PHYS 259 W2024 - Lab 07 - Charge to mass ratio

PHYS 259, Lab 7, Winter 2024 University of Calgary Department of Physics and Astronomy PHYS 259, Winter 2024 Lab 7: Charge-to-Mass Ratio of the Electron In this lab, you will use concepts learned in class about the motion of charged particles in electric and magnetic fields to measure the charge-to-mass ratio e/m of the electron. The apparatus, shown to the right, consists of an electron gun that expels elec- trons at a set speed and a configuration of Helmholtz coils that provide an approximately uniform mag- netic field. You will analyze each piece of the setup, both conceptually and theoretically, to gain an under- standing of how each component works and how both components work together. You will then take mea- surements from pre-recorded videos to determine the charge-to-mass ratio and compare it to the accepted value based on independently known values for e and m . Learning Goals: To understand how charged particles are accelerated by electric and magnetic fields. Preparation: Halliday, Resnick, and Walker, “Fundamentals of Physics” 12th edition, Wiley: 28.1, 28.4, 29.1, and 29.5. Pre-Lab Assignment: Complete Checkpoint 1 prior to arriving at the lab. Your TA will check for completeness. Any student who does not complete the pre-lab assignment will not receive credit for the first checkpoint. Equipment: 4 videos of e/m apparatus (259 Lab 7 Video e m apparatus, 259 Lab 7 Video e m current, 259 Lab 7 Video - e m voltage, and 259 Lab 7 Video e m measurements), computer, Excel template. 1

PHYS 259, Lab 7, Winter 2024 1 The Apparatus Components There are two main components to this apparatus: the electron gun that accelerates electrons to a set velocity, and the Helmholtz coils that provide an approximately uniform magnetic field in which the electrons move in a circular path. Before starting, watch the video 259 Lab 7 Video e m apparatus to make sense of what is being referenced in the questions below. The Electron Gun The electron gun consists of a heating element that makes up the cathode, and an anode such that the potential difference between the two is Δ V . As the element heats up, electrons are liberated and are then free to move. Focus on the motion of a single electron in the e/m apparatus: the actual apparatus uses a beam of electrons. Question 1 : Given that the electric potential between the cathode and anode is Δ V , how much electric potential energy does an electron have when it is just liberated from the heating element? Question 2 : Using energy conservation, find an expression for the exit speed v of the electron in terms of the gun voltage Δ V and the charge-to-mass ratio e/m . Assume it starts from rest. Question 3 : Which physical parameter of the electron gun do you need to control in order to change the electron exit speed? 2

PHYS 259, Lab 7, Winter 2024 The Helmholtz Coils, continued Question 7 : What is an expression for the magnitude of the net force acting on an object of mass m moving in a circle of radius r at constant speed v ? Question 8 : Assuming that the magnetic field strength between the coils is B , what is the magnitude of the magnetic force acting on an electron moving with speed v ? Question 9 : Use your previous two answers to express r in terms of v , B , and e/m . In class you saw that the magnetic field strength B a distance z along the axis of a coil of N loops of current I and radius R (don’t get this confused with the electron path radius r ) is given by B = μ 0 NIR 2 2 ( R 2 + z 2 ) 3 / 2 . Helmholtz coils are in a configuration such that there are two separate coils to consider (as considered in Question 4), and each coil is located at a distance z = R/ 2 from the midpoint. Question 10 : Use the above expression to derive a relationship between the current in the two coils (of N loops each) and the magnetic field strength at the halfway point between them. Simplify your answer as much as you can. 4

PHYS 259, Lab 7, Winter 2024 Question 11 : What are the two variables in Question 9 that affect the radius of the electron path r and how can each be varied? B: A: CHECKPOINT 2: Before moving on to the next part, discuss your results as a group, then have your TA evaluate your answers. 2 The Complete e/m Apparatus Question 12 : In the figure below label the following components of the e/m apparatus: power switch, voltage dial, voltage display, current dial, current display, Helmholtz coils, heating element, e/m glass tube 5

PHYS 259, Lab 7, Winter 2024 Question 15 : What happened to the radius of the electron beam when the voltage was kept constant at 250 V and the current increased or decreased? Do your observations agree with what you previously predicted should happen in Question 13? Question 16 : Now watch the video: 259 Lab 7 Video e m voltage. What happened to the radius of the electron beam when the current in the coils was kept constant at 1.5 A and the voltage increased or decreased? Do your observations agree with what you previously predicted should happen in Question 13? You should notice that changing either the voltage or the current will change the radius of the electron beam. When taking measurements, it is easiest to set the voltage, then for each value of the voltage the current (equivalently B ) is adjusted to keep the radius constant; i.e., r is constant while Δ V and B vary. Question 17 : Using your answer to Question 14, rearrange it to find an equation for a line Y = MX that can be plotted to find e/m . What are Y , M , and X in terms of the physical parameters Δ V , r 2 , B 2 , and e/m ? B: A: CHECKPOINT 3: Before moving on to the next part, discuss your results as a group, then have your TA evaluate your answers. 7

PHYS 259, Lab 7, Winter 2024 4 Data Acquisition Now you will record data and analyze it to extract the charge-to-mass ratio. In order to do so, the parameters of the Helmholtz coils need to be specified. Each coil contains N = 132 loops of wire, and each loop has a radius of R = 15 . 0 cm. The accompanying Excel template you will use already has these values as inputs, but this is where they come from. Watch the video: 259 Lab 7 Video e m measurements and follow the steps below: 1. Download the accompanying Excel template from D2L and save it to your computer. 2. The voltage dial on the apparatus is initially set to 200 V and the current adjusted until the diameter of the electron beam is 8.0 cm. Record the values of the voltage and the current in the appropriate fields in the template. 3. The voltage is increased by 15 V and the current adjusted until the diameter of the electron beam is 8.0 cm. Record the values of the voltage and the current in the appropriate fields in the template. 4. Carry out this procedure for 9 more 15 V increments (the diameter of the electron beam is 8.0 cm for all of them) and fill in the relevant fields of the Excel template. Question 18 : The Excel template generates a plot of Δ V vs B 2 . Does your data form a line or a parabola? Which should it form and why? Question 19 : The generated graph also contains a best fit line. Which fit parameter is related to e/m ? Question 20 : How can you calculate the ratio e/m from this fit parameter? 8