Brown University PHYS 0060 Physics Department LAB B -190

Size: px

Start display at page:

Download "Brown University PHYS 0060 Physics Department LAB B -190"

Cameron Scott
5 years ago
Views:

1 Physics Department LAB B -190 THE FORCE OF A MAGNETIC FIELD ON A MOVING ELECTRIC CHARGE DETERMINATION OF THE RATIO OF CHARGE TO MASS, e/m, FOR ELECTRONS References: H.D. Young, University Physics, Eleventh Edition,Chapters 27,28. Theory: Electric and magnetic fields exert forces on charged particles. A stream of electrons can be focused into a narrow beam by an electric lens, and accelerated by passing through a region of electric potential difference V, which determines the velocity given to the electrons. The magnitude of the velocity v acquired by the electron (of charge e and mass m) in passing through the accelerating potential V is given by: Gain in Kinetic Energy of the electron = Work done on the electron by the electric field arising from V. ½ mv 2 = Ve (1) Solving for v, we obtain the expression v = 2Ve m (2) The direction of the velocity vector is toward the positive potential, since the electron has negative charge. In this experiment, a moving charge e, with velocity v, passing through a region of constant magnetic field B, is acted on by a force, which is constant in magnitude and always perpendicular to the electron s direction of motion. This force has the magnitude F B = Bev (3) Such a uniform force at right angles to the velocity vector leads to uniform circular motion with radius R. From Chapter 3 we recall that the force in this case (centripetal force) is given by F C = mv 2 R (4)

2 F B must equal F C, and we obtain from (3) and (4) Bev = mv 2 / R (5.1) Be = mv / R. (5.2) Now, using Eq. (2) to eliminate v from (5), we obtain RBe = m 2Ve m Squaring both sides of this equation and solving for e/m we obtain R 2 B 2 e 2 /m 2 = 2Ve/m, e 2V = 2 2 (6) m R B From Eq. (6) we can see that measuring the radius R of the electron trajectory, given a known accelerating potential V and a known magnetic field B, will give us the ratio of two important fundamental constants, the charge divided by the mass of the electron. Two apparatus set-ups are available to do this experiment. The geometries of these set-ups are different, and the methods of determining R are different, but both set-ups include a variable potential V to accelerate a beam of electrons, and a variable magnetic field perpendicular to the direction of motion of the electron. In both cases the magnetic field is provided by a pair of Helmholtz coils. The Helmholtz coils consist of two identical coils placed coaxially and wired so that they are in series, and their individual magnetic fields are oriented in the same direction. The unique requirement of the Helmholtz configuration is that the coils must be separated from each other by a distance exactly equal to the common radius of the coils. Helmholtz showed that under this condition the field between the coils is uniform in strength over a significant range. In the central region of a pair of Helmholtz coils of radius a with n turns of wire each, and with a current I flowing; the flux density B is 32πnI (10 B = 5 5a 7 ) (7) with all MKS units, B has the units of weber/m

3 Given the coil characteristics the field can be determined by measuring the current I in the coils. THE EXPERIMENT IS TO BE DONE BY BOTH METHODS, AND THE RESULTS COMPARED. NOTE THAT OPERATING CONDITIONS (CURRENT AND VOLTAGE) ARE VERY DIFFERENT FOR THE TWO KINDS OF SETUPS: OBSERVE THE LIMITS

4 E 1 : Cenco Model (Specific Charge of Electron Apparatus) E 2 : Power Supply for the Filament (part of, Model 71267) E 3 : Cenco Model V: Voltmeter (we can use the Agilent U1232A Digital Multi-meter) R: Agilent U1232A Digital Multimeter

5 Set-Up 1 The basic apparatus is shown in Figure 1. An evacuated glass bulb (at right) contains a heated filament (wire loop) from which electrons are emitted, an electrical focusing lens (the grid), and a horizontal circular plate at positive voltage V toward which the electrons are attracted. The electron beam passes upward through a central hole in the plate (H) and into the hemispherical drift space above. External controls arc the adjustable accelerating voltage (V) connected between filament and plate, and a voltage divider in the form of a variable resistance R by means of which a variable fraction of V can be applied between filament and grid. The electric field lines between filament and plate are shaped by the grid voltage and cause focusing of the electron beam. External coils supply a magnetic field in the region above the plate, this magnetic field exerts a force perpendicular to the direction of motion of the electrons in the beam. The top surface of the horizontal plate has a coating, much like that on the inside front face of a TV tube, which will fluoresce (glow) when and where electrons strike it, so that the diameter of the electron beam s circular orbit is shown as the distance between the plate s central hole (through which electrons move into the drift space) and the glowing spot (where they stop after going through a semicircle). To facilitate measurement of the orbit diameter, the metal plate is inscribed with circles of radius 0.5 cm, 1.0 cm, 1.5 cm and 2.0 cm (Figure 2)

6 Experimental Procedure - Set-Up 1: PRECAUTIONS: o Always be sure voltage is off before touching connections. o The accelerating voltage V should never exceed 110 volts DC. o The current I in the Helmholtz coils should never exceed 5 amperes. o Both the accelerating voltage V and the filament voltage should be kept off except when measurements are being made. (The apparatus has limited total lifetime.) 1. Measure the average radius of the Helmholtz coils, and record the number of turns in each coil in your apparatus. These values are slightly different for the different sets of coils, so be sure to obtain these values for the set up you use. 2. Connect the electrical controls as in Figure 1. With V set at 100 volts and with no field on focus the beam by means of the grid potential knob (0-80 Volts) until the narrowest possible beam is obtained. (When V is changed subsequently the grid voltage will change in proportions so the beam will probably stay focused, but you can try refocusing by means of the variable resistance R in the grid control potentiometer at any time it seems advisable.). Make a Table in your lab notes to record the readings to follow. The values of V, I, and 2R (orbit diameter) should be recorded for each setting, together with your estimated uncertainties of the measured values. 3. With V still set at 100 volts, see whether you can bring the electron spot to rest on the smallest ring, without exceeding 5 amperes in the field coils. If not, reduce V until you can. Keeping V constant, vary I to bring the spot to rest on each of the four rings in succession. Record I, R, and V each time (four sets of values). 4. Reduce V to about volts and see if you can bring the spot to the outermost ring by an appropriate setting of I. If not, raise V until you can. Keeping V constant at this value, repeat the measurement described in Step Keeping V as in Step 4, reverse the direction of the current in the field coils and note what happens to the electron beam. Repeat Step 3 measurements

7 Analysis and Discussion - Set-Up 1 : From your experimental values of V, I and R, using equations (6) and (7), calculate e/m for each setting, obtain the average and its standard deviation. Comment on whether your estimated uncertainties in your individual V, I and R values account for the spread in values of e/m. Compare the results for Steps 4 and 5, to see the effect of the change in magnetic field direction. Comment on this. Taking the standard deviation into account, compare with the accepted value of e/m (1.76 x Coulomb/kilogram). (Be careful of units.) Compare with your results from Set-Up 2 Set-Up 2 As in Set-Up 1, the glass deflection tube used in this experiment contains a filament heated by passing a current through it, so that electrons are thermally liberated from the coated surface of the filament. The freed electrons are accelerated and guided through a small slit. Also as in Set-Up 1, a pair of Helmholtz coils is used to create the magnetic field B. The direction of motion of the beam is horizontal instead of vertical (Figures 3,4,5), but this is not an essential difference. However, the method of observing the beam and determining the radius R of the electron trajectory is different. In this setup the amount of deflection obtained for a given field B is observed on a screen. This screen consists of a flat mica sheet, one side of which is coated with a luminescent material. The mica sheet is held at an angle of 15 degrees with respect to the electron beam, and furnishes a clear and measurable trace of the beam if the room light is sufficiently dimmed that the luminescence is made visible. A grid (in 1 cm blocks) is inscribed on the screen (called a graticle). The path of the electron is visible on the graticle, and the grid provides for reading the horizontal and vertical coordinates at various points on this path. To find the radius R from these coordinates, we take the origin of our coordinate system as the exit aperture of the anode (in essence, the end of the electron gun). Then a circle which passes through this origin, and through either the points (x, y) or (x, -y), has the radius given by R = ( x y ) 2y (8)

8 Thus radii are obtained readily from measurements on the graticle, but remember to convert to meter units from the centimeter units of the graticle scale and to take the origin at the end of the electron gun. Experimental Procedure Set-Up 2 PRECAUTIONS: o Always be sure voltage is off before touching connections. o Maximum Accelerating voltage 5000 V (normal range 2000 V V). o Current I in Helmholtz coils should be less than 1 amp. o Note: Each coil for Set Up 2 has an average radius a = m, and n = 320 turns. 1. Connect the apparatus as shown in Fig. 3. Make a Table to show V, I, (x 1, y 1 ), (x 2, y 2 ), (x 3, y 3 ) for each setting, together with estimated uncertainties. 2. For two settings of V, and I chosen to give convenient trajectories, measure 3 points on each trajectory (2 trajectories, 2 values of V and I; 3 points per trajectory). 3. For two settings of V, and I chosen to give convenient trajectories, measure 3 points on each trajectory (2 trajectories, 2 values of V and I; 3 points per trajectory)

9 Analysis and Discussion Set-Up 2 Calculate an average R and its standard deviation at each setting of V and I from Eq. (8), using each of your three points (x 1, y 1 ), (x 2, y 2 ), (x 3, y 3 ) on the trajectory to give one value of R. From your experimental values of V, I and R, using Equations (6) and (7), calculate e/m for each setting of V and I, and obtain the average and its standard deviation. Do your estimated uncertainties account for the spread in individual values of e/m? Compare the results of the two measurements at the same settings but opposite magnetic field direction. Compare your overall average e/m with the accepted value of e/m (1.76 x Coulomb/kilogram), taking the standard deviation into account. Remember to compare your results from Set-Up 1 with your results from Set Up

Laboratory 14: Ratio of Charge to Mass for the Electron

Laboratory 14: Ratio of Charge to Mass for the Electron Introduction The discovery of the electron as a discrete particle of electricity is generally credited to the British physicist Sir J. J. Thomson