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# Rice PHYS 102 - Deflection of Electrons

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Deflection of Electrons Every statement in physics has to state relations between observable quantities. E. Mach (1838-1916) OBJECTIVES To determine the effect of electric and magnetic fields on a beam of electrons. To measure the charge to mass ratio of an electron. THEORY In this exercise we ask how a stream of negatively charged particles responds to electric and magnetic fields, and thereby determine the speed of the particles and their charge to mass ratio. Consider a beam of particles with charge e and mass m, moving at speed u through a vacuum. If there is a uniform electric field ! E and uniform magnetic field ! B , The force on each particle is given by ! F = e! E + e! u !! B (1) We might choose to produce the ! E field with a voltage applied to two parallel plates, and to produce the ! B with a coil outside the vacuum chamber. In either case, the fields are calculable from the geometry, shown in Fig. 1, and measurable voltages and currents. For simplicity, we consider only one field at a time, and calculate the deflection of the beam when we turn on each field. The electric field causes a constant transverse acceleration during the time the particles are between the plates. The force acting is simply eE, and the time is !t = Lp/u, so the particles acquire a velocity u1, perpendicular to u, given by u1=Fm!t =emELpu (2) After leaving the plates the particles travel in a straight line to the end wall of the tube. The path makes an angle ! with the original direction, resulting in a displacement DE given by DE= LEtan!= LEu1u (3)Deflection of Electrons 2 The electric field can be expressed in terms of the voltage Vd applied to the plates by E = Vd/a. Using this result in Eq. 2 and 3, we get DE=emLELpau2Vd (4) Since the tube wall glows where the beam hits, we can measure DE with a ruler. (There is also a small displacement that occurs within the plates, given by eE(!t)2/2m. This is only about 5% of DE for the geometry we will use, and is neglected.) From Eq. 1, the magnetic force is constant and perpendicular to ! u . The particles will move in a circle whose radius may be found by equating the magnetic force to the centripetal force needed to move in a circle of radius R euB =mu2R (5) or R =mueB (6) !Luu1aDEELDBRR!"BLp Fig 1 Arrangements for electric or magnetic deflection of charged particles. The magnetic field is uniform over the entire distance LB, while the electric field exists only between the plates.Deflection of Electrons 3 Referring to Fig. 1 again, the total deflection DB is given by DB = R - R cos! (7) For small angles, we can use the approximations LB ! R! and cos! ! 1 - (!2/2) to combine Eq. 6 and Eq. 7 into DB=emLB22uB =emLB22uGI (8) In the last part we expressed the field B by B = GI, where I is the current in the coils and G is a geometric factor that will be given later. Eqs. 4 and 8 are our main results. They tell us that graphs of DE vs Vd and DB vs I will be straight lines with specified slopes. Calling the slopes sE and sB respectively, we can solve for the two unknowns u and e/m, to arrive at u =2LpLEsBaGLB2sEandem=4LpLEsB2aG2LB4sE (9) These two equations will be used for analyzing our data. EXPERIMENTAL PROCEDURE Figure 2 is a sketch of the vacuum tube that we will use, showing the electrical connections needed. Electrons are boiled out of the cathode with a heater. Various electrodes, labeled as grids and anodes, form the stream of electrons into a focused beam. The voltage labeled C- is adjusted to provide the proper focus, while the B+ and C- voltages together determine the energy and speed of the electrons. After leaving the accelerating and focusing electrodes, the beam passes between two sets of deflecting plates which are arranged to deflect the beam in orthogonal directions. As shown in Fig. 2, we will use only the second set, nearer the screen. The other set is connected to the anode so it will not charge up and produce undesirable deflections. After the deflection plates, the beam travels a distance LE to the end of the tube, where it causes a special material to glow at the point of impact. The coils which produce the magnetic field will be discussed later. Table 1 contains a summary of the constants you will need. Before proceeding with the experiment, there is a hazard you must be aware of. The power supply used here produces dangerously high voltage. Because of this it has two controlDeflection of Electrons 4 switches. The AC POWER switch turns on the supply and the low voltage outputs. It may be left on throughout the experiment. The DC ON/STANDBY switch controls the high voltage. This switch must be at STANDBY when you make any changes in the wiring. Failure to follow this procedure may result in a dangerous electrical shock. It is good practice to set the switch to STANDBY any time you are not actually making measurements. Check that the tube and deflection circuits are connected as shown in Fig. 2. Turn on the AC POWER switch and set the deflection voltage to zero. Allow the heater to warm up for about a minute, and then turn on the DC with the DC ON switch. The VOLT METER switch connects the meter on the power supply to either the B+ VOLTS (red scale) or the C- VOLTS (black scale). Note that the B+ and C- potentials are relative to the COM (i.e. common) terminal. This means that, for DeflectionPlate 26.3 ACC-COMB+GNDH. V.SupplyDMM+30GND-30HeaterCathodeGrid 1Grid 2Anode 1Anode 2DeflectionPlate 1YellowGreenBlueRedYellowBeamScreenWhiteWhite0CT Fig. 2. Electrode configuration and wiring diagram for the cathode ray tube (CRT). Wiring to the left of the dashed line is already in place inside the tube mounting. The voltage divider circuit driving the deflection plates is contained in a separate box which must be connected to GND as shown. Connect leads to the DMM terminals marked V! and COM and set the large knob to point at the V with the solid and dashed lines over it.Deflection of Electrons 5 example, anode 1 will be at a potential B+ more negative than anode 2 (which is at ground potential). Set the B+ to about 250 V, and then adjust the C- to get the sharpest possible spot on the

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