The the The Meaning Behind the Symbols Common in Symbols in Starring… h p c a s Ω λ υ I π Q S e L James Lincoln, Tarbut V’ Torah Irvine, CA, 2010 Introduction & Motivation Often times in Physics we (teachers and students) get so deep into solving a “problem” that the pursuit becomes entirely mathematical. We can sometimes forget that the formulas we are working with actually stand for physical things. This project is my attempt at an antidote for that, and as a reminder that the symbols were chosen for a reason; they actually symbolized something. Introduction First recall that the formulas do not actually exist in nature. They are only a tool that assists in the understanding. I often have to remind my students that there are no formulas in the real world. That the symbols are representing real objects and interactions; and that in the end it is the result of those interactions that generates a predictable reality. It is naïve to believe that the formulas themselves are controlling reality, or that the physical world is obeying the laws we have written down. On the contrary, we are merely attempting to approximate behaviors with these algebraic formulae. Some Quick and Easy Ones - 1 The symbol “equals” or “ = ” represents two segments of the same length. “…to avoide the tediouse repetition of these woordes: is equal to: I will sette [a pair of] lines of one lengthe, thus: = bicause noe 2 thinges can be moare equalle…” -Robert Recorde, 1543 ( The originator of “ = ” ) Are these two equals? Source: E=mc2, Bodanis, pg 25 Some Quick and Easy Ones - 2 •In 1637, Rene Descartes in his La Geometrie used letters toward the end of the alphabet, especially x, for unknowns, and letters near the beginning of the alphabet for given quantities, ie a, b, c. ▪Descartes introduced the equation ax + by = c, which is still used to describe the equation of a line. Source: Return of Descartes Johnson, pg 145 http://veling.nl/anne/templars/ constants.html Some Quick and Easy Ones - 3 The symbol “ π ” was popularized by Euler who used it in the year 1748 * in Introductio in Analysin Infinitorum. It was chosen because it is the first letter of the greek word for perimeter. περιμετρου ( “perimetron” ) Source: http://www.maths.tcd.ie/pub/HistMath/People/Euler/RouseBall/RB_Euler.html * Note it was first used by W. Jones, 1706 Some Quick and Easy Ones - 4 “Ask me a question in Latin and I will answer you in Greek.” - J. Robert Oppenheimer Some of the symbols slip from one language to another... v=λ υ A wave’s velocity is equal to the wave’s λength multiplied by the υumber of times it passes per second. Source: Old Text: An Introductory Course in College Physics, Black pg 577 Some Quick and Easy Ones - 5 The letter “ I ” has not always been the symbol for Current. In fact, “ C ” was also an earlier convention. As an example, Maxwell in 1876 writes Ohm’s Law: “ R = E/C ” At other times, he would use to the cross-cultural symbol letter “ I ” which he says represents the “Intensity” of the current. (Intensité is the French word for intensity.) Here he is discussing Faraday’s Lines of Force. I have a hypothesis that this “intensité” is actually related to way to measure current before there were galvanometers. When people actually had to feel the “Intensity of the pain” themselves in order to measure it. Any Ideas, sources? Source: The Scientific Papers of James Clerk Maxwell, V2, pg 534 V1, pg 182 Q: the symbol for Everything The everpresent the letter “Q” in Physics Equations almost always has its origin in the French word “ quantité.” For example: The Quantity of heat: Q=mcT ( Carnot ) The Quantity of charge: Q = CV (Coulomb) The Quantity of Motion q = m*v ( Descartes ) The Quantity of speed ( Planck ) q = s/T The Quantity of displacement? q (Hamilton?) The Quantity of Mass & Force Net Q = Q a (?) A few symbols I haven’t solved yet: s for distance? … Source: s = spacio, spatio, latin ? no source, Descartes, Einstein? p for momentum= perpetuum, perpetuity ? Source: no source, Hamilton? L for anguLar momentum = ??????? Source: no source, Hamilton? Do you know the answers? A More Thorough Discussion is required of the following symbols S – entropy c – speed of light a = ∑ F/m Ω – Ohm’s Unit k – Boltzmann h – Planck Here we go……. Der Entropie und Rudolf Clausius: S Rudolfo ClaSiuS – inventor of Entropie Der Entropie von Rudolf Clausius: S “I propose to name the quantity “ S ” the entropie of the system, after the Greek word [τροπη trope], the transformation. I have deliberately chosen the word entropie to be as similar as possible to the word energie: the two quantities to be named by these words are so closely related in physical significance that a certain similarity in their names appears to be appropriate.” - Rudolf Clausius, 1865, Clausius’ reason for choosing “ S ” is not provided There are three possible reasons for this choice… Source: Mechanical Theory of Heat, Clausius [1850-1865] Why did Clasius choose S for Entropie? Guess #1 Clausius named it after Sadi Carnot. This is a reasonable guess, especially because he often refers to him as “ S . Carnot ” in his writing. Also, much of Clausius is based closely around Carnot’s. It seems even more reasonable when we find out that he even invented his own Entropie unit, the Claud, “ Cl. ” (no joke) Source: Mechanical Theory of Heat, Clausius [1850-1865] Why did Clasius choose S for Entropie? Guess #2 Clausius chose it for the word summa, the integral. This seems less likely, and only possible because of his previous formulas of Entropie which often contain integrals. Source: Mechanical Theory of Heat, Clausius [1850-1865] Why did Clasius choose S for Entropie? Guess #3 - My Guess Clausius chose it as a symbol to match Entropie with the other symbols that are already associated. Remember Clausius said, “ I have deliberately chosen the word entropie to be as similar as possible to the word energie.” What letter would help make Entropie fit in with Energie? p – Pressure Q – Quantity of Heat R – Ideal Gas Konstante S – available T – Temperature U – Internal Energy hmmmmmm….? v – Volume W – Work (Clausius both used L and W) Source: Mechanical Theory of Heat, Clausius [1850-1865] NEXT! The speed of light, c The history of using “c” as the symbol for the speed of light in Physics was chronicled by Kenneth Mendelson in the article “The story of c” published in the Nov. 2006 Am. J. Physics. I will be referring heavily to that article and adding a few bits of extra insight; especially with Einstein. Mendelson found that c was first introduced by Wilhelm Weber (German), who recognized that it was a constant but did not connect it to light or EM waves. Also, Weber’s “ c ” was actually off by a factor of √2 . “Nevertheless it is from Weber that the letter c eventually came to denote the speed of light.” [Mendelson] the speed of light, c “Weber did not say why he chose c.” It is unlikely that he chose it to represent the word “celerity” or “celeritas” as is often believed. ( This idea was introduced by Asimov, with no historical context). In fact, Weber’s papers show that he originally assigned this constant the letter “a” then later switched to c. Not knowing it was the speed of light. It is possible he was using c to represent Constante. Often this German word is spelled with a K, Konstante, however in Weber’s original papers it is spelled with a “c”. (This flip-flop is not unusual). Reference: Isaac Asimov – C for Celeritas, Mag of Fantasy Sci Fi 1959 What about E=mc2 Initially, Einstein would follow Maxwell’s notation, using V for the speed of light (probably because he referred directly to Maxwell’s work so frequently). Einstein originally wrote “if a body emits energy L then its mass decreases by L / V2 .” Where he called L the available energy to be lost of radiation. [Source: Einstein, 1905 #4] How do you feel about L = m V2 ? ….. NOTE: The first person to use c exclusively for the speed of light was Max Abraham* (also German) in 1903, he also published a popular Electromagnetism textbook in 1904, which – translated to English – became and stayed popular up to the 1950’s. * Abraham studied under Max Planck What about E=mc2 Q. So why in 1907 did Einstein switch from L to ε and V to c ? It wasn’t just because it was the first letter of his name. A. Einstein made a new friend, someone who used c for the speed of light, and used ε for Energy. ………………Can you Guess who that friend was? Source: Einstein, Walter Isaacson What about E=mc2 Q. So why in 1907 did Einstein switch from L to ε and V to c ? It wasn’t just because it was the first letter of his name. A. Einstein made a new friend, someone who used c for the speed of light, and used ε for Energy. ………………Can you Guess who that friend was? Source: Einstein, Walter Isaacson Max Planck Die Universalkonstanten von Max Planck und von Ludwig Boltzmann Mr. h Planck, 1901 when he wrote ε=hν Mr. k Boltzmann, 1844-1906 note the equation S = k log W In 1899 Planck solves the equation for the Black Body Spectrum & finds two distinct constants in the solution. At first he calls them α and β Later he redoes the solution and republishes, changing the letters. He solves for the values of the constants, and interprets their units, calling both of them either “natural” or “universal” constants. Source: Planck 1900, Planck 1901 The Boltzmann Konstante He recognizes one of the constants from Boltzmann’s work. (Boltzmann and Planck corresponded frequently). It is the gas constant R divided by Avogadro’s Number NA. k = R / NA Planck gives it a new symbol, k, probably for konstante. …we continue… Source: Planck 1900, Planck 1901 The Planck Constant There’s also that other constant - a new one -he doesn’t recognize. Initially, Planck didn’t know what to call his new “Naturconstanten.” [Source:tp://isi.cbs.nl/NLet/NLet081.htm ] Thus, he named it h for hilfsgröße (auxiliary quantity) and made no further interpretation at this time. [Source:tp://isi.cbs.nl/NLet/NLet081.htm ] “In this situation, Planck had no reason to question the continuity of the resonator energy.” [Source: pg 75 Darrigol from C to Q Numbers ] Therefore, Planck probably did not fully understand that h was the Quantum of Action until Einstein explained it in 1906. [Source: Planck's Theory of Radiation and the Theory of Specific Heat" ] The Planck Constant ================= In this house taught MAX PLANCK The discoverer of the elementary Quantum of Action h from to 1889 1928 ================= Fun Fact: In German “h-bar” is Called “h-cross.” Retranslating Newton’s 2nd Law Newton wrote his second law in sentences, not equations. He also wrote the Principia in Latin. Latin does not translate very well into English. Therefore, some translations will differ slightly. Some will be more helpful than others. Retranslating Newton’s 2nd Law Here is one translation of N 2nd that I found to be enlightening: The alteration of motion* is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed. * where “Motion” = mass x velocity Source: Principia, Translated by Andrew Motte Retranslating Newton’s 2nd Law The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed. Essentially, a ~ F . The constant of proportionality is obviously 1/m. Thus: a = F/m can be read not as “acceleration” but as change or “alteration” of motion. Why is this helpful? Source: Principia, Translated by Andrew Motte Retranslating Newton’s 2nd Law Why is this helpful? If you tell students that force is proportional to “ a ” the alteration of motion - they will quickly understand that forces can do more than just speed things up. Forces can change direction – an alteration. Forces can reduce speeds – an alteration. Forces can add perpendicular components of velocity. It eliminates the confusion of “Does accelerating mean getting faster?” Is this ever an issue in your classroom? Ohm’s Unit Let’s do some quick unit analysis on Ohm’s Law. What is an Ohm? From Ohm’s law you might think that it is a Volt per Ampere. V / I = R Very well, then what is an ampere? What is a Volt? An Amp is 1 Coulomb/sec A Volt is a Joule/Coulomb. Very well, then by plugging, an Ohm is a Joule*sec/Coulomb2. This makes an Ohm……………. The Resistible G.S. Ohm Ohm’s Unit Makes an Ohm = 1 kilogram meter2/sec/ (.624*1019 e) ( where e = 1 electronic charge ) That does not look very standard. Where did we get this “Ohm?” Georg Simon Ohm (1789-1854) lived before Thompson and Millikan and therefore could not have possibly known the electronic charge. The Ohm is a derived unit. This is how he derived it in 1827* *source: metrication timeline . The Original Ohm Unit “Taking wire of different material, but of the same thickness, he found that the following lengths possessed equal conductivities: copper 1000, gold 574, silver 356*, zinc 333, brass 280, iron 174, platinum 171, tin 168, lead 97.” The measurements were made by using a torsion balance with a constant current running through it, via by Ampere’s attraction of two parallel currents. (Cajori) *An experimental error. Silver is actually a better conductor that copper, the real value is actually 1057 Source: A History of Physics Cajori The Original Ohm Unit It is important to note that Ohm’s Unit of resistance, “1000 units” of no. 10 [ American Standard Guage] copper wire*, when the unit is a foot results in 1 Ohm of resistance. 1 Ohm I do not know if Ohm’s unit was the “foot” (although it is possible because Germany didn’t go metric until the European “Convention du Mètre” in 1875. Hence there actually should be an apostrophe in the unit Ohms. From now on you may wish to refer to it as “Ohm’s Units for Resistance” or simply “Ohm’s.” *Note we now define the Ohm using Mercury rather than copper, because copper changes resistance quickly when heated. ~ FIN ~ Thank you. James Lincoln: [email protected] Some Quick and Easy Ones The letter “ G ” was of course first measured by Cavendish, but he did call it that. Instead he usd the symbol k, just like everyone else. G is a good example of when someone establishing a very good measurement assigns their own symbol to a constant. G was measured precisely in ________ in a german paper called _____________. (Popularized by Planck? Check constants of the universe book). Source: http://www.maths.tcd.ie/pub/HistMath/People/Euler/RouseBall/RB_Euler.html * Note it was first used by W. Jones, 1706 Introduction and Major Points Formulas were not always present throughout the history of physics. Consider Hooke’s Law as he states it: Latin: “ Ut tensio, sic vis ” ( As the extension, so the force ) Introduction and Major Points Change in motion are proportionate to the motive force impressed and takes place on a the straight line along which that force is impressed. Where previously Newton defined motion as the product of mass and velocity, ie momentum. Newton would also use segments for symbols, being highly geometrical in his writing For example, his formula for centripetal acceleration was written as bd2/sb. Introduction and Major Points Newton also often spoke in terms. Consider his statement of the second Law: Mutationem motus proportionalem esse vi motrici impressae et fieri secundum lineam rectam qua vis illa imprimitur. Change in motion are proportionate to the motive force impressed and takes place on a the straight line along which that force is impressed.