Misconceptions in Deriving the Poynting Vector: History and Physics

Before I get started, I feel that I should warn you that this video is VERY fast and dense with high level physics. I basically go through the physics as if you were students in an advanced physics course on Electricity and Magnetism at an elite university. I am not saying that to discourage anyone from watching this video, only to warn you so that you don’t feel overwhelmed. What I would advise if you are interested, and I hope you are interested, is to watch the video just to get an overview of what I am saying, without checking the validity of every statement. Then, go back and check on individual things that you want to examine further. Now back to the video. 

John Henry Poynting¹ was inspired to create Poynting’s vector by James Clerk Maxwell’s work over three long papers and a textbook on what is known as “Maxwell’s Equations” or, at the time the “Maxwell-Faraday Equations” (as Maxwell’s equations were inspired by the theories of Faraday). In the “Maxwell-Faraday equations” Maxwell created two types of magnetic fields: a “magnetic induction” \textbf{B} and a “magnetic intensity” or a “magnetic force” \textbf{H} which are related to each other in uniform media by a magnetic coefficient called the permeability or \textbf{B} = \mu \textbf{H}. ² However, it wasn’t particularly clear from Maxwell’s papers which magnetic field, \textbf{B} or \textbf{H} was the main magnetic field parallel to the electric field. For example, in Maxwell’s famous 1864 paper Maxwell represented light as following a wave equation of the “magnetic induction” \textbf{B} – implying that light is an \textbf{E} - \textbf{B} wave (or maybe just a \textbf{B} wave) and that \textbf{B} is the main magnetic field. Then, in 1873, Maxwell seemed to have changed his mind when he wrote in his textbook that linearly polarized light was composed of a vibrating electric “electro-motive force” \textbf{E} and a perpendicular “magnetic force” \textbf{H}. However, in his diagram, he labeled the vibrations as “electric displacement” which is given the letter \textbf{D} versus the “magnetic force” \textbf{H}. Tragically, Maxwell died in 1879 before he could clarify his theories more.

Anyway, Maxwell’s work, especially his 1873 book, inspired Poynting, who was a former student of Maxwell’s to publish a paper in early 1884  “On the Transfer of Energy in the Electromagnetic  Field.” ³ In that paper Poynting stated that there was a power density that depended on the electric intensity \textbf{E} field times the magnetic intensity \textbf{H} field times the sine of the angle between them, or, in modern vector math \textbf{S} = \textbf{E} \times \textbf{H}. This was immediately appealing to scientists as Poynting’s equation for the energy density loss seemingly settled the debate about the fields towards the conclusion that light is a vibrating \textbf{E} - \textbf{H} wave. Poynting’s viewpoint was also amplified when, in 1900 Poynting and his good friend J. J. Thomson (who used Maxwell’s equations to discover the electron) co-wrote a very popular undergraduate textbook on physics the first volume of which, was still being used with undergraduates in 1947!⁴

Fast forward one year to 1948. That was when the scientist Arnold Sommerfeld became the first physicist that I know of to insist that \textbf{B} is the main magnetic field parallel to the electric field \textbf{E}. However, despite that, Sommerfeld still believed in the Poynting vector and that light is a vibrating \textbf{E} - \textbf{H} wave. ⁵

As a side note, the delightful engineer Charles Proteus Steinmetz seemed to have used \textbf{E} - \textbf{B} as the main magnetic field as well as championed the term “magnetic field” over “magnetic lines of force” as early as 1897 but he was ignored by most physicists.⁶

Anyway, Sommersfeld’s book was translated into English in 1952, a few months after his death and it ignited a big debate about the nature of the magnetic field that was still raging in 1961-1963. That was when a 43 year old charismatic physics teacher at CalTech on the cusp of winning a Nobel Prize named Richard Feynman taught a 2 year long class to undergraduates. These lectures were recorded and made into a wildly popular textbook which can still be found in the bookshelves of most physicists to this very day. 

Feynman agreed with Sommerfeld that, “\textbf{B} and \textbf{E} are physically the fundamental fields” and it was only “some confusion in the past” which caused people to think that “\textbf{H} was ‘the magnetic field’.”⁷ However, unlike Sommerfeld, Feynman believed in both the Poynting vector and that light is an \textbf{E} - \textbf{B} wave instead of an \textbf{E} - \textbf{H} wave. Feynman created this miracle by writing a new equation for the Poynting vector, “something quite new” as he put it.⁸  Instead of the Poynting vector equaling to the \textbf{E} field cross the \textbf{H} field it was equal to the \textbf{E} field cross the \textbf{B} field times the speed of light in a vacuum squared times the electric coefficient in a vacuum. The strange thing is that these equations are equivalent if, and only if, the material is in a vacuum. 

Anyway, after the popularity of Feynman’s lectures, and Feynman’s insistence on how the Poynting Vector was strange but true increased interest in the Poynting vector in textbooks and peer-reviewed papers, although most people wrote it as a simpler form of \textbf{E} \times \textbf{B} / \mu_0 which is mathematically equivalent to how Feynman wrote it (ie. only true in a vacuum).⁹  Then, in the late 1990s a famous scientist and, yes, textbook writer named John “Dave” Jackson wrote a series of papers promoted using a sleeve of surface charges to allow the Poynting vector to point down the wire in a direct current circuit, differing from the DC theory promoted by Poynting, Sommersfeld and Feynman where the Poynting vector points into the wire.¹⁰ In the last 5 or 6 years the Jackson model and the Poynting/Sommerfeld/Feynman model of the Poynting Vector in DC systems has become particularly popular among YouTubers looking for unusual physics to discuss which brings us to today. 

But before you spend too much time trying to understand these models let me just say that using the Poynting Vector to explain physical phenomena is … well, not to be too blunt about it but it is dumb. It is crazy. It is obviously nuts, and screwy and even absurd. These are not my words, they are Richard Feynman’s. Feynman himself admitted – well, as I mentioned Feynman’s lectures were recorded so listen to a few comments of his about the Poynting vector over a 7 minute tape for yourself:!!¹¹ Even when he had time to reflect on his words, as in his famous textbook, he wrote explicitly, “How absurd it gets!”¹²

But why? Why would Richard Feynman call the Poynting Vector all these names? And if the Poynting vector is illogical, why do we use it? That is why I want to make this video. In this video I will discuss the misunderstandings in John Poynting’s derivation of the poynting vector, then Arnold Sommersfield’s derivation of the poynting vector and the poynting equation, and, then finally, problems in Richard Feynman’s derivations.  Ready? Let’s go!

Hello, my name is Kathy Joseph and yes, I love physics.  I use the history of the development of science to teach and understand it on a deeper level. I have come to believe that physics is not in the equations and concepts but in how we got the equations and concepts from original documents. From my studies, I have found that occasionally there is a false theory that is believed and used for many years by almost all physicists before someone, usually a person outside of academia, determinedly points out the fallacy. 

For example, in 1777 the great chemist Lavoisier postulated that heat is an indestructible fluid.¹³ Over the years, this “caloric theory” was used to explain chemical and thermodynamic systems, like Carnot’s theories,¹⁴ decades after the experiments of Count Ruhmford’s seemed to demonstrate that this was false.¹⁵ Then, in the 1840s, a beer brewer named James Prescott Joule started giving relentless public talks about his theory that heat is just a form of energy that can be generated by friction. Eventually, this was mentioned in a paper by an Irish scientist named William Thomson¹⁶ and then used by a German scientist named Clausius to create the first law of thermodynamics. I have a whole video about the history of the first law of thermodynamics¹⁷ if you want to have more details. ¹⁸

Or, in another example, in 1902 the physicist Phillip Lenard explained away the results of the photo-electric effect as atoms having a “fuse” where the energy of the emitted electron was independent of the energy of the light that impinged upon it, which he mentioned in his Nobel prize acceptance speech in 1906.¹⁹  For many years this was considered settled science²⁰ even though in 1905 a patent clerk 3rd class had the radical idea of taking Max Planck’s quantization of light energy literally instead. This is arguably the real origin of quantum mechanics. Eventually, Einstein’s idea replaced Lenard’s. And in 1921 Einstein won a Nobel Prize for this model, specifically, I think, as a way to insult Lenard and his lackey Johannes Stark, who, by this time, were well on their way to becoming anti-science Nazis. Once again, I have a video on the history and physics of the photo-electric effect if you are interested in more details.

I am not saying that I think like Einstein. I am merely saying that there is a history of smart physicists twisting themselves into knots to promote weak models if they came from intelligent scientists that they admire. This is not a conspiracy, merely a side effect of physicists being real human beings. We have this feeling that if someone is a brilliant, outside-the-box thinker like Levoisier or young Lenard or Einstein or Poynting or Sommerfeld or whoever that they cannot make a mistake. But logically, of course, that isn’t true. Every once and a while a talented scientist gets confused and promotes a weak idea bundled with some brilliant ones and this illogical idea permeates into mainstream science. This is actually a good thing as all new discoveries are in its way an indictment of old theories. If any scientist was perfect, then scientific development would be over.

Anyway, through my studies I have come to believe that the Poynting vector is such a weak model. It is erroneously derived and it doesn’t work to explain any physical phenomena. In other words, when it comes to the Poynting Vector, I agree with Feynman when he wrote in his textbook: “How absurd it gets!”.²¹ But unlike Feynman I am willing to stand up and say that I disagree with an absurd, obviously nuts, and crazy theory that makes no sense to me even if famous physicists promoted it. Which brings me to…

Part 1: Misunderstandings in Poynting’s Derivation of the Poynting Vector

As I mentioned previously, John Henry Poynting derived his famous equation from the model and mathematics of Maxwell’s work from 1873. In this, like Maxwell, Poynting used these hard to read squiggly German letters and extra 4 pi’s which we now ignore and a few different letters like the capital letter K for the permittivity rather than the Greek letter \epsilon – none of which are important but I thought it was worth mentioning so you would know why Poynting wrote his equations the way he did. Note that for the most part I will use modern letters and ignore 4\pi’s in Poynting’s and Maxwell’s equations as they just get in the way. 

Short comment about the 4\pis and why we now ignore them. According to an Italian scientist named Giovani Giorgi, Heaviside believed that Maxwell arbitrarily added “the unnecessary factor of 4\pi” because he wanted the electric force to be proportional to the charges divided by the distance squared. “While, rationally it ought to be” proportional to the charges divided by 4\pi r^2 so that it is inversely proportional not to the distance to the charges, but, instead to “the area of the sphere” surrounding the charge.²² The reason I mention this letter of Giovani Giorgi’s is because in it he describes to Oliver Heaviside his theory of a “rational units of electromagnetism” otherwise known as SI units.²³ Someday, I will do a history of SI units, it is really cool.

Back to John Henry Poynting. In Poynting’s work Poynting perpetuated a mistake of Maxwell’s. In Maxwell’s final book he defined the energy density as 1/2ED²⁴ and the magnetic density as 1/8 \pi BH.²⁵  (Note that Maxwell was brilliant but so different from anyone before him that basically no one could understand him without literally years of study. Even his close friend and mentor William Thomson aka Lord Kelvin couldn’t get far enough to truly proofread his work so everything is chock full of mistakes like adding a 4\pi to the magnetic density and forgetting to add one to the electric density because he felt compelled to add an extra 4\pi to the equation to the displacement. I mean, he wrote his equation for the speed of an electromagnetic wave \left( \nu = \sqrt{1/\epsilon\mu} \right) in three ways in three different documents, and all of them are incorrect due to those darn 4\pi’s!) Don’t get me wrong, I ADORE Maxwell but his work will give anyone a headache. Anyway as \textbf{B} equals \mu \textbf{H} and Maxwell routinely replaced \textbf{D} with \epsilon\textbf{E}/4\pi, Poynting and Hertz (and others) defined the total energy density to be \frac{1}{8\pi} \epsilon E^2 + \frac{1}{8\pi} \mu H^2. Poynting then derived his famous equation by starting with his version Maxwell’s equation for the energy per volume \frac{1}{8\pi} \epsilon E^2 + \frac{1}{8\pi} \mu H^2 and then taking the time derivative to get the power per volume to get \frac{1}{4\pi} \epsilon E \frac{dE}{dt} + \frac{1}{4\pi} \mu H \frac{dH}{dt}.

The 4\pi business isn’t that important, but what is important are the hidden assumptions that the magnetic permeability \mu and the electric permittivity \epsilon are not dependent on time and using D = \epsilon E as if it was universally true. 

Let me explain. As I said, \mu H \frac{dH}{dt} only equals the time derivative of the energy density \left(\frac12 \mu H^2\right) if the magnetic coefficient (\mu) is a constant with respect to time. But, as mathematically explained by the delightful Charles Proteus Steinmetz in 1890,²⁶ the fact that the magnetic field, \textbf{B} in a piece of iron takes a certain amount of time to be magnetized when in a magnetic intensity \textbf{H} causes heat to be produced in iron in alternating fields. This delay in magnetic reaction is called hysteresis, Greek for lag behind. (Note that in this original paper, Steinmetz confusingly labeled the energy loss with the letter \textbf{H}, this has nothing to do with magnetic \textbf{H} – sorry). Anyway, Steinmetz’s work means that not only does the magnetic coefficient,\mu , depend on time, but also that a changing magnetic coefficient has a power dependance. 

As you might expect, I have a similar problem with \epsilon E \frac{dE}{dt} equaling the time derivative of the electric energy density \left(\frac12 ED\right). For one, the electric coefficient is due to the electric properties of the non-conductor (which Faraday called the dielectric) and it is not independent of time. See, the displacement field from the charges go through a dielectric, they push the atoms in it to be a bit polarized, with one end more positive and one more negative which reduces the electric field in the material by a coefficient called the relative coefficient \epsilon_r²⁷ where the total coefficient equals the relative coefficient times the vacuum coefficient. By the way, the coefficient in a vacuum 0 has nothing to do with the electric properties of a vacuum, as a vacuum \epsilon_0 has nothing in it. It is just there to make the units of displacement (Amps per meter squared)²⁸ match the units of Electric fields (Newtons per Coulomb).²⁹ It doesn’t have any physical significance, it is just like the constant k in Coulomb’s law. In fact, the constant \epsilon_0 is equal to 1/(4\pi k). I have discovered that Oliver Heaviside agreed with me on this point as, in 1902 when Giovanni Giorgi wrote that the coefficients in a vacuum “express a physical truth” Oliver Heaviside wrote in the margins, “No. Bad Logic.”³⁰

Anyway, this polarization of the atoms in the dielectric is not instantaneous which means that the electric coefficient is a function of time. An example of that is in a microwave oven where a vibrating electromagnetic wave causes the polarized water atoms in your food to spin which converts into heat. According to modern textbooks, this “dielectric heating” can be derived from Maxwell’s equation and is a function of the frequency of the electromagnetic wave, the Electric field squared and what is called the imaginary part of the relative dielectric constant.³¹ But even in ordinary situations, the dielectric constant depends on the frequency of the light through it, which means that the dielectric constant has a time dependance.

Finally, I object to stating that \textbf{D} is equivalent to \epsilon \textbf{E} in all situations.  For example, as Maxwell noted in his 1873 book, in conductors the current density in the wire is equal to the electric field times the conductivity \left(\textbf{j} = \sigma \textbf{E}\right)³² which is called the microscopic Ohm’s law. However, if \textbf{D} = \epsilon \textbf{E} than by simple division \textbf{E} = \textbf{D} / \epsilon. However if \textbf{E} = \textbf{D} / \epsilon is the full equation for the electric field everywhere then \textbf{E} = \textbf{D} / \epsilon is true inside a conductor. This would necessitate that the current density \left(\textbf{j} = \sigma \textbf{E}\right) is the same as \left(\textbf{j} = \sigma \textbf{D}/\epsilon\right). Note that the permittivity of a conductor is basically infinite which means that the current in a conductor has to be zero in all cases, which is of course ridiculous.^{33 34} In addition, there is a third equation for the electric field, E:

E = - \operatorname{Grad} \varphi - \frac{dA}{dt} = \operatorname{Grad} V - \frac{dA}{dt}, \quad \text{where } \varphi = -V

is the potential or the internal energy per charge³⁵ and \textbf{A}, called either the magnetic momentum (Maxwell) or the vector potential (Heaviside), is related to the magnetic field \textbf{B} with the function \textbf{B}= curl \textbf{A}. I like to call the equation E = \operatorname{Grad} V - \frac{dA}{dt} the universal electric field equation, as it is universally true, whereas \textbf{E} = \textbf{D} / \epsilon and \textbf{E} = \textbf{j} / \epsilon are true only in a dielectric or a conductor respectively and only true when the magnetic field is constant. I will explain this in more detail in future videos but if you want a preview I recently gave a colloquium at the Physics department of UC Davis on the “Evolution of Maxwell’s equations” and the PowerPoint and my voiceover was recorded and can be linked to in the description of the video.³⁶ I think that from now on I will just do my physics talks on the whiteboard as, no matter how dynamic I try to make it, I find pure PowerPoint talks static and I worry it discourages people from asking questions. 

Anyway, my belief in the universal electric field equation and the time dependency of the electric and magnetic coefficients is why I do not agree that the time derivative of \frac{1}{2} ED + \frac{1}{2} BH is \epsilon E \frac{dE}{dt} + \mu H \frac{dH}{dt}. Instead it equals the far less elegant: \frac{1}{2} E \frac{dD}{dt} + \frac{1}{2} D \frac{dE}{dt} + \frac{1}{2} B \frac{dH}{dt} + \frac{1}{2} H \frac{dB}{dt}.   

However, this was the faulty logic that Poynting used to derive that the “the energy flows …[where] the amount crossing unit area per second of this plane is equal to electromotive intensity x magnetic intensity x sine included angle… [with] the direction of flow … being in right handed order.”³⁷ In other words, Poynting wrote that this energy flow vector equals \textbf{E} \times \textbf{H} because of a faulty derivation of the time derivative of the energy density.

I should mention that just because there are difficulties in the original derivation of a function, does not mean that scientists discount the equation, if the equation is found to be useful for practical applications and someone else can derive it from fundamental concepts. For example, in 1848, William Thomson, one year before he promoted James Joule’s ideas that heat was a form of energy, derived that the absolute zero is 273 degrees below zero in Celsius using the idea that heat was an indestructible caloric.³⁸ Despite this error in Thomson’s original derivation, William Thomson aka. the Lord of Kelvin was honored with the temperature scale named after him.  

This is not an insult to John Henry Poynting or James Clerk Maxwell. Maxwell’s 1873 work and Poynting’s 1884 work predated Steinmetz’s theories from 1890 on changing magnetic permeability due to hysteresis, and far predated equations on microwave dielectric heating, so it makes sense that they would miss it. Heck, their work even predated the creation of the universal electric field equation E = \operatorname{Grad} V - \frac{dA}{dt} , as Maxwell added another term of \nu \times B to his universal electric field equation or E = \nu \times B - \operatorname{Grad} \varphi - \frac{dA}{dt}. It was only after Lorentz moved the \nu \times B to the force equation F/q = E + \nu \times B or F = qE + q\nu \times B that we ended up with the true universal equation for the electric field E = \operatorname{Grad} V - \frac{dA}{dt}. (Note that it wasn’t as simple as that, as of course it wasn’t. I explain it a bit in my talk at UC Irvine if you want more details)³⁹

But what about Sommerfeld who knew all these things? That leads me to…

Part 2: Misunderstandings in Sommerfeld’s Derivation of the Poynting Vector

As I said before, Sommerfeld was the first major physicist to declare that \textbf{B} bot \textbf{H} is the magnetic field. Sommerfeld noted in the preface how every equation (aside from energy) with both magnetic fields and electric fields either had \textbf{B}s and \textbf{E}s or had\textbf{H}s and \textbf{D}s⁴⁰ concluding with, “the fact that \textbf{B} and \textbf{E}, and \textbf{H} and \textbf{D}, belong together follows unambiguously from the theory of relativity, in which the quantities c\textbf{B} and -i\textbf{E}, and \textbf{H} and -ic\textbf{D}, respectively, are coupled together in a six-vector (antisymmetric tensor).”⁴¹ Note that Maxwell’s energy equations \frac12 \textbf{DE} and \frac12 \textbf{HB}⁴² only match electric fields (\textbf{D} and \textbf{E}) or match magnetic fields (\textbf{H} and \textbf{B}). And the power per volume lost in a DC circuit \textbf{j} - \textbf{E} matches the electric current per area \textbf{j} and the electric \textbf{E} field. It is only Poynting’s vector (\textbf{S} = \textbf{E} \times \textbf{H}) which matches an electric \textbf{E} field with a magnetic \textbf{H} field. 

As a side note, Sommerfield has been called the most underrated physicist of all time due to the fact that he was so influential and inspirational that he was nominated for a Nobel Prize a jaw dropping 84 times, more than anyone who didn’t win.⁴³  I have a theory that Sommerfeld didn’t win due to being an anti-Nazi German scientist who didn’t win before Hitler came to power. I wonder how much a 1937 full page Nazi article on the danger of non-Jewish people who have a “Jewish spirit” affected this. Especially as they stated explicitly that “The Jewish spirit is probably most clearly recognizable in the field of physics,” and then complained bitterly of Sommerfeld’s (and Max Planck’s) work in helping their talented students into professorships even if they were Jewish⁴⁴, but that is just a theory. By the way, in my video on why Heisenberg volunteered to make a Nuclear bomb for Hitler and in my video on what I think really happened between Heisenberg and Niels Bohr in Copenhagen I mistakenly said that the Nobel Prize winner Johannes Stark wrote that article, but he actually only publicly agreed with it, it came from the top brass instead. 

Anyway, despite deciding that \textbf{B} is the magnetic field, he still believed in the Poynting vector, and this is how he derived it. Sommerfeld began with Faraday’s law of induction and took the scalar multiplication of \textbf{H} to both sides of the equation. Then, he took Ampere’s law with Maxwell’s addition and added the scalar multiplication of \textbf{E} to both sides of the equation. Combining equations he got this equation  \left(H \cdot B' + E \cdot D' + E \cdot j = E \cdot \operatorname{Curl} H - H \cdot \operatorname{Curl} E\right). Finally he used a math relationship to get the final equation that \dot{H} \cdot B' + E \cdot D' + E \cdot j plus the divergence of E \times H equals zero.⁴⁵ By the way, this equation, called the Poynting theorem, is completely legitimate and universally true  which is why it has occasionally been used effectively in physics problems. 

Then Sommerfeld made the same mistake that Poynting made. Namely, Sommerfeld stated that H \cdot \frac{dB}{dt} equals the time derivative of \frac12 \textbf{HB} (ignoring that the permeability can be a function of time) and that D \cdot \frac{dE}{dt} equals the time derivative of \frac12 \textbf{DE} (ignoring that the permittivity can be a function of time and that \textbf{D} = \epsilon \textbf{E} is only true in a dielectric with a constant magnetic field).

In other words, the equation Summersfield derived known as the Poynting’s theorem, while true, does NOT represent the collective change in energy over time. In fact, I would argue that it is the universal electric field equation⁴⁶ that represents the first law of thermodynamics rather than this meaningless math relation of Sommersfield’s/Poynting’s. 

Please understand that I have nothing but the utmost respect for Professor Sommerfeld. He was a world-class physicist, one of the most inspiring teachers that ever lived as well as a moral person in a very challenging time. But as with every scientist great and small I don’t believe that anyone is infallible. Also, it would be ironic to complain too much about Sommerfeld for making these mistakes as I made the same for the majority of my life. I mean just check out my video on deriving Maxwell’s equation that I made just 2 years ago.⁴⁷

I literally just want to go back in time and yell at myself. NO, E doesn’t equal D/\epsilon if you are in a conductor or the magnetic field is changing. Also, we don’t replace all ’s with \epsilon_0’s because a vacuum or air at room temperature and pressure “is so common that often we are given Maxwell’s laws for this case as if it was universal.” No, Kathy of 2 years ago, that is not why we write Maxwell’s laws with all those \epsilon_0’s and \mu_0’s. We write them because of how Richard Feynman approached the derivation of Poynting’s vector. Which brings me to…

 Part 3: Misconceptions in Feynman’s Lectures

I would like to start with a bit about my personal history with Feynman’s lectures. When I was a Junior at the University of Chicago (called a 3rd year), I was conned by my boyfriend to spend a whopping $120 for the hardback version of Feynman’s lectures in Physics. Then, considering how much he and others talked up Feynman’s lectures, especially the part on Electricity and Magnetism, I tried to work through the book. Although I really really tried, I could not make heads or tails of it and gave up after a couple of weeks. I’m sure many of you have had a similar experience. Luckily, I was in my fourth year of advanced undergraduate Physics work (I had doubled up) and had an excellent E&M teacher (thanks Herr Muller) who used a decent textbook of Wangsness’s “Electromagnetic Fields” (although I prefer Griffith’s E&M book) so I knew I could do the material. However, I didn’t think to blame Feynman, after all he was supposed to be the best teacher who ever lived. Instead, I figured I couldn’t learn without a great teacher, and gave up on self-improvement through textbooks. Which meant that for many, many years that I was out of school, I didn’t learn any new physics. Luckily, I continued to learn from people, from investigating things on the internet, and from obsessively thinking about everything that I love, especially things like music and movies. So many movies. 

Let me explain. Warning: this is going to get a little personal and dark. See, I was born to a family dealing with extreme trauma as I had a 3 year old sister who had drowned in a pool the year before I was born. After that my father was determined to push the family through this tragedy while continuing to be a busy lawyer so he focused on me to bring my family through as I was the only non-traumatized person in the family. Now that may sound horrific, but actually I had a lovely and creative childhood (aside from school which I was mixed on: I loved learning but I hated the endless busywork and stifling of originality). Anyway, the main way my dad and I communicated was through movies and we had a particular fondness for musicals. We would rate them, evaluate them, improve them in our minds and argue endlessly about them. My dad was an incredibly skilled lawyer and knew the power of truthful storytelling – but he also knew that fiction and humor was usually the best window to learn that art. That is also how I learned that storytelling and kind humor was the best way for me to help my family and friends. Also, my dad would say that “a lawyer’s only currency is their honesty” and would never exaggerate, even in family stories, a practice that I follow as well (aside from when I am having a tiff with my hubby). Because of that, we were always searching for the juiciest true stories and using our skills from examining movies to tell them to the best of our abilities. 

When I was a kid I was taken to the science museum called the Exploratorium for a school trip. At the Exploratorium they have tons of kiosks with different hands-on experiments. I did the “do and notice” and then read the “what is going on” section. I immediately thought it was cool and grabbed a friend to show her. That is how I found that teaching physics was so joyful. By the time I got to college I learned that advanced physics has no busywork and tons of creative debate so I was in heaven. However, I then found the research part of graduate school to be isolating and pointless and went through three different graduate programs before becoming a high school physics teacher. Luckily, I found that teaching high school was amazing. I learned so much from them. My god, you can tell with a room of 34 young teens when they get it and when they don’t. And when they get it you can feel the delight.

However, around 8 years ago after taking a year off for maternity leave my new principal said that she didn’t need me and if I insisted on returning I would be forced to teach freshman Algebra instead of physics!  I know! It all worked out for the best but I am still mad. Anyway, that is how I decided to take a break from teaching and instead to try to write a book about the history of electricity to help non-scientists get an understanding of basic electrical distribution through its history (by the way, my mom came up with the title).  Honestly, I only started this channel to get people to buy my book. But then I fell in love with the medium. I can talk about whatever I want for as long as I want to do. I can follow any tangent that interests me for as long as it interests me. I really don’t understand how more physicists don’t want to make YouTube videos, the freedom is fantastic. Even better than teaching high school. Plus, putting my material in a visual medium and getting such great feedback and pushback has improved my book and my later videos and made me a far better physicist. I think of each video as a mini-movie or maybe a mini-musical, with the discovery process like the dialogue and the equations like the songs.  (By the way, I am not the only one who sees music in physics. For example, in 1948 Einstein recalled that when he first heard Niels Bohr’s model of atoms he said that it, “the highest form of musicality in the sphere of thought.”⁴⁸) 

In truth, I see music and movies in everything, not just physics. Since I study movies and music so intensely I end up using references to them all day every day. That is why most of my videos are so cinematic – it is how I think. [I even made a commercial free version of this section of the video with even more musical and movie references]. In October of 2022, I published my first book and have the detailed outline on five more. Honestly, my biggest problem now is focusing on one or two projects when I have dozens that I want to develop. 

One of the surprises of following those odd tangents is that, with a lot of effort and historical context, I could understand the core points and derivations of some of the most notoriously difficult original concepts in physics and engineering. Heck, I could learn and explain subjects like electrical distribution and radio and astronomy where I have never taken a class (Danke Shane Herr Muller). Even so, I was terrified to study Maxwell’s papers – after all, if Feynman’s derivations were preferable and I couldn’t understand Feynman then what hope did I have to understand Maxwell?  

But then, around 2 years ago I had the opportunity to visit The Spark Museum in Washington State, where the owner let me see some original historical papers – most of which I was too scared to touch. Let’s take a visit.⁴⁹ So that is why I started to study the history and physics of the Faraday cage. It was in that study where I discovered that, in 1837 Micheal Faraday created the idea of electric fields and the concept of a dielectric because of his creation of the Faraday Cage! Not only that, but by delving into Faraday’s 1837 work I started to feel that I truly understood Gauss’ law for the first time. That is why I decided to take the plunge into understanding Maxwell’s derivation of his laws one equation at a time. It turned out to be the perfect method for me. Although I thought I would make a video about each equation, it turned out that one was enough to crack open the whole can of worms wide open. Finally, I felt able to tackle the math Maxwell used called quaternions and how it was transformed through Maxwell’s equations into vector calculus. Then, as I understood Maxwell’s odd language and quirks, I could also decipher not only Maxwell but other early physicist’s electricity and magnetism papers like Hertz, Heaviside, Lorentz, Minkowski, Einstein, and of course Poynting. 

However, when I went back to Feynman’s lectures I still couldn’t figure out why it was so strange. It just made no sense. Then it finally hit me. Feynman’s lectures, to put a fine point on it, are absolute gibberish. I know what you have been told, I have been told the same thing, but we were misinformed. 

I mean just compare how Sommerfield, who recall was the first major physicist to declare that \textbf{B} not \textbf{H} was the magnetic field, made his argument in 1948 compared to Feynman’s weak declaration of the same point 15 years later.  Sommerfield gave a list of arguments and included that he was contradicting his friend Max Planck’s published opinion and started an international debate on the subject. Then Feynman just pretends that the debate over the magnetic field didn’t even occur aside from some nameless “people” who, as he mentioned as an aside in Chapter 36, were confused by some simple math, “to think that \textbf{H} was ‘the magnetic field.’”⁵⁰ By the way, here is a little list of a few people who thought that \textbf{H} was the main magnetic field: John Henry Poynting, Heinrich Hertz, Oliver Heaviside, Hendrik Lorentz, Hermann Minkowski, Max Planck,⁵¹ Niels Bohr,⁵² and of course that simpleminded nobody Albert Einstein.  

In addition, Feynman only introduced the \textbf{H} field and the permeability of a vacuum \mu_0 in chapter 36 as a convenient math shorthand to explain ferromagnetic or highly magnetic materials. And then he argue against the use of the permeability of a vacuum \mu_0 as just “one more constant to keep track of.”⁵³ Otherwise he replaced every instance of \textbf{H} with \epsilon_0 c^2\textbf{B} and every instance of \textbf{D} with \epsilon_0 \textbf{E} even though that is only true in a vacuum with a constant magnetic field. 

That is how Feynman not only changed the Poynting vector to such a strange form⁵⁴ (which isn’t that upsetting to me as I believe it is a weak model) and also how he changed Gauss’s law, Ampere’s law,⁵⁵ Energy density,⁵⁶ and, in an act of pure insanity, Feynman just drops the equation for the velocity of a electromagnetic wave in a medium of \nu \equiv \sqrt{\frac{1}{\epsilon \mu}}.⁵⁷ It is just gone, kaput, vanished. (Yes, you heard right, Feynman just completely ignores Maxwell’s equation that demonstrates the relation between electromagnetic light and the electric and magnetic properties of a medium). 

But how did Feynman justify changing what are considered some of the most fundamental equations in Physics in his textbook? He didn’t. He didn’t give *any* explanation – just stated it like it was a fact. Well, that isn’t completely true, he did mention a bit about changing equations in his lectures in his introduction. He outright denied he made any changes: writing, “In the first part of the course, dealing with electricity and magnetism, I couldn’t think of any really unique or different way of doing it—of any way that would be particularly more exciting than the usual way of presenting it. So I don’t think I did very much in the lectures on electricity and magnetism.” ⁵⁸

Just for comparison, here is how Sommerfeld represented Maxwell’s laws in 1948, this is how a very popular Physics book from Bleaney and Bleaney represented Maxwell’s laws in 1957⁵⁹ and this is how Feynman presented them in 1963. Note that Faraday’s law of induction and Gauss’s law for a vacuum (the ones with \textbf{B}’s and \textbf{E}’s) are exactly the same in all three cases but that the other laws are completely different for Feynman. Just take a moment to compare and contrast, it is astonishing isn’t it?

Even worse, Feynman created a new and terrible method of teaching and learning Physics. Feynman expressed it clearly when he said that his method of teaching, “ is also completely opposite to the historical approach in which one develops the subject in terms of the experiments by which the information was obtained.” He said that his reason for stripping physics of its motivation was “we have only a limited time to acquire our knowledge” but don’t worry, Feynman added, any student can learn all about the development of physics on their own simply and easily ”by reading the Encyclopedia Britannica, which has excellent historical articles on electricity and on other parts of physics.”⁶⁰ No, Feynman, the historical approach isn’t wasteful, it is fundamental. It is transformational. It is Physics. 

And then there is how Richard Feynman evaluated his own lectures in the introduction. He admitted that this book is a disaster. No, really. Feynman wrote that, “the way the course was given, there wasn’t any feedback from the students to the lecturer to indicate how well the lectures were going over. This is indeed a very serious difficulty, and I don’t know how good the lectures really are. The whole thing was essentially an experiment…The question, of course, is how well this experiment has succeeded. My own point of view… is pessimistic. I don’t think I did very well by the students… I think that the system is a failure.”⁶¹ On that point, I agree. I completely agree.

Now I am sure that you have a ton of questions. I would ask you to please, please, please for the love of Faraday please…. Ask someone else! Reddit has a lovely /physics channel (follow their rules please!), there are websites and discord servers and tons of online geek hangouts stock full of people just dying to debate physics. You will be amazed. And keep bugging them until your questions are answered. If you subscribe to a YouTube science channel or personally know a physicist or advanced engineer ask them what they think. I want this to be peer reviewed beyond the level that any scientific idea has ever been peer reviewed. I have a link to my script with lots of  footnotes so you can cut and paste and ask specific questions. Like “is it true that Feynman changed all of Maxwell’s equations and then pretended he didn’t?” or “is it true that \textbf{D} = \epsilon \textbf{E} only works for a dielectric in a constant magnetic field and E = -\operatorname{Grad}(\varphi) - \frac{dA}{dt} is the universal equation for the electric field?” or whatever you want to know. Nothing would help me more than you questioning people’s assumptions. 

Also, if you want to use any of this information or the information in any of my videos to help you create a podcast, a wikipedia update (and PLEASE update wikipedia), an article, a video, or a peer-reviewed paper, please do! I include the links for just such a reason. As long as you don’t plagiarize me and believe in what you are saying don’t worry about using my ideas or research. In fact, I want you to use my ideas and research. I mean, I would like credit in places where it is appropriate, but that isn’t my primary motivation. I literally just want to fight ignorance, cruelty and elitism and have fun doing it. What could top that?

Finally, to my fellow science devotees throughout the world, those of you who, like me, love physics and engineering, aren’t you sick and dismayed with the terrible rise of cruel pseudoscience, fear of science and outright anti science beliefs? Aren’t you done with wringing your hands about the ignorance in the world?  Don’t you want to prove that physics is not illogical, elitist and “screwy, screwy, inherently screwy”? Well, now it is your time to shine. Print out the script, read every citation, get together with your smartest and weirdest friends and stay up all night dissecting my ideas to the bone. And once you have made up your mind, whether you agree with me or not, put it out there in the world.  Remember, don’t be scared of change. As the mighty Faraday said in his seventh Research in Electricity which remember was the inspiration for Maxwell’s equations⁶²: “it is the great beauty of our science… that advancement in it, whether in a degree great or small, instead of exhausting the subject of research, opens the doors to further and more abundant knowledge, overflowing with beauty and utility.”⁶³

For example, if you use the universal electric field equation, not only do you dispel the need for the “crazy”⁶⁴ Poynting vector, but you also get a new equation for light. Rather than separate equations for the electric field  (del squared \textbf{E} equals mu epsilon times the second time derivative of \textbf{E}), and a similar equation for the magnetic field⁶⁵ you get one intertwined electromagnetic equation (minus the gradient of the divergence of \textbf{E} minus del squared \textbf{K} equals mu epsilon times the second time derivative of \textbf{E} + \textbf{K}) where \textbf{K} is just the time derivative of the magnetic momentum \textbf{A}, which is why I am calling it the magnetic impulse. Now tell me, do you agree? Do you disagree? Can you model this equation with computers? Can you use this equation to derive other equations? Let’s do some new physics baby! This is going to be so much fun.

Obviously, I am incredibly busy. However, I am also beyond excited to get the word out. So, if you want to arrange a filmed colloquia at your University⁶⁶ – contact me and let’s set something up.

In addition, I don’t want to ignore the artists, comedians and artistic engineers. I love talking to artists – you all are so beautifully inquisitive.  Honestly, I need you. We need to convince everyone, physicists and non-physicists alike, that physics can be kind and welcoming. Because we have seen what happens when you let cruel Nobel Prize winners take over.  

Note that I absolutely hate elitist snobs. They are insufferable. I have no interest in how smart you are, how much you know about any subject or how much money you have. Richard Feynman was very smart. Phillip Lenard, an actual Nazi, was even smarter (and a far better textbook writer). Instead, I care if you are kind. If you are creative. If you are generous. If you are inquisitive. If you are equally as interested in learning from people as you are at sharing your passions with them. 

I mean, if as Voltaire said “Anyone who can make you believe absurdities can make you commit atrocities”⁶⁷ then isn’t the opposite true as well, “Anyone who can help you light the spark of knowledge can help illuminate the goodness in your heart”? 

OK, as my queen put it in “Hairspray”.  So that is my video. I want to apologize for this being my first video in over a year.  I had some health issues although I am doing much better now. Also, this may sound strange but it turns out that when I am working on something big I kind of cocoon. It becomes very hard to communicate for a while. I know it sounds weird but… I’m weird so, I am sorry about being so bad at communication. This is especially true for my long suffering Patrons. Thank you patrons. 

This is a reminder that I posted the script for this video on my website www.KathyLovesPhysics.com and I included links to all of my citations so you can check them for yourself as well as my email to contact me. Remember as Jonathan Larson wrote so eloquently in his musical Rent about the joy of the artistic or bohemian life, “The opposite of war isn’t peace, it’s creation,” So, get cracking. 

 And, as always, stay safe and curious, my friends. 

---

1. Royal Society (Great Britain), Philosophical Transactions of the Royal Society of London (1885) p. 101
    https://archive.org/details/philosophicaltr07britgoog/page/n100/mode/2up? ↩︎
2. Maxwell “A Treatise on Electricity and Magnetism” (1873) vol 2 p. 236-7 A treatise on electricity and magnetism : Maxwell, James Clerk, 1831-1879 : Free Download, Borrow, and Streaming : Internet Archive ↩︎
3. Royal Society (Great Britain), Philosophical Transactions of the Royal Society of London (1885) p. 101 https://archive.org/details/philosophicaltr07britgoog/page/n100/mode/2up? ↩︎
4. Sir J.J. Thomson, A University Text Book Of Physics Vol I (1947) p. 10 https://archive.org/details/in.ernet.dli.2015.78170/page/n9/mode/2up? ↩︎
5.  A. Sommerfeld, Electrodynamics Lectures on theoretical physics,. Vol.III (1952) 
   https://archive.org/details/electrodynamicsl0003arno_e7b9/page/36/mode/2up? ↩︎
6. C.P. Steinmetz, Electric currents, Alternating, History / General (1897) p. 129
   Theory and Calculation of Alternating Current Phenomena – Google Books
   ↩︎
7. M.A. Gottlieb and R. Pfeiffer, California Institute of Technology (1964, 2006, 2013), “After equation 36.32”  The Feynman Lectures on Physics Vol. II Ch. 36: Ferromagnetism (caltech.edu) ↩︎
8. M.A. Gottlieb and R. Pfeiffer, California Institute of Technology (1964, 2006, 2013) “27-5” The Feynman Lectures on Physics Vol. II Ch. 27: Field Energy and Field Momentum ↩︎
9. D.J. Griffiths, (David Jeffery), 1942- author, Introduction to electrodynamics” (2017)
   https://archive.org/details/introductiontoel0000grif/page/358/mode/2up? ↩︎
10. J. D. Jackson, Surface charges on circuit wires and resistors play three roles. Citation: American Journal of Physics 64, 855 (1996); doi: 10.1119/1.18112 p. 855
    View online: http://dx.doi.org/10.1119/1.18112? ↩︎
11. Feynman uses slightly different language in the audio recording. Some highlights “crazy” 35:54, “screwy, screwy, intuitively screwy” 38:22 “nuts” 41:33 “this dumb thing” 42:12 The Feynman Lectures on Physics Playlist CH 27 ↩︎
12. M.A. Gottlieb and R. Pfeiffer, California Institute of Technology (1964, 2006, 2013) “Feynman lectures, Book 2, chapter 27-5” The Feynman Lectures on Physics Vol. II Ch. 27: Field Energy and Field Momentum (caltech.edu) ↩︎
13.  Lavoisier changed the name from “igneous fluid” to “caloric” in 1783 p. 19 “This is what had determined me, in the memoir that I published in 1777 (1), to designate it under the name of igneous fluid and matter of heat….We have consequently designated the cause of heat, the eminently elastic fluid that produces it, by the name of caloric.” translated from (Note, link says it is not secure so view at your own risk).
    Oeuvres de Lavoisier. Tome premier. Traité élémentaire de chimie. – Antoine-Laurent LAVOISIER (1743-1794) ↩︎
14. Carnot, Sadi, 1796-1832; Clapeyron, E. (Emile), 1799-1864 “Reflections on the motive power of fire” Mémoire sur la puissance motrice de la chaleur. English; Clausius, R. (Rudolf), 1822-1888. Ueber die bewegende Kraft der Wärme. English (1960) https://archive.org/details/reflectionsonmot00carn/page/n31/mode/2up? ↩︎
15. B. Count, Philosophical Transactions of the Royal Society of London, Volume 88, Issue 88 (1798) p. 89 IV. An inquiry concerning the source of the heat which is excited by friction ↩︎
16. W. Thomson, Transactions of the Royal, (1849) p. 543 [PDF] XXXVI.—An Account of Carnot’s Theory of the Motive Power of Heat;with Numerical Results deduced from Regnault’s Experiments on Steam. | Semantic Scholar ↩︎
17. Carnot, Sadi, 1796-1832; Clapeyron, E. (Emile), 1799-1864 “Reflections on the motive power of fire” Mémoire sur la puissance motrice de la chaleur. English; Clausius, R. (Rudolf), 1822-1888. Ueber die bewegende Kraft der Wärme. English (1960) ↩︎
18. K. Joseph, First Law of Thermodynamics: History of the Concept video (2020) https://youtu.be/a9c7u-FM-Wc?si=NYq9mnd4O3W8p7kF ↩︎
19. P. E. A. Vonlenard, Noble Lecture, On Cathode Rays, p. 123 (1906)
    Philipp E. A. Lenard – Nobel Lecture ↩︎
20. Philipp Lenard and the Photoelectric Effect, 1889-1911 Author(s): Bruce R. Wheaton Source: Historical Studies in the Physical Sciences, Vol. 9 (1978), p. 300 Philipp Lenard and the Photoelectric Effect, 1889-1911 ↩︎
21. M.A. Gottlieb and R. Pfeiffer, California Institute of Technology (1964, 2006, 2013), Feynman lectures, Book 2, chapter 27-5. The Feynman Lectures on Physics Vol. II Ch. 27: Field Energy and Field Momentum (caltech.edu) ↩︎
22. Giorgi Rational Units of Electromagnetism (1902) p. 5 Giovanni Giorgi – Original paper | IE ↩︎
23. International Electrical Congress, Transactions of the International Electrical Congress (1904) https://archive.org/details/transactionsint06conggoog/page/136/mode/2up? History of the S ↩︎
24. J.C. Maxwell, 1831-1879; Niven, W. D. (William Davidson), Sir, 1842-1917” A treatise on electricity and magnetism” (1881) https://archive.org/details/electricityndmag02maxwrich/page/n275/mode/2up? ↩︎
25. J.C. Maxwell, 1831-1879; Niven, W. D. (William Davidson), Sir, 1842-1917” A treatise on electricity and magnetism” (1881) p. 249 and 251. ↩︎
26. Steinmetz, “On the Law of Hysteresis” (1892) p. 52 Transactions of the American Institute of Electrical Engineers – Google Books ↩︎
27. Faraday described this as, “a certain polarized state of the particles, into which they are thrown by the electrified body sustaining the action, the particles assuming positive and negative points or parts, which are symmetrically arranged with respect to each other and the inducting surfaces or particles.”Michael Faraday, “Experimental Researches in Electricity – Series 11” (Dec 21, 1837) Proceedings of the Royal Society vol 128 (1838), p. 20 paragraph [1298] https://doi.org/10.1098/rstl.1838.0002 ↩︎
28. A. Sommerfeld, Electrodynamics Lectures on theoretical physics,. Vol.III (1952) https://archive.org/details/electrodynamicsl0003arno_e7b9/page/8/mode/2up? ↩︎
29. A. Sommerfeld, Electrodynamics Lectures on theoretical physics,. Vol.III (1952) https://archive.org/details/electrodynamicsl0003arno_e7b9/page/6/mode/2up? ↩︎
30. Giorgi Rational Units of Electromagnetism (1902) p. 5 Giovanni Giorgi – Original paper | IEC ↩︎
31. R. J. Meredith, Engineers’ Handbook of Industrial Microwave Heating (1998) p. 22 Engineers’ Handbook of Industrial Microwave Heating – Google Books ↩︎
32. Maxwell “A Treatise on Electricity and Magnetism” (1873) vol 2 p. 236-7 A treatise on electricity and magnetism : Maxwell, James Clerk, 1831-1879 : Free Download, Borrow, and Streaming : Internet Archive ↩︎
33. Royal Society, Philosophical Transactions of the Royal Society of London (1885) https://archive.org/details/philosophicaltr07britgoog/page/n100/mode/2up? ↩︎
34. A. Das, Lectures on Electromagnetism (2013) p. 59 Lectures on Electromagnetism – Google Books ↩︎
35. A. Sommerfeld, Electrodynamics Lectures on theoretical physics,. Vol.III (1952) https://archive.org/details/electrodynamicsl0003arno_e7b9/page/146/mode/2up? ↩︎
36. D. Whiteson, University of California Irvine Daniel Whiteson’s Zoom Meeting – Zoom ↩︎
37. Royal Society, Philosophical Transactions of the Royal Society of London (1885) https://archive.org/details/philosophicaltr07britgoog/page/n86/mode/2up? ↩︎
38. Cambridge Philosophical Society Proceedings (1848) p. 104 Mathematical and physical papers ↩︎
39. D. Whiteson, University of California Irvine Daniel Whiteson’s Zoom Meeting – Zoom ↩︎
40. A. Sommerfeld, Electrodynamics Lectures on theoretical physics (1952) “Maxwell’s equations”: https://archive.org/details/electrodynamicsl0003arno_e7b9/page/18/mode/2up? Lorentz’s force at  https://archive.org/details/electrodynamicsl0003arno_e7b9/page/238/mode/2up? ↩︎
41. A. Sommerfeld, Electrodynamics Lectures on theoretical physics (1952) https://archive.org/details/electrodynamicsl0003arno_e7b9/page/n9/mode/2up? ↩︎
42. A. Sommerfeld, Electrodynamics Lectures on theoretical physics (1952) https://archive.org/details/electrodynamicsl0003arno_e7b9/page/26/mode/2up? ↩︎
43. A. Sommerfeld, Nobel Prize Case Study.Nobel Prize Case Study: Sommerfeld_NobelCaseStudy.pdf ↩︎
44. Werner Heisenberg Exhibition: 1The German Physics – Attack on Heisenberg. “The Black Corps” ↩︎
45. A. Sommerfeld, Electrodynamics Lectures on theoretical physics (1952) pp. 25-29 https://archive.org/details/electrodynamicsl0003arno_e7b9/page/24/mode/2up? ↩︎
46. A. Sommerfeld, Electrodynamics Lectures on theoretical physics (1952) https://archive.org/details/electrodynamicsl0003arno_e7b9/page/146/mode/2up? ↩︎
47. K. Joseph, Maxwell’s Equations Explained: Supplement to the History of Maxwell’s Equation video (2023) https://youtu.be/aFYKKSoXC5Y?si=hmZOk2Egobglm32j&t=576 ↩︎
48. Einstein, Albert (Autobiographical Notes) “Albert Einstein: Philosopher-Scientist” (1949) p. 47 https://archive.org/details/alberteinsteinph0000unse/page/46/mode/2up? ↩︎
49.  https://youtu.be/B0JlGI9qfEo  ↩︎
50. M.A. Gottlieb and R. Pfeiffer, California Institute of Technology, Chapter 36.3 (1964, 2006, 2013)
    The Feynman Lectures on Physics Vol. II Ch. 36: Ferromagnetism (caltech.edu) ↩︎
51. Max Planck, Henry L Brose, Theory of Electricity and Magnetism “Introduction to theoretical Physics” (1949). https://archive.org/details/theoryofelectric0003maxp/page/10/mode/2up? ↩︎
52. Niels Bohr, On the Quantum “Theory of Line-spectra, Part 1” (1918)  P. 80 On the Quantum Theory of Line-spectra – Niels Bohr – Google Books ↩︎
53. M.A. Gottlieb and R. Pfeiffer, California Institute of Technology (1964, 2006, 2013).  The Feynman Lectures on Physics Vol. II Ch. 36: Ferromagnetism ↩︎
54. M.A. Gottlieb and R. Pfeiffer, California Institute of Technology (1964, 2006, 2013).  The Feynman Lectures on Physics Vol. II Ch. 27: Field Energy and Field Momentum vs. A. Sommerfeld, “Electrodynamics Lectures on theoretical physics Vol.III” (1952)
     https://archive.org/details/electrodynamicsl0003arno_e7b9/page/26/mode/2up? ↩︎
55. vs. A. Sommerfeld, “Electrodynamics Lectures on theoretical physics Vol.III” (1952)
    https://archive.org/details/electrodynamicsl0003arno_e7b9/page/18/mode/2up? ↩︎
56. M.A. Gottlieb and R. Pfeiffer, California Institute of Technology (1964, 2006, 2013) The Feynman Lectures on Physics Vol. II Ch. 27: Field Energy and Field Momentum vs. A. Sommerfeld, “Electrodynamics Lectures on theoretical physics Vol.III” (1952)
    https://archive.org/details/electrodynamicsl0003arno_e7b9/page/26/mode/2up? ↩︎
57. A. Sommerfeld, “Electrodynamics Lectures on theoretical physics Vol.III” (1952)
    https://archive.org/details/electrodynamicsl0003arno_e7b9/page/34/mode/2up? ↩︎
58. R.P. Feynman, The Feynman lectures on physics. 1-3 (1964)
     https://archive.org/details/r-p-feynman.-the-feynman-lectures-on-physics-vol-1-3/page/3/mode/2up? ↩︎
59. B.I. Bleaney (Electricity and Magnetism) P. 236-7 (1957) https://archive.org/details/electricitymagne0000bibl_h8i3/page/236/mode/2up? ↩︎
60. M.A. Gottlieb and R. Pfeiffer, California Institute of Technology (1964, 2006, 2013) “Section 2-1” The Feynman Lectures on Physics Vol. II Ch. 2: Differential Calculus of Vector Fields Feynman The Feynman Lectures on Physics Playlist S2 4:50+ (edited for time) ↩︎
61. R. Phillips, The Feynman lectures on physics, pages 1-3, (1964)
     https://archive.org/details/r-p-feynman.-the-feynman-lectures-on-physics-vol-1-3/page/3/mode/2up? ↩︎
62. J.C. Maxwell and Sir J.J. Thomson, “A treatise on electricity and magnetism” (1892)
     https://archive.org/details/atreatiseonelec02thomgoog/page/n14/mode/2up? ↩︎
63. W. Bowyer and J. Nichols for Lockyer Davis, printer to the Royal Society P. 122 (1834)  Philosophical Transactions, Giving Some Account of the Present Undertakings… – Google Books ↩︎
64. M.A. Gottlieb and R. Pfeiffer, California Institute of Technology, “27-5” (1964, 2006, 2013)  The Feynman Lectures on Physics Vol. II Ch. 27: Field Energy and Field Momentum ↩︎
65. David Griffiths Electrodynamics, page 392 (2019) https://archive.org/details/davidgriffithselectrodynamics/page/n391/mode/2up? ↩︎
66. D. Whiteson, University of California Irvine  Daniel Whiteson’s Zoom Meeting – Zoom ↩︎
67. W. Olson, The Origins of a Warning from Voltaire, (2020)  The Origins of a Warning from Voltaire | Cato Institute ↩︎