Ratio of Charge to Mass (e/m) for the Electron

Transcription

1 Objective: In this experiment you will determine the ratio of charge to mass (e/m) of the electron, by measuring the deflecting of electrons as they move through a magnetic field. Apparatus: e/m apparatus (EM-1N), 500V high voltage (accelerating voltage) power supply, 6V (filament) power supply, 15V regulated DC (Helmholtz coil) power supply, banana plug patch-cords, magnetic compass, and bar magnet. Theory: J.J Thompson is credited with the discovery of the electron. In 1897 Thompson used a cathode ray tube and an applied magnetic field to determine the charge to mass ratio of the electron. In this experiment we will use a modern version of Thompson s apparatus to repeat his measurement. Figure 1 e/m apparatus (Narika Scientific) Figure 1. shows the structure of the e/m apparatus. A uniform magnetic field is produced in the center of a Helmholtz coil. A Helmholtz coil consists of two circular coils mounted parallel to each other, and along a common axis. These coils are electrically connected in series. When a voltage is applied to the Helmholtz coil an electrical current will flow in the same direction in both coils. The resulting B-field from each coil is oriented in the same direction, and a uniform

2 magnetic field is created between the coils. The discharge tube used for measuring e/m is placed within the uniform magnetic field created by the Helmholtz coils. The discharge tube contains an electron gun and is filled with low-pressure Helium. The electron beam emitted by the electron gun collides the low pressure Helium, so that the electrons path appears as a luminous trace. Terminals for attaching external power supplies are located on the front panel of the e/m device, as well as a knob for adjusting the current through the Helmholtz coil. Features of the e/m Device: The discharge tube incorporates two special features. Low-pressure helium gas is sealed within the tube. Helium produces light when excited by collisions with electrons. The discharge tube also contains a built-in scale for measuring the diameter of the path traced by the electron beam. The light produced when electrons collide with the low-pressure Helium illuminates the graduations on the scale. Coil Specifications: Number of turns/windings per coil: 130 Internal Radius: R = 0.150m Discharge Tube Specifications: Filament Heater: 6.3V at 0.4A Maximum Operating Accelerating Voltage: 500V at 10mA 1. The Electron Gun: Figure 2 illustrates the Electron Gun used to accelerate electrons by means of an applied electromagnetic field. The Cathode K (Figure 2) is heated by a 6V, filament releasing thermal electrons. The released electrons are accelerated by the electric potential (V) applied between the Anode P and the Cathode K. The velocity (v) of an electron accelerated by the electric potential (V) applied at the anode, can be calculated using the Law of the Conservation of Energy, as follows (ignoring the initial velocity of the electron as it is discharged from the cathode): (1) Where m (kg) is the mass of electron and e [C] is the elementary electrical charge of the electron. The above expression can be written as

3 Figure 2. The Electron Gun (Narika Sceintific) 2. Electron s Motion in a Magnetic Field: Electrons that travel with a velocity (v), through a uniform magnetic field (B), experience the Lorentz Force (FB = -e(v x B)). In the special case where the motion of the electrons is perpendicular to the magnetic field lines, the electrons experience a Lorentz Force described by the expression In this special case the Lorentz Force constrains the electrons to move in a circular path in the plane perpendicular to the magnetic field. Recall from mechanics that an object with mass (m) that travel in a closed circular path with radius (R) experience the Centripetal Force, where the magnitude of the FC given by Equating the Lorentz force with the Centripetal Force we find that

4 This expression can be rearrange to give The above expression can be squared and substituted into the equation (1) for the kinetic energy of of the electron, to give.. (2) This is an expression for the charge to mass ratio (e/m) of the electron, given in terms of the accelerating voltage (V), the magnetic field (B) and the radius (R) of the electron s path. 3. Helmholtz Coil: Two circular coils of equal radius are mounted, in parallel and along a common axis. When an electrical current is applied to both coils, so that the current is in the same direction for each coil, a uniform magnetic field is formed between the coils. According to Bio-Savart's Law, the magnetic flux density (B) of the magnetic field between two turns of wire of radius (r), carrying a current I (A), is given by the following formula: When there are N turns of wire in each coil, the intensity of the magnetic field is multiplied N times. Given that the permeability of vacuum is μ= 4π/10 7, the intra-coil magnetic flux density is expressed as: As the coils of the apparatus are constructed with 130 turns of wire, with a radius of 0.150m; the formula given above can be applied to determine B between the coils to find that: (3)

5 Setting up the e/m Apparatus: To configure the e/m apparatus for use: one will need the following items 1. Filament power supply (6 volt, AC or DC) 2. Accelerating Voltage power supply (0-500V) suitable for vacuum tubes 3. Regulated DC power supply unit, (0-15V) 4. Magnetic compass Connect each power supplies to the e/m unit as follows: 1. Connect the 6V power supply to the Heater Terminal on the front panel of the e/m unit, either AC or DC voltage may be used (see Fig. 3). 2. Connect the 0-500V DC B-power supply to the B-power terminal on the front panel of the e/m unit (see Fig. 3). Caution: The Red Terminal P is positive (+), while the Black Terminal K is negative (-)(see Fig. 3) 3. Connect the DC (0-15V) power supply to the coil power supply terminals on the front panel of the unit. Once again the Red Terminal P is (+), while the Black Terminal K is negative (-)(see Fig. 3). Caution: A regulated DC power supply (0-15V) should be used for this purpose. When using this type of power supply, turn the knob for the coil power fully counter-clockwise.

6 Initial Observation: Figure 3: Wiring Diagram (1) Set the accelerating voltage of the B-power supply as well as the coil power supply to the minimum ( turn the power supply knobs all the way to the left). Turn on the filament power (the 6V unit). (2) When the cathode glows red, gradually raise the voltage of the B-power supply as you observe the discharge tube. A glowing electron beam will gradually appear at just under 200V. Caution: If the B-voltage is raised too high, the discharge tube will be damaged! Be careful when raising the B-voltage, do not raise the applied voltage above 500! (3) Gradually increase the voltage of the coil-power supply. Increasing the voltage in the coils will raise the intensity of the magnetic field, until the electron beam is bent into a circular pattern. When the path of the beam becomes circular, use a magnetic needle to check the polarity at the magnetic field formed between the Helmholtz coils. This will make it possible to confirm the relation of the magnetic field with the direction of Lorentz force, and the sign of the charge on the electron. Note: If the electron gun is not pointed perpendicular to the magnetic field, the electron beam will trace a spiral path rather than a circular one. In this case, loosen the mounting screw of the discharge tube and rotate the position of the tube until the electron beam traces a circular path.

7 Measurement Procedures: 1. Part 1: Using the voltmeter and ammeters that are integrated into the power supplies, set the Helmholtz Coil current (I) to some fixed value (say one amp). Note the current control knob on the e/m apparatus controls the Helmholtz Coil current, not by adjusting the power supply. Now measure the radius of the electron beam for five different electron, accelerating voltages or B-voltage settings, between V. Caution: If the B-voltage is raised too high, the discharge tube will be damaged! Be careful when raising the B-voltage, do not raise the applied voltage above 500V. This data is to be recorded in Table 1. Given that e/m is proportional to 1/R 2 you should try and determine R as best you can. 2. Part 2: Using the voltmeter and ammeters that are integral to the power supplies, set the electron accelerating voltage or B-Voltage to some fixed value (say 300V). Now measure the radius of the electron beam for five different current settings of the Helmholtz Coils. This data is to be recorded in Table 2. Pre-lab Questions: (1) 1) What is the accepted value of the charge on the electron? 2) What is the accepted value for the mass of the electron (in MKS units)? 3) Consider a e/m measurement performed when the accelerating voltage or B-voltage is set to 300V, the coil current is at 1.48A, and the path of the electron is found to be 100mm in diameter. What is the value for e/m in this case?

8 Data: Table 1: Constant I (I = ) V R R 2 Table 2: Constant V (V= ) I 2 R 1/R 2

9 Note: All units must be MKS. Analysis: 1. For a fixed or constant current I, equation (2) can be rewritten as. Notice that the graph of voltage V as a function of R 2, describes a straight line i.e. Where y = V, X= R 2 and the slope m is equal to.. Plot the data from Table 1, and fit the resulting curve to a straight line. Obtain the slope of this line, and solve for e/m. Note for a constant current I, you will need to use equation (3) to determine the value of the B-field given the current I. (e/m) = 2. For a constant voltage (V), the graph of I 2 as a function of 1/R 2 also plots as a straight line. Plot the data from Table 2, and fit the resulting curve to a straight line. Use the slope to determine the value of e/m. (e/m) = 3. Compare the values of e/m obtained in parts 1 and 2 with the accepted value of e/m = x [C/kg]. What is the percent difference between your experimental value and the accepted value of e/m, as found in Part 1=.

10 What is the percent difference between your experimental value and the accepted value of e/m, as found in Part 2=. Questions: 1) What two sources of error are most likely to affect your results? 2) Suppose protons were emitted from the e/m tube, rather than electrons. How would this affect the experiment? 3) Compute the velocity of an electron that has been accelerated through a difference of potential of 100 volts. Express your answer in meters per second. 4) Determine the mass of the electron using your value for e/m and the accepted value of the electron s charge, e =1.602x10-19 C. How does your value for the mass compare with the accepted value of the electron s mass?