Physics Documents

John Denker

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The following were written mainly to answer questions that came up on Phys-L, the Forum for Physics Educators. There is also a secure version of this page.

  1. Spreadsheets for solving Laplace's equation. Suitable for students. Demonstrating gauge invariance. Demonstrating conservation of charge. Calculating the capacitance of oddly-shaped multi-electrode capacitors.
  2. The definition of capacitance, including the capacitance matrix for multi-terminal capacitors.
  3. An explanation of why there is fundamentally only one kind of charge, not two kinds of charge.
  4. A discussion of gorge versus charge. When people speak of the ``amount of charge '' in battery or capacitor, usually it is a misnomer. It would be better to speak of the gorge. We can gorge and disgorge the device. Charge is conserved; gorge is not.
  5. How to make a non-polarized capacitor using two polarized (e.g. electrolytic) capacitors back-to-back.
  6. A discussion of the notorious Two-Capacitor Problem. Contrary to what you may read in the literature, it is quite possible to very efficiently transfer energy and charge (or rather gorge) from one capacitor to another.
  7. A discussion of the electrophorus, and other variable-geometry capacitors.
  8. Tides. Why are the typical tides twice a day? Why are some tides once a day? Making hands-on models and/or mathematical models of the tide-producing potential.
  9. A discussion of Galileo's celebrated interrupted pendulum (also known as the stopped pendulum) and the related loop-de-loop maneuver.
  10. The real laws of thermodynamics. First law defined as conservation of energy. Second law defined as paraconservation of entropy. Entropy defined in terms of statistics.
  11. A discussion of whether a reaction will proceed spontaneously or not. This is related to the question of whether a reaction is reversible or irreversible. The fundamental criterion involves the total entropy, which is sometimes related to the system's free energy or free enthalpy.
  12. The importance of the black-box approach. Also some discussion of reality -- Are waves "real"? Is energy "real"? Also some discussion of reductionism.
  13. Conservation as related to Continuity and Constancy. Continuity of world-lines for charge, energy and other conserved quantities. This includes a video illustrating steady, conservative flow without constant density.
  14. A non-sneaky derivation of Euler's equation -- force and momentum-flow in fluid dynamics.
  15. A discussion of How to teach -- and how to learn -- general thinking skills, including critical thinking. The point is that thinking is important, portable, and learnable.
  16. The idea of successive refinement illustrated in a simple situation, namely building a model of the collision between two carts.
  17. A discussion of exploring a maze using only local information ... which serves as a metaphor for how research is done. There is also an online app that runs in the browser, allowing you to try your hand at exploring a maze.
  18. A more general discussion of the principles of teaching and learning.
  19. It is important to realize that words acquire meaning from how they are used ... not from some pithy dictionary-style definition.
  20. The spiral approach to thinking and learning.
  21. A checklist of techniques, useful for general-purpose problem-solving.
  22. Cause and Effect. Why it is important to think carefully about causation, and how to go about it.
  23. A discussion of hard versus soft evidence, why argument from authority is unscientific, and why it is necessary to challenge seemingly-well-established facts.
  24. On a more positive note, a discussion of scientific methods. Notice that I didn't say ``the'' scientific method.
  25. A discussion of changing only one variable at a time, which is not actually a very good idea. Usually you can get more and better information if you change multiple variables at a time.
  26. The definition of hypothesis .
  27. An essay on Truth in Contrast to Knowledge and Belief.
  28. A discussion of the notorious fallacy called argument from no evidence.
  29. A discussion of the so-called theory of intelligent design.
  30. A discussion of breadth, depth, and interdisciplinary connections.
  31. A rant about story problems including ill-posed problems, and the importance of not always following instructions.
  32. An introduction to probability. Topics include fundamental notions of probability measure, random walks, and convergence of distributions.
  33. An simple experiment that involves probability: tossing tacks.
  34. The ``Margin of Error'' of Polls -- Sampling Error, Bernoulli Processes, and Random Walks
  35. A discussion of how to report measurement uncertainties -- which is related to the heavily flawed notion of significant digits or significant figures.
  36. A simple Uncertainty Calculator (Crank Three Times) and a fancy Uncertainty Calculator (Monte Carlo) Given inputs with error bars, and a formula, it calculates the output and its error bars. The formula may be almost any mathematical expression, or a multi-step sequence of expressions. It can handle the correlations that arise when there are multiple variables. It runs in the browser, so you do not need to download anything. You should probably start with the documentation, which offers numerous live demonstrations of the calculator. There is some simple documentation (Crank Three Times) and fancy documentation (Monte Carlo).
  37. A discussion of data analysis, especially the risks of preprocessing data before modeling it.
  38. A discussion of data visualization. This includes visualizing the data while the experiment is still under way.
  39. A discussion of how to define mass.
  40. A careful definition of weight, definition of gravitational force, definition of gravity, definition of g, et cetera.
  41. Some key ideas needed in preparation for understanding gravitational waves, and for avoiding some misconceptions.
  42. An introduction to the ideas of force, momentum, torque, and angular momentum.
  43. A discussion what we mean by volt and voltage.
  44. A no-nonsense discussion of Kirchhoff's Circuit «Laws» i.e. Kirchhoff's Voltage «Law» and Kirchhoff's Current «Law».
  45. Some discussion of the response function of a damped harmonic oscillator, specifically an RLC circuit. In particular: there are two notions of bandwidth that arise, so you have to be careful.
  46. Some perspectives on thermal noise in resistors : Johnson noise and the Nyquist formula.
  47. How to think about the physics and mathematics of combining resistors in parallel.
  48. Some useful formulas for finding the center of mass of an aircraft (or similar object) in terms of the observed weight on each wheel.
  49. A careful definition of vapor, gas, and fluid.
  50. A discussion and definition of motion in a rotating reference frame, including ideas such as centrifugal force, plus better ideas such as centrifugal field and centrifugal acceleration.
  51. An inconclusive discussion of what is a fictitious force (or pseudo force).
  52. Can You Feel Gravity? What you feel is not explainable just by the gravity at your own location; what you feel is due to the difference between gravity at your location and gravity in distant parts of the world.
  53. A discussion of Kepler's second law which is also known as Kepler's equal-area law. This is related to conservation of angular momentum. This includes an explanation of why the classic picture from the Principia is a swindle, based on the provably wrong assumption that the average force is equal to the instantaneous force in the non-limiting case.
  54. A Simple Home-Made Accelerometer that can be made in about 20 minutes with ordinary materials: a lead weight, two rubber bands, a dowel rod, and some bailing wire.
  55. Some examples showing the value of the equivalence principle in practical situations. This includes a discussion of what ``Horizontal'' means during acceleration.
  56. A discussion of inertial reference frames, Newtonian reference frames, freely-falling reference frames, unaccelerated reference frames et cetera.
  57. The definition of anode and cathode, and a discussion of why these terms are usually not worth memorizing.
  58. A discussion of the definition of electrical resistance and its relationship to Ohm's law.
  59. Spectral Data for FD&C Food Coloring Dyes.
  60. Jodi Smith's new book on Wax Resist Decorated Eggs aka pysanky.
  61. Jodi Smith's book on Medieval Dyes -- plants, materials, dyeing procedures et cetera.
  62. My book on how to fly an airplane, including a chapter on how a wing works.
  63. A pattern for making a paper airplane that flies well.
  64. A detailed discussion of the question, What makes the car go? -- i.e. how to balance the energy budget and momentum budget.
  65. A discussion of why we need more than one definition of kinetic energy and more than one work / kinetic energy theorem. This is related to our notions of Momentum, Force times Time, and Force dot Distance.
  66. A discussion of Why the Sky is Blue.
  67. A brief discussion of localization, which explains What Makes White Things White and How Electrical Insulators Really Work. Also some discussion of what makes black things black.
  68. A discussion of how to evaluate research projects or other creative, risky-but-worthwhile endeavors.
  69. A diagram and explanation of the analemma and the equation of time.
  70. Instructions for how to tell time by the stars.
  71. An introduction to rapidities and boosts, and some insights on the structure of spacetime.
  72. A introduction to vectors, with an emphasis on the physical significance. A vector exists as a thing unto itself, independent of the choice of reference frame, and therefore not having -- nor needing -- any unique decomposition into components.
  73. A Welcome to Spacetime It emphasizes, at an introductory level, the idea that special relativity is the geometry and trigonometry of spacetime. It also emphasizes the ways in which special relativity simplifies and unifies a great many results that would otherwise need to be learned separately.
  74. A discussion of kinetic energy, including the non-relativistic limit, the non-relativistic limit, and everything in between. This includes suggestions for how to formulate things in a way that is numerically well behaved.
  75. A somewhat more technical discussion of the geometry and trigonometry of spacetime ... in particular an inquiry into how literally we can take the idea that time is the fourth dimension.
  76. A discussion of odometers and clocks in introductory special relativity. In particular, we use rulers that are not Lorentz-contracted and clocks that are not time-dilated. This does not require much beyond high-school notions of geometry, trigonometry, and vectors.
  77. A discussion of Velocity, Speed, Acceleration, and Deceleration , et cetera. Compare next item.
  78. A discussion of velocity and acceleration in spacetime, including the important dissimilarity between 3-velocity and 4-velocity. Compare previous item.
  79. A discussion of relativistic acceleration of an extended object, i.e. an object with some large size (L) undergoing a large acceleration (a), such that La/c2 is large compared to unity. Key ideas include hyperbolic motion, spacetime geometry including the center of the hyperbolas, and a generalization of centrifugal force.
  80. Some spacetime diagrams and some discussion to help with Visualizing the Liénard-Wiechert Potentials.
  81. Some fine points of Fourier Transforms and Spectrum Analyzers, including techiques for normalizing the abscissas and ordinates of Fourier transforms. Also a discussion of why you might want to increase the resolution of discrete Fourier transforms.
  82. A collection of tips and techniques for doing useful things with spreadsheets, including some slightly non-obvious things.
  83. A discussion of linear least-squares fitting, also known as linear regression, using a spreadsheet or otherwise, including the case of multiple fitted parameters, and including the case where the basis functions are nonlinear (even though the fitted function remains a linear combination of the basis functions). Examples include using the linest(...) spreadsheet function to fit a quadratic, or to fit a Fourier series.
  84. A discusion of Fitting an Exponential to Noisy Data. This could apply to radioactive decay, or to the growth of yeast in bread or beer, or to the spread of disease in some local area.
  85. A discussion of nonlinear least-squares fitting ... in particular, the procedure for estimating the uncertainty of the fitted parameters.
  86. An overview of higher math (including algebra, geometry, statistics, logic, etc.) including a discussion of why it is relevant in the real world.
  87. The smart version of the quadratic formula -- which serves as an illustration of basic notions of numerical methods.
  88. A discussion of logarithmic units in general, and of why a a centineper (cNp) is better than a percentage in many situations.
  89. Measuring Reaction Time by Dropping a Ruler. This is an easy experiment that provides some interesting information. Analyzing the data requires applying some basic physics.
  90. A quiz on the basic algebra and trig skills you ought to have in preparation for an introductory physics class.
  91. A brief discussion of positive and negative numbers, including the idea that the negative of a negative is a positive ... or perhaps better, the opposite of the opposite is the original thing.
  92. How to represent a line using a position vector (i.e. anchor point) plus a velocity vector (aka direction vector) ... and calculate the intersection of two such lines.
  93. An Introduction to Clifford Algebra.
  94. If you know about complex numbers, and a little bit about vectors, you can use that to jump-start your understanding of Clifford Algebra. So here is a side-by-side comparison of complex numbers and Clifford Algebra.
  95. A discussion of N-Dimensional Rotations, Including Boosts. Includes a review of various ways to represent rotations, including Clifford algebra, matrices, Rodrigues vectors, and/or Euler angles.
  96. An exercise using vectors to calculate direction (or heading) along a great circle from point a to point b. Mentions Clifford Algebra in passing.
  97. How to calculate the area of parallelograms and the volume of parallelepipeds using wedge products (Clifford Algebra).
  98. An exercise checking the correspondence between the Clifford Algebra formulation of electromagnetism and the old-fashioned vector formulation of Maxwell's equations.
  99. Another exercise, calculating the magnetic field of a long straight wire from scratch, using the Clifford Algebra formulation.
  100. The force and work associated with a current in a wire in a magnetic field.
  101. The microscopic origins of the magnetic field of a current-carrying wire, in terms of bivectors and a space-time diagram, explaining magnetism in terms of electrostatics plus relativity.
  102. A correct, modern analysis of the Faraday Rotor Experiment. This is commonly called Faraday's paradox but of course it is not really a paradox.
  103. How to Make Antimatter -- An Exercise using Four-Vectors.
  104. A discussion of pressure, degeneracy, exchange energy (exchange force), neutron stars, etc.
  105. A discussion of the ideal gas law and adiabatic gas law, plus osmosis, osmotic pressure, and osmotic flow
  106. A fluid has pressure everywhere, not just at tangible boundaries.
  107. There is a mathematical theorem that says vortex lines are endless. They either go on forever, or form closed loops. If something looks like half a vortex loop, it cannot possibly be what it seems, and it's almost certainly worth your trouble to figure out what's actually happening.
  108. A careful derivation (actually two derivations) of Bernoulli's principle aka Bernoulli's equation aka Bernoulli's theorem. In particular, we find that the equation directly describes the enthalpy (not energy) of the fluid parcel. The equation applies just fine to compressible fluids, which is good thing, because there are no incompressible fluids.
  109. A puzzle about the inertia of a cube, illustrating qualitative reasoning, and illustrating the geometrical and physical significance of a tensor, with applications to the Wigner-Eckart theorem.
  110. The famous Twelve Coins Puzzle with a discussion involving Design-of-Experiment, Information Theory, and Communication Theory.
  111. An analysis of the famous Twenty Questions game, including a method for winning 100% of the time. The analysis is a good illustration of information theory.
  112. Pierre's Puzzle -- which asks about the symmetry of electromagnets and permanent magnets (such as compass needles) -- solved using bivectors.
  113. A riddle: Why does a jet engine turn the right way?
  114. A related bit of physics: sailing upwind ≡ sailing directly downwind, faster than the wind.
  115. A physics lesson without words. Explain what you see. How sure are you? How do you know?
  116. An analysis of clocks that use pulses of light to keep time.
  117. A special relativity puzzle, involving the infamous twins, one of whom goes on a trip.
  118. A puzzle illustrating some points about general relativity, namely how various forms of energy contribute as sources of the gravitational field.
  119. Setting the Alarm Clock -- A Story about Symmetry and Information. Thermodynamics and information theory provide results that are independent of mechanism.
  120. An illustration of Liouville's phase-space theorem as applied to the light passing through a thin lens.
  121. Some diagrams illustrating Liouville's theorem, including motion in two dimensions (i.e. Keplerian circular motion).
  122. Basic Properties of a Symplectic Integrator including why it is useful, and how it conserves phase space.
  123. How to build a table-top model of Straight Lines in a Curved Space -- Geodesics, General Relativity, and Embedding Diagrams.
  124. A discussion of the expansion of the universe, which addresses some fundamental questions about what we mean by distance.
  125. Here are some diagrams that may help you visualize a non-grady field, i.e. a non-conservative field, i.e. one that is not the gradient of any potential. More precisely, this shows how to visualize an inexact one-form.
  126. A simulation illustrating some of the factors that increase and decrease the entropy of a system. Entropy is a statistical measure of how much you don't know about the system.
  127. A discussion of negative temperatures in a spin system.
  128. Temperature : Definition and Fundamental Properties.
  129. The basic properties of differential forms.
  130. Some rough notes on formulating thermodynamics in terms of differential forms.
  131. A discussion of why we sometimes observe sublimation and sometimes instead observe melting followed by evaporation. This is easily explained in terms of a tradeoff of energy versus entropy. The same ideas provide a nice explanation for the freezing-point depression when an impurity is added to the liquid.
  132. A pictorial representation of partial derivatives, including a discussion of what is a ``direction'' in terms of pointy vectors, differential forms, et cetera. This gives us not just a geometric interpretation of partial derivatives, but actually a topological interpretation.
  133. A nice way to draw the periodic table of the elements, as a cylinder with bulges in 3D.
  134. A discussion of the relationships between three ideas, namely the Aufbau principle, isoelectronic correspondence, and ionization.
  135. A discussion of how to balance chemical reaction equations, including charge-balance as well as atom-balance.
  136. A discussion of how to balance chemical reaction equations systematically, or more generally how to solve any system of N linear equations in N+m unknowns. This topic is known as linear algebra. Methods include Gaussian elimination (which can be carried out using only pencil and paper) matrix inverse methods, and pseudo-inverse methods.
  137. A perl program to parse chemical formulas and produce the list of elements and the amount of each.
  138. A classroom demo of catalysis using gelatin and cysteine protease enzymes.
  139. Some graphs of pH versus concentration for various pKa values ... including weak acids and strong acids, as well as intermediate-strength acids, which are particularly interesting.
  140. Some hints on how to do basic math calculations, including long multiplication and long division.
  141. A discussion of why students, especially in an introductory course, should be given the best evidence, not the most ancient evidence. To say it another way, one should not use the history of science to organize or motivate the study of science, especially in an introductory course. The true history of science is advanced topic, suitable for those who already have a good grasp of science and a good grasp of historical methods. Studying the false history of science is worse than useless.
  142. An introduction to atoms. This includes a discussion of the notions of atom, atomic number, nucleus, proton number, molar mass, nuclide, isotope, neutron number, nucleon number, and baryon number . Also a deprecation of outdated and/or confusing terms such as atomic weight, atomic mass, atomic mass number, and mass number .
  143. An overview of some easily-understandable evidence that our world is governed by the laws of quantum mechanics.
  144. In some cases a 4π rotation gets you back where you started, but a 2π rotation does not. This is known as the Dirac String Trick.
  145. A deprecation of the alleged distinction of "chemical" versus "physical" changes.
  146. How to think about the specific heat capacity and enthalpy, including the latent heat of a first-order phase transition.
  147. A discussion of how to draw molecules and chemical bonds ... like Lewis dot diagrams, except not wrong. It turns out that Lewis dot diagrams have no firm theoretical basis, and despite some successes have many failures.
  148. An introduction to waves. This includes an answer (or non-answer) to the question of What is a Wave?
  149. Various ways to make models and pictures of atomic wavefunctions (aka atomic orbitals). This includes an animation, i.e. a java applet that adds dots one by one, gradually building up a picture of the probability distribution, showing the position of an electron within the wavefunction.
  150. A movie of the earth as it spins on its axis and orbits around the center of mass of the earth/moon system
  151. A discussion of Fields, and Excitations in the Fields. This also discusses the so-called wave/particle duality and argues that there's no such thing.
  152. A discussion of why atomic physics says that electrons hate each other and pair up only as a last resort (Hund's rule #1) whereas high-school chemistry deals almost exclusively with molecules that have all their valence electrons paired up. Why pairs -- Or not?
  153. A discussion of how quarks combine to form mesons. This includes a discussion of why there are nine lightweight pseudoscalar mesons. It even explains why it is possible to pick out 8 of the 9 and call them an octet ... although I'm not convinced this is worth the trouble.
  154. A discussion of the correct direction of the arrow representing a dipole moment, in molecules and otherwise.
  155. A discussion of why in the first excited level of a dye molecule, the triplet (T1) always has lower energy than the singlet (S1). This turns out to be a thinly-disguised version of Hund's rule #1. We explain why Hund's rule applies to molecules, not just atoms.
  156. A discussion of what happens to the amplitude, power, and quantum-mechanical probability when you add waves. This includes explaining why the proverbial rule 1 & 1 makes 2 is only valid in the classical limit.
  157. A discussion of the quantum harmonic oscillator -- energy versus temperature.
  158. A discussion of coherent states, also known as Glauber states. This includes a discussion of how and why not all waves are quantized. There's also a movie of a squeezed state.
  159. A terse review of the concept of quantum-mechanical spin.
  160. Some notes on static electricity aka contact electrification.
  161. How an electrical battery works.
  162. An overview of the chemical reactions in a lead-acid battery and how they reputedly work --- including some unanswered questions.
  163. Some notes on how to take care of a swimming pool.
  164. Some words about how to understand the Boundary between Quantum Mechanics and the Classical Limit.
  165. An introduction to scaling laws, including non-dimensional scaling.
  166. An discussion of dimensional analysis.
  167. A discussion of units, including how to use them and how to think about them in physical and algebraic terms.
  168. A discussion of units of dimension one, also known as dimensionless units.
  169. A discussion of the Secchi disk pattern, which illustrates the fact that boundaries have zero width and therefore exhibits some interesting scaling properties.
  170. A discussion of thermal wave packets including the observation that the thermal de Broglie length does not really behave like a wavelength. It has more to do with the envelope-size of the wave packet.
  171. A discussion of The Exchange of Identical and Possibly Indistinguishable Particles and how that relates to the Pauli exclusion principle.
  172. A discussion of what happens as we approach exhaustion of fossil resources of energy, including coal, oil, and uranium-235.
  173. A computer-based lock-in amplifier. This is not a software simulation, but a real, fully functional lockin using PC hardware (including the audio interface, aka ``sound card''). It is good for measuring phasors, and for synchronous detection of tiny signals.
  174. A perl program to calculate your local barometric pressure, based on weather reports. This is useful if you don't own a precision barometer, and don't want to buy one. Also a spreadsheet to calculate your local barometric pressure, based on weather reports.
  175. A discussion of barometric pressure and pressure altimetry, including aircraft altimeters. This includes a discussion of how the Kollsman window (altimeter setting) works.
  176. Some examples of weird terminology, where the name of the thing does not provide a good description of the thing.
  177. Print your own spacetime diagram paper: blue, blue-only, red, red-only, redblue.

    Also triangular-ruled graph paper.

  178. A discussion of various situations where we need to plot something that isn't a function, and/or where the concept of ``axes'' is unhelpful.
  179. A demonstration that you can write high-level graphics-intensive code that runs entirely in the browser. The objective is maximally convenient portability. Your can write code that is "mostly" compatible with conventional visual python. That's a high-level language with a high-level graphics library. I call this approach glorpy. Your code then gets compiled and executed, all in the browser.
  180. Comments on the ``California Standards Test'' in physics, and the process by which the test is constructed, including the underlying ``Science Content Standards for California Public Schools''.
  181. A discussion of electrical power grid physics and engineering including some thoughts about the 15 August 2003 northeast blackout.
  182. Utility is the best mnemonic. Also, some people are luckier than others ... for a reason.
  183. Another story about why some people are luckier than others.
  184. An HTML technique for adding decorations to symbols ... decorations such as a dot (perhaps to indicate a time derivative) or an overbar (perhaps to indicate an average) or an arrow (to indicate a vector).
  185. Physics Books. Recommended as a "starter kit" for a college library.
  186. A book review:
    Tom M. Apostol, Calculus.
  187. A book review:
    Philip Keller, The New Math SAT Game Plan.
  188. Not yet a book review, but some notes about the book:
    Chabay and Sherwood, Matter and Interactions.
  189. A book review:
    Serway & Faughn, Holt Physics.
  190. A book review:
    Paul G. Hewitt, Conceptual Physics.
  191. A book review:
    Paul Zitzewitz, Glencoe Physics: Principles and Problems.
  192. A book review:
    Arnold Arons, Teaching Introductory Physics.
  193. A brief book review:
    Barton & Black, An Introduction to Practical Physics For Colleges and Schools (1922).
  194. A book review:
    Millikan & Gale, Practical Physics (1906-1922).
  195. Directory listing. Miscellaneous physics-related diagrams, spreadsheets, et cetera.
  196. jsd home page.
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