# LAB 8: Electron Charge-to-Mass Ratio

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1 Name Date Partner(s) OBJECTIVES LAB 8: Electron Charge-to-Mass Ratio To understand how electric and magnetic fields impact an electron beam To experimentally determine the electron charge-to-mass ratio. OVERVIEW In this experiment, you will measure e/m, the ratio of the electron s charge e to its mass m. Given that it is also possible to perform a measurement of e alone (the Millikan Oil Drop Experiment), it is possible to obtain the value of the mass of the electron, a very small quantity. If a particle carrying an electric charge q moves with a velocity v in a magnetic field B that is at a right angle to the direction of motion, it will experience the Lorentz force: F = q v x B, (1) which, because of the vector product, is always perpendicular to both the magnetic field and the direction of motion. A constant force that is always perpendicular to the direction of motion will cause a particle to move in a circle. We will use this fact to determine e/m of the electron by measuring the radius of that circle. To this end we must: produce a narrow beam of electrons of known energy, produce a uniform magnetic field, find a way to measure the radius r of the circular orbit of the electrons in that magnetic field find the relation between that radius and the ratio e/m. We discuss these problems in order. The Electron Beam When one heats a piece of metal, say a wire, to 1000 K or beyond, electrons will "boil off" from its surface. If one surrounds the wire with a positively charged electrode, an anode, the electrons will be attracted to it and move radially outward as indicated in Fig. 1. On their way to the anode they will acquire a kinetic energy Fig. 1: Electron gun. Electrons are produced by the hot cathode at the center. A voltage V between the cathode and anode accelerates the electron through a hole in the anode, so the electrons have kinetic energy ev.

2 Here, V is the potential difference, or voltage, between the heated filament, called the cathode, and the anode. Most of the electrons will strike the anode. However, if one cuts a narrow slit into the anode, those electrons that started out toward the slit will exit through it as a narrow beam with a kinetic energy Ek. The Magnetic Field According to Ampere's law a wire carrying a current I is surrounded by a magnetic field B, as shown in Fig. 2. If the wire is bent into a circle the field lines from all sides reinforce each other at the center, creating an axial field (see Fig. 3). Usually one will not use a single loop of wire to create a field but a coil with many turns. Fig. 2. Magnetic field of a straight wire. Fig. 3. Magnetic field of a wire loop. If one uses two coaxial coils of radius R that are a distance d apart, as shown in Fig. 4, the field at the center point between the coils will be nearly homogeneous. H. von Helmholtz ( ) realized that when d = R, the result is a particularly homogeneous field in the central region between the coils. In fact, both db/dx and d 2 B/dx 2 vanish at the center, where x is the distance along the axis from the middle point. Since that time Helmholtz coils have been used when there is a need for homogeneous magnetic fields.

3 Fig. 4. Magnetic field B of a pair of Helmholtz coils. R is the radius of the coils and d is the spacing between them. One can show that the field in the center of a Helmholtz coil is given by where I is the current flowing through both coils, R is their mean radius, N is the number of turns of wire in each coil, and µ0 = 4!x10-7 Tm/A is the permeability constant.

4 The Electron Orbit Experiments like the one that you will perform have been used to measure the mass of charged particles with great precision. In these experiments the particles move in a circular arc whose beginning and end are measured very accurately in a near perfect vacuum. For our simple experiment we cannot go to such lengths and a simple expedient has been used to make the electron orbit visible. The bulb surrounding the electron source is filled with an inert gas. When the electrons collide with the gas atoms, the atoms emit light so that one can follow the path of the electron beam. These collisions will diminish the accuracy of the experiment but it remains adequate for our purposes. A particle moving in a circle of radius r must be held there by a centripetal force: In our case, that centripetal force is the Lorentz force, equation (1), so we can write: This equation contains the velocity v, which we can eliminate by using the relationship between the electron kinetic energy and the accelerating voltage from equation (2): Therefore: In this equation, V is the voltage between cathode and anode and r is the mean radius of the circular electron orbit, both of which can be measured, and B is the magnetic field through which the electrons pass. We know the magnetic field at the center of the Helmholtz coil, which can be obtained from a measurement of the current through the coils, the dimension of the coils and the number of turns. The magnetic field does not change very much away from the center of the coils. Equipment: Voltmeter Ammeter Electron beam tube and Helmholtz coils 6.3V A.C. filament power supply to heat beam tube filament and produce free electrons Adjustable 0-300V D.C. power supply for electron beam acceleration in beam tube Adjustable 0-2A D.C. power supply for current to Helmholtz coils to produce magnetic field Computer running Graphical Analysis or some other graphical software

5 Activity 1-1: Field Produced by the Helmholz Coils 1. First, record the number of turns and the radius of the Helmholtz coils you are using: Number of Turns (N) = Radius of Coil (R) = 2. Now, using the formula given in the Overview, calculate the value of the magnetic field at the center of the coils. Enter the value below. Bcenter = 3. The Earth s magnetic field is approximately 10-4 T. Discuss how much difficulty the Earth s magnetic field will cause in your experiment. Question 1-1: In what direction should the apparatus be aligned to minimize the effects of the Earth's magnetic field? Explain. (We can t really realign the apparatus this way, but you should think about it ) Is there a way to do the experiment so that you can completely cancel the effects of the earth s field? 4. Turn on the power supplies. Allow the filament of the electron tube to warm up for a minute or so. Set the current I in the Helmholtz coils to some value around 1.0A. When the filament is hot, slowly turn up the accelerating voltage Va and observe the path of the electrons. (DO NOT EXCEED 300V.) If the path of the beam is not perpendicular to the magnetic field, grasp the lowest part of the base of the tube and rotate it, being VERY CAREFUL not to pull or break any wires. 5. Vary the accelerating voltage while keeping the coil current constant. Again, DO NOT EXCEED 300V. Question 1-2: Now that you have observed the beam, and given that you know the sign of the electron s charge, what is the direction of the magnetic field produced by the coils? How do you know? For fun, you can change the polarity by switching the leads, just to see what happens. Question 1-3: What do you observe as the anode voltage is changed while keeping the coil current is constant? Explain why this occurs.

6 6. Vary the current in the coils (not to exceed 2.0A) and observe the bending of the beam. Reverse the current to the coils and observe the effect. (Then go back to the original polarity.) Observe the path of the electrons when the tube is rotated slightly away from the perpendicular position and record your observations. Question 1-4: What happens to the beam of electrons as the coil current is increased? Question 1-5: What causes the phenomena that you just described in Question 1-4? 7. Set the coil current and the anode voltage so that the electron beam has a radius of approximately half of the tube radius. Use the bar magnet to see how it affects the electron beam. We prefer that you not place the bar magnet above the glass tube, but you may move it around the bottom and the sides. Question 1-6: Describe what you observe as you move the bar magnet around. Can you produce a helical path for the electron? Can you see why you want to keep spurious magnetic fields away from the electron beam? 8. Make sure you place the bar magnet at the other end of the table from your apparatus to minimize any possible effects. Activity 1-2: Measurement of the Electron Charge-to-Mass Ratio This part of the experiment is broken up into two sections: measuring the radii of electron trajectories with fixed anode voltage by varying the magnetic field, or with fixed magnetic field and varying anode voltage. You should refer to the figures on the next page for further explanation. Part 1: Orbit Radius vs. B, Fixed Anode Voltage 9 Position the electron beam perpendicular to the magnetic field. 10 Now set Va at a value of!175v (or greater, if you are unable to see an entire circle of electron beam) and record Va. Keep Va at this same value for all of this section. The magnetic field B is varied by changing the current I. 11 The mean radius r of the electron beam's circular path is determined by measuring the positions of the inner and outer edges of the beam, x 1, x 2, x 3, x 4. (See figure on next page.) When you measure these values, try to line up the image of the beam in the mirror with the actual beam, so as to be sure you are viewing the scale perpendicularly to avoid reading errors due to parallax. The inner radius r 1 and outer radius r 2 of the beam are then given by:

7 12 The current should be set at! 1A (or the lowest value to give a full circular beam). Measure the values Va, I, x 1, x 2, x 3, x 4 and tabulate them using Graphical Analysis or DataStudio. 13 Change I and repeat part 4. for two additional I values (not to exceed 2.0A). (Each student in the group should take readings at least once.) When this is complete, each group should have a table of the measured radius for each of 3 different values of I, all for the same value of Va.

8 Definition of Measurement Variables:

9 Part 2: Orbit Radius vs. Anode Voltage, Fixed B 1 Set I back to one of the values used in Part 1. Keep I at this same value for all of Part 2. 2 Make and record sets of independent measurements of Va, I, and x 1, x 2, x 3, x 4 as in Part 1. for a total of three Va values different from the value used in part B. (Do not exceed Va = 300V.) 3 When this is complete, each group should have measurements for each of 4 different values of Va (including the value from Part 1) in a data table, all for the same value of I. We recommend you use Graphical Analysis or DataStudio for tabulation; this will make later analysis much easier. Activity 1-3: Analysis For the two different sets of measurements, we can manipulate the formulae given above to generate straight lines, which definitely makes the fitting easier. For the case with constant anode voltage and varying B, we can re-write equation (3) to look like so a fit of 1/r vs B should give you something proportional to e/m. Likewise, rearranging equation (3) in a different way gives and you can do a similar fit. 1 Plot your data for both Part 1 and Part 2 in something like the manner suggested above. 2 For each plot, perform a linear fit, and include the parameters on the plot. 3 Print the plots out to turn in with your lab report. 4 Extract e/m from each of the plots. The accepted value is 1.7 x C/kg. How does yours compare? Questions: 1 There were many assumptions that were made in obtaining equation (3) which are not exactly true. List as many of these assumptions as you can (within reason) and estimate which of them would cause the greatest systematic error in your measurement. 2 Many students will answer that equation (3) does not account for special relativity. Calculate the speed of electrons for a typical accelerating voltage in this experiment. Relativistic effects are only important if the electron is moving at a significant fraction of the speed of light. Are they important here? 3 What effect could the presence of the earth s magnetic field have had on your results? End of Lab Checklist: Please turn off all power supplies and multimeters when you are finished.

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