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

Transcription

1 Determining the Charge to Mass Ratio (e/m) for an Electron Introduction In order to determine the charge to mass ratio (e/m) for an electron we create a beam of electrons by heating a metal filament in an evacuated glass bulb by passing current through it. The heated filament liberates electrons which are then focused into a beam. This is accomplished by applying an accelerating potential V A, i.e. by placing a plate at a positive potential, V A at a certain distance from the filament so that the negatively charged electrons are attracted towards it (See Fig. 1). The electrons gain kinetic energy when they move towards the positive plate so that by using energy conservation (because there are no other forces acting on the electron) between the filament (point 1) and the positive plate (point ) we get: (1) (1a) () 0 + > ) K! + U! = K! + U! ( Work ( 1 ) = Kinetic energy ( eva ) = 1/ (mv ) + 0 where: v is the velocity of the electron when it reaches the plate, and V A is the accelerating potential. The electrons, having acquired a velocity v on reaching the positive plate, now pass through a hole in the plate and enter a region where there is a magnetic field,. The magnetic field exerts a force, F on the electrons which is perpendicular to the direction of the velocity of the electrons and is given by: (3) F = qv where, is the magnetic field, and q = -e = Coulombs, for an electron. This causes the electrons to change the direction of their velocity, although not the magnitude, and thereby move along a circular path, with the magnetic force, F acting towards the center of the path (See Fig. ). Electrons (or any particle) moving along a circular path experiences an acceleration, a which points radially inward. This R acceleration is given by: a = R v / R (4)

2 where R is the radius of the path. y Newton s second law, (5) F = ma R Substituting (3) and (4) into (5) for velocity v from (), and rearranging we get: (6) e m = V R A where: V A = Accelerating voltage, R = Radius of the trajectory of the electron, and = Magnetic field. Generating the Magnetic Field () The magnetic field in the coils which is used to deflect the electrons is generated by sending an electric current through a pair of coils called Helmholtz coils. The coils are separated by a distance which is half their diameter and the magnetic field, at a point midway between them is given by: 8µ NI H = 0 (5 5) a (7) where: N = Number of turns in the Helmholtz coil, I H = Current in the Helmholtz coil, a = Radius of coil, and 7 µ = 4π 10 T.m/A 0 The direction of the magnetic field, generated using the Helmholtz coils is: or IN if current is CLOCKWISE. or OUT if current is COUNTERCLOCKWISE

3 Filament e- Positive Plate Accelerated Electron eam (1) () V A Fig

4 FIG

5 Procedure: Set-up: (See Fig. 3) Note: All circuit connections have been made. Do not disconnect any wires. (1) Ensure that the circular knobs on both the KILOVOLT DIGIRAMP (Model TEL 813) as well as on the LOVOLT DIGIRAMP (Model TEL 800) are turned all the way down i.e. all the way counterclockwise. () Turn ON the KILOVOLT DIGIRAMP Model TEL 813 and the LOVOLT DIGIRAMP (Model TEL 800) (ON/OFF switch is on the right rear side). (3) Press and release the blue button under kv on the right hand side of the front panel of the TEL 813. This makes the instrument display the value of the accelerating voltage, V A in kilovolts. (4) Press and release the blue button under A on the right hand side of the front panel of the TEL 800. This makes the instrument display the value of the Helmholtz coil current, I H in Amperes. (5) The TEL 813 is the supply for the accelerating voltage, V A, as well as the filament supply. Slowly increase the accelerating voltage, V A, by turning the knob on the instrument in the clockwise direction. A blue beam on electrons should emerge which is along the axis of the tube. (6) Set the accelerating voltage, V A, to.00 kv (i.e. 000V). Record this value, V A on the Data & Calculations sheet. (Note: you must multiply the reading of the TEL 813 display by 1000 to convert from kv to V). (7) Increase the current in the Helmholtz coil by slowly turning the knob on the right hand side of the front panel of the TEL 800 in the clockwise direction. The beam in the tube must begin to bend upwards. If it bends downwards interchange the wires in the sockets of the TEL 800. This reverses the direction of the current in the Helmholtz coils and hence the direction of the magnetic field produced, thereby causing the beam to bend in the opposite direction (i.e. upwards). Decide on a location of a point on the scale inscribed top right edge of the screen in the tube. The location of such a point can be specified by L which denotes the distance of the point from the top of the screen. (Note: The graduations on the screen begin at a distance of 0.01m from the top corner of the screen i.e. they extend from L=0.01m, near the top of the screen to L=0.07m, near the right corner of the screen. You account for this in mind while recording L.) Record the current in the Helmholtz coils, I H in Amperes, as read off the display of the TEL

6 (8) With accelerating voltage, V A, fixed repeat (7) for two other points at different values of L on the screen. Record L in each case. (See Data & Calculations). (9) Reduce the current in the Helmholtz coil, I H all the way down to zero, by turning the knob on the right hand side of the TEL 800 all the way in the counter clockwise direction. The beam should one again appear horizontal with respect to the screen. (10) Repeat (6) through (9) for two other values of accelerating voltage, V A, which are higher than.00kv. (e.g. 3.00kV and 4.00kV)

7 Notation used: SET UP-I Measured Quantities: Quantity Unit Definition a=0.068 meters Helmholtz coil radius (given) N= Number of turns in Helmholtz coil (given) L meters Distance of point F (on the top right edge of the screen from where the beam passes) with respect to top of screen (See Fig. 6) V A Volts Accelerating voltage of the beam of electrons ΔV A Volts Uncertainty in V A I H Amperes Current in Helmholtz coil ΔI H Amperes Uncertainty in I H Calculated Quantities: Quantity Formula Unit Definition R meter Radius of the trajectory of 1 ( L ) the electron beam Δ e/m R = [ ] (0.08 L) 8 NI H = 0 = ki H (5 5) a Δ Δ = 8 N( I ) 0 H = k( ΔI ) (5 5) a H e V A = m R µ Tesla Magnetic field created by Helmholtz coil µ Tesla Uncertainty in C/Kg Charge to mass ratio of an electron Δ(e/m) See elow** C/Kg Uncertainty in (e/m) 8µ N This is a constant which does not 0 3 k = = T / A change for all the runs on Set up-ii (5 5) a and hence needs to be calculated only once e e ΔVA Δ Δ( ) = ( ) + 4 m m VA 7 Constant: µ = 4π 10 T m / A

8 Data: SET UP-I Note: you should include a copy of this sheet in your report. The original you should staple into your lab book. Coil Radius, a=0.068m. Numbers of turns in coil, N=30 turns Δ (e/m) e/m Δ= k (ΔIH) =k IH Δ IH IH R L ΔVA VA (C/Kg) (C/Kg) (Tesla) (Tesla) (Amp) (Amp) (meters) (meters) (Volts) (Volts)

9 Calculations: These calculations and discussion should appear in your report. Mean (e/m) = C/Kg. RMS [(e/m)] = C/Kg. = Mean [ (e/m)] = C/Kg. Actual (e/m) = X c/kg.!!!!!!!!!!!"!! Comparison of experimental Mean(e/m) with Actual(e/m) [in the light of the RMS(e/m). Is Mean (e/m)- Actual (e/m) < δ(e/m,δ (e/m),3δ (e/m)? Determining whether experimental uncertainties (Mean [ (e/m)]) accounts for spread of values (RMS(e/m)) Is Mean [ (e/m)] > or < RMS(e/m)? METHOD : ALANCED ELECTRIC AND MAGNETIC FIELDS In this part of the lab you will perform the experiment similar to Thomson s original e/m experiment. The Teltron 811 hivolt bias will provide a voltage across parallel plates which will cause the electron beam to deflect upward. The Helmholtz coils can be configured to provide a magnetic which will deflect the beam downward. When the electric and magnetic forces are equal (balanced) the electron beam will pass through the apparatus undeflected. When this condition is met the ratio e/m can easily be determined. THEORY ( Method ) When the electric (qe) and magnetic forces (qv) are balanced we have (8.) qe y = qv x y Then Ey (9) v x = y

10 1 y the conservation of energy mv = ev Substituting v x from equation ( 9 ) we have: e Ey (10) = m V A y A where V is the accelerating potential. A V ut E = p y d where V p is the parallel plate voltage and d is the parallel plate separation 3 ( d = 8 10 m) and we know from the earlier experiment y = ki H where k is T/A. Thus (11.) e 1 V P m d = V A k I H A straightforward application of the rules of error analysis yields: (1) e e ΔV ΔV A p Δ ( ) 4 Δ = m m VA Vp Procedure (Method ) Set-up: (See Fig. 4) Note: All circuit connections have been made. Do not disconnect any wires. (1) Ensure that the circular knobs on both the KILOVOLT DIGIRAMP (Model TEL 813) as well as on the LOVOLT DIGIRAMP (Model TEL 800) are turned all the way down i.e. all the way counterclockwise. () Turn ON the KILOVOLT DIGIRAMP Model TEL 813 and the LOVOLT DIGIRAMP (Model TEL 800) (ON/OFF switch is on the right rear side). (3) Press and release the blue button under kv on the right hand side of the front panel of the TEL 813. This makes the instrument display the value of the accelerating voltage, V A in kilovolts. (4) Press and release the blue button under A on the right hand side of the front panel of the TEL 800. This makes the instrument display the value of the Helmholtz coil current, I H in Amperes. (5) The TEL 813 is the supply for the accelerating voltage, V A, as well as the filament supply. Slowly increase the accelerating voltage, V A, by turning the knob on the

11 instrument in the clockwise direction. A blue beam on electrons should emerge which is along the axis of the tube. (6) Set the accelerating voltage, V A, to.00 kv (i.e. 000V). Record this value, V A in DATA and Calculations: Thomson Method. (Note: you must multiply the reading of the TEL 813 display by 1000 to convert from kv to V). (7) Plug in the Tel 811 Unit. This will put a 300 Volt potential across the parallel plate. The electron beam will then curve upwards due to the electric field acting on the electrons. The manufacturer states that this value is calibrated to 1%. (8) Increase the current in the Helmholtz coil by slowly turning the knob on the right hand side of the front panel of the TEL 800 in the clockwise direction. The beam in the tube must begin to bend downwards. If it bends upwards interchange the wires in the sockets of the TEL 800. This reverses the direction of the current in the Helmholtz coils and hence the direction of the magnetic field produced, thereby causing the beam to bend in the opposite direction (i.e. downwards).adjust the current until the beam is undeflected (null deflection). Record the current in the Helmholtz coils, I H in Amperes, as read off the display of the TEL 800. (9) Reduce the current in the Helmholtz coil, I H all the way down to zero, by turning the knob on the right hand side of the TEL 800 all the way in the counter clockwise direction. (10) Repeat (6) through (9) for four other values of accelerating voltage, V A, which are higher than.00kv. (e.g..00 kv,.50kv, 3.00 kv and 3.50kV). (11) Determine m e from equation 11. Calculate this value in the lab, compare with its current accepted value. Plot V P with slope d k calculated. m e 1 I versus H, whose graph will then be a straight line V A. Since V p,k, and d are known, the e/m ratio can easily be

12

13 Notation used: SET UP-II Measured Quantities: Quantity Unit Definition a=0.068 meters Helmholtz coil radius (given) N= Number of turns in Helmholtz coil (given) L meters Distance of point F (on the top right edge of the screen from where the beam passes) with respect to top of screen (See Fig. 6) V A Volts Accelerating voltage of the beam of electrons ΔV A Volts Uncertainty in V A I H Amperes Current in Helmholtz coil ΔI H Amperes Uncertainty in I H Calculated Quantities: Quantity Formula Unit Definition E y V E V/m Electric field between the p y parallel plate d 8µ NI Tesla Magnetic field created by H = 0 = ki H Helmholtz coil (5 5) a Δ Δ Δ = µ N( I ) Tesla Uncertainty in 0 H = k( ΔI ) (5 5) a H e/m = 1 V C/Kg Charge to mass ratio of an e p electron m V A d k I H Δ(e/m) See elow** C/Kg Uncertainty in (e/m) 8µ N This is a constant which does not 0 3 k = = T / A change for all the runs on Set up- (5 5) a II and hence needs to be calculated only once e e ΔV ΔV A p Δ ( ) 4 Δ = m m VA V p 7 Constant: µ = 4π 10 T m / A

14 VA (Volts) ΔVA (Volts) Data Setup # Data and Calculations: Thomson Method Note: you should include a copy of this sheet in your report. The original you should staple into your lab book. Coil radius = a = m. Numbers of turns in coil, N = 30 turns. VP ΔVP IH Δ IH =k IH Δ= k (ΔIH) (Volts) (Volts) (Amps) (Amps) (Tesla) (Tesla) (e/m) (C/Kg) Δ (e/m) (C/Kg)

15 Results & Discussion (1) Determine the Mean (e/m) and RMS(e/m) from the e/m values in the tables find the % difference between the experimental mean and the actual value of e/m. Also compare them in the light of the uncertainty (See Data & Calculations). () Determine the mean uncertainty i.e. Mean[ Δ (e/m)] from the Δ( e/m) values in table. Does this experimental uncertainty account for the spread of values reflected in the rms(e/m)? (See Data & Calculations). (3) The Thomson method of determining e/m is essentially a velocity selector, for each accelerating voltage V only electrons with a certain velocity will be undeflected, A determine that velocity for each of your accelerating potentials. Do you have to worry about relativistic corrections for any of these velocity values?