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

Brown University
Physics Department
Physics 0040
Lab 3
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 VA, i.e. by placing a plate at a
positive potential, VA 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 2) we get:
(1)
0  ( Work (1 2 ) )  Kinetic energy  0
(1a)
0  (eVA )  1 / 2(mv 2 )  0
(2)
where: ‘v’ is the velocity of the electron when it reaches the plate, and ‘VA’ 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, ‘B’. The
magnetic field exerts a force, FB on the electrons which is perpendicular to the direction
of the velocity of the electrons and is given by:

 
FB  qv  B
(3)
where, ‘B’ is the magnetic field, and ‘q’ = -e =  1.602  10 19 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, FB acting
towards the center of the path (See Fig. 2). Electrons (or any particle) moving along a
circular path experiences an acceleration, a R which points radially inward. This
acceleration is given by:
a R  v2 / R
(4)
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Lab 3
where R is the radius of the path.
By Newton’s second law,
FB  ma R
(5)
Substituting (3) and (4) into (5) for velocity ‘v’ from (2), and rearranging we get:
2V
e
 2 A2
m R B
(6)
where: VA = Accelerating voltage,
R = Radius of the trajectory of the electron, and
B = Magnetic field.
Generating the Magnetic Field (B)
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 B, at a point
midway between them is given by:
8 NI
B 0 H
(5 5 )a
(7)
where: N = Number of turns in the Helmholtz coil,
IH = Current in the Helmholtz coil,
a = Radius of coil, and
 0  4  10 7 T.m/A
The direction of the magnetic field, B generated using the Helmholtz coils is:
 or IN if current is CLOCKWISE.
 or OUT if current is COUNTERCLOCKWISE.
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Filament
Physics 0040
Lab 3
Positive
Plate
e(1)
Accelerated Electron Beam
(2)
V A
Fig. 1
a=v2/R
F
a
B
B (out of paper)
R
Deviated trajectory
(with magnetic field)
Filament
Accelerating
Plate
Undeviated trajectory
(no magnetic field)
Fig. 2
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FIG.3
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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
2813) as well as on the LOVOLT DIGIRAMP (Model TEL 2800) are turned all the
way down i.e. all the way counterclockwise.
(2) Turn ON the KILOVOLT DIGIRAMP – Model TEL 2813 and the LOVOLT
DIGIRAMP (Model TEL 2800) (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 2813. This makes the instrument display the value of the accelerating
voltage, VA in kilovolts.
(4) Press and release the blue button under ‘A’ on the right hand side of the front panel
of the TEL 2800. This makes the instrument display the value of the Helmholtz
coil current, IH in Amperes.
(5) The TEL 2813 is the supply for the accelerating voltage, VA, as well as the filament
supply. Slowly increase the accelerating voltage, VA, 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, VA, to 2.00 kV (i.e. 2000V). Record this value, VA on
the Data & Calculations sheet. (Note: you must multiply the reading of the TEL
2813 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 2800 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 2800. 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, IH in Amperes, as read off the display of the TEL 2800.
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(8) With accelerating voltage, VA, 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, IH all the way down to zero, by turning the
knob on the right hand side of the TEL 2800 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, VA, which are
higher than 2.00kV. (e.g. 3.00kV and 4.00kV).
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Physics 0040
Lab 3
SET UP-I
Notation used:
Measured Quantities:
Quantity
Unit
a=0.068
meters
N=320
--L
meters
VA
ΔVA
IH
ΔIH
Volts
Volts
Amperes
Amperes
Calculated Quantities:
Quantity
R
B
ΔB
e/m
Formula
1 (0.082  L2 )
R
[
]
2 (0.08  L)
8 NI
B  0 H  kI H 
(5 5 )a
8 N (I H )
B  0
 k (I H ) 
(5 5 )a
2V
e
 2 A2
m R B
Δ(e/m)
k 
Definition
Helmholtz coil radius (given)
Number of turns in Helmholtz coil (given)
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)
Accelerating voltage of the beam of electrons
Uncertainty in ‘VA’
Current in Helmholtz coil
Uncertainty in IH
See Below**
80 N
 4.17  103 T / A
(5 5 )a
2
Unit
meter
Definition
Radius of the trajectory of
the electron beam
Tesla
Magnetic field created by
Helmholtz coil
Tesla
Uncertainty in ‘B’
C/Kg
Charge to mass ratio of an
electron
C/Kg
Uncertainty in ‘(e/m)’
This is a constant which does not
change for all the runs on Set up-II
and hence needs to be calculated
only once
2
 VA 
 B 

  4 
 B
 VA 
7
Constant:  0  4  10 T  m / A
e
e
 ( )  ( )
m
m
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(Volts)
(Volts)
±
±
±
ΔVA
VA
(meters)
L
(meters)
R
(Amp)
IH
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
(Tesla)
ΔB= k
(ΔIH)
±
(Tesla)
B=k IH
±
±
(Amp)
Δ IH
(C/Kg)
e/m
±
±
±
±
±
±
±
±
±
(C/Kg)
Δ (e/m)
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=320 turns
Brown University
Physics Department
Physics 0040
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Brown University
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Lab 3
Calculations: These calculations and discussion should appear in your report.
Mean (e/m) = _________________________ C/Kg.
n
RMS [(e/m)] = _______________________ C/Kg. = 
1
 e  e 
   
 m  m 
(n  1)
2
Mean [∆(e/m)] = ______________________ C/Kg.
Actual (e/m) = 1.758 X 10+11c/Kg.

Comparison of experimental Mean(e/m) with Actual(e/m) [in the light of
RMS(e/m).
the
Is Mean (e/m) - Actual (e/m)   (e/m, 2 (e/m), 3 (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 2: BALANCED 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 2811 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 2 )
When the electric (qE) and magnetic forces (qvB) are balanced we have
(8.)
Then
qE y  qv x B y
(9)
vx 
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Ey
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Brown University
Physics Department
Physics 0040
Lab 3
1 2
mv  eVA where VA is the accelerating potential.
2
Substituting vx from equation ( 9 ) we have:
By the conservation of energy
2
Ey
e

m 2VA B y2
(10)
But E y 
Vp
d
where V p is the parallel plate voltage and d is the parallel plate separation
d  8  10 m and we know from the earlier experiment
3
B y  kI H where k is 0.00417
T/A. Thus
1
e

m 2V A
(11.)
 V P2

 d 2 k 2 I H2





A straightforward application of the rules of error analysis yields:
2
(12)
2
2
 V p 
e
e  VA 
 B 



 4
( ) 
 4

 V 
m
m  VA 
 B 
 p 
Procedure (Method 2)
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
2813) as well as on the LOVOLT DIGIRAMP (Model TEL 2800) are turned all the
way down i.e. all the way counterclockwise.
(2) Turn ON the KILOVOLT DIGIRAMP – Model TEL 2813 and the LOVOLT
DIGIRAMP (Model TEL 2800) (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 2813. This makes the instrument display the value of the accelerating
voltage, VA in kilovolts.
(4) Press and release the blue button under ‘A’ on the right hand side of the front panel
of the TEL 2800. This makes the instrument display the value of the Helmholtz
coil current, IH in Amperes.
(5) The TEL 2813 is the supply for the accelerating voltage, VA, as well as the filament
supply. Slowly increase the accelerating voltage, VA, by turning the knob on the
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Physics Department
Physics 0040
Lab 3
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, VA, to 2.00 kV (i.e. 2000V). Record this value, VA in
DATA and Calculations: Thomson Method. (Note: you must multiply the reading
of the TEL 2813 display by 1000 to convert from kV to V).
(7) Plug in the Tel 2811 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 2800 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 2800. 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, IH in Amperes, as read off the display of the TEL 2800.
(9) Reduce the current in the Helmholtz coil, IH all the way down to zero, by turning the
knob on the right hand side of the TEL 2800 all the way in the counter clockwise
direction.
(10) Repeat (6) through (9) for four other values of accelerating voltage, VA, which are
higher than 2.00kV. (e.g. 2.00 kV, 2.50kV, 3.00 kV and 3.50kV).
e
from equation 11. Calculate this value in the lab, compare with its
m
1
current accepted value. Plot I H2 versus , whose graph will then be a straight line
VA
(11) Determine
with slope
VP2 m
. Since V p ,k, and d are known, the e/m ratio can easily be
2d 2 k 2 e
calculated.
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Physics Department
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Lab 3
SET UP-II
Notation used:
Measured Quantities:
Quantity
Unit
a=0.068
meters
N=320
--L
meters
VA
ΔVA
IH
ΔIH
Definition
Helmholtz coil radius (given)
Number of turns in Helmholtz coil (given)
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)
Accelerating voltage of the beam of electrons
Uncertainty in ‘VA’
Current in Helmholtz coil
Uncertainty in IH
Volts
Volts
Amperes
Amperes
Calculated Quantities:
Quantity
Ey
B
ΔB
e/m
Formula
Ey 
80 NI H
 kI H 
(5 5 )a
8 N (I H )
B  0
 k (I H ) 
(5 5 )a
1
e

m 2V A
 V p2

 d 2 k 2 I H2





See Below**
80 N
 4.17  103 T / A
(5 5 )a
2
Definition
Electric field between the
parallel plate
Tesla
Magnetic field created by
Helmholtz coil
Tesla
Uncertainty in ‘B’
C/Kg
Charge to mass ratio of an
electron
C/Kg
Uncertainty in ‘(e/m)’
d
B
Δ(e/m)
k 
Vp
Unit
V/m
This is a constant which does not
change for all the runs on Set upII and hence needs to be
calculated only once
2
2
 Vp 
 VA 

B


  4 
  4



B
 VA 
 Vp 
7
Constant:  0  4  10 T  m / A
e e
( ) 
m m
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300
300
±
300
300
±
±
±
±
300
±
ΔVP
±
±
±
(Volts)
(Volts)
(Volts)
(Volts)
VP
ΔVA
VA
(Amps)
IH
±
±
±
±
±
(Amps)
Δ IH
(Tesla)
B=k IH
DATA AND CALCULATIONS: THOMSON METHOD
±
±
±
±
±
(Tesla)
ΔB= k
(ΔIH)
Brown University
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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).
(2) Determine the mean uncertainty i.e. Mean (e/m) from the  (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 VA only electrons with a certain velocity will be undeflected,
determine that velocity for each of your accelerating potentials. Do you have to worry
about relativistic corrections for any of these velocity values?
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