The electron beam used to study motion of charged particles in electric and/or magnetic fields. CHARGED PARTICLES IN FIELDS Physics 41/61 Fall 01 1 Introduction The precise control of charged particles in electromagnetic fields is crucial to a wide range of advances in experimental physics. From large scale experiments like those that study subatomic particles at CERN to small table-top experiments that trap a single ion, physicists around the world use electromagnetic fields to control and manipulate matter. In experiments large and small, the precise control of the particles with the fields is necessary in order for the data to be the best possible. In today s experiment, we will study the basics of motion of charged particles in fields by observing a beam of electrons in an electric field, a magnetic field, and combined electric and magnetic fields. The ideas you will encounter form the basis of experimental particle control using electromagnetic fields. By the end of this lab, you should be able to predict the path of an electron (or any other charged particle) in a constant electric or magnetic field and understand how the two cases differ. You should understand how the physical variables you control (voltages and currents) relate to the electric and magnetic field strengths in the apparatus and the electron kinematics. Equipment The setup for this experiment is a bit more involved than the ones you have used in previous experiments. The equipment includes a cathode ray tube with an inclosed beam gun and electrostatic deflection plates (for electric field control), as well as external Helmholtz coils (for magnetic field control). All of the apparatus should be assembled for you when you arrive in lab, and we ask you not to move any of the wires before, during, or after the experiment. The two exceptions are the red jumper cables and the leads to the Extech power supply. Both will be discussed in Section 5. Electron beam tube with grid (TEL55) Tel Atomic TEL 813 high voltage power supply University of Michigan High Voltage Power Supply (UMHVPS) Extech power supply 1
V d plug V a plug Deflection plates Mica sheet V d plug Electron filament Electron accelerating anode (V a ) Anode slit to define beam Figure 1: Tel-Atomic TEL 55 Cathode Ray Tube. Undeflected electron beam 1 y (cm) 10 9 8 7 6 5 4 3 Anode slit at (x,y) = (0,0) 1 x (cm) Figure : Diagram of CRT grid for measuring electron trajectories (not to scale). Set of insulated HV power leads (red, black, blue: TEL500 or equivalent) Pair of matched solenoid electromagnet coils (Helmholtz coil pair: TEL50 or equivalent) Stand for CRT and Helmholtz coil pair (TEL501 or equivalent) Meterman 15XP digital multimeter to read Helmholtz coil current.1 Deflection Tube 1 The deflection tube (Tel Atomic TEL 55) contains an electron gun which emits a narrow ribbon of electrons (cathode rays) within an evacuated, clear glass tube. A 6 V alternating current heats up a wire filament which excites and liberates electrons from the filament s metal surface. These free electrons are attracted to the positively-charged high-voltage anode and they accelerate toward it. A slit at the far end 1 Adapted, in part, from Tel Atomic deflection tube (TEL 55) instruction manuals.
of the anode (left end of anode as shown in Figure 1) forms a beam of electrons which travels into the main spherical section of the tube. The electrons (cathode rays) are intercepted by a flat mica sheet, one side of which is coated with a luminescent screen and the other printed with a centimeter graticule screen (Figure ). Thus, the path followed by the electrons is made visible. The mica sheet is held at 15 to the axis of the tube by two electric-field deflecting plates.. Helmholtz Coils The Helmholtz coils are used to provide a relatively uniform magnetic field in the center of the tube. The two coils are wired in series so that the field from each solenoid adds together in the same direction to yield a uniform magnetic field in the region between them. Note that although the magnetic field is in the horizontal direction, it will produce a deflection in the up/down (vertical) direction (see Figure 5). When using the right hand rule," remember that it refers to positive charges, while our beam is composed of negative charges. The diagrams shown in this write-up correspond to a negatively charged particle like the electron. Positive charges would be deflected in the opposite direction..3 Extech Power Supply The Extech power supply should be familiar from several previous experiments. In this experiment, we will use it to control the current delivered to the Helmholtz coils, I B, and thus the magnetic field that they produce, B..4 UMHVPS This power supply should be familiar from the electrostatics experiment. In this experiment, it will control the voltage V d across the plates on the top and bottom of the mica screen, and thus the electric field produced between them, E..5 TEL 813 The TEL 813 power supply, shown in Figure 3, controls the electron gun in the back of the deflection tube. Specifically, it controls the anode voltage V a. This is the potential difference across which electrons are accelerated before emerging as our electron beam. The display on the front can either indicate the voltage being applied or the current being delivered. We are interested in the voltage. In order to display the voltage, press the kv" so that the red kv" indicator lights up. When using the TEL 813, you should not exceed the recommended voltage of 3 kv for long periods of time. WARNING Turn off power supplies when changing leads or polarity. Turn down voltages when not making measurements. Do not touch exposed metal plugs when high voltage is present. Do not remove leads by pulling on wires; instead, grasp the insulated banana plug. Have your instructor examine the setup before you turn on the power supplies. 3
anode voltage Figure 3: The Tel Atomic TEL 813 high voltage power supply provides the anode voltage, V a. For the experiments which do not use the electric deflection: Make sure to turn off this power supply! GP-303 1 V DC DM-33 A Add these jumper cables only when you do the parts that do not use electric deflection. Never turn on this UMHVPS when the jumpers are connected. UMHVPS I B - + Electrostatic deflection plate lead (V d ) TEL 813 CRT Anode lead (V a ) Filament leads - + Electrostatic deflection plate lead (V d ) Figure 4: A circuit diagram of the experimental setup. Note carefully the adjustments in the wiring required when not using the electrostatic deflection plates (see Section 5) and when using electrostatic deflection (see Sections 4 and 6); specifically, note the addition and removal of jumpers. See Fig. 7 for a picture of the actual setup. 4
E F B B v e- F E Figure 5: A snapshot of the forces acting on an electron in crossed fields. At this position, F E and F B have equal magnitudes and opposite directions. 3 Theory Before starting this lab, you should be familiar with the following physical concepts. If you need to review them, or if you haven t yet discussed them in your lecture course, consult the indicated sections in Young & Freedman, University Physics. Force on a charged particle due to a magnetic field, 7.4-5 Projectile motion, 3.3 Uniform circular motion, 3.4 Magnetic field due to current in a circular loop, 8.5 3.1 Electrons traveling in a uniform Electric Field Looking at the Lorentz Force equation, ) F = q( E + v B (1) we see that if an electron is traveling in a uniform electric field (with no magnetic field) it will experience a constant acceleration. Recall from your mechanics class that if you have an object traveling at some velocity in the x direction and experiencing an acceleration in the y direction, it will have a parabolic trajectory. The series of equations governing the motion are similar to those of gravitational acceleration: x = v x0 t + x 0 = v e t + x 0 and y = 1 a yt + v y0 t + y 0. 5
In order to determine the velocity of the electrons, we assume that the electrons leave the cathode (filament) of the electron gun at rest. Therefore the velocity they have at the anode is governed by the anode voltage, V a and the conservation of energy: PE i = q e V a = 1 m ev e = KE f () There is no initial velocity or displacement in the y direction (v y0 = 0, y 0 = 0). If we consider the case of only having an electric field, F = m e a y = q e E. We can combine this force equation and the two parabolic trajectory motion equations to give: y = 1 [ ]( ) qe x x0 E (3) m e You will observe the parabolic deflection of the electron beam. Using your data and the equations above, you will derive an equation for determining the electric field between the plates as a function of variables we can measure (x,y,v a, etc.). 3. Electrons traveling in a uniform Magnetic Field The magnetic field in Teslas (1 T = 1 weber/m ) is directly proportional to the coil current being supplied by the Extech power supply: v e B = ( 4.3 10 3 T/A ) I B (4) ) The Lorentz Force equation, F = q( E + v B, tells us that if a charged particle is in motion in only a uniform magnetic field (with no electric field), it will experience a force in a direction perpendicular to its current trajectory. You should recall from your Mechanics course that an acceleration at right angles to velocity will create uniform circular motion with the magnitude of force given by F circular = ma r = mv R. For the motion of an electron in a uniform magnetic field, we know the magnitude is F = q e v e B, where q e and v e are the charge and velocity of the electron. We can then equate this with the uniform circular motion and solve for R: R = m ev e q e B Using Equation, we can see that v e = qe V a m e, and can thus modify Equation 5 to be in terms of quantities we can directly measure: V a m e R = q e B (6) For circles passing through the origin (which is at the exit aperture slit of the anode) and the points (x,y), the radius of curvature is given by: R = x + y (7) y You can derive this relationship by applying the Pythagorean theorem to the geometry shown in Figure 6. (5) 6
Electromagnetic deflection (circular trajectory) R R y x 1 10 9 8 7 6 5 4 3 y Electron beam passing through anode slit at (0,0) 1 Figure 6: An example of a circular trajectory. 3.3 Electrons traveling in balanced Electric and Magnetic Fields If an electric field of strength E is applied simultaneously and perpendicularly to an magnetic field B, so that the two deflections are in the same plane but opposite in direction (see Fig. 5), this can yield a balance of forces. To achieve balanced force, the fields must satisfy: and therefore the velocity must obey F E (= q e E) = F B (= q e v e B), (8) v e = E B (9) We can measure the velocity using this equation since we will determine the strength of the magnetic field from the current in the coils and we will determine the magnitude of the electric field in the electrostatic deflection experiment. 4 Experiment: Electric Field Deflection The purpose of this experiment is to observe the parabolic trajectory of the electron beam in an electric field, and to use this data to determine the magnitude of the electric field. Check that the red jumper cables that short the inputs to the UMHVPS are not attached and turn on the UMHVPS high voltage power supply in order to establish a potential difference V d across the two deflecting plates. On the UMHVPS deflection plate HV supply, the voltage readout will display the proper value of V d. Recall that a pair of parallel conductive plates charged to some potential difference will create an electric field between the plates. A method for determining the electric field is to observe the deflection of the electron beam. This method has one significant advantage. The trajectory of the beam depends only on the beam s velocity, the beam s position, and the magnitude of the electric field at that point in space. You should also note that the deflection plates do not extend all the way to the x = 0 point due to the spherical shape of the apparatus. For this reason, we need to determine the point at which the electrons 7
Figure 7: Examples of the experimental setup for electrostatic deflection, magnetic deflection, and crossed fields. The TEL 813 power supply is used for the anode voltage (Va ), the UMHVPS power supply is used for the deflection plate voltage (Vd ), and the Extech power supply is used for the current in the Helmholtz coils IB. See Fig. 4 for a circuit diagram of the setup. 8
.0 1.6 V = 3 kv d y (cm) 1. 0.8 V = kv d 0.4 0 0 30 40 50 60 70 (x-x 0) (cm ) Figure 8: A sample plot with two sets of data sharing the axes. enter the electric field (the region between the parallel plates). Though it varies slightly between the different setups, we will assume x 0 = cm. Set the anode voltage, V a, to 3 kv. Make sure the kv button is pressed and the red kv" indicator is lit, so that the TEL 813 power supply will display the voltage; see Figure 3. Using two different potential voltages (V d = kv and V d = 3 kv) carefully record the x and y values of eight points along this beam path. Use the same x coordinates for both V d values and remember that points with large x values are preferred. When you have recorded your points, turn off the UMHVPS power supply. Plot y vs. (x x off ) using both sets of your data. Find the slope for each line on your plot. An example plot is given in Fig. 8. Combine Equations and 3 by solving for the velocity in one of the equations and substituting this into the other equation. Rearrange your new equation until you have an expression for the electric field in terms of the slope of your graph. (Hint: What do we expect for the slope of a graph of y vs. (x x 0 ) if Equation 3 is correct?) Use your equation to determine the magnitude of the electric field for each of your V d values. There is a simpler way to determine the magnitude of the electric field between the plates if we assume they form a parallel plate capacitor. For an ideal parallel plate capacitor, the electric field can easily be computed from the voltage difference between the two plates V d and their separation d: E = V d /d. If you do not remember what constitutes an ideal parallel plate capacitor, it may be useful to refer to the Capacitance experiments you performed earlier in the semester. Record d, the separation distance between the plates. Use this value to calculate E = V d /d. Compare 9
this to the value you measured from your graph. You should find a large difference between the two. What physical property makes the V d /d calculation less reliable? 5 Experiment: Magnetic Field Deflection In this experiment we will study the motion of electrons in a uniform magnetic field and verify that they travel in a circle, with a radius given by Equation 6. Because we do not want an electric field, turn off the UMHVPS and connect jumper cables to short the inputs to the power supply. This will ensure that the deflection plates are grounded so that no charge will build up; thus, the electric field between the plates will be zero. Set V a to 3 kv. Observe the path of the beam. It should be undeflected (is it?). Turn on the Extech power supply to turn on the magnetic field from the Helmholtz coils. Set I B to 0.150 A or so. Is the electron trajectory visually different than in the electric field case? Spend a short time adjusting the settings of the different power supplies in use. Observe, with reference to the mica screen, that: 1. with V a fixed, the radius of the electron path, R, decreases with an increase in coil current I B and hence the magnetic field, B);. with I B fixed, the radius increases with an increase in anode potential V a, and hence a higher electron velocity; 3. the path of the electron beam is approximately circular, the deflection being in a plane perpendicular to the direction of the magnetic field, B. With V a = 3 kv, adjust I B until the beam has a "nice" coordinate (e.g. x = 8.0 cm, y = 1.0 cm rather than x = 8.3 cm, y = 0.85 cm). Choosing a point with a large x value will help reduce measurement error. Use this coordinate to calculate R in SI units using Equation 7. Using Equation 6 and the known values of m e and q e, calculate R in SI units. Compare this to your previous value. We will look at both upward and downward deflection of the electron beam. For each given value of V a, tune I B until the beam passes through your coordinate (or (x, y)) and record this value. Repeat the measurements for the same V a s but with the magnetic field direction reversed. The easiest way to reverse the direction of the magnetic field is to reverse the direction of the current through the coils by exchanging the leads to the Extech power supply. Your observations should confirm that an electron of mass m e and charge q e = e moving at right angles to a magnetic field B will experience a central deflecting magnetic force, F B, constraining it to a circular path with an inward radial acceleration, a r, in accordance with the equations in Section 3.. Using your upward deflection data for V a = 3 kv, calculate B from Equation 4. Then, along with the known value of m e = 9.109 10 31 kg, calculate q e. Compare your measured value to the known value of q e = 1e = 1.60 10 19 C. In the next lab, we will use all of the data in the table to perform a statistical analysis. Do you see an obvious difference between the upward and downward deflections? If so, briefly describe it. Note: Not all tables will have this difference. Why do we take measurements with the magnetic field in two different directions? Hint: Is there any additional magnetic field we have neglected? Compare the radius measured with the coordinates on the mica screen to the radius calculated from the measurements of the individual parameters in Equation 6. 10
Electron beam Figure 9: Due to the nonuniformity of the electric field between the electrostatic plates used in this experiment, the magnetic field cannot be made to exactly cancel the electric field at all points along the mica grid. 6 Experiment: Crossed Fields In this experiment, we will measure the velocity of the electrons by balancing the forces from the electric and magnetic fields. Remove the jumper cables shorting the UMHVPS inputs. Set the anode voltage, V a, to 3000 V. For each of the V d listed, establish a balanced deflection condition (see Fig. 9) by adjusting I B. You may need to reverse the leads on the Extech power supply to accomplish this. Record I B. Copy your values of E (based on the slope) from section 4. Calculate B, v e, and v e /c. Use c =.998 10 8 m/s. Is the trajectory of your electron beam a straight line? You probably observed something like Figure 9. Why is it impossible to achieve a perfectly straight line using this setup? Using Equation, the known value of m e = 9.109 10 31 kg, and the value of q e that you measured in the previous section, calculate v e. Compare this to the value measured with your V d = 3 kv data. Using V d = 3 kv, calculate F E, F B, and F gravity. Discuss. Is gravity a significant factor in this experiment? 11