F
UNDAMENTAL
C
ONSTANTS
7.1 I
Over the next two days you will investigate various properties of electrons. The first day will be investigating the Charge to Mass ratio of the electron and the second day will look at the
Photoelectric Effect . As a result of these two experiments you will then be in a position to present your own experimental values for some important Fundamental Constants .
7.2 C
M
R
E
7.2.1 O VERVIEW
Faraday in 1833 through a careful series of electrolysis experiments demonstrated the quantisation of charge. Thomson ended a heated debate among physicists when he was able to identify the cathode rays seen in low pressure gas discharges as caused by negatively charged particles. These particles, electrons , were accepted as being fundamental constituents of matter, firmly banishing the notion that the atom was indivisible and fundamental. Thomson was able to measure the charge to mass ratio of this newly baptised particle with an experiment that made use of the same principles as the one you are about to perform. The value of e / m that Thomson determined was 1.0

10 11 C/kg. This is to be compared with the currently accepted value of (1.7588028

0.0000054)

10 11 C/kg. Since the properties of the electron depend almost solely on these two quantities an accurate determination of their value is sought. It would be of interest to compare the value of e / m that you calculate with Thomson’s first effort!
7.2.2 T HEORY
If a beam of electrons is accelerated through a known potential then their kinetic energy is given by the expression
E k
 mv
2
2
 eV
Eq. 7.1 where m is the mass of the electron, e is the charge of the electron and V is the potential.
If a magnetic field is applied perpendicular to the velocity of the electrons, they will then experience a Lorentz force mutually perpendicular to both the field, B , and the electron direction of motion. The magnitude of this force is
F
 evB Eq. 7.2
Since the electrons always experience a force perpendicular to their motion they move in a circle and thus experience a centripetal force with a magnitude given by
2
F
 mv r
Eq. 7.3 where r is the radius of the circle described by the el ectrons’ path.
Pre-lab Question (a) What path would the electrons follow if their velocity were not totally perpendicular to the magnetic field (say at 45 o )?
Pre-lab Question (b) Using the equations above determine a formula for the ratio e/m in terms of the applied field strength, B , the radius of the circle, r , and the accelerating potential, V.
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7.2.3 A PPARATUS
The apparatus are displayed schematically below.
Figure 7.1 The apparatus used to determine e/m. a) Helmholtz Coils
The magnetic field for this experiment is generated by a pair of Helmholtz coils with the magnitude of the field generated at a volume in their centre given by
/ 2
B



4
5


3
N
 a
0
I Eq. 7.4 where N is the number of turns in each coil,

0
= 4

10 -7 Tm/A is the permeability of free space, I is the current through the coils and a is the radius of the coils. In this case the radius of the coils is 150 mm and each coil has 130 turns. b) The Cathode Ray Tube
The cathode ray tube is a 130 mm diameter helium filled glass bulb. This contains at its base the electron gun consisting of a heated cathode to produce the electrons that are then accelerated towards the anode. The anode is made of a wire grill so that the electrons may pass through it and enter the body of the tube. The helium gas is retained at a pressure of
10 -2 Torr and is ionised by the electrons as they pass through it. As the helium ions recombine, they emit light, allowing the radius of the electron beam to be determined.
Pre-lab Question (c) What happens to the kinetic energy of the electrons as they ionise the helium? How will this affect the quality of your results? c) Mirrored Scale
A mirrored scale is attached to the rear coil and will be illuminated when power is provided to the apparatus. This may be used to align the beam and its image and so eliminate errors due to parallax.
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ONSTANTS
d) Cathode Ray Tube Power Supply
A power supply is provided for the cathode ray tube. This provides 6V AC to heat the cathode and a variable DC voltage to accelerate the electrons. e) Helmholtz Coil Supply
Current to the Helmholtz coils is supplied by a current regulating power supply capable of providing 2A at small voltages.
Pre-lab Question (d) Why must both of these power supplies be kept as far from the cathode ray tube as possible?
7.2.4 E XPERIMENT

Ensure that both the power supplies are connected as shown in the diagram.
Before switching the supplies on check that the high voltage control and the current controls (supply and apparatus) are set to be minimum.

Switch on the supplies and allow the heater to warm up. Increase the accelerating potential until the electron beam can be seen. Turn the current control on the apparatus to maximum and adjust the current in the Helmholtz coils until the beam traverses a circle. If the beam travels in a spiral consult your demonstrator.

Adjust the accelerating potential to 300V and adjust the Helmholtz coil current until the radius of the beam is 4.0cm. Note the current, voltage and left and right radii.
Question (a) Why are the two radii measurements different? How will this affect the way you measure the radius?

Repeat this procedure at 5V intervals until the beam no longer traverses a circle
(around 230V the beam will collapse). Try as much as is practicable to adjust the current to keep the radius constant at 4.0 cm.
Detail the methods you employ to increase the accuracy of your results.
7.2.5 A NALYSIS
Enter your data into the computer and using SigmaPlot calculate the value of e/m for each voltage. Plot your results against potential.
Question (b) How can you explain the shape of this graph?
A better method for determining the value of e/m is to calculate the field variable. The field variable,

, evolves naturally from an attempt to accommodate the energy loss of the electrons due to their interactions with the gas. We may fit the function
V
 e m
   Eq. 7.5 where e/m and

are parameters and V and

= [ (Br ) 2 /2]

10 11 are variables. This allows a unique value of e/m to be determined while allowing for the effective potential loss due to the presence of the gas with the parameter

. Plot

against V on another graph and overlay the fit. Hence determine a value for e/m and compare it to the accepted value.
Question (c) Given the ionisation energy of He, how many e-He collisions occur during the electron path through the tube?
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7.3 T
P
E
7.3.1 O VERVIEW
At the middle of the nineteenth century the wave theory of light was yet unchallenged.
However, in 1887, Hertz noticed that a spark induced in his circuit was stronger when he deliberately illuminated the detector with UV light. Although his study concentrated on the properties of electromagnetic radiation, his was the first observation of the photoelectric effect . The liberation of electrons from the clean surfaces of metals under the action of radiant energy is known as the photoelectric effect. His collegue, Lenard, later confirmed that the sign of the emitted carriers was negative. However, it wasn’t until Einstein’s revolutionary postulate in 1905, that all the observed features of this effect were reconciled with mainstream theory.
7.3.2 T HE F AILURE OF C LASSICAL T HEORY AND E INSTEIN
’
S P OSTULATE
Experimentalists found that electrons ejected from the surface in this way had small, but finite speeds ranging from zero to some maximum value. By making the collecting plate negative with respect to the illuminated plate, they could measure the force required to stop the most energetic of electrons. This is how the stopping potential, V
0
, is defined.
Pre-lab Question (e) Cite instances where classical theory failed to account for the observed features of this phenomenon. You may wish to consult the suggested references described at the end of these notes.
Einstein extended the quantum theory of Planck to the radiation field itself, hypothesising that the light existed in quanta of energy with the magnitude h

.
Thus the mechanism of the photoelectric effect became as follows: an electron absorbs a photon of energy h

and attains enough energy to escape the surface. Energy not used to overcome the binding of the electron to the atom becomes kinetic energy. Thus the following obtains,
E k
 eV
0
 h
 
W
0
Eq. 7.6
Where E k is the kinetic energy of the photoelectrons and W
0 is the work function particular to the material that is illuminated. The work function is the energy required for the electron to escape the surface.
7.3.3 A PPARATUS a) The Mercury Vapour Lamp
The spectral lamp you will use is a high intensity (100 W) mercury vapour lamp. The mercury is in the form of vapour. Atoms are ionised by passing a current through the vapour. The recombination of electrons and mercury ions produces a light spectrum composed of discrete
wavelengths. The emitted frequencies are displayed in Table 7.1.
Table 7.1 Frequencies for the light spectrum issued from the Mercury lamp
C OLOUR F REQUENCY (10 14 Hz)
Yellow
Green
Blue
Violet
Deep Violet
5.19
5.49
6.88
7.41
8.22
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ONSTANTS
b) The Diffraction Grating
The light from the lamp will be dispersed using a transmission diffraction grating and focussed with a lens. The grating equation is a sin
 m
= m

, where a is the distance between the ruled grooves, m is the order of interference and
 m is the angle through which the order is deflected. A diagram illustrates this below. Note that each order is resolved in space, as is each spectral line.
Figure 7.2 The transmission diffraction grating with the orders of interference shown.
c) The Photoelectric cell
You are supplied with a photoelectric cell for measuring the stopping potential. This consists of a photodiode in which a photocurrent is induced, and electronics for measuring the stopping potential. The photoelectrons generated at the cathode are collected at the anode.
This photocurrent charges a capacitance associated with the photodiode and the electronics.
When the potential on this capacitance reaches the stopping potential no more current flows.
The voltage between the anode and cathode stabilises. The stopping potential is then recorded as the voltage across the anode and cathode. An ultrahigh input impedance (>10 12

) unitary gain amplifier allows the measurement of this voltage with negligible charge leakage. d) Filters
Three Filters are required for this experiment:
 a variable transmission filter that allows accurate adjustment of the light intensity between 20% and 100%
 a yellow filter to prevent ambient light sources (UV from overhead flourescent lights and also the violet line from the 3 rd order spectrum) from interfering with measurements of the stopping potential for the yellow line
 a green filter for use with the green line for the reasons cited above.
All of these filters have magnetic strips for easy attachment to the photocell.
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7.3.4 E XPERIMENT

A representation of the apparatus is given in Figure 7.3 with the features you
will need to be familiar with identified.
Figure 7.3 The apparatus for the observation of the Photoelectric effect.

Allow the lamp to warm up for five minutes. This should give you the opportunity to identify the orders of interference produced by the transmission grating. Focus a spectral line of the 1 st order onto the white reflective mask of the photocell by moving the lens and grating back and forth. Fix the lens in place when you have obtained a sharp image.

Roll the light shield of the photocell (black tube) back and rotate the cell until the line is centred on the aperture of the photodiode. Tighten the locking screws on the photocell and grating.

Place the transmission filter on the front of the photocell to allow 100% of the light to enter the photodiode. If you have chosen the yellow or the green spectral lines remember to use the appropriate filters. Connect the voltmeter to the output of the photocell and switch the photocell on. Zero the photocell by pushing the red discharge button. Record the stopping potential after a minute or so.

Zero the photocell again and record the time taken to reach the maximum potential. Repeat this procedure for the 80% transmission and the other possible percentages of the original intensity.

Re-adjust the apparatus for another spectral line and repeat the above procedure.

Explain what you observe in terms of the operation of the photocell.
7.3.5 V ARIATION OF S TOPPING P OTENTIAL WITH F REQUENCY
Remove the transmission filter. Adjust the apparatus for all of the spectral lines you can observe (both 1 st and 2 nd order lines on both sides) and record the stopping potential for each line. Remember to use the appropriate filters. Consider also the affects of other light sources, for instance, overhead lighting.
Question (d) What can you do to minimise these affects?
Average your results in SigmaPlot. Fit your data with Einstein’s equation. Determine h/e and
W
0 from the parameters of this fit. Compare your result with the expected value of h/e= (4.135708

0.000014)

10 -15 Js/C.
Question (e) Do your results confirm quantal mechanics or classical mechanics?
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ONSTANTS
7.3.6 C ALCULATION OF THE F UNDAMENTAL C ONSTANTS
The third fundamental constant is introduced in the electron spin resonance experiment and is the Bohr magneton (
 b
= e  / 2 m e
). The value of this constant has been measured as
(9.274096

0.000065)

10 -24 J/T. You can now use your results from this ( h/e ) and the previous practical exercise ( e/m ) and the result for the Bohr magneton cited now to calculate the values of h , m e , and e . You should determine errors for each of these quantities so that you can effect a meaningful comparison with the accepted values of these constants.
7.4 U
D







 h =6.6262

10 -34 Js e =1.6022

10 -19 C m e
=9.1096

10 -34 kg e/m =(1.7588028

0.0000054)

10 11 C/kg

0.000014)

10 -15 Js/C h/e=


0 b
(4.135708
= 4

10 -7 Tm/A
=(9.274096

0.000065)

10 -24 J/T
Ionisation energy of He = 24.58eV
7.5 R
Modern Physics by Serway, Moses and Moyer. Sections 2-5.
Fundamentals of Physics by Haliday and Resnick. Section 43-3
61