MSCD/UCD Physics Laboratories Lab II e/m The Measurement of e/m PURPOSE The objectives of this experiment are to measure the ratio between the charge and the mass of electrons, and then to find the mass of the electron using the accepted value of e. NOTE The e/m apparatus is extremely expensive. It takes time to replace them all. We currently have five new e/m apparatuses. Please be careful when using them. Thank you. THEORY The e/m for Electrons Experiment uses a sealed, glass enclosure with very low internal pressure. Inside is an incandescent source of electrons. Near the source of electrons are electrodes (circular capacitor plates) for setting up a voltage potential. This accelerates the electrons into a beam of known potential difference V and energy ev per electron. The last electrode that the electrons in the beam 'see' is a stainless steel plate. The bottom capacitor plate has a small hole through which electrons pass, moving downward inside of the glass enclosure. After the electron beam exits the cone-shaped electrode, it enters a region in which the electric field (but not necessarily the magnetic field) is zero. During manufacture, the evacuated glass enclosure is back-filled with a small amount of helium gas. Its vapor pressure is sufficiently high. This means that an appreciable number of electrons will collide with and thereby positively ionize helium atoms during the electrons motion. These positive ions exist along the path of the beam itself. The presence of the positive ions does two important things. It helps focus the electron beam, keeping it a compact pencil of trajectories. As electrons are recaptured by the excited helium ions, the atom emits photons of which the red, blue-green, and violet are visible to the human eye. The combination of these four wavelengths of light (the violet has two component wavelengths) is seen as a pale bluish light emanating at the location of the atom/ion. The net result of this is that the beam provides some self-focusing and the position of the electron path is indicated by the pale blue light emitted by those atoms that have experienced a collision with an electron (thereby removing that electron from the beam). The beam is quite faint; NC-3604 is equipped with special room light dimmers to lower the ambient light level. Flashlights are in the drawer. The path of the electrons can be altered by applying a magnetic field. Helmholtz coils are used in the experiment to do this. Whenever a charge e moves in a magnetic field, that moving charge is acted upon by a force F = e v x B. where the x denotes a special product between two vectors called the vector (or, sometimes, the cross) product. Here, v is the velocity vector and B the magnetic field vector. 27
e/m Lab II MSCD/UCD Physics Laboratories One way of finding F is to recall that it is perpendicular to the plane formed by v and B. F is also in a direction opposite (because e is negative) to the direction in which the thumb of the right hand points as its fingers curl from v to B. The magnitude of the force on the electron is F = e vb sin Θ = e v B, and points to the center of the tube, and Θ is the angle between v and B (Θ = 90 in this experiment). F = evb is also known as the Lorentz Force. So, the particle moves in a circular path which makes the Lorentz Force equal to the centripetal force. The centripetal force from Newton s Second Law is or ma = m v 2 R e = F, m v 2 R e = e v B. (1) F B v Here, m is the mass of the particle, v is its velocity, and R e its path. the radius of Path of electron In this e/m experiment, the electrons are projected into the magnetic field by an electron gun that causes all of the electrons to fall through the same potential difference V (Volts). The potential energy is converted to kinetic energy so the electrons each acquire a kinetic energy e V, 1 2 m v2 = e V. (2) Solve this for v, plug into equation (1) and rearrange (let d = 2R e for the diameter of the orbit) to yield e m = 8 B 2 V d 2. (3) A pair of Helmholtz coils is mounted so that the glass enclosure is near its center. Near the center of the coils the magnetic field is assumed to be constant in magnitude and direction. The apparatus is set up before the students arrive for the laboratory session. The magnitude of this magnetic field is given by B = 3 4 2 µ o N I 5, r coil (4) where µ o = 12.57 x 10-7 (NA -2 ), N = number of turns / coil = 130, I = coil current (A), and r coil = coil radius (m). 28
MSCD/UCD Physics Laboratories Lab II e/m Combining equations (3) and (4) gives: e m = 125 8 r coil 2 µ o 2 N 2 I 2 V d 2. (5) V is the potential difference accelerating the electrons; I is the current producing the magnetic field. Equation (5) suggests that if I is held constant then V / d 2 would also hold constant and one could easily find e / m. EQUIPMENT e / m apparatus from Daedalon Corporation. Magnetic compass Flashlight PROCEDURE The apparatus used in this experiment is similar to what was used to first measure e/m by J. J. Thomson in 1897. The left knob varies the potential difference across which the electrons are accelerated into a beam. This potential difference is measured and displayed in the digital voltmeter. 0-500 VDC is being supplied! The right-hand knob controls the Helmholtz coil current. The amount of current is indicated by the right-hand ammeter. Values to 3 amperes are available. A glass measuring rod has been conveniently placed inside the glass tube. The measuring rod is coated with fluorescent markings; this causes the mark to glow when struck by the electron beam. Diameters can be measured this way in increments of 0.5 cm. 1. Measure coil radius (r coil ) in meters. You need to average several measurements because the coil is not a true circle. 2. Make two data tables. Each data table will have the same constant coil radius (r coil ) listed at the top. Each table should also have the constant coil current (I ); this will be different for each data table. Four column titles should be as follows: V (V) d (m) d 2 (m 2 ) V / d 2 (Vm -2 ). 3. Turn on the apparatus. It will take 30 seconds for the displays to clear. 4. Do NOT turn the Voltage Knob! The electron-baker element needs to first warm up for about three minutes. Please do not leave the unit on longer than necessary! 5. Find the magnetic field direction of the coils. While the electron-baking element is warming-up, turn the Current Control Knob (right knob) until about 2 Amps is traveling through the coils. Place a magnetic compass near the center of one of the coils. Which way does the south-seeking (green) pole of the compass point? Please do not leave the coil current on longer than necessary! Thank you. Sketch a bird s eye view of the coils and map the magnetic field lines with their directions. You can find the direction of the field lines by moving the compass from one coil center around the outside of the coils and back to the center of the other coils. 29
e/m Lab II MSCD/UCD Physics Laboratories Bird's Eye View of e/m Apparatus Helmholtz Coil Glass Bulb Move compass in this path from coil center around the outside to coil center. 6. Turn off the coil current as soon as possible! Turn the Current Control Knob counter clockwise until the ammeter reads zero. 7. Dim the lights via the switches to the right of the front dry erase board. 8. Accelerate electrons into a beam. Adjust the Voltage control knob (left knob) until a beam is visible. (You may need 100+ Volts to see the beam.) 9. Set coil current to 1.5 Amperes. 10. Re-adjust accelerating voltage until the beam obtains the smallest measurable circle. You can read the marking on the glass rod to match either a 0.05 or 0.055 m diameter beam. 11. Record voltage V and beam diameter d. 12. Increase voltage until the next 0.5 cm mark is reached. 13. Repeat steps 11 and 12 until the beam is too large for the measuring rod. You should have at least 11 entries when you are through. 14. Begin your second data table recordings. Readjust coil current to either 2 or 2.5 Amperes and record I. 15. Repeat steps 10 through 12. You will not obtain as many readings for the higher current. ANALYZING THE DATA 1. Calculate d 2 and V / d 2 for each entry of each data table. A spreadsheet might be very helpful at this point! 2. Find two averages of V / d 2, one average for each data table. 3. Calculate e / m twice, once for each data table using the known current and the average V / d 2. 30
MSCD/UCD Physics Laboratories Lab II e/m 4. Average the two e / m values. 5. Compare your average e / m to the accepted value by finding the percent discrepancy. The accepted value: e / m = 1.76x10 11 C/kg. 6. Find the mass of the electron. Use the accepted value of the charge of an electron (1.6 x 10 19 C) and your average e / m. Compare to the accepted value (9.11 x 10 31 kg) by taking the percent discrepancy. QUESTIONS 1. Was the ratio V / d 2 constant within each data table as expected? 2. If not, how did it change as d increased? 3. Is B constant inside the Helmholtz coils? Support your answer with your data. You may also want to compare the derivations of equations (3) and (4). (In other words, from where did each equation come?) 4. Your coil was not a perfect circle. Consider Data Table 1. If you change the r coil in your calculation for e / m to either the smallest or highest in the range of r coil, how much does e / m change? Try the calculations with the smallest and largest coil radius measurements. 5. When you are facing the front of the e / m apparatus, which way does the current travel, clockwise or counter-clockwise? CONCLUSIONS Record your comments regarding results, uncertainties, and procedures. 31
e/m Lab II MSCD/UCD Physics Laboratories 32