by Ciro Santilli (@cirosantilli, 34)
Physics (like all well done science) is the art of predicting the future by modelling the world with mathematics.
And predicting the future is the first step towards controlling it, i.e.: engineering.
Ciro Santilli doesn't know physics. He writes about it partly to start playing with some scientific content for: OurBigBook.com, partly because this stuff is just amazingly beautiful.
Ciro's main intellectual physics fetishes are to learn quantum electrodynamics (understanding the point of Lie groups being a subpart of that) and condensed matter physics.
Every science is Physics in disguise, but the number of objects in the real world is so large that we can't solve the real equations in practice.
Luckily, due to emergence, we can use uglier higher level approximations of the world to solve many problems, with the complex limits of applicability of those approximations.
Therefore, such higher level approximations are highly specialized, and given different names such as:
As of 2019, all known physics can be described by two theories:
Unifying those two into the theory of everything one of the major goals of modern physics.
Figure 1.
xkcd 435: Fields arranged by purity
. Source. Reductionism comes to mind.
Figure 2.
Physically accurate genie by Psychomic
. Source. This sane square composition from: www.reddit.com/r/funny/comments/u08dw3/nice_guy_genie/.
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The approach many courses take to physics, specially "modern Physics" is really bad, this is how it should be taught:
This is likely because at some point, experiments get more and more complicated, and so people are tempted to say "this is the truth" instead of "this is why we think this is the truth", which is much harder.
But we can't be lazy, there is no replacement to the why.
Related:
This is the only way to truly understand and appreciate the subject.
Understanding the experiments gets intimately entangled with basically learning the history of physics, which is extremely beneficial as also highlighted by Ron Maimon, related: there is value in tutorials written by early pioneers of the field.
"How we know" is a basically more fundamental point than "what we know" in the natural sciences.
In the Surely You're Joking, Mr. Feynman chapter O Americano, Outra Vez! Richard Feynman describes his experience teaching in Brazil in the early 1950s, and how everything was memorized, without any explanation of the experiments or that the theory has some relationship to the real world!
Although things have improved considerably since in Brazil, Ciro still feels that some areas of physics are still taught without enough experiments described upfront. Notably, ironically, quantum field theory, which is where Feynman himself worked.
Feynman gave huge importance to understanding and explaining experiments, as can also be seen on Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979).
Video 1.
'Making' - the best way of learning science and technology by Manish Jain (2018)
Source.
Everyone is beginner when the field is new, and there is value in tutorials written by beginners.
For example, Ciro Santilli felt it shocking how direct and satisfying Richard Feynman's scientific vulgarization of quantum electrodynamics were, e.g. at: Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979), and that if he had just assumed minimal knowledge of mathematics, he was about to give a full satisfactory picture in just a few hours.
Other supporters of this:
In Physics, in order to test a theory, you must be able to extract a number from it.
It does not matter how, if it is exact, or numerical, or a message from God: a number has to come out of the formulas in the end, and you have to compare it with the experimental data.
Many theoretical physicists seem to forget this in their lectures, see also: Section "How to teach and learn physics".
You don't need to understand the from first principles derivation of every single phenomena.
And most important of all: you should not start learning phenomena by reading the from first principles derivation.
Instead, you should see what happens in experiments, and how matches some known formula (which hopefully has been derived from first principles).
Only open the boxes (understand from first principles derivation) if the need is felt!
E.g.:
  • you don't need to understand everything about why SQUID devices have their specific I-V curve curve. You have to first of all learn what the I-V curve would be in an experiment!
  • you don't need to understand the fine details of how cavity magnetrons work. What you need to understand first is what kind of microwave you get from what kind of input (DC current), and how that compares to other sources of microwaves
  • lasers: same
Physics is all about predicting the future. If you can predict the future with an end result, that's already predicting the future, and valid.
Videos should be found/made for all of those: videos of all key physics experiments
This shows that viewing electromagnetism as gauge theory does have experimentally observable consequences. TODO understand what that means.
In more understandable terms, it shows that the magnetic vector potential matters where the magnetic field is 0.
Video 1.
The Quantum Experiment that ALMOST broke Locality by The Science Asylum (2019)
Source.
Classic theory predicts that the output frequency must be the same as the input one since the electromagnetic wave makes the electron vibrate with same frequency as itself, which then irradiates further waves.
But the output waves are longer because photons are discrete and energy is proportional to frequency:
The formula is exactly that of two relativistic billiard balls colliding.
Therefore this is evidence that photons exist and have momentum.
Video 1.
Compton Scattering by Compton Scattering (2017)
Source. Experiment with a caesium-137 source.
Video 2.
L3.3 Compton Scattering by Barton Zwiebach (2017)
Source.
No matter how hight the wave intensity, if it the frequency is small, no photons are removed from the material.
This is different from classic waves where energy is proportional to intensity, and coherent with the existence of photons and the Planck-Einstein relation.
Video 1.
Photoelectric effect by UCSB Physics Lecture Demonstrations (2021)
Source.
The key thing in a good system of units is to define units in a way that depends only on physical properties of nature.
Ideally (or basically necessarily?) the starting point generally has to be discrete phenomena, e.g.
What we don't want is to have macroscopic measurement artifacts, (or even worse, the size of body parts! Inset dick joke) as you can always make a bar slightly more or less wide. And even metals evaporate over time! Though the mad people of the Avogadro project still attempted otherwise well into the 2010s!
Standards of measure that don't depend on artifacts are known as intrinsic standards.
A dimension in a system of units is something like length, weight or time, without considering how to assign numerical values ot them, which requires units of measurement such as the meter, kilogram or second.
Talking about dimensions can be useful when explaining new derived units without worrying about the exact units involved. See e.g. this table: en.wikipedia.org/w/index.php?title=Lumen_(unit)&oldid=1233810964#SI_photometric_units
The key is to define only the minimum number of measures: if you define more definitions become over constrained and could be inconsistent.
Learning the modern SI is also a great way to learn some interesting Physics.
Video 1. Source.
Great overview of the earlier history of unit standardization.
Gives particular emphasis to the invention of gauge blocks.
web.archive.org/web/20181119214326/https://www.bipm.org/utils/common/pdf/CGPM-2018/26th-CGPM-Resolutions.pdf gives it in raw:
  • the unperturbed ground state hyperfine transition frequency of the caesium-133 atom is 9 192 631 770 Hz
  • the speed of light in vacuum c is 299 792 458 m/s
  • the Planck constant h is 6.626 070 15 × J s
  • the elementary charge e is 1.602 176 634 × C
  • the Boltzmann constant k is 1.380 649 × J/K
  • the Avogadro constant NA is 6.022 140 76 × mol
  • the luminous efficacy of monochromatic radiation of frequency 540 × 1012 Hz, Kcd, is 683 lm/W,
The breakdown is:
  • actually use some physical constant:
    • the unperturbed ground state hyperfine transition frequency of the caesium-133 atom is 9 192 631 770 Hz
      Defines the second in terms of caesium-133 experiments. The beauty of this definition is that we only have to count an integer number of discrete events, which is what allows us to make things precise.
    • the speed of light in vacuum c is 299 792 458 m/s
      Defines the meter in terms of speed of light experiments. We already had the second from the previous definition.
    • the Planck constant h is 6.626 070 15 × J s
      Defines the kilogram in terms of the Planck constant.
    • the elementary charge e is 1.602 176 634 × C
      Defines the Coulomb in terms of the electron charge.
  • arbitrary definitions based on the above just to match historical values as well as possible:
    • the Boltzmann constant k is 1.380 649 × J/K
      Arbitrarily defines temperature from previously defined energy (J) to match historical values.
    • the Avogadro constant NA is 6.022 140 76 × mol
      Arbitrarily defines the mol to match historical values. In particular, the kilogram is not an exact multiple of the weight of an atom of hydrogen.
    • the luminous efficacy of monochromatic radiation of frequency 540 × 1012 Hz, Kcd, is 683 lm/W
      Arbitrarily defines the Candela in terms of previous values to match historical records. The most useless unit comes last as you'd expect.
Made possible by the Kibble balance.
Starting in the 2019 redefinition of the SI base units, the elementary charge is assigned a fixed number, and the Ampere is based on it and on the second, which is beautiful.
This choice is not because we attempt to count individual electrons going through a wire, as it would be far too many to count!
Rather, it is because because there are two crazy quantum mechanical effects that give us macroscopic measures that are directly related to the electron charge. www.nist.gov/si-redefinition/ampere/ampere-quantum-metrology-triangle by the NIST explains that the two effects are:
Those effect work because they also involve dividing by the Planck constant, the fundamental constant of quantum mechanics, which is also tiny, and thus brings values into a much more measurable order of size.
Unit of mass.
Defined in the 2019 redefinition of the SI base units via the Planck constant. This was possible due to the development of the Kibble balance.
Whichever problem you present a German, they will look for a mechanical solution to it!
Figure 1. Source.
The Kibble balance is so precise and reproducible that it was responsible for the 2019 redefinition of the Kilogram.
Figure 1.
NIST-4 Kibble balance
. Source.
It relies rely on not one, but three macroscopic quantum mechanical effects:
How cool is that! As usual, the advantage of those effects is that they are discrete, and have very fixed values that don't depend either:
  • on the physical dimensions of any apparatus (otherwise fabrication precision would be an issue)
  • small variations of temperature, magnetic field and so on
One downside of using some quantum mechanical effects is that you have to cool everything down to 5K. But that's OK, we've got liquid helium!
The operating principle is something along:
Then, based on all this, you can determine how much the object weights.
Video 1.
How We're Redefining the kg by Veritasium
. Source.
Video 2.
The Kibble Balance, realizing the Kilogram from fundamental constants of nature by Richard Green
. Source. Presented in 2022 for a CENAM seminar, the Mexican metrology institute. The speaker is from the Canadian metrology institute
Video 3.
The Watt balance and redefining the kilogram by National Physical Laboratory
. Source. Nothing much, but fun to hear Kibble talking about his balance in beautiful English before he passed.
Candela is Lumen density per solid angle. A sphere emitting 1 Candela uniformly in all directions produces 4π Lumen total power.
1 Watt equals 683 Lumens of light power at wavelength 555 nm. At other wavelengths 1 Watt is less Lumens as it takes into account the sensitivity of the average human eye.
Candela is lumen density per solid angle.
Named after radio pioneer Heinrich Hertz.
Mega-Hertz, i.e. a million Hertz.
Video 1.
How a quartz watch works by Steve Mould (2017)
Source. Mentions feedback loop loop with the quartz tuning fork for the piezoelectricity and an amplifier. Also mentions the choice of 32768 Hertz () as the first power of 2 that is outside of the human hearing range, and then how a frequency divider is used to reduce the frequency to get the second counter.
Video 1.
How an atomic clock works, and its use in the global positioning system (GPS) by EngineerGuy (2012)
Source. Shows how conceptually an atomic clock is based on a feedback loop of two hyperfine structure states of caesium atoms (non-radioactive caesium-133 as clarified by the Wikipedia page). Like a quartz clock, it also relies on the piezoelectricity of quartz, but unlike the quartz clock, the quartz is not shaped like a tuning fork, and has a much larger resonating frequency of about 7 MHz. The feedback is completed by producing photons that resonate at the right frequency to excite the caesium.
Video 2.
Inside the HP 5061A Cesium Clock by CuriousMarc (2020)
Source.
A similar model was used in the Hafele-Keating experiment to test special relativity on two planes flying in opposite directions. Miniaturization was key.
Contains a disposable tube with 6g of Caesium. You boil it, so when it runs out, you change the tube, 40k USD. Their tube is made by Agilent Technologies, so a replacement since that opened in 1999, and the original machine is from the 60s.
Detection is done with an electron multiplier.
youtu.be/eOti3kKWX-c?t=1166 They compare it with their 100 dollar GPS disciplined oscillator, since GPS satellites have atomic clocks in them.
Video 3.
Quick presentation of the atomic clock at the National Physical Laboratory (2010)
Source. Their super accurate setup first does laser cooling on the caesium atoms.
Uses the frequency of the hyperfine structure of caesium-133 ground state, i.e spin up vs spin down of its valence electron , to define the second.
International System of Units definition of the second since 1967, because this is what atomic clocks use.
TODO why does this have more energy than the hyperfine split of the hydrogen line given that it is further from the nucleus?
A series of systems usually derived from the International System of Units that are more convenient for certain applications.
Currently an informal name for the Standard Model
Chronological outline of the key theories:
As of the 20th century, this can be described well as "the phenomena described by Maxwell's equations".
Back through its history however, that was not at all clear. This highlights how big of an achievement Maxwell's equations are.
Unified all previous electro-magnetism theories into one equation.
Explains the propagation of light as a wave, and matches the previously known relationship between the speed of light and electromagnetic constants.
The equations are a limit case of the more complete quantum electrodynamics, and unlike that more general theory account for the quantization of photon.
The equations are a system of partial differential equation.
The system consists of 6 unknown functions that map 4 variables: time t and the x, y and z positions in space, to a real number:
  • , , : directions of the electric field
  • , , : directions of the magnetic field
and two known input functions:
  • : density of charges in space
  • : current vector in space. This represents the strength of moving charges in space.
Due to the conservation of charge however, those input functions have the following restriction:
(1)
Equation 1.
Charge conservation
.
Also consider the following cases:
  • if a spherical charge is moving, then this of course means that is changing with time, and at the same time that a current exists
  • in an ideal infinite cylindrical wire however, we can have constant in the wire, but there can still be a current because those charges are moving
    Such infinite cylindrical wire is of course an ideal case, but one which is a good approximation to the huge number of electrons that travel in a actual wire.
The goal of finding and is that those fields allow us to determine the force that gets applied to a charge via the Equation "Lorentz force", and then to find the force we just need to integrate over the entire body.
Finally, now that we have defined all terms involved in the Maxwell equations, let's see the equations:
(2)
Equation 2.
Gauss' law
.
(3)
Equation 3.
Gauss's law for magnetism
.
(4)
Equation 4.
Faraday's law
.
(5)
Equation 5.
Ampere's circuital law
.
You should also review the intuitive interpretation of divergence and curl.
(1)
Equation 1.
Lorentz force
.
A little suspicious that it bears the name of Lorentz, who is famous for special relativity, isn't it? See: Maxwell's equations require special relativity.
Force between two wires carrying an electric current.
Caused by the Lorentz force.
For numerical algorithms and to get a more low level understanding of the equations, we can expand all terms to the simpler and more explicit form:
(1)
Equation 1
As seen from explicit scalar form of the Maxwell's equations, this expands to 8 equations, so the question arises if the system is over-determined because it only has 6 functions to be determined.
As explained on the Wikipedia page however, this is not the case, because if the first two equations hold for the initial condition, then the othe six equations imply that they also hold for all time, so they can be essentially omitted.
It is also worth noting that the first two equations don't involve time derivatives. Therefore, they can be seen as spacial constraints.
TODO: the electric field and magnetic field can be expressed in terms of the electric potential and magnetic vector potential. So then we only need 4 variables?
Static case of Maxwell's law for electricity only.
The "static" part is important: if this law were true for moving charges, we would be able to transmit information instantly at infinite distances. This is basically where the idea of field comes in.
Video 1.
Coulomb's Law experiment with torsion balance with a mirror on the balance to amplify rotations by uclaphysics (2010)
Source.
Video 1.
Understanding Electromagnetic Radiation! by Learn Engineering (2019)
Source. Shows animations of a dipole antenna which illustrates well how radiation is emitted from moving charges and travels at the speed of light.
In the standard formulation of Maxwell's equations, the electric current is a convient but magic input.
Would it be possible to use Maxwell's equations to solve a system of pointlike particles such as electrons instead?
The following suggest no, or only for certain subcases less general than Maxwell's equations:
This is the type of thing where the probability aspect of quantum mechanics seems it could "help".
TODO it would be awesome if we could de-generalize the equations in 2D and do a JavaScript demo of it!
In the context of Maxwell's equations, it is vector field that is one of the inputs of the equation.
Section "Maxwell's equations with pointlike particles" asks if the theory would work for pointlike particles in order to predict the evolution of this field as part of the equations themselves rather than as an external element.
After the 2019 redefinition of the SI base units it is by definition exactly Joules.
The voltage changes perpendicular to the current when magnetic field is applied.
Figure 1.
Hall effect experimental diagram
. Source. The Hall effect refers to the produced voltage , AKA on this setup.
An intuitive video is:
Video 1. Source.
The key formula for it is:
(1)
where:
Applications:
  • the direction of the effect proves that electric currents in common electrical conductors are made up of negative charged particles
  • measure magnetic fields, TODO vs other methods
Other more precise non-classical versions:
In some contexts, we want to observe what happens for a given fixed magnetic field strength on a specific plate (thus and are also fixed).
In those cases, it can be useful to talk about the "Hall resistance" defined as:
(1)
So note that it is not a "regular resistance", it just has the same dimensions, and is more usefully understood as a proportionality constant for the voltage given an input current:
(2)
This notion can be useful because everything else being equal, if we increase the current , then also increases proportionally, making this a way to talk about the voltage in a current independent manner.
And this is particularly the case for the quantum Hall effect, where is constant for wide ranges of applied magnetic field and TODO presumably the height can be made to a single molecular layer with chemical vapor deposition of the like, and if therefore fixed.
A different and more elegant way to express Maxwell's equations by using the:
instead of the:
There are several choices of electromagnetic four-potential that lead to the same physics.
E.g. thinking about the electric potential alone, you could set the zero anywhere, and everything would remain be the same.
The Lorentz gauge is just one such choice. It is however a very popular one, because it is also manifestly Lorentz invariant.
Alternative to the Lorentz gauge, but less used in general as it is not as nice for relativity invariance.
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