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Teaching guide: Turning points in physics
This teaching guide aims to provide background material for teachers preparing students for the Turning Points in physics option of our A-level Physics specification (7408). It gives teachers more detail on specification topics they may not be familiar with and should be used alongside the specification. This guide is not designed to be used as a comprehensive set of teaching notes.
Contents Page
Introduction 2
Section 1 The discovery of the electron (specification reference 3.12.1) 3
a) Cathode rays 3
b) The specific charge of the electron 5
c) Millikan’s determination of the charge of the electron 9
Section 2 Wave particle duality (specification reference 3.12.2) 10
a) Theories of light 10
b) Matter waves 17
c) Electron microscopes 18
i) the transmission electron microscope 18 ii) the scanning tunneling microscope 20
Section 3 Special relativity (specification reference 3.12.3) 22
a) Frames of reference 23
b) The Michelson-Morley experiment 23
c) Time dilation and proper time 25
d) Length contraction and proper length 27
e) Evidence for time dilation and length contraction 28
f) Mass and energy 30
Appendix 34
A Suggested experiments and demonstrations 34
Introduction
Turning Points in physics is intended to enable key developments in physics to be studied in depth so that students can appreciate, from a historical viewpoint, the significance of major conceptual shifts in the subject, both in terms of the understanding of the subject and in terms of its experimental basis.
Each of the three sections within the course represents a different approach to the subject. The discovery of the electron and the determination of e/m and e took the subject forward through important experimental work by Thomson and Millikan. Further work on electron beams led to support for relativity and the discovery of electron diffraction. Recent 'Millikan' experiments have unsuccessfully sought evidence for fractional charge.
The section on wave particle duality follows the changes in the theory of light arising from experimental discoveries over two centuries from Newton to Einstein, and links up to the prediction and discovery of matter waves. Recent important developments in electron microscopy illustrate the unpredicted benefits of fundamental research.
Einstein's theory of special relativity overthrew the concept of absolute motion. The null result of the Michelson-Morley experiment presented a challenge to accepted ideas of space and time which was not solved until Einstein put forward his revolutionary theory of special relativity.
Each section offers opportunities to consider the relationship between experimental and theoretical physics. The nature of scientific proof can be developed in each section in line with the philosophy of science; no amount of experimentation can ever prove a theory fully, yet one experiment can be sufficient to overthrow it.
Some of the electrons pulled out of the gas atoms do not recombine and are attracted to the anode and therefore move away from the cathode – hence the term ‘cathode rays’ was used to describe them. These electrons move towards the anode and cause excitation by collision of gas atoms in the tube. The positive column of glowing gas is due to de-excitation of these excited gas atoms. The processes of recombination and de-excitation result in the emission of photons of visible and ultraviolet light.
Thermionic emission
Thermionic emission is a much simpler way of producing an electron beam than using a discharge tube. When a metal is heated, some of the electrons that move about freely inside the metal (referred to as ‘free’ or ‘conduction’ electrons) gain sufficient kinetic energy to leave the metal at its surface. In practice, the metal is a wire filament which is heated by passing an electric current through it. The filament or ‘cathode’ is at one end of an evacuated glass tube with a metal plate or ‘anode’ nearby, as shown in Figure 2.
Figure 2 Thermionic emission
The electrons emitted from the filament are attracted to the anode by connecting a high voltage power supply between the anode and the cathode, with the anode positive relative to the filament. Because there are no gas molecules in the tube to stop the electrons, the electrons are accelerated to the anode where some of them pass through a small hole to form a narrow beam.
The work done by the potential difference (pd) V between the anode and the cathode on each electron = e V , where e is the charge of the electron.
The kinetic energy of each electron passing through the hole = 12 𝑚𝑣^2 where v is
the speed of each electron at this position.
Since the work done on each electron increases its kinetic energy from a negligible value at the cathode, then the speed, v , of each electron leaving the anode is given by
1 2 𝑚𝑣
(^2) = 𝑒𝑉
For the above equation to apply, the speed of the electrons must be much less than the speed of light in free space, c. Students will not be expected in this section to use the relativistic expression for kinetic energy.
b) The specific charge of the electron
Students should know what is meant by the specific charge of the electron and should be able to understand the principles underlying its determination through the use of the relevant equations in the Data and Formulae Booklet, namely:
𝐹 =
𝑒𝑉 𝑑
𝐹 = 𝐵𝑒𝑣
𝑟 =
𝑚𝑣 𝐵𝑒 1 2 𝑚𝑣
(^2) = 𝑒𝑉
As outlined below, given a diagram and description of suitable apparatus and sufficient relevant data, they should be able to use one of more of the above equations to calculate the speed of the electrons in a beam and/or determine the specific charge of the electron from the given data. They should also be able to describe one method, including the data to be collected, to determine the specific charge of the electron. They should appreciate why electron tubes in general need to be evacuated and why the tube in Figure 3 needs to contain a gas at low pressure.
They should also appreciate that the specific charge of the electron was found by Thomson to be much larger than the previous largest specific charge, namely that of the hydrogen ion.
(i) Using a magnetic field to deflect the beam
(a) The radius of curvature r of the beam in a uniform magnetic field of flux density B may be measured using the arrangement shown in Figure 3 or using a ‘Teltron tube’ arrangement in which a straight beam enters the field and is deflected on a circular arc by a magnetic field.
Details of how to measure the radius of curvature in each case are not required.
arrangement of the fields is referred to as ‘crossed fields’. When the beam is straightened out (ie undeflected), the forces due to the crossed fields are balanced.
Figure 4 Balanced forces due to crossed E and B fields
As the magnetic force ( Bev ) on each electron is equal and opposite to the electric force ( eE) , the speed of the electrons passing through undeflected is given by the equation v = E/B from eE = Bev.
Note that E =
pd between the deflecting plates plate separation
The measured values of v, B and r can then be substituted into the equation
r = mv/Be and the value of e/m calculated.
(iii) Using an electric field to deflect the beam and a magnetic field to balance the deflection
Figure 5 Deflection by a uniform electric field
a) The deflection, y, of the beam at the edge of the plates is measured for a measured plate pd, VP.
The time taken by each electron to pass between the plates,
t =
plate length 𝐿 electron speed 𝑣
The measurement of the speed of the electrons is explained below.
The acceleration, a , of each electron towards the positive plate can then be
calculated from the measured deflection y using the equation y = 12 a t 2.
The value of e/m can then be determined as the acceleration, a , of each electron towards the positive plate, a = F/m = e V/md where V is the plate pd and d is the plate separation. Hence e/m = ad /V.
b) The speed, v, of the electrons is measured directly using a magnetic field B perpendicular to the beam and to the electric field to straighten the beam out. As explained before, the speed of the electrons passing through undeflected is given by the equation v = E/B (from eE = Bev where E = pd between the deflecting plates/plate separation). Hence the speed can be calculated if E and B are measured when the beam is undeflected.
(ii) A droplet falling vertically with no electric field present
Figure 7 An oil droplet falling at terminal speed
The droplet falls at constant speed because the drag force on it acts vertically upwards and is equal and opposite to its weight. Using Stokes’ Law for the drag force FD = 6π ηrv therefore gives 6π ηrv = mg.
Assuming the droplet is spherical, its volume = 4 r^3 /3 hence its mass m = its
density × its volume = 4 r^3 /3.
Hence 6π ηrv = 4 r^3 /3 which gives r^2 = ρg
η v 2
9 .
Thus the radius can be calculated if the values of the speed v , the oil density
and the viscosity of air are known.
Section 2 Wave particle duality
a) Theories of light
Students should be able to use Newton’s corpuscular theory to explain reflection and refraction in terms of the velocity or momentum components of the corpuscles parallel and perpendicular to the reflecting surface or the refractive boundary. They should be able to explain reflection and refraction using wave theory in outline. Proof of Snell’s law or the law of reflection is not expected. Newton’s ideas about refraction may be demonstrated by rolling a marble down and across an inclined board which has a horizontal boundary where the incline becomes steeper. They should know why Newton’s theory was preferred to Huygens’ theory.
They should also be able to describe and explain Young’s fringes using Huygens’ wave theory and recognise that interference cannot be explained using Newton’s theory of light as it predicts the formation of two fringes corresponding to the two slits. In addition, they should know that Huygens explained refraction by assuming that light travels slower in a transparent substance than in air, in contrast with Newton’s assumption that its speed is faster in a transparent substance. They should appreciate that Newton’s theory of light was only rejected in favour of wave theory long after Young’s discovery of interference when the speed of light in water was measured and found to be less than the speed in air
Measuring the speed of light.
Early scientists thought that light travelled at an infinite speed. However, the earliest reliable experiments involving astronomical observations (Römer, 1676) suggested that light had a finite speed. In 1849, Fizeau obtained a value that differed by only 5% from that now accepted using a terrestrial method. The arrangement used is shown in Figure 8.
Figure 8 Fizeau’s experiment for determining speed of light
The principle was to determine the time taken for a beam of light to travel to a mirror and back to the observer. The time was measured using a toothed wheel that was rotated at high speed. Pulses of light were transmitted through the gaps in the wheel. At low speeds of rotation, light from the source passed through a gap and then passed through the same gap on its return so the observer could see the light. As the speed of rotation increased there came a time when the returning beam found its path blocked by the adjacent tooth. As this was the same for all the gaps the observer did not now see any reflected beam. For example, no light would have been seen when light that passes through gap 0 finds its path blocked by tooth a on return, light passing through gap 1 would be blocked by tooth b and so on. When the speed was doubled, the light passing through 0 could pass though gap 1 on return so the reflected beam was once again observable.
theory demonstrates the power of a theory that predicts observable effects that
agree with experimental observations. Students should be aware that o is a
constant that relates the strength of an electric field in free space to the charge
that creates it and that o is a constant that relates the magnetic flux density of
a magnetic field to the current that creates it.
Hertz’s discovery of radio waves and their properties draw on previous knowledge of polarisation and stationary waves. Hertz showed that radio waves are produced when high voltage sparks jump across an air gap and he showed they could be detected using:
either a wire loop with a small gap in it,
or a ‘dipole’ detector consisting of two metal rods aligned with each other at the centre of curvature of a concave reflector.
Students should be aware that Hertz also found that the radio waves are reflected by a metal sheet and discovered that a concave metal sheet placed behind the transmitter made the detector sparks stronger. Hertz also produced stationary radio waves by using a flat metal sheet to reflect the waves back towards the transmitter. Students should be able to explain why stationary waves are produced in this way and should be able to calculate the speed of the radio waves from their frequency and the distance between adjacent nodes. Hertz also discovered that insulators do not stop radio waves and he showed that the radio waves he produced are polarised. Students should be able to explain why the detector signal changes in strength when the detector is rotated about the line between the transmitter and the detector in a plane perpendicular to this line.
The ultraviolet catastrophe
The ultraviolet catastrophe refers to the disagreement between practical measurements of the energy intensity at different wavelengths from a black body at a given temperature and the theoretical predictions using classical physics.
A black body is one that emits all wavelengths of radiation that are possible for that temperature. The dotted lines in the graph in Figure 9 show the practical variation of energy intensity E emitted at different wavelengths with wavelength
for a particular temperature.
Figure 9 Black body radiation curve
The solid line shows the prediction made using classical physics. This predicts that most of the energy would be emitted at short (ultraviolet) wavelengths and would become infinite at very short wavelengths. Measurements indicate however that there is a peak as shown by the solid line. This peak occurs at a shorter wavelength as the temperature increases (Wein’s Law).
The problem was resolved when Planck suggested that:
radiation is emitted in quanta (which we now call photons) the quantum is related to a single frequency an energy quantum is hf so high frequency radiation is emitted in larger ‘chunks’ of energy.
Using the idea of energy ‘quanta’ as packets of energy Planck was able to develop a theory to explain the shape of the observed spectrum. The proposition by Planck also laid the foundations to solve other areas of physics where classical physics was failing to explain what was being observed.
Photoelectricity was one such phenomenon and the photon theory became firmly established when Einstein’s used the photon theory of light to explain the observations.
Photoelectricity
Note that a study of the photoelectric effect is in the Physics core (section 3.2.2.1) so this knowledge will be assumed.
It is included here to complete the study on the nature of light.
Just as students are expected to be aware of the main differences between wave theory and corpuscular theory, students should know why wave theory was rejected in favour of the photon theory and they should be aware of the
emission is stopped because the maximum kinetic energy has been reduced to zero and they should be able to recall and explain:
why the stopping potential is given by eV S = hf -
why the graph of VS against f is a straight line with a gradient and intercepts as shown in Figure 10.
Figure 10 Stopping potential v frequency
They should appreciate how the stopping potential may be measured using a potential divider and a photocell and how the measurements may be plotted to
enable the value of h and the value of to be determined.
The first measurements were obtained by RA Millikan and gave results and a graph as shown in Figure 10 above, that confirmed the correctness of Einstein’s explanation and thus confirmed Einstein’s photon theory of light. Einstein thus showed that light consists of photons which are wavepackets of electromagnetic radiation, each carrying energy hf , where f is the frequency of the radiation. Einstein was awarded the 1921 Nobel Prize for physics for the photon theory of light which he put forward in 1905, although it was not confirmed experimentally until 10 years later.
Students should know that the photon is the least quantity or ‘quantum’ of electromagnetic radiation and may be considered as a massless particle. It has a dual ‘wave particle’ nature in that its particle-like nature is observed in the photoelectric effect and its wave-like nature is observed in diffraction and interference experiments such as Young’s double slits experiment.
b) Matter waves
Students should know from their AS course that de Broglie put forward the hypothesis that all matter particles have a wave-like nature as well as a particle- like nature and that the particle momentum mv is linked to its wavelength by the equation:
m v × = h where h is the Planck constant.
De Broglie arrived at this equation after successfully explaining one of the laws of thermal radiation by using the idea of photons as ‘atoms of light’. Although photons are massless, in his explanation he supposed a photon of energy hf to have an equivalent mass m given by mc^2 = hf and therefore a momentum
mc = hf/c = h/ where is its wavelength.
De Broglie’s theory of matter waves and equation ‘momentum × wavelength = h ’ remained a hypothesis for several years until the experimental discovery that electrons in a beam were diffracted when they pass through a very thin metal foil. Figure 11 shows an arrangement.
Figure 11 Diffraction of electrons
Photographs of the diffraction pattern showed concentric rings, similar to those obtained using X-rays. Since X-ray diffraction was already a well-established experimental technique for investigating crystal structures, it was realised that similar observations with electrons instead of X-rays meant that electrons can also be diffracted and therefore they have a wave-like nature. So de Broglie’s hypothesis was thus confirmed by experiment. Matter particles do have a wave- like nature.
The correctness of de Broglie’s equation was also confirmed as the angles of diffraction were observed to increase (or decrease) when the speed of the electrons was decreased (or increased).
Figure 12 Transmission electron microscope
Figure 12 shows an outline of the transmission electron microscope in which the beam of electrons passes through the sample to form an image on a fluorescent screen. The electrons are produced by thermionic emission from a filament wire and are accelerated through a pd of between 50 to 100 kV. The system of magnetic lenses in the latest TEMs is capable of l07 times magnification at most, enabling images no smaller than about 0.1 nm to be seen.
Students should know the effect of the magnetic lenses on the electrons transmitted through the object as shown in Figure 12. Electrons passing near the gap in the soft iron shield are deflected towards the axis of the microscope; electrons passing through the centre are undeflected. Hence a magnetic lens can be compared with an optical convex lens which deflects light rays passing near the edge towards the centre and allows light rays through the centre without change of direction. The condenser lens deflects the electrons into a wide parallel beam incident uniformly on the sample. The objective lens then forms an image of the sample. The projector lens then casts a second image onto the fluorescent screen.
They should know that the amount of detail in the image is determined by the resolving power which increases as the wavelength of the electrons decreases. They should also know that the wavelength becomes smaller at higher electron speeds so that raising the anode potential in the microscope gives a more detailed image. In addition, they should know that the amount of detail possible is limited by lens aberrations (because the lenses are unable to focus electrons from each point on the sample to a point on the screen since some electrons are moving slightly faster than others) and sample thickness (because the passage of electrons through the sample causes a slight loss of speed of the electrons which means that their wavelength is slightly increased, thus reducing the detail of the image).
For a given anode potential, as outlined earlier, students should be able to calculate the de Broglie wavelength of the electrons using the de Broglie
equation ( 2 )
m e V
h ^ λ ; for example, the electrons in the beam of a TEM
operating at 80 kV would have a de Broglie wavelength of about 0.004 nm. In addition, students should appreciate that:
in theory, electrons of such a small wavelength ought to be able to resolve atoms less than 0.1 nm in diameter
in practice in most TEMs, electrons of such a small wavelength do not resolve such small objects for the reasons outlined above.
Note, the TEAM 0.5 electron microscope at the US Lawrence Berkeley Laboratory is the most powerful electron microscope in the world. Aberration correctors developed at the laboratory are fitted in the 80 kV microscope, enabling individual atoms to be seen.
ii) The scanning tunneling microscope
The scanning tunneling microscope (STM), invented in 1981, gives images of individual rows of atoms. Students should know that the STM is based on a fine- tipped probe that scans across a small area of a surface and that the probe's scanning movement is controlled to within 0.001 nm by piezoelectric transducers. They should be aware that the probe is at a small constant potential, with the tip held at a fixed height of no more than 1 nm above the surface so that electrons 'tunnel' across the gap. They should also know that if the probe tip moves near a raised atom or across a dip in the surface, the tunneling current increases or decreases respectively due to a respective decrease or increase of the gap width. They should also know that:
in constant height mode, the change of current is used to generate an image of the surface provided the probe’s vertical position is unchanged
