Determination of the Charge to Mass Ratio of the Electron

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Richard Plews

1




















D
etermination 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±0.03
x10
11
Ckg
-
1
. The
various d
isagreements between the theoretical and experimental results do not exceed
± 2%

and is

attributed
largely
to

voltmeter precision
.
Richard Plews

2

Introduction


There were a number of results gathered over the years by cathode ray tube researchers, J. J.
Thomson was a pio
neer 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 particle
s. The force due to an
electric field is given by


F
E

= 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


F
B

= Bev

(2)

Where B is the magnetic flux density and v is the vel
ocity of the charged particle.













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


E
e

= eV
a

(4)

Where V
a

is the voltage provided to the gun. Assuming all energy provid
ed is converted to
Kinetic energy it can be said that


(5)


Rearranging (5) and substituting (4) into this gives


(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 coi
ls 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 othe
r particles. The cathode ray gun emits electrons from a filament when a
voltage is applied.

Magnetic
field

Charge

Force

Fig 2: Force due to magnetic field

Charge

Electric Field

Force

Fig 1: Force due to elec
tric field

Richard Plews

3


The apparatus was set up as shown in figure 4. The Electron gun voltage, V
a,

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 th
en 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 elec
tric 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).

Fig. 4: Apparatus diagram

Power

supply


Power

supply


Power

supply


Parallel E
-
field
plates

2x Parallel aligned
Helmholtz coils

Cathode ray
tube

+

1/1000

Divider

Fig. 3: Thomson tube components

e
-

Evacuated glass
envelope

Helmholtz coils

Deflection plates

Cathode ray
tube


Richard Plews

4

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 th
e 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 be
en 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.




This gave a final value of
1.724x10
11
Ckg
-
1
. The error was then calculated with error
propagation methods to give an error of 2.64x10
9
Ckg
-
1
.






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

Richard Plews

5

Conclusion


The experiment
al result lies within 2% of the

actual (data book) value of the e/m
ratio

(
1.79x10
11

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 beca
use 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 significan
t 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
,

coul
d have been a cause for concern
. The coherence of the results can only s
uggest 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.

M
ore repeat sets of data could have been obtained should mor
e time have been available,
a
lthough
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