Download Determination of the Charge to Mass Ratio of the Electron

Richard Plews
Determination of the Charge to Mass
Ratio of the Electron
Student: Richard Plews
Tutor: A.Aruliah
Course 1B40
19th December 2005
Abstract
An experiment was conducted to calculate the charge to mass ratio of an electron
using J.J. Thomson’s 1897 method by means of opposing forces due to electric and
magnetic fields on a beam of electrons from a cathode ray tube. The results proved to
be accurate and highly precise, with a final e/m value of 1.724±0.03x1011 Ckg-1. The
various disagreements between the theoretical and experimental results do not exceed
± 2% and is attributed largely to voltmeter precision.
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Richard Plews
Introduction
There were a number of results gathered over the years by cathode ray tube researchers, J. J.
Thomson was a pioneer in this area of research in the 1800s.
The e/m ratio that he calculated is important because it is as far as Thomson could get with
his cathode ray tubes. Knowledge of the value of 'e' or of 'm' (the charge and mass of an
electron respectively) would be needed to get to the other once you knew e/m, which
Thomson did know. More information about this subject can be found in "The Discovery of
Subatomic Particles" by Steven Weinberg, 1983.
Both magnetic and electric fields exert forces on charged particles. The force due to an
electric field is given by
FE = Ee
(1)
Where E is the electric field intensity and e is the charge of an electron, while the force due to
magnetic field is given by
FB = Bev
(2)
Where B is the magnetic flux density and v is the velocity of the charged particle.
Force
Electric Field
Charge
Charge
Magnetic
field
Force
Fig 1: Force due to electric field
Fig 2: Force due to magnetic field
By placing the magnetic and electric fields at right angles, an electron travelling through both
fields can continue travelling in an undeterred path by tweaking both fields to have an equal
and opposing pair of forces so the resulting force is zero. For nil deflection It is required that
Ee = Bev
(3)
In this experiment, the electron is provided by an electron gun with energy
Ee = eVa
(4)
Where Va is the voltage provided to the gun. Assuming all energy provided is converted to
Kinetic energy it can be said that
eVa 
1 2
mv
2
(5)
Rearranging (5) and substituting (4) into this gives
e
v2
E2


m 2Va 2 B 2Va
(6)
This means that the e/m ratio can be determined by varying the electric field intensity and
magnetic field intensity so that both field forces cancel.
Method
The main component of the experiment is the e/m Thomson tube, comprised of the elements
shown in figure 3. The Helmholtz coils are set parallel at a distance of 6.9cm apart, this is
achieved by checking the distance between the coils at the top and sides of the coils to ensure
an evenly distributed B-field between them. The electric field is provided by two plates of
opposite charge and is contained within an evacuated glass envelope to minimize collisions of
the electrons with other particles. The cathode ray gun emits electrons from a filament when a
voltage is applied.
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Richard Plews
Helmholtz coils
Evacuated glass
envelope
e
-
Cathode ray
tube
Deflection plates
Fig. 3: Thomson tube components
The apparatus was set up as shown in figure 4. The Electron gun voltage, Va, was set to 3kV
when passed through a 1/1000 potential divider. When turned on without any field applied,
the beam of electrons is made visible on a luminescent plate parallel to the beam as a green
line. The electric field was then turned on with a potential difference of 50V, causing the
electrons to accelerate downwards, this force is then eliminated with an equal and opposing
force by activating the B-field with the Helmholtz coils. The values of the current required to
bring the electron beam deflection back zero were recorded. This process was repeated for
varying strengths of the electric field up to 300Vin increments of 25V, and two more repeats
were taken to reduce anomalous results. The results given were tabulated and used to create a
straight line graph, from which the gradient was used to calculate a value for the e/m ratio
using equation (6).
2x Parallel aligned
Helmholtz coils
Cathode ray
tube
+
Parallel E-field
plates
1/1000
Divider
Power
supply
Power
supply
Power
supply
Fig. 4: Apparatus diagram
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Richard Plews
Results
The average of the three current readings was used to plot the graph.
The error bars were created by subtracting the minimum value of the
B field current from the maximum value and dividing by two for
each voltage. The graph shows the expected linear trend, by
increasing the voltage, the current required to result in zero
deflection was increased proportionally, however the trend line
applied using statistical methods by software does not intercept at 0,
which means there must have been a systematic error present as the
points are all consistent with the gradient of the trend line.
Rearranging equation 6 and substituting in the appropriate formula
for B-field strength and E-field strength, it can be shown that using
the gradient, M, we can calculate the e/m ratio.
e 4.49  10 M

m
Va
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E-field p.d/V mean B-field I/A
50
54.00
75
76.00
100
100.33
125
127.33
150
149.67
175
166.67
200
188.33
225
213.67
250
234.33
275
258.67
300
281.00
Figure 5: Results table
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This gave a final value of 1.724x1011 Ckg-1. The error was then calculated with error
propagation methods to give an error of 2.64x109 Ckg-1.
300
Helmholtz Coil Current/mA
250
200
150
100
50
0
0
50
y = 0.9015x + 10.427
100
150
200
Deflector Plate Voltage/V
4
250
300
Richard Plews
Conclusion
The experimental result lies within 2% of the actual (data book) value of the e/m ratio
(1.79x1011 Ckg-1) which is a relatively accurate result and reinforces the theory expectations,
however this lies just outside the calculated error range of the results. This is because of the
high precision in the results in comparison to the true accuracy. It is possible this has occurred
due to poorly calibrated apparatus. The largest contribution to the error was the digital
voltmeter display, which could read only to 3 significant figures resulting in an error of ±5V
on all voltage readings, which were then taken into account upon calculating the overall error
on the e/m ratio. Despite the data book value being outside the range of the calculated errors,
it is the product only of having minute error size, otherwise the results can be considered
precise and accurate.
The way the trend line does not intercept at 0, as the theory suggests it would have done,
could have been a cause for concern. The coherence of the results can only suggest this is
caused by a systematic fault, either with the way the data was recorded, or with the equipment
provided. There were initial problems with recording data caused by the spread of the electron
beam, which were later resolved by altering the way ‘nil deflection’ was defined, which may
have led to such a result.
More repeat sets of data could have been obtained should more time have been available,
although it is likely this would only decrease the already near-insignificant error on the e/m
ratio more than changing the calculated value itself. A more effective improvement on the
method would be to attempt the same experiment with different sets of similar equipment, to
detect any possible anomalies in any of the several devices used.
References
Physics for Scientists and Engineers - Serway Jewett, Brooks Cole, 2004
The Discovery of Subatomic Particles - Steven Weinberg, W.H. Freeman and Company, 1983
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