PICUP Wien (E x B) Filter Exercise 2 JavaScript HTML5 Applet Simulation Model

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About

Intro Page

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 2

EXERCISE 2: COMPUTING THE TRAJECTORY

Solve the equations of motion to obtain the trajectory of the Li ion from Exercise 1 while it traverses the field region from to m.

(a) On separate graphs, plot , , and 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?

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)

# 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()

Translations

Code	Language		Translator	Run

Credits

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

Overview:

This document provides a briefing on the "PICUP Wien (E x B) Filter Exercise 2 JavaScript HTML5 Applet Simulation Model," an open educational resource designed for students to understand the behavior of charged particles in mutually perpendicular electric and magnetic fields. The resource focuses on the Wien filter, also known as a velocity selector. It guides students through computational analysis and simulation of particle trajectories and the filter's operation.

Main Themes and Important Ideas/Facts:

The Wien (E x B) Filter: The core concept is the Wien filter, which utilizes perpendicular electric ( $\vec{E}$ ) and magnetic ( $\vec{B}$ ) fields to select charged particles based on their velocity. The resource aims to help students understand how these fields interact with charged particles and how a specific velocity can be selected when the electric and magnetic forces balance.

The description explicitly mentions the simulation of the " $\vec{E} \times \vec{B}$ " (Wien) filter.

Educational Exercises: The resource is structured as a set of exercises to facilitate student learning. Exercise 2, which is the focus of the provided excerpts, specifically deals with "COMPUTING THE TRAJECTORY" of a charged particle.

The "About" section states, "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 Learning Objectives clearly outline the skills students will develop, including:
Generating equations of motion (Exercise 1).
Calculating particle trajectories (Exercise 2).
Producing 2D and 3D plots of trajectories (Exercise 2).
Simulating the operation of a Wien filter (Exercise 3).
Determining the range of transmitted velocities and the effect of filter geometry (Exercise 4).

Numerical Integration: The simulation and analysis rely on numerical integration to solve the equations of motion for the charged particle under the influence of the electromagnetic fields.

The "About" section notes, "The solutions to these equations are obtained through numerical integration..."
The provided Python code (ExB_Filter_Exercise_2.py) explicitly uses the scipy.integrate.odeint routine for this purpose.

Particle Trajectory Analysis: Exercise 2 specifically requires students to solve the equations of motion for a Li+ ion traversing the field region and to analyze its trajectory. This involves:

Plotting the x, y, and z positions of the ion versus time.
Plotting the trajectory in 3D space.
Investigating how changes in initial kinetic energy affect the trajectory.
Comparing the initial and final kinetic energies of the ion.
The exercise description asks, "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?"
It also asks, "What is the kinetic energy of the ion at the end of its trajectory? How does it compare to its initial energy?"

Simulation Model: The resource includes a JavaScript HTML5 applet simulation model that students can interact with. The provided iframe embed code suggests that a visual simulation is available for students to observe the behavior of charged particles in the Wien filter.

The title of the resource itself indicates the presence of a "JavaScript HTML5 Applet Simulation Model."
The "Embed this model in a webpage:" section provides the code to integrate the interactive simulation.

Python Implementation: Alongside the applet, a Python script (ExB_Filter_Exercise_2.py) is provided, which performs the numerical integration and generates plots. This offers an alternative or supplementary approach for students to compute and visualize the particle trajectories.

The "Available Implementation" section lists "Python."
The provided code includes comments explaining the physical setup (z-axis aligned with the filter, magnetic field along +x, electric field along -y) and the parameters used in the simulation (charge q, mass m, kinetic energy KE_eV, electric field Ey, magnetic field Bx, filter length L, initial velocity direction u).

Learning Objectives: The resource clearly states the intended learning outcomes for students who engage with the exercises. These focus on developing skills in applying fundamental physics principles to predict and analyze the motion of charged particles in electromagnetic fields, as well as using computational tools for simulation and visualization.

The "Learning Objectives" section lists four key abilities students will gain.

Key Code Insights:

The Python script initializes physical parameters for a Li+ ion and the electric and magnetic fields.
It defines a derivs function that calculates the time derivatives of the particle's position and velocity based on the Lorentz force law.
The scipy.integrate.odeint function numerically solves these differential equations to obtain the particle's trajectory over time.
The script then extracts and prints information such as initial and final velocities and kinetic energies, as well as generates plots of position versus time and the 3D trajectory using pylab.
The code calculates a "pass velocity" (vzpass = -Ey/Bx), which is the velocity component along the z-axis for which the electric and magnetic forces cancel out, resulting in undeflected motion.

Target Audience and Level:

The resource is designed for "First Year and Beyond the First Year" university-level physics students studying electromagnetism.

Potential Uses:

Interactive learning tool for understanding the principles of the Wien filter.
Computational physics exercise involving numerical integration of equations of motion.
Visualization of charged particle trajectories in electromagnetic fields.
Laboratory activity or homework assignment to reinforce concepts related to the Lorentz force.

Credits and Licensing:

The resource is credited to Fremont Teng and Loo Kang Wee, based on codes by E. Behringer. The content is licensed under a Creative Commons Attribution-Share Alike 4.0 Singapore License, promoting open access and sharing for educational purposes. Commercial use of the underlying EasyJavaScriptSimulations Library requires a separate license.

Related Resources:

The page includes a long list of other interactive physics and mathematics simulations available on the platform, indicating a rich collection of open educational resources.

This briefing document summarizes the key aspects of the PICUP Wien (E x B) Filter Exercise 2 resource, highlighting its pedagogical goals, technical implementation, and potential applications in physics education.

Wien (E x B) Filter Study Guide

Key Concepts

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.
Electric Force: The force experienced by a charged particle in an electric field, given by $\vec{F}_E = q\vec{E}$ , where $q$ is the charge and $\vec{E}$ is the electric field.
Magnetic Force (Lorentz Force): The force experienced by a moving charged particle in a magnetic field, given by $\vec{F}_B = q(\vec{v} \times \vec{B})$ , where $q$ is the charge, $\vec{v}$ is the velocity of the particle, and $\vec{B}$ is the magnetic field.
Equations of Motion: Mathematical expressions that describe the motion of an object under the influence of forces, often involving Newton's second law ( $\vec{F} = m\vec{a}$ ).
Numerical Integration: Approximate methods for finding the solution to differential equations, especially when analytical solutions are difficult or impossible to obtain. The provided code uses odeint from the scipy.integrate library.
Particle Trajectory: The path followed by a particle as it moves through space.
Cartesian Components: The $x$ , $y$ , and $z$ components of a vector, such as force, velocity, or position.
Kinetic Energy: The energy of motion, given by $KE = \frac{1}{2}mv^2$ , where $m$ is the mass and $v$ is the speed of the particle.

Study Questions

What is the fundamental principle behind the operation of a Wien filter? Explain how the electric and magnetic forces are utilized for velocity selection.
Describe the orientation of the electric and magnetic fields in the Wien filter simulation described in the source. In which direction is the filter axis aligned?
How are the equations of motion for a charged particle in mutually perpendicular electric and magnetic fields derived? What are the key variables involved?
Explain the role of numerical integration in determining the trajectory of the charged particle in this simulation. Why is this approach necessary?
What are the learning objectives outlined for the set of exercises related to the Wien filter? Focus on the objectives specifically addressed in Exercise 2.
In the provided Python code, identify the lines where the electric and magnetic field components are defined. What are their initial values?
What does the variable vzpass represent in the Python code, and how is it calculated? What is its physical significance?
Explain how the initial velocity components of the charged particle are determined in the Python code. What factors influence these values?
Describe the purpose of the derivs function in the Python code. What does it calculate and return?
How do the plots generated in Exercise 2 (position vs. time and 3D trajectory) help in analyzing the motion of the charged particle through the Wien filter?

Quiz

A Wien filter uses what type of fields to select charged particles based on their velocity? Briefly describe how these fields interact with a charged particle.
For a particle to pass undeflected through a Wien filter with perpendicular electric and magnetic fields, what must be the relationship between the electric and magnetic forces acting on it? Explain the condition for zero net force.
According to the provided code, what are the assumed directions of the magnetic field and the electric field within the Wien filter? Along which axis is the filter aligned?
The Python code uses odeint from scipy.integrate. What is the general purpose of this function in the context of the simulation?
What physical quantity does the variable qoverm represent in the code, and why is it a useful parameter in the equations of motion for a charged particle in electromagnetic fields?
Explain what the "pass velocity" (vzpass) signifies for a charged particle entering the Wien filter along the z-axis. How is it related to the electric and magnetic field strengths?
The code initializes the kinetic energy of the particle in electron volts (eV). Why is it then converted to Joules (J) later in the script?
What information can be gained by plotting the x, y, and z positions of the ion as a function of time as done in Exercise 2a?
Exercise 2b asks about reducing the initial kinetic energy. How would a decrease in kinetic energy affect the initial velocity magnitude of the ion?
In Exercise 2c, the kinetic energy of the ion at the end of its trajectory is compared to its initial energy. What could cause a change in the kinetic energy of the ion as it passes through the filter?

Quiz Answer Key

A Wien filter uses perpendicular electric and magnetic fields. The electric field exerts a force parallel (or antiparallel) to the field, while the magnetic field exerts a force perpendicular to both the velocity of the particle and the magnetic field direction.
For undeflected passage, the electric and magnetic forces must be equal in magnitude and opposite in direction. This results in a net force of zero, allowing the particle to move in a straight line.
The code assumes the magnetic field is along the +x-direction and the electric field is along the -y-direction. The axis of the filter is aligned with the z-axis.
The odeint function is used for numerical integration of a system of ordinary differential equations given an initial value. In this simulation, it solves the equations of motion to find the particle's trajectory over time.
qoverm represents the charge-to-mass ratio of the particle. It simplifies the equations of motion as the acceleration of the particle is directly proportional to this ratio and the electromagnetic forces.
The "pass velocity" is the velocity along the z-axis that a charged particle must have to experience zero net force in the x-y plane due to the balanced electric and magnetic forces. It allows the particle to pass through undeflected.
The initial kinetic energy is given in eV, a convenient unit at the atomic/particle level. However, for calculations within the code using standard physics formulas, SI units like Joules are required.
Plotting the position components versus time reveals how the particle's displacement in each direction evolves over the duration of its passage through the filter, showing the influence of the fields on each component of motion.
Reducing the initial kinetic energy would decrease the magnitude of the initial velocity of the ion, as kinetic energy is directly proportional to the square of the velocity.
A change in the kinetic energy could be caused by the electric field doing work on the charged particle. If the particle is deflected in the direction of (or opposite to) the electric force, its speed and hence kinetic energy will change.

Essay Format Questions

Discuss the principles behind velocity selection in a Wien filter. How do the magnitudes and directions of the electric and magnetic fields determine the velocity of the particles that pass through undeflected? Consider the forces acting on a charged particle and the conditions for zero net deflection.
Explain the process of modeling the trajectory of a charged particle in a Wien filter using numerical integration. What are the advantages and limitations of this approach compared to analytical solutions? Refer to the provided Python code and discuss the role of the derivs function and the odeint solver.
Analyze the effect of changing the initial kinetic energy of the charged particle on its trajectory through the Wien filter. Based on the simulation exercise, how does reducing or increasing the initial kinetic energy influence whether a particle passes through the filter undeflected? Discuss the implications for the filter's velocity selection capabilities.
Describe the significance of the "pass velocity" in the context of a Wien filter. How is this velocity mathematically determined by the electric and magnetic field strengths? What happens to particles with velocities significantly different from the pass velocity?
Evaluate the use of the provided JavaScript HTML5 applet simulation model as an educational tool for understanding the behavior of charged particles in electromagnetic fields and the operation of a Wien filter. What are the benefits of such a simulation for student learning compared to theoretical analysis alone? Consider the learning objectives outlined in the source material.

Glossary of Key Terms

Charged Particle: A subatomic particle (e.g., electron, proton, ion) that carries an electric charge.
Electric Field ( $\vec{E}$ ): A region of space around an electrically charged object in which a force would be exerted on other electrically charged objects. Measured in Newtons per Coulomb (N/C).
Magnetic Field ( $\vec{B}$ ): A field of force produced by moving electric charges or magnetic dipoles that exerts a force on other moving charges or magnetic dipoles. Measured in Tesla (T).
Force ( $\vec{F}$ ): An interaction that, when unopposed, will change the motion of an object. Measured in Newtons (N).
Velocity ( $\vec{v}$ ): The rate of change of an object's position with respect to time and a frame of reference. It is a vector quantity, having both magnitude (speed) and direction. Measured in meters per second (m/s).
Trajectory: The path followed by an object moving through space as a function of time.
Numerical Integration: A set of algorithms for computing the numerical value of a definite integral, and by extension, for solving differential equations numerically.
Cartesian Coordinates: A coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. In three dimensions, a third coordinate indicates the distance from a third mutually perpendicular plane.
Kinetic Energy (KE): The energy that an object possesses due to its motion. Measured in Joules (J) or electron volts (eV).
Charge (q): A fundamental property of matter that can be either positive or negative. Measured in Coulombs (C).
Mass (m): A fundamental property of an object that measures its resistance to acceleration when a net force is applied. Measured in kilograms (kg).
Equations of Motion: Mathematical equations that describe the physical behavior of a system in terms of its motion as a function of time.
Simulation: The use of a model (e.g., a computer program) to represent the behavior or characteristics of a physical or abstract system.
Open Educational Resources (OER): Teaching, learning, and research materials that are in the public domain or have been released under an open license that permits no-cost access, use, adaptation, and redistribution by others with no or limited restrictions.
Applet: A small application program, especially one designed for relatively simple, single-purpose tasks, often designed to be embedded within a document or other application.

Sample Learning Goals

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Frequently Asked Questions: PICUP Wien (E x B) Filter Simulation

1. What is a Wien (E x B) filter and how does it work?

A Wien filter is a device that uses perpendicular electric ((\vec{E})) and magnetic ((\vec{B})) fields to select charged particles based on their velocity. The electric field exerts a force (\vec{F}_E = q\vec{E}) on a charged particle ((q)), while the magnetic field exerts a force (\vec{F}_B = q(\vec{v} \times \vec{B})), where (\vec{v}) is the velocity of the particle. By carefully orienting these fields to be mutually perpendicular, and tuning their magnitudes, particles with a specific velocity will experience equal and opposite electric and magnetic forces, resulting in no net deflection and allowing them to pass through the filter. Particles with velocities greater or less than this selected velocity will experience a net force and be deflected.

2. What is the purpose of the PICUP Wien (E x B) Filter Exercise 2 JavaScript HTML5 Applet Simulation Model?

This simulation model is designed as an educational tool to help students understand the principles behind the Wien filter. It allows students to computationally analyze the motion of a charged particle in perpendicular electric and magnetic fields. Through interactive exercises, students can:

Determine the forces acting on the particle.
Obtain and solve the equations of motion using numerical integration.
Visualize the particle's trajectory in 2D and 3D plots.
Simulate the operation of a Wien filter and investigate how its geometry affects the range of transmitted velocities.

3. What are the learning objectives of this simulation exercise?

Upon completion of this set of exercises, students should be able to:

Generate equations predicting the Cartesian components of the force acting on a charged particle in perpendicular electric and magnetic fields and derive the corresponding equations of motion.
Calculate the trajectory of a charged particle by solving these equations of motion, often through numerical methods.
Produce two-dimensional and three-dimensional plots visualizing the calculated trajectories.
Simulate the operation of a Wien (E x B) filter.
Determine the range of particle velocities that can pass through the filter without deflection and understand how the filter's geometry influences this velocity selection.

4. What physical principles are involved in the operation of a Wien filter?

The operation of a Wien filter relies on the fundamental principles of electromagnetism, specifically the forces exerted on a moving charged particle by electric and magnetic fields (Lorentz force). The key is the balance between the electric force ((\vec{F}_E = q\vec{E})) and the magnetic force ((\vec{F}_B = q(\vec{v} \times \vec{B})). When these forces are equal in magnitude and opposite in direction, the net force on the charged particle is zero, leading to no deflection. This condition is velocity-dependent, allowing the filter to select particles with a specific velocity.

5. How does the simulation help in understanding the trajectory of a charged particle in the filter?

The simulation utilizes numerical integration (specifically the odeint routine in the provided Python code) to solve the second-order linear differential equations that describe the motion of a charged particle under the influence of the electric and magnetic fields. By computing the particle's position and velocity at small time intervals, the simulation generates data that can be plotted to visualize the trajectory in space and the individual components of motion (x, y, and z positions) as a function of time. This allows students to observe how the forces affect the particle's path and to compare their predictions with the simulated results.

6. What is the significance of the "pass velocity" ((v_{zpass} = -E_y/B_x)) mentioned in the code?

The "pass velocity" ((v_{zpass})) represents the specific velocity in the z-direction (assuming the filter axis is aligned with z, the magnetic field along +x, and the electric field along -y as per the code's assumptions) for which the magnetic force and the electric force on a positively charged particle will exactly balance each other in the y-direction. For a particle with this velocity, the net force in the y-direction (the direction of deflection) will be zero, ideally allowing the particle to pass through the filter undeflected.

7. How does changing the initial kinetic energy of the ion affect its trajectory through the Wien filter, according to Exercise 2?

Reducing the initial kinetic energy of the ion will change the magnitude of its initial velocity. Since the magnetic force depends on the velocity of the particle, altering the kinetic energy will disrupt the balance between the electric and magnetic forces for particles that would have otherwise passed undeflected. As indicated in the exercise questions, reducing the kinetic energy significantly (by factors of 100 or 10,000) is expected to cause greater deflection of the ion from its intended path. Particles with lower initial velocities will experience a stronger influence from the electric and/or magnetic fields, leading to curved trajectories and likely preventing them from traversing the filter without being deflected.

8. Does a charged particle gain or lose kinetic energy as it passes through an ideal Wien filter that selects it?

In an ideal Wien filter that successfully selects a particle (i.e., the particle passes through undeflected because the electric and magnetic forces are balanced), the net work done on the particle by the fields is zero. The electric force does work, but the magnetic force, being always perpendicular to the velocity, does no work. For a particle to pass through without deflection, the trajectory should be along a straight line. In such a case, there is no net force in the direction of motion, and therefore no net work done, implying that the kinetic energy of the selected particle remains constant as it passes through the ideal filter. The provided code and exercise also explore scenarios where particles are deflected, and in those cases, the final kinetic energy might differ from the initial kinetic energy due to the net work done by the electric field.

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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: 6273