(picture of muon-g2 experiment at Fermilab)
Question 9:
A charge is brought towards a neutral molecule, which becomes polarized and becomes an induced dipole. The electric force on this induced dipole by the charge is proportional to which power of (1/r)? (r = distance between the centers)
Answer: 5
This example illustrates the force between an ion and an induced dipole. Other chemical bonds are the ion-ion, covalent and hydrogen bonds.
More complicated forces between molecules are the forces between dipole-dipole (or higher power multipoles) whether induced or permanent. These are generally grouped as Van der Waals forces.
In fact, the strength of these forces tends to vary with (1/r) to the power of 7 and they are important in explaining many physical phenomena, such as the increasing boiling points of the relatively inert noble gases as the atomic size increases.
Books for further reading:
1. Wayne Saslow, Electricity, Magnetism, and Light, Academic Press, 2002.
2. Michel Daune, Molecular Biophysics, Oxford University Press, 1999.
Question 8: When a stretched rubber band is heated by warm air from a hair-blower, does it expand, stay the same length or contract? Answer: Contracts. This behaviour is opposite that expected of a typical metal, which expands. Rubber is made of long polymer chains. When heat is given, the entropy S of rubber increases. This causes the polymer chains to curl up because there are many more of such curled configurations than straight chains. This tendency to curling of the polymers causes the rubber band to contract. The Gibbs energy for this curling: ∆𝐺 = ∆𝑈 + 𝑃∆𝑉 − 𝑇∆𝑆 ≈ −𝑇∆𝑆 < 0 The change in Gibbs energy is negative, and hence the curling is spontaneously driven, and in fact part of the Gibbs energy can be used to do work to lift a small weight tied to the lower end of the rubber band. You can get an idea of the entropic force component of the contraction by measuring the change in the force per unit temperature change (see Dill and Bromberg or Thomas Nordlund). Another way to understand this rubber band phenomenon is to model the polymer as a freely jointed chain, where you can get a version of Hooke's law after simplification of the Langevin's function (Howard). This example partly explains why proteins (made up of amino-acids) are folded. DNA helices too are supercoiled. Other forces that help drive this curling up of long molecules include hydrogen bonds and interactions with other molecules, some called "hydrophobic" forces. Results from lysosome denaturation experiments show that there is an entropic component (Daune). This can be said to be the thermodynamical "biological equivalent" of Hamilton's principle of least action in physics.
Books for further reading: 1. Joon Chang Lee, Thermal Physics, World Scientific, 2002. 2. Daniel V Schroeder, An Introduction To Thermal Physics, Addison Wesley, 2000. 3. Donald A McQuarrie and John D Simon, Molecular Thermodynamics, University Science Books, 1999. 4. Berg, Tymoczko, Stryer, Biochemistry, W H Freeman & Co, 2002. 5. Ken A Dill and Sarina Bromberg, Molecular Driving Forces, Statistical Thermodynamics In Chemistry and Biology, Garland Science, 2003. 6. Jonathon Howard, Mechanics of Motor Proteins and the Cytoskeleton, Sinauer Associates, 2001. 7. Michel Daune, Molecular Biophysics, Oxford University Press, 1999. 8. Meyer B Jackson, Molecular and Cellular Biophysics, Cambridge University Press, 2006. 9. Thomas M Nordlund, An Introduction to Biophysics, CRC Press, 2011.
10. David Goodstein, Thermal Physics, Energy and Entropy, Cambridge University Press, 2015.
Question 7: Can a stationary neutron be affected by a magnetic field? Answer: yes. We know that a moving charged particle can be affected by a magnetic field. But since the neutron has no electric charge and said to be stationary, it shouldn't be, should it? For a stationary particle to be affected by a magnetic field, it must have a magnetic dipole moment. An electron has an intrinsic spin which "produces" a magnetic moment and the value of this moment is approximately the Bohr magneton (which actually refers to the magnetic moment of an orbital electron the lowest orbital energy level in the Bohr's atomic model and is roughly 9.3 x 10⁻²⁴/T).
A proton (a fermion with spin 1/2) has a positive charge and a magnetic moment, which is 2.79 x nuclear magneton (based on the charge and mass of the proton). But strangely, a neutron also has a magnetic moment about 2/3 that of a proton. It was not entirely clear what was contributing to this moment if the neutron is electrically neutral, until Quantum Chromodynamics and the Standard Model were proposed. In the Standard Model, the neutron is made up of smaller components udd, 2 down quarks and 1 up quark with fractional electric charges. Another way of qualitatively explaining the magnetic moment of the neutron is through Yukawa's theory of meson (pion) exchange. A neutron can momentarily transform into a proton and a negative pion, before reverting to a neutron. Due to the virtual pion minus and the proton, there is a magnetic moment accounted to the neutron. Books for further reading: 1. Francis Bitter, Nuclear Physics, Addison-Wesley, 1950. 2. Povh, Rith, Scholz, Zetsche, Rodejohann, Particles and Nuclei, Springer, 2015. 3. Stephen Battersby (ed), Why The Universe Exists, How Particle Physics Unlocks The Secrets Of Everything, New Scientist, 2017. 4. Carlos A Bertulani, Nuclear Physics in a Nutshell, Princeton University Press, 2007.
Question 6:
Chinese lunar spacecraft Chang'e5 recently returned to earth orbit in Dec 2020 from the moon with samples of lunar rocks. The path of the spacecraft from the Earth to the Moon and back is essentially in a figure of 8, similar to some of the Apollo missions. Why is the flight path designed in such a way?
Answer:
The flight path is designed to conserve fuel as it aims to take a lower potential energy path.
For a typical lunar or planetary transfer trajectory, as when the spacecraft is launched from the almost circular earth orbit at perigee in a typical Hohmann transfer, it travels out initially in a highly eccentric elliptical orbit to reach the moon or planet (eg. Mars).
This lunar transfer can be designed to use the gravitational effect of the moon, to swing the craft round from the opposite side of the moon and head it back to the earth with little propulsion needed. This is called the circumlunar free-return orbit which is in the figure of 8.
Interestingly, due to the gravitational pulls of the earth and the moon, there are points along the line joining the earth and the moon, at which the gravitational forces from the earth and moon largely cancel, giving points with minimum potential energy.
For example, there is one such point called the Lagrange point L1 between the earth and the moon. This result was first derived by Euler and Lagrange in what is called the restricted three-body problem (ie. the system of 3 objects interacting via a "gravitational" force) under certain assumptions made of circular earth and moon orbits.
For another example where such low energy points were useful: Chang'e5 test spacecraft service module was sent to the earth-moon L2 lunar Lissajous orbit earlier on January 2015, where it could remain without much propulsion and saved on fuel, and used the remaining 800 kg of fuel to test maneuvers key to future lunar missions.
There are also low energy (or low ∆v, v for velocity) trajectories called manifold superhighways that, due to the changing interactive gravitational effects of the sun, earth, moon or other planetary bodies relative to the spacecraft, allow the craft to be captured by gravity of adjacent bodies to travel through space with minimal propulsion.
One such was successfully used by the first Japanese lunar probe called Hiten, launched in 1990. Due to the failure of its smaller orbital unit, this ballistic capture trajectory was conceived to allow the main Hiten probe instead to enter the lunar orbit, requiring only a perturbation sufficiently small to be achievable by the Hiten's thrusters. This low energy trajectory saved the Hiten's overall mission but required five months instead of the usual three days to bring it to the moon.
On 16 Nov 2022, NASA launched the Orion space capsule to the moon in the Artemis I space mission programme.
The Artemis Accords establish a framework for cooperation in the civil exploration and peaceful use of the Moon, Mars, and other astronomical objects and were signed by Australia, Canada, Italy, Japan, Luxembourg, the United Arab Emirates, the United Kingdom, the United States, Ukraine, South Korea, New Zealand, Brazil, Poland, Mexico, Israel, Romania, Bahrain, Singapore, Colombia, France and Saudi Arabia.
The Artemis I trajectory is the well-understood figure of 8:
(Picture taken from the New York Times)
Recently on 14 July 2023, India launched the Chandrayaan-3 mission spacecraft consisting of three modules — orbiter, lander, and rover which will take 14 days to reach the moon. The propulsion module is the one that will take the lander and the rover to the moon through a series of increasing velocity monoeuvres before a direct lunar transfer trajectory. This module does not land on the moon and instead settles on a parking orbit of 100 km x 100 km around the moon. The lander and rover, on the other hand, will separate from the propulsion module to land on the moon.
(image taken from Indian Space Research Organization)
Books for further reading:
1. Alexander J Hahn, Basic Calculus of Planetary Orbits and Interplanetary Flight, Springer, 2020.
2. Vernon D Barger, Martin G Olsson, Classical Mechanics A Modern Perspective, McGraw-Hill, 1995.
Question 5:
During the COVID-19 pandemic, many airports use thermal scanners to detect persons with fever. How do these scanners work?
Answer:
Interestingly, based on Wien's law, a person with a normal body surface temperature of 37 C (310 K), if assumed to be an emitter of blackbody thermal radiation, will emit radiation with the peak at 9.4 um wavelength (which is in the mid infrared region of the electromagnetic spectrum).
And partly based on Stefan's law, a person is thought to radiate thermal radiation at the net power of about 100W! (Refer to Fleisch's book mentioned below). Which goes to show how much cooling air-conditioners would need to work in a large crowded mall to keep the temperature low and stable.
Both these laws are used in estimating the surface temperatures of stars (see the book by Fleisch and Kregenow mentioned below for a simple and clear explanation and calculation. More detailed calculations can be found in other astrophysics texts).
(figure taken from Dan Maoz, vide infra)
A "contactless" forehead or ear thermometer collects and focuses this infrared radiation through a silicon lens (that is relatively transparent to infrared) on to a thermocouple detector, which converts the thermal energy to electrical potential. If a fixed area of infrared radiation flux is allowed to enter, and assuming the emitter is a blackbody, the radiation flux intensity should be proportional to the 4th power of the body temperature according to Stefan's law.
In which case in theory at least, it should be easy to measure the body temperature with good resolution, because the electric potential should in a simplistic way roughly be proportional to the 4th power of the body surface temperature.
However, in practice, this accuracy is dependent on various factors related to the emitting body, the transmission of radiation and the detector itself, such as the body surface properties which may not be constant for all people; so for the infrared thermometer, the electric potential is calibrated against a database of human body temperatures to provide a more accurate measurement. In a similar way, a thermal scanner detects such infrared radiation using pixel detectors.
There are some limitations of such infrared thermal radiation detectors.
a. they measure surface temperatures rather than the inner or core temperature.
b. the readings are highly affected by water vapour or liquid water, due to the fact infrared radiation is highly absorbed by water molecules.
Looking at the absorption spectrum of electromagnetic radiation by water, one can see that water is largely transparent to visible light but absorbs infrared radiation by several orders of magnitude. This absorption is largely due to the vibrational energy levels of the O-H bonds and explains why the microwave oven is effective for heating of foods.
The infrared radiation thermometer typically gives temperature readings much lower than the true skin temperature when the skin is moist than when it is dry.
Books for further reading:
1. Daniel Fleisch and Julia Kregenow, A Student's Guide to the Mathematics of Astronomy, Cambridge University Press, 2013.
2. Dan Maoz, Astrophysics in a Nutshell, Princeton University Press, 2016.
Question 4:
Perfume sprays or drug nebulizers for asthma treatment produce a fine mist of liquid droplets. How do they work?
Answer:
For the conventional atomizer or nebulizer, there are 2 parts to this operation.
Firstly, the atomizer or nebulizer uses a narrowing air nozzle (placed adjacent to a small collection of liquid) to produce a small low pressure air space just at the nozzle which draws in the adjacent liquid.
This is explained by the Bernoulli principle. This states that when any fluid mass velocity is increased, the fluid pressure drops (assuming that there is no gravitational potential component, and that the fluid is incompressible and inviscid). This is the fluid version of the principle of conservation of energy.
When the velocity is increased, the kinetic energy is increased. This extra kinetic energy is drawn from the work done to/by the fluid, which in this case is the due to the pressure difference across the region of fluid velocity change. The greater the velocity change, the greater the pressure difference.
So in the atomizer or nebulizer, by pushing the air rapidly through a narrowing nozzle, the air velocity is increased particularly at the nozzle tip, creating a minute low air pressure region which draws the liquid adjacent to the nozzle into, and to mix with, the air jet - producing a small amount of liquid-air mixture at the nozzle tip.
Secondly, as the liquid-air jet at the nozzle tip collides with the normal pressure air immediately further out, turbulence is generated which causes the liquid-air jet mixture to be dispersed into small droplets, forming a mist.
In fact, the Bernoulli principle explains why 2 hanging balloons will paradoxically "attract" each other when you blow a rapid gust of air through the small space between them.
For a nebulizer, apart from this conventional design, there is another based on the principle of agitation using high-frequency vibrations arising from piezoelectric crystals.
Books for further reading:
1. R Shankar, Fundamentals of Physics Volume 1, Yale University Press, 2014, chapter 20.
Question 3:
At what length of a simple pendulum will the period of swing (T) be 1 second?
Answer:
Based on the standard formula for the simple pendulum given in most textbooks,
the length would be approximately 25 cm. However, this formula is based on an assumption that the amplitude of swing is very small, probably less than 15 degrees from the vertical. A more accurate approximation (up to second order in amplitude), obtained using Jacobian elliptic function/complete elliptic integral of the first kind to derive, is:
where
is the maximum angle or amplitude.
The elliptic functions are very interesting and are related to the arc-length of the ellipse. They are also "oscillatory" like the trigonometric functions, but in general, they have 2 different periods in the real and imaginary axes respectively (unlike the singly-periodic trigonometric functions). These doubly-periodic functions can be visualised in the complex plane, such as a modular graph of the elliptic function sn(u), with x-axis as the real axis and y-axis as the imaginary axis, shown below:
(taken from Gino Moretti's book as mentioned below)
Books for further reading:
1. John L Synge and Byron A Griffith, Principles of Mechanics, McGraw-Hill, 1959, chapter 13.
2. L.D. Landau and E.M. Lifshitz, Mechanics, Course of Theoretical Physics Volume 1, Butterworth Heinemann, 1976, p 26.
3. E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, Cambridge University Press, 1927, chapters 20 and 22.
4. Gino Moretti, Functions of a Complex Variable, Prentice-Hall, 1964, chapter 9.
Question 2:
Why do the hurricanes, when viewed from above, in the Northern hemisphere swirl in an anti-clockwise direction (picture on the left) while those in the Southern hemisphere swirl in a clockwise direction (picture on the right)?
Answer:
This is due to the rotation of the earth around its north-south axis.
In view of the rotational (non-inertial) frame of reference, fictitious forces become apparent, in order to explain the motion based on Newton's second law.
In this case, when the winds in the Northern hemisphere move downwards in the direction from the north pole to the equator, but due to the curvature of the earth's surface slightly slanted to the axis of rotation of the earth, an observer moving with the rotation of the earth will see it as if it appears to be deflected by a fictitious force, called the Coriolis force, tending to push the wind to its right side.
This "force" generally appears whenever there is (1) a rotational frame of reference with angular velocity ω, and (2) a moving particle with velocity v within it. It is given by the twice mass times the vector product of ω and v, ie. 2mω ⨯ v.
For the Southern hemisphere, the velocity is opposite in direction and the Coriolis force therefore swings the winds in the opposite direction. For a mass particle moving at the equator, the axis of rotation of the earth and the wind velocity are parallel, and hence the vector product is zero, and thus no Coriolis force is evident.
Books for further reading:
1. David Morin, Introduction to Classical Mechanics, Cambridge University Press, 2008.
Satellite image of the hurricane Surigae (Bising) reaching eastern Philippines (right of picture) on 18 April 2021, showing the typical anti-clockwise spiral of the Northern Hemisphere.
Cyclone Yaas as seen about to make landfall on 26 May 2021 in the north-eastern Orissa state of India, showing the same anti-clockwise north-hemispheric swirl.
Question 1:
How does one determine the shapes of the atomic orbitals, such as are illustrated below?
Answer:
For the simplest hydrogenic atoms (ie. single electron atoms), these are obtained from solving the Schroedinger's equation for a spherically symmetric (central) potential 1/r. The wavefunctions ψ (or rather more accurately, the ⎪ψ⎪² ) represent the probability density of where the electron will be and the orbitals are the graphs of these wavefunctions that contain 95% of the probability that the electron is found inside, plotted in the polar co-ordinates.
Typically, these wavefunctions are the product of 2 parts - the radial part and the spherical harmonic part ie. ψ❨r,𝜃,𝜙❩=R❨r❩Y❨𝜃,𝜙❩. Solving for such ψ gives rise to 3 parameters for the hydrogenic atom, n, l and m, which are the principal quantum number, the orbital momentum quantum number and the magnetic quantum number, respectively.
One limitation of this hydrogenic atom model is the exclusion of spin-orbital coupling.
Another is the difficulty of extending this to multi-electrons atoms. A good approximation to multi-electrons atoms is the Hartree-Fock model which extends the Molecular Orbital theory to the many electrons case. It approximates each additional electron field as if it is arising from the centre, sums all energy linearly (assuming that the electrons behave independently) and iteratively averages the individual electron fields and calculates until the solution reaches stability. As electrons are fermions obeying the Pauli principle, the overall multi-electron wavefunction is expressed as an anti-symmetric determinant product of individual electrons wavefunctions called a Slater determinant.
An example of a 2-electrons Slater determinant is (where σ is the spatial wavefunction, α and β are the spin wavefunctions and the numbers refer to the 2 individual electrons):
Books for further reading:
1. Ira Levine, Quantum Chemistry, Prentice Hall, 2000, p 134.
2. David McIntyre, Quantum Mechanics, Pearson Addison Wesley, 2012, chapter 8.
3. Mark Fox, A Student's Guide to Atomic Physics, Cambridge University Press, 2018, chapter 2.
4. Steven Weinberg, Foundations of Modern Physics, Cambridge University Press, 2021, chapter 5.
5. Keith Hannabuss, An Introduction to Quantum Theory, Oxford University Press, 1997, chapter 4.
6. Frank L. Pilar, Elementary Quantum Chemistry, Dover Publications, 1990, chapter 5.
7. John P. Lowe and Kirk A. Peterson, Quantum Chemistry, Elsevier Academic Press, 2006, chapter 4.
8. H.Haken and H.C. Wolf, The Physics of Atoms and Quanta, Springer, 2004, chapter 19.
9. Thomas Engel, Quantum Chemistry & Spectroscopy, Pearson Prentice Hall, 2010, chapters 13 and 15.
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