Experiment 7: The Electron s Charge to Mass Ratio

Transcription

1 Chapter 9 Experiment 7: The Electron s Charge to Mass Ratio 9.1 Introduction Historical Aside Benjamin Franklin suggested that all matter was made of positively and negatively charged fluids that flowed under the influence of electric forces. In the 1800s English chemist John Dalton established that matter was instead made of particles that he and the ancient Greeks called atoms. We learned to make vacuum tubes, and in 1869 German physicist Johann Hittorf observed that cathode rays could jump across the vacuum between two electrodes. J. J. Thomson, using a device similar to a cathode ray tube discovered the electron and measured the ratio of the size of its electric charge (e) to its mass (m e ). Thomson s experiment was concerned with observing the deflection of a beam of particles in a combined electric and magnetic field. Its result established: 1) the existence of the electron; 2) the fact that the electron has a mass (m e ); 3) the fact that the electron has an electric charge ( e); 4) that both charge and mass are quantized; 5) that the ratio of e/m e is constant; 6) that all electrons might be identical; 7) that e/m e = C/kg. In this lab we will repeat Thomson s measurement by observing the deflection of an electron beam by a magnetic field. 131

2 General Information The electron s charge to mass ratio is the first datum ever measured about the fundamental particles and is important in and of itself; however, the method of using a beam of particles in experiments is at least as important. Without this method we would know nothing about the nuclei or the unstable fundamental particles. The electrons orbiting around nuclei are what matter is made from, what allows chemists to develop new compounds, and what allows biological phenomena to be explained. The electrons oscillating in a radio transmitter s antenna enable them to transmit information. The controlled flow of electrons through a semiconductor allows a computer to process data; it is a beam of electrons that allows specimens to be seen with an electron microscope. It is with a beam of very high energy electrons that scientists have established that nucleons are made up of components which are called quarks and gluons. Possibly, these new components of matter will shape the way mankind will live 100 years from now in a way similar to that in which Thomson s discovery is responsible for the way we are living today. There are two basic physical phenomena which play a significant role in the experiment carried out in this lab: the existence of a magnetic field associated with an electric current and the deflection of a moving charged particle in a magnetic field. Before we discuss the experiment itself we will briefly review these two phenomena and we will discuss the fact that the earth has its own magnetic field Magnetic Forces The forces between electric currents are called magnetic forces; because, the same phenomenon accounts for the forces acting between magnetic materials such as pieces of magnetized iron. William Gilbert, Queen Elizabeth I s physician, noted that a magnet has two poles at which magnetic effects seem to be concentrated. He also showed that like poles repel each other, whereas unlike poles attract each other. Today we explain the forces between magnetic poles in exact analogy with the electrostatic forces between charges (2 nd lab in Physics 136-2) by introducing a magnetic field represented by field lines. Figure 9.1: A vector schematic diagram illustrating the vector directions for the Biot-Savart law. 132

3 As we shall see later, the relation between magnetic field lines and magnetic forces is more complicated than in the case of electrostatic forces. Historical Aside In 1820 Hans Christian Oersted observed that a magnetic field was created by an electric current; during a lecture he observed that a magnetic compass was deflected from pointing north when he completed a nearby circuit and caused current to flow. This stimulated French mathematician Andre-Marie Ampere to study the phenomenon more closely. In addition to Ampere s observation that two currents interact via a magnetic force (circa 1820), these studies also led to Ampere s law of magnetostatics. Parallel currents are attracted and anti-parallel currents repel. Ampere also published a magnetic force law similar to Coulomb s law of electrostatics using electric currents as sources; however, the Biot-Savart law below seems to be more fundamental. Michael Faraday s law of induction (published in 1831) was later proved to be due to the magnetic force on electric charges in conductors due to the conductor s motion. The combined electric and magnetic forces on a moving charge in both fields is called the Lorentz force though it was published earlier by Oliver Heaviside in 1889 and perhaps by James Clerk Maxwell himself in Moving electric charges create a magnetic field. In the case of a continuous electric current this field is described by the Biot- Savart Law in Equation (9.1) reported by J. B. Biot and F. Savart to the French Academy. db = µ 0I 4π dl ˆr r 2 (9.1) Figure 9.2: A picture illustrating Right Hand Rule #1. The vector db is tangent to the magnetic field lines, which in the case of a straight wire ( dl ) conductor are represented by concentric circles in a plane perpendicular to the conductor. ˆr = r is a unit vector directed from the current element toward r the position where db is computed. Figure 9.2 shows such a field and also illustrates (together with Figure 9.5) the right hand rule, which gives the relation between the directions of the 3 vectors db, dl, and ˆr. If the thumb of the right hand is pointed along the direction of the current I, then the fingers curl in the direction of B. To increase the strength of B in a given volume, one usually uses a solenoid. In this case the B generated by each winding of the coil will add up inside the solenoid to the B of the other windings, as shown in Figure 9.3. The same figure also shows the similarity between the fields of a solenoid and of a permanent magnet. Notice that the magnetic field inside the solenoid is nearly uniform. (This is true only when the length 133

4 of the solenoid is large compared to its diameter.) The solenoid is the classic device for generating a magnetic field. It is the analog of the parallel plate capacitor, which is the classic device for generating a uniform electric field throughout a given volume. Magnetic fields of a specific shape can be generated by a system of coils. Two parallel coils separated by a distance equal to the radius of the coils (Figure 9.4) are known as Helmholtz coils. They are frequently used because they generate a magnetic field that is uniform over an appreciable region about its midpoint. We will be using a system of Helmholtz coils to carry out this lab s experiment. If each one of the coils, with radius R, has N turns and carries a current I, then the field at the center of the system is, Table 9.1: Symbol definitions for the Helmholtz coil s magnetic field equation. Figure 9.3: Illustrations of the magnetic field lines generated by current flowing through a solenoid coil and by a bar magnet. B C = µ 0 NI R (9.2) Let us now look at the force exerted by a magnetic field on a moving charged particle. It was the American physicist H. A. Rowland who first observed that a particle with charge q and velocity v moving in a magnetic field B will be subjected to a force F. The direction of this force is perpendicular to v and to B. If a particle enters a volume with an electric field E and a magnetic field B then the total force on the particle is given by the Lorentz law F = qe + qv B. (9.3) 134

5 What is the Biot-Savart law? What is the Lorentz force law? Why are Helmholtz coils useful in physics experiments? What relation between radius, position, and separation distinguishes Helmholtz coils from other coil pairs? R I R y R I x z B Figure 9.4: A schematic diagram of one pair of Helmholtz coils. The radius, R, of both coils equals the distance between the centers. Helmholtz coils generate uniform magnetic fields near the geometric center at the origin. This is an extremely important relation partly because it connects mechanics (force F) to electromagnetism (fields E and B). The magnetic force is the cross product of two vectors. Figure 9.5 reminds you once more of the right-hand rule which is defined for positive charge. You must be familiar with this rule in order to carry out this experiment. As mentioned in the introduction, in this experiment we will observe the deflection of electrons in a magnetic field. To simplify the experiment we will choose the direction of the electron beam (qv) to be perpendicular to B. In this case the force acting on the electrons (each with charge q) is simply, F mag = qv B. (9.4) According to the right-hand rule #2, F mag is perpendicular to v and B; consequently the electrons would move in a circle as shown in Figure 9.6 if they had a positive charge. Since they have a negative charge, the electrons will actually rotate in the opposite direction from that shown in the figure. (Remember from mechanics: if F v, then it must be circular motion.) The centripetal force responsible for motion in a circle instead of a line is given by m v 2 r = m a r = F mag = qvb. (9.5) We can calculate the radius of curvature of the circular motion of a charged particle moving perpendicularly to a Figure 9.5: A picture of the right hand rule needed to deduce the direction of the magnetic force on positive charge moving in a magnetic field. 135

6 magnetic field using r = m q v B. (9.6) This simple relation is the basic equation that we will use to carry out the measurement of e/m e. By measuring the radius of curvature r of an electron beam s trajectory, by knowing each electron s velocity, and by having the beam deflected by a magnetic field, we can calculate the ratio q/m. Before we describe how to do this experiment, there is one last topic to cover: the earth s magnetic field! This field will have an effect on our electron beam, hence we must properly take account of it - otherwise, our measurement of e/m e will be wrong. Describe the angles between the electron s velocity, the magnetic force, and the magnetic field The Earth s Magnetic Field The earth s magnetic field is the field of a magnetic dipole, which means that it is equivalent to the external field of a huge bar magnet. The lines of force of such a field are directed not towards the geographic poles but rather towards the magnetic poles. (The magnetic north pole is located near the geographic south pole.) They are also directed (except at the equator) towards or away from the center of the earth - as shown in Figure 9.8. The intensity of the field at the surface is on the order of one Gauss. Sediments of magnetic materials (iron, cobalt, nickel) can drastically change the local pattern of this field which has been carefully mapped, most recently with the use of satellites. After centuries of research, the earth s magnetic field remains one of the best described and least understood of all planetary phenomena. The history of the earth s magnetic field has been traced back 3.6 million years, and it has been established that during this time the earth s field has reversed nine times. To establish such a fact two elements were necessary: the magnetic memory of volcanic rocks and the presence in the same rocks of atomic clocks that begin to run just when their magnetism is acquired. The memory elements themselves are magnetic domains, tiny volumes of magnetic material in which magnetism is uniform. These bodies consist of iron and titanium oxide. At temperatures above a few hundred degrees (depending upon the chemical composition) these domains are not ferromagnetic. When a domain cools it becomes magnetized in the direction of the surrounding magnetic field. The atomic clocks that record the time of the 136

7 lava solidification are based on the radioactive decay of potassium 40 into argon 40. This radioactive decay (transformation of potassium into argon) takes place similar to the decay of an RC circuit (see 5 th lab last quarter) but with a much longer time constant or half-life. The argon is trapped within the crystal structure of the minerals and if the minerals are not heated or changed in some way, it accumulates there. The amount of trapped argon is a function of the amount of potassium present and the length of time since the decay and entrapment process began. The potassium-argon dating method has now been successfully applied to rocks from nearly 100 magnetized volcanic formations, with ages ranging from the present back to 3.6 million years ago; nine earth magnetic field reversals were observed during this time. You should not worry about the earth s magnetic field changing during your experiment; the data of volcanic rocks shows that it takes about 5000 years for a field reversal to complete once it has begun. You must be aware, however, that there is an earth magnetic field which affects this experiment. Figure 9.6: An illustration of the cyclotron motion resulting from a positive charge moving in a magnetic field. All other forces are assumed to be negligible. How do we know that earth s magnetic field has reversed itself? How do we know how long ago these geologic changes occurred? Radius vs. Magnetic Field Measure the ratio of the electron s charge to the electron s mass in C/kg. above, the basic relation for this measurement is given by Equation (9.6) As described 137

8 Figure 9.7: A cutaway schematic illustration of our charge to mass ratio apparatus. r = m q This relation tells us that in carrying out the measurement of e/m e we need three basic elements: 1) a beam of electrons with known velocity. 2) a magnetic field (uniform over the region where the electrons will describe a circular trajectory). 3) a way to observe the electron s path, so that we can measure the radius of curvature. Figure 9.7 and Figure 9.10 show schematic drawings of the equipment used to measure e/m e. v B. If v F, what kind of motion results? Why doesn t the speed change? 9.2 The Apparatus The apparatus needed to measure e/m e consists of an electron gun to generate a beam of electrons with known velocity, a pair of Helmholtz coils to generate a known uniform 138

9 S I N (a) (b) (c) Figure 9.8: An illustration of Earth s magnetic field (a). Note the similarity to the dipole field of a current loop (b) or a bar magnet (c). Earth s field protects us from energetic charged particles emitted by the Sun. magnetic field, a thin argon gas to make the electron beam visible, and calibrated position pegs to allow us to measure the radius of the electron s circular trajectory The Electron Gun We generate a beam of electrons with an electron gun. It is shown schematically in Figure 9.9. A large current heats a thin filament so that electrons boil off. The filament is biased to a large negative potential to accelerate the electrons toward ground. A grid of thin wires with even more negative potential repels the electrons and allows us to control the electron beam s current. An anode is connect to earth ground to accelerate the electrons that pass the grid. The electrons are accelerated to a final velocity that converts the electrons electric potential energy qv into their kinetic energy 1 2 ev 2 m e v 2 = ev and v =. (9.7) m e Ordinarily, the electrons would strike the anode and transfer their kinetic energy to the anode s atoms causing the anode to heat up. The anode in the electron gun has a hole in it, however, that allows some of the electrons to continue past into the thin gas of argon atoms. Figure 9.10 shows a view of the electrical connections for the equipment used. The electron gun is controlled by 2 knobs both on the blue Power Supply. The DC V knob sets the accelerating potential, and the DC V knob sets the current in the filament, thereby controlling the electron beam current I anode. How can we know the velocity of the electrons? 139

10 9.2.2 The Magnetic Field The magnetic field is generated by a set of Helmholtz coils. Equation (9.2) gives the value of the field at the center of the system. The number of turns for each coil is N = 72. With a ruler you can measure the radius of the coils and/or the distance between them and then N calculate the value of c = µ 0. The magnetic field is aligned along the axis of the R coil system, and its direction is determined by the right-hand rule (see Figure 9.2). As mentioned previously, the earth also has a magnetic field B e, which cannot be neglected in this experiment. Each apparatus has been individually aligned, with the help of a compass needle, in such a way that the field of the coil is parallel with the earth s field but in the opposite direction of the earth s field. Can you demonstrate this? Consequently, the magnetic field B T that will deflect the electrons is B T = ci ± B e. (9.8) where the ± sign depends upon the directions of the current in the coil and the earth s field. The knob labeled CURRENT, on the current power supply (see Figure 9.10), will allow you to vary the strength of B T, while the meter above the knob gives a reading of the current I through both coils. How do we generate a magnetic field? Is this the only field we need to worry about? Electron Trajectory Electrons are infinitesimally small objects (radius< cm) that cannot be seen by the naked eye. In order to observe their trajectory without blocking their path, the electron gun is installed in a glass enclosure (25 cm diameter) which contains low pressure argon (Ar) gas. The electrons (with 50 ev kinetic energy) will excite the Ar atoms (requiring 2 ev) which then emit an orange light. The electron trajectory can be observed (in a darkened room) as a ray of orange light emanating from the electron gun. The glass bulb also contains, along one of its diameters, a set of pins. The distances from the anode slit of the electron gun to each of these pins are: m, m, m, m and m. Some of these pins are still covered with a fluorescent material which emits light when struck by the electron beam; but most of this material has been burned off over the years. By varying the CURRENT control knob you change the current I in the Helmholtz coils, producing different values of B and forcing the electrons to describe different orbits. Certain values of B will allow the electron beam to strike the calibrated pins. Knowing B and the radius of 140

11 Table 9.2: A summary of the equations describing the electron s motion. These will be combined to evaluate a measured e/m e. Filament Current Filament e - e - e e - - e- e - e - e - e - e - E E Beam Focus Accelerating Voltage Figure 9.9: A schematic diagram showing the electrical connections and electron motion for our electron gun. the electron beam s trajectory will allow you to determine e/m e provided the accelerating potential is known. The equations summarized in Table 9.10 are used in calculating e/m e. We can combine these equations and write ci ± B e = B T = m e v er = 1 r m e e 2 V e m e = 1 2 V me. r e In this equation there are two unknowns: (e/m e ) that we want to measure and the strength of earth s magnetic field, B e. It will require a minimum of two measurements (r l and r 2 for values of V and I to determine both unknowns). The measurement will be carried out by varying I, measuring r, and plotting B C = ci on the vertical axis and 1 on the r 141

12 Accelerating Voltage Electron Gun Control Voltmeter Current Supply Magnetic Field Current Beam Focus Filament Heater Figure 9.10: An illustration of the electrical connections between our e/m e apparatus and the power supplies. horizontal axis. The data must fit a straight line with so that intercept = B e and slope = e m e = 2 V m e e 2 V (slope) 2. (9.9) Make a linear-least-squares fit to the data and calculate the limit of uncertainty. How can we extract our measured e/m e from measurements of accelerating voltage, Helmoltz coil current, and trajectory radius? 9.3 Procedure Measure the diameter of the Helmholtz coils with the meter stick. To calculate the magnetic field of the center of the coils you need, according to Equation (9.2), the radius R as seen in Figure 9.4, the number of windings N = 72, and the current in each coil. B C = µ 0 NI R 142

13 Express this equation as B C = ci and calculate the value of the constant as well as its units using µ 0 = 4π 10 7 T m/a. 1. The equipment is wired up according to the diagram in Figure Turn the power on in all three units. Voltmeter, Black Laboratory Power Source, and Blue Power Source. 3. Turn up the DC control to 8 V, check to see that there is an orange glowing spot in the heater of the electron gun and then wait for 10 minutes.. 4. Turn the DC V control to about 100 V and note the reading on the voltmeter. You should see the electron beam as a thin orange line emerging from the anode slit. Observe the orange beam in the tube. It should look like it is slightly bent downward. Why? 5. Turn up the coil current using the Current control on the Laboratory Power Source and you should see the electron beam bend into a circle. If the electron beam does not stay flat in a circle, rotate the glass tube slightly on its axis until the electron beam forms a flat circle that hits the pins of the lattice in the tube. The tips of the pins were coated to fluoresce when the beam hits them, but this coating may no longer exist on your tube. 6. Dial down the voltage to 70 V. Do you still have a beam? If not, raise the voltage until you have a beam. You want to have the lowest possible voltage to maximize your chances of hitting the closest peg. Your equipment is now ready to take data Trajectory Radius vs. Magnetic Field Vary the current in the Helmholtz coils (Current knob) until the outer edge of the electron beam matches the outer edge of each pin in the tube. You may not have ample current to bend the beam enough to hit the pins closer to the electron gun. Three pins would be sufficient to get a value of e/m e. Record in your lab book the accelerating voltage (voltmeter) of the electron gun and the Helmholtz coil current settings (ammeter) for each measurement corresponding to an electron orbit of radius r. Don t vary the accelerating voltage, but note any fluctuations. The distances from the anode to the pins (the diameter of the electron orbit) are m, m, m, m, and m. Calculate the field B C. Use Vernier Software s Graphical Analysis 3.4 (Ga3) to help you graph your data. A suitable Ga3 configuration can be downloaded from the lab s website at 143

14 Your notebook and write-up should include a plot of B vs. (1/r) for these measurements. Be sure Ga3 reports the uncertainties in slope M ± δm and intercept b ± δb; if not, rightclick the parameters box, Fit Properties..., and select Show Uncertainties. Using the fit parameters and other measurement(s), calculate e/m e in C/kg; also give the the expected tolerance limits using ( ) ( ) ( e e δv δ = me me V ) 2 ( ) 2 2 δm +. (9.10) M It would be wise to use the formulas in Section to derive this result. (Equa- Show that the dimensions of the relation you are using to calculate e/m e tion (9.9)) are indeed C/kg. An important relation that allows you to relate units of mechanics (such as kg, m, s) to units of electricity and magnetism (such as A, V, s) is the energy relation A V s = C V = J = N m = kg m2 s 2. Calculate the earth s magnetic field (in Gauss) at the location of your e/m e apparatus; this is given by the intercept of the line fitted through your data points and the B C axis. Sketch for both measurements: 1) the direction of the electron beam, 2) the direction of the centripetal force and the Lorentz force acting on the electron beam, 3) the direction of the current in the Helmholtz coils, and 4) the direction of the field; indicate if it was the measurement with (B e + B C ) or with (B e B C ). The beam of electrons also represents a current and consequently must generate a magnetic field. Does this field point in the same or in the opposite direction to the field generated by the Helmholtz coils? Remember that an electric current is defined for positive charge carriers Acceleration vs. Magnetic Field 1) Set the voltage control to 100 V. 2) Adjust the current to have the beam hit a convenient peg. 3) Record the readings in your lab book. 144

15 4) Increase the voltage by 10 V. 5) Find the new current that steers the beam back to the same peg. 6) Record the second set of data. 7) Repeat for several more voltage readings. 8) Calculate the magnetic field from the current as before using Equation (9.2). 9) Plot V vs (B C B e ) 2. 10) Obtain the slope of the line and note that the slope is equal to ) Obtain a value for e/m e with error limits and compare it to the value obtained in the first experiment. 12) Comment on the agreement or lack of agreement. Note that in this case ( ) ( ) ( e e δm δ = me me M ( e m ) r 2. ) 2 ( ) 2 2 δr +. (9.11) r Why is it necessary to perform Section before it is possible to perform Section 9.3.2? 9.4 Analysis First, check the consistency between your two measurements of e/m e. Label the two measurements and their uncertainties with subscripts 1, 2. e s.c. = 1 e 2 (9.12) m e1 m e2 and [ ( )] e1 2 [ ( )] e2 2. σ s.c. = δ + δ (9.13) m e1 m e2 The best estimate ρ ± δ is the average of your two measurements ρ 1 ± δ 1 and ρ 2 ± δ 2 ρ = ρ 1 δ 1 + ρ 1 δ 1 1 δ (9.14) δ 1 where each measurement is weighted by the reciprocal of its uncertainty. The uncertainty, δ 145

16 in this average is 2 δ = 1 δ (9.15) δ 2 Compare the average ρ ± δ of your two measurements to the accepted best estimate of the science community e/m e = C/kg = ρ (e/m e ). (9.16) What should you expect to see if e/m e was not a constant? Would the electrons velocities depend upon accelerating voltage if they had no mass? Would the electrons accelerate at all in your apparatus if they had no electric charge? Do your comparisons agree better than 2 σ? Worse than 3 σ? What other sources of error have we omitted from consideration before now? Are all of our assumptions valid? For example, are the electrons that boil off of the filament at rest before we accelerate them? Can you think of other assumptions that are not quite valid? Have you made ancillary observations that support our hypothesis? For example, do the data points form the straight line predicted by our theory? What other observations support our hypothesis? 9.5 Conclusion Can you conclude that your ratio of e/m e is constant? Is e/m e sufficiently important to science that we should report it in our Conclusions? Do our observations support our hypothesis? Do they contradict it? Answer these questions after allowing for all of your observations. Report your answers in a short paragraph or two. Can you think of any applications for what we have observed? How might we improve the experiment? 146