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.
- 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.
definition of capacitance, including the capacitance matrix for
- An explanation of why there is fundamentally only one kind of charge, not two kinds of charge.
- 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.
- 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
- A discussion of the
electrophorus, and other variable-geometry capacitors.
- 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
- A discussion of Galileo's celebrated interrupted pendulum (also known as the
stopped pendulum) and the related loop-de-loop maneuver.
- The real laws of thermodynamics.
First law defined as conservation of energy. Second
law defined as paraconservation of entropy. Entropy defined in terms
- 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.
- The importance of the black-box
approach. Also some discussion of reality -- Are
waves "real"? Is energy "real"? Also some discussion of reductionism.
- 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
- A non-sneaky derivation of
-- force and momentum-flow in fluid dynamics.
- 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.
- The idea of successive
refinement illustrated in a simple situation, namely
building a model of the collision between two carts.
- 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.
- A more general discussion of the principles of teaching
- It is important to realize that words acquire meaning from
how they are used ... not from some pithy dictionary-style
spiral approach to thinking and learning.
- A checklist
of techniques, useful for general-purpose problem-solving.
- Cause and Effect.
Why it is important to think carefully about causation, and
how to go about it.
- A discussion of hard versus soft
evidence, why argument from authority
is unscientific, and why it is necessary to challenge
- On a more positive note, a discussion of scientific methods. Notice that I
didn't say ``the'' scientific
- 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.
- The definition of hypothesis
- An essay on Truth
in Contrast to Knowledge and Belief.
- A discussion of the notorious fallacy called argument from no evidence.
- A discussion of the so-called theory of intelligent design.
- A discussion of breadth, depth,
and interdisciplinary connections.
- A rant about story problems including
ill-posed problems, and the importance of
not always following instructions.
- An introduction to
probability. Topics include fundamental notions of probability measure, random walks, and
convergence of distributions.
- An simple
experiment that involves probability: tossing tacks.
- The ``Margin
of Error'' of Polls -- Sampling Error, Bernoulli Processes, and Random
- A discussion of how to report measurement
uncertainties -- which is related to the heavily flawed notion of
digits or significant figures.
- 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).
- A discussion of data analysis, especially the risks of preprocessing data
before modeling it.
- A discussion of data
visualization. This includes visualizing the
data while the experiment is still under way.
- A discussion of how to define
- A careful definition of weight,
definition of gravitational force,
definition of gravity,
definition of g,
- Some key ideas needed in preparation for
understanding gravitational waves, and for avoiding some misconceptions.
- An introduction to the ideas of force, momentum,
torque, and angular momentum.
- A discussion what we mean by volt and voltage.
- A no-nonsense discussion of Kirchhoff's Circuit «Laws» i.e. Kirchhoff's Voltage «Law» and Kirchhoff's Current «Law».
- 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.
- Some perspectives on thermal noise in resistors : Johnson noise
and the Nyquist formula.
- How to think about the physics and mathematics of combining
resistors in parallel.
- Some useful formulas for finding the center of mass of an aircraft (or
similar object) in terms of the observed weight on each wheel.
- A careful definition of
vapor, gas, and fluid.
- 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
- An inconclusive discussion of what is a fictitious force (or pseudo force).
- 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.
- 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.
- 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
- Some examples showing the value of
the equivalence principle in practical situations. This includes
a discussion of what
``Horizontal'' means during acceleration.
- A discussion of inertial
reference frames, Newtonian reference frames, freely-falling
reference frames, unaccelerated reference frames et cetera.
- The definition of anode
and cathode, and a discussion of why these terms are
usually not worth memorizing.
- A discussion of the definition of
and its relationship to Ohm's law.
- Spectral Data for FD&C Food Coloring Dyes.
- Jodi Smith's new book on Wax Resist Decorated Eggs aka pysanky.
- Jodi Smith's book on Medieval
Dyes -- plants, materials, dyeing procedures et cetera.
- My book on how to fly an airplane,
including a chapter on how a wing
- A pattern for making a paper airplane that flies well.
- A detailed discussion of the question,
What makes the car go? -- i.e.
how to balance the energy budget and momentum budget.
- 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.
- A discussion of
Why the Sky is Blue.
- 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.
- A discussion of how to
evaluate research projects
or other creative, risky-but-worthwhile endeavors.
- Instructions for how to
tell time by the stars.
- An introduction to
rapidities and boosts,
and some insights on the structure of spacetime.
- 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.
- 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
- 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.
- 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
- 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.
- A discussion of Velocity, Speed,
Acceleration, and Deceleration , et cetera. Compare next
- A discussion of velocity and
acceleration in spacetime, including the important dissimilarity
between 3-velocity and 4-velocity. Compare previous item.
- 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
- Some spacetime diagrams and some discussion to help with Visualizing
the Liénard-Wiechert Potentials.
- 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
- A collection of tips and
techniques for doing useful things with spreadsheets, including
some slightly non-obvious things.
- 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
- 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.
- A discussion of nonlinear
least-squares fitting ... in particular, the procedure for
estimating the uncertainty of the fitted parameters.
- An overview
of higher math (including algebra, geometry, statistics, logic,
etc.) including a discussion of why it is relevant in the
- The smart
version of the quadratic formula -- which serves as an
illustration of basic notions of numerical
- A discussion of logarithmic units in general, and of why a a centineper (cNp) is
better than a percentage in many situations.
Reaction Time by Dropping a Ruler. This is an easy experiment
that provides some interesting information. Analyzing the data
requires applying some basic physics.
- A quiz on the basic algebra and trig skills you ought to have in
preparation for an introductory physics class.
- 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.
- An Introduction to Clifford Algebra.
- 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
- 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.
- An exercise using vectors to calculate direction (or heading)
along a great circle from point a to point b. Mentions
Clifford Algebra in passing.
- How to calculate the area of
parallelograms and the volume of
parallelepipeds using wedge products (Clifford Algebra).
- An exercise checking the correspondence between the
Clifford Algebra formulation of
electromagnetism and the old-fashioned vector formulation of
- Another exercise, calculating the magnetic field of a
long straight wire from scratch, using the Clifford Algebra
- The force and work associated with a current in a wire in a magnetic
- 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
- A correct, modern analysis of the
Rotor Experiment. This is commonly called
paradox but of course it is not really a paradox.
- How to Make Antimatter -- An
Exercise using Four-Vectors.
- A discussion of pressure,
degeneracy, exchange energy (exchange force), neutron stars, etc.
- A discussion of the ideal gas law
and adiabatic gas law, plus osmosis, osmotic pressure, and osmotic flow
- A fluid has pressure everywhere, not just at tangible boundaries.
- 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.
- A careful derivation (actually two derivations) of
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.
- 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.
- The famous Twelve Coins Puzzle
with a discussion involving Design-of-Experiment, Information Theory,
and Communication Theory.
- 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.
- Pierre's Puzzle -- which
asks about the symmetry of
electromagnets and permanent magnets (such as compass needles) --
solved using bivectors.
- A riddle: Why
does a jet engine turn the right way?
- A related bit of physics:
sailing upwind ≡ sailing directly downwind, faster than the
- A physics lesson without
Explain what you see. How sure are you? How do you know?
- An analysis of clocks that use pulses of light to keep time.
- A special relativity puzzle, involving the infamous
twins, one of whom goes on a trip.
- A puzzle illustrating some points about
general relativity, namely how various forms of energy contribute as
sources of the
- Setting the Alarm
Clock -- A Story about Symmetry and Information.
Thermodynamics and information
theory provide results that are independent of mechanism.
- An illustration of Liouville's phase-space theorem as
applied to the light passing through a thin lens.
- Some diagrams illustrating
Liouville's theorem, including motion in two dimensions
(i.e. Keplerian circular motion).
- Basic Properties of a Symplectic Integrator including
why it is useful, and how it conserves phase space.
- How to build a table-top model of Straight
Lines in a Curved Space -- Geodesics,
General Relativity, and Embedding Diagrams.
- A discussion of the expansion of the universe, which addresses some fundamental
questions about what we mean by distance.
- 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.
- 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
- A discussion of negative
temperatures in a spin system.
- Temperature : Definition and Fundamental Properties.
- The basic properties of differential forms.
- Some rough notes on formulating thermodynamics in
terms of differential forms.
- 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.
- 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
- A nice way to draw the
periodic table of the elements, as a cylinder with bulges in 3D.
- A discussion of the relationships between three ideas, namely
the Aufbau principle, isoelectronic
correspondence, and ionization.
- A discussion of how to
balance chemical reaction equations, including charge-balance
as well as atom-balance.
- 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.
- A perl program to
parse chemical formulas and produce the list of elements and the
amount of each.
- A classroom demo of catalysis
using gelatin and cysteine protease enzymes.
- 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.
- Some hints on how to do basic math calculations, including long
multiplication and long division.
- 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.
- 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
- An overview of some easily-understandable evidence that our world is governed by the laws
of quantum mechanics.
- 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
- A deprecation of the alleged distinction of
"chemical" versus "physical" changes.
- How to think about the specific heat capacity and enthalpy, including the latent heat of a
first-order phase transition.
- 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.
- An introduction to waves. This includes an answer (or non-answer)
to the question of What is a Wave?
- 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.
- A movie of
the earth as it spins on its axis and orbits around the center of mass
of the earth/moon system
- 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.
- 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?
- 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.
- A discussion of the correct direction of the arrow representing a dipole moment, in
molecules and otherwise.
- 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.
- 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.
- A discussion of the quantum harmonic
oscillator -- energy versus temperature.
- 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
- A terse review of the concept of
- Some notes on
static electricity aka
- How an electrical battery works.
- An overview of the chemical reactions in a lead-acid battery and how they reputedly
work --- including some unanswered questions.
- Some notes on how
to take care of a swimming pool.
- Some words about how to understand the Boundary between
Quantum Mechanics and the Classical Limit.
- An introduction to scaling laws,
including non-dimensional scaling.
- An discussion of dimensional analysis.
- A discussion of units, including how to use them and how
to think about them in physical and algebraic terms.
- A discussion of units of dimension one, also known as
- A discussion of the Secchi disk pattern, which illustrates the
fact that boundaries have zero width and therefore exhibits
some interesting scaling properties.
- 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.
- A discussion of The Exchange of
Identical and Possibly Indistinguishable Particles and how that
relates to the Pauli exclusion principle.
- A discussion of what happens as we approach
fossil resources of energy, including coal, oil, and uranium-235.
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.
- 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.
- A discussion of barometric pressure and pressure altimetry, including aircraft
altimeters. This includes a discussion of how the Kollsman window
(altimeter setting) works.
- Some examples of weird terminology, where the name of the
thing does not provide a good description of
- Print your own spacetime diagram paper:
Also triangular-ruled graph paper.
- A discussion of various situations where we need to plot
something that isn't a function, and/or where the concept of ``axes''
- 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.
- 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''.
- A discussion of electrical power grid
physics and engineering including some thoughts about the 15 August 2003 northeast
- 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).
- Physics Books.
Recommended as a "starter kit" for a college library.
- A book review:
Tom M. Apostol,
- A book review:
The New Math SAT Game Plan.
- Not yet a book review, but some notes about the book:
Chabay and Sherwood,
Matter and Interactions.
- A book review:
Serway & Faughn,
- A book review:
Paul G. Hewitt,
- A book review:
Glencoe Physics: Principles and Problems.
- A book review:
Teaching Introductory Physics.
- A brief book review:
Black, An Introduction to Practical Physics For Colleges and
- A book review:
Millikan & Gale,
Practical Physics (1906-1922).
- Directory listing.
Miscellaneous physics-related diagrams, spreadsheets, et cetera.
- jsd home page.