a certain velocity. When these two forces are balanced (qE = qvB), the net force on the particle is zero, and it passes through the filter undeflected. Particles with velocities greater or smaller than this selected velocity will experience a net force and be deflected.

7. What is the significance of the condition $v_z >> v_x, v_y$ in the context of this exercise?

The condition $v_z >> v_x, v_y$ implies that the charged particle is primarily moving along the z-axis, which is the direction of its initial entry into the field region of length L. This assumption can simplify the initial analysis by allowing approximations where the transverse velocity components ( $v_x, v_y$ ) are initially small compared to the longitudinal velocity ( $v_z$ ). However, the exercises also aim to show how these smaller velocity components and the resulting forces in the x and y directions lead to deviations from simple straight-line motion along the z-axis, necessitating numerical methods for accurate trajectory prediction.

8. How does the charge-to-mass ratio (q/m) affect the motion and filtering of particles in a Wien filter?

The charge-to-mass ratio (q/m) is a crucial factor determining the behavior of charged particles in electric and magnetic fields. In the equations of motion, the acceleration is directly proportional to q/m multiplied by the forces per unit charge (E and v x B). For a Wien filter, the velocity selected for undeflected passage (v = E/B) is independent of q/m. However, for particles with different q/m ratios but the same kinetic energy, their velocities will differ, leading to different degrees of deflection within the filter. Furthermore, when analyzing the detailed trajectories and the range of velocities transmitted, q/m plays a significant role in how strongly the electric and magnetic forces influence the particle's path. The provided example with Li+ and N++ ions illustrates how different q/m ratios result in different accelerations under the same field conditions.

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Details: Written by Fremont; Parent Category: 05 Electricity and Magnetism; Category: 08 Electromagnetism; Published: 30 April 2018; Created: 30 April 2018; Last Updated: 22 March 2025; Hits: 5484

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