Deflection of Electrons by Electric and Magnetic Fields


 Dorcas Tucker
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1 Physics 233 Expeiment 42 Deflection of Electons by Electic and Magnetic Fields Refeences Loain, P. and D.R. Coson, Electomagnetism, Pinciples and Applications, 2nd ed., W.H. Feeman, 199. Intoduction An electon with chage e in an extenal electic field E expeiences a foce F =! e E. (1) Note that the acceleation is in the diection opposite that of the field. In an extenal magnetic field, B, the foce on the electon is given by F =! e v x B (2) whee v is the velocity of the electon. Note that the acceleation depends on the magnitudes of the velocity and the magnetic field, the angle between them, and is pependicula to both. To analyze the motion of the electons we will use a cathode ay tube (CRT). The CRT is the basic component of oscilloscopes and television sets. The pincipal pats of the CRT used in this expeiment ae shown in Fig. 1. F Figue 1 Side View of the 5DEP1 cathode ay tube. The dimensions ae in cm. The sepaation of the X deflection plates is unknown.
2 The indiectly heated cathode has a coating of baium and stontium oxides which emit a highdensity electon steam. These electons ae then acceleated and focussed by the anode and gid stuctue shown. Afte they have passed the second anode, the electons ae not subjected to any moe foce in the z diection since the sceen is at the same potential as the anode and the electons may be consideed to be a "beam" which goes fom the anode to the sceen. The electons will all have the same velocity, given by 1 2 mv 2 = e ( V anode! V cathode ) = e V acc (3) whee V acc is the "acceleating voltage". Eq. 3 may be obtained fom the law of enegy consevation, o, altenatively, fom integating Eq. 1. A voltage applied to the X o Y deflection plates of the CRT poduces an electic field between the plates which will exet a foce on the electons (Eq. 1) as they pass though. As a esult the beam is deflected by this field. The "deflection sensitivity" is defined as the atio of the deflection of the electon beam at the sceen of the CRT to the deflection voltage, V defl. Television sets use magnetic deflection as it has some pactical advantages ove electic deflection. A pai of coils is mounted on the side of the CRT. A cuent passing though the coils poduces a magnetic field which exets a foce on the electons (Eq. 2). Fo this situation the deflection sensitivity is defined as the atio of the deflection of the electon beam at the sceen of the CRT to the cuent in the deflection coils, I defl. Pelab Questions 1. Deive a fomula fo the deflection sensitivity fo deflection by an electic field. Using values of the dimensions given in Fig. 1 compute the deflection sensitivity fo this appaatus. The flae on the plates makes an exact calculation of the deflection sensitivity difficult. A simple appoximation is to assume that the electic field is equal to the voltage divided by the plate sepaation at all points between the plates. The aveage electic field ove the whole 1.8 cm is then given by E = V defl / d eff, whee d eff =.175 cm fo the Y plates of the 5DEP1. We suggest you use this value. 2. Calculate the speed of an electon acceleated though a potential diffeence of 1 volts. How long does it take this electon to tavel fom the second anode to the sceen of the CRT? Calculate the tansvese velocity this electon gains if the voltage on the Y plates is 5 volts. 3. The magnetic field poduced by the deflection coils is not unifom along the tube axis. Thus the calculation of the deflection sensitivity is moe difficult fo magnetic as opposed to electic deflection. The deflection sensitivity fo deflection by a spatially vaying field is deived in the Appendix. The vaiation of the magnetic field along the axis of the CRT fo this appaatus is given in Table 1. Calculate the deflection sensitivity by making a numeical integation of the magnetic field along the axis of the tube. 4. Veify Eq. 4, see below. You will need to assume some quantities ae small compaed to othes, and to make an appopiate appoximation.
3 Appaatus 5DEP1 cathode ay tube chassis DC powe supply Digital multimete plastic ule, venie calipes Expeiment 1) Examine the sample cathode ay tube. Compae it with Fig. 1 and identify the component pats. 2) In any expeiment it is a good idea to become familia with you equipment. Spend some time finding out what the vaious contols (focus, acceleating potential, and bightness) do. Obseve the effects of applying voltages to the X and Y deflection plates of the cathode ay tube. 3) Detemine the deflection sensitivity due to an electic field applied acoss the Y deflection plates. Compae you calculated and expeimental deflection sensitivities. Show how the deflection sensitivity vaies with the acceleating voltage. 4) Detemine the deflection sensitivity due to a magnetic field poduced by a cuent in the magnetic deflection coils. Choose a single value of the acceleating voltage fo this measuement. Explain you easons fo you choice of acceleating voltage. Compae you expeimental obsevations with the esult you calculated above. 5) You pobably have noticed that the spot on the 5DEP1 moves hoizontally when the acceleating potential is vaied. This deflection is due to the motion of the electons in the eath's magnetic field and is given by: x = B v L 2 2 e 2V acc m (4) whee x is the amount of hoizontal deflection, B v is the vetical component of the eath's field, L is the distance fom the second anode to the sceen, and V acc is the acceleating potential. Measue x as a function of V acc and detemine B v. UG2/2
4 Table 1. The magnetic field in Gauss poduced by a cuent of 1 ma in the magnetic deflection coils fo Chassis #5. Values of the field ae quoted as a function of distance fom the second anode along the axis of the CRT. The diection of the magnetic field is pependicula to the axis of the CRT. Owing to small diffeences in constuction and assembly the field poduced by the coils on Chassis #2 is slightly smalle. Multiply the values fo Chassis #5 by.91 to obtain the magnetic field poduced by a cuent of 1 ma in the coils fo Chassis #2. Distance fom second anode (cm) Magnetic field (Gauss) Chassis
5 Appendix: Calculation of Deflection Sensitivities The aim of this appendix is to detemine the deflection sensitivities of the CRT fo both electic and magnetic fields. As illustated below, the electons leave the second anode with velocity v in the z diection and ae subsequently deflected by a tansvese foce F(z) whee F(z) = e E(z) electic fields = e v B(z) magnetic fields Let u(z) be the tansvese velocity and!(z) be the tansvese deflection at z. We assume that as they leave the second anode exit (at z = ), the electons have both zeo tansvese velocity and deflection (i.e., u() = and!() = ). Thei subsequent motion is detemined by Newton's Second Law: m du(z) dt The time dependence of u is of no inteest and so we use the chain ule to wite = F(z) (A1) du(z) dt = du(z) dt = v du(z) wheeby Eqn. (A1) becomes mv du(z) Integating Eqn. (A2) and setting u() =, we obtain u(z) = 1 mv F( z!)d z! The deflection!(z) can now be obtained fom the elationship z = F(z) (A2) " (A3) u(z) = d!(z) dt = d!(z) dt = v d!(z) (A4)
6 Using Eqn. (A4) in Eqn. (A3), we integate once moe and set!() = to get!(z) = 1 z $ z " ' F( z "")d z " mv # # 2 & ) d z " % ( (A5) whee z! and z!! ae dummy vaiables. The integal in Eqn. (A5) is taken ove the shaded aeas in the ( z!, z!!) plane as shown in the diagam. The student can show that by intechanging the ode of integation (i.e., by integating with espect to z" fist and making use of the fact that F is a function of z"" only) Eqn. (A5) can be witten as which becomes!(z) = 1 z $ z ' mv 2 # & # d z ") F( z "")d z "" % ( z " "!(z) = 1 z mv 2 $ (z " z ##)F( z ##)d z ## (A6) The quantity of inteest is the total deflection at the sceen. If L is the distance fom the anode exit to the sceen we then have fo the total deflection "(L) = 1 mv 2 L $ (L # z)f(z) (A7) If the foce F(z) is known in eithe analytical fom (as in the case of the electic field) o in numeical fom (as in the case of the magnetic field) then Eqn. (A7) can fom the basis fo computing the deflection! sensitivity.
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