I wrote the following documents as a result of questions related to courses I have taught or interesting issues discussed on the PHYS-L email list. Most are in PDF; a couple are in HTML. Comments and corrections are invited. Occasionally I refer to specific textbooks or acknowledge my sources in a highly abbreviated fashion; no slight or plagiarism should be inferred by any omissions--you should instead assume I am merely being lazy about credit. Within each category, they are listed in reverse chronological order of writing.
- Four Conditions that a Physically Meaningful Spatial Wavefunction must Satisfy
A wavefunction must be both normalizable and smooth. Each of these requirements leads to a pair of conditions that restricts the acceptable solutions of the Schrödinger equation so that we know what kinds of general wavefunction forms to write down in each region of space.
- Rotating Disk Puzzle
A disk rolls without slipping around another disk of different size. How many rotations does the first disk make until two points on the disks that were initially coincident come back into contact with each other?
- Area of an Ellipse in Polar Coordinates
To describe an ellipse in polar coordinates and integrate its area, it is helpful to introduce the eccentric anomaly.
- Phase and Group Velocity of Matter Waves
A discussion of various calculations of the phase and group speeds for a monoenergetic beam of particles in free space.
- Uniqueness of Brachistochrone Solution
A subtle but elegant proof that a cycloid is the unique analytic functional shape of track that has minimum descent time between given initial and final points along a frictionless track starting from rest.
- Fastest Descent along a Ramp and Horizontal Track
A particle slides frictionlessly starting from rest down a ramp and then along a horizontal track. For fixed vertical and horizontal distances between the starting and ending points, what ramp angle minimizes the total travel time of the particle?
- Pressure Exerted by a Rotating Cylinder of Fluid
A cylindrical can of water is rigidly rotating about its vertical axis of symmetry, such that water makes contact with the top surface of the can. What pressure does the water exert on that surface?
- Inertia Ball
A weight is hung from a fixed support by a light string. An identical string hangs from the bottom of the weight. If you pull slowly on the lower string, the upper string breaks first. But if you jerk the lower string, it breaks first. I analyze this well-known demo both analytically and numerically using Hooke's law, Newton's second law, and kinematics.
- Polar Form of an Ellipse
Algebra is used to derive the polar form of an ellipse, the relation between the semilatus rectum and angular momentum, and the construction of an ellipse by looping a string around two thumbtacks and a pencil moving in such a way as to keep the string taut. Other than definitions, the only needed ingredients are the rectangular form of an ellipse, and conservation of angular momentum and mechanical energy.
- Magic Newton's Cradle
Three balls are arranged so they make 1D elastic collisions. If the balls have relative masses of 1, 0.236, and 1 in order, then sending the first ball in to impact the others at rest will eventually lead to the third ball coming out with all the initial momentum, after the middle ball has bounced back and forth making four collisions with the two end balls.
- Velocity of an Initially Stationary Target after a Projectile Impacts it Head On
The final velocity of an initially stationary target that is impacted head on by a projectile varies linearly with the coefficient of restitution.
- The Twirling Rope
A rope hanging from one point is set into rotation about that point. A nonlinear second order differential equation is derived that describes the equilibrium shape of the rope. It is solved numerically for one example set of parameters.
- Floating Cork in an Accelerating Elevator
Does a cork float higher or lower in a beaker of water when it is accelerating upward compared to when the elevator is stationary?
- Direct Harmonic Balance
Several different published methods to find the frequency of oscillation of a nonlinear oscillator are compared for five different example problems.
- The Marble Loop-the-Loop on Angle Track
A marble rolls without slipping on a 90-degree angle bracket bent into a loop-the-loop. From what minimum height must the ball start to just make it around?
- Gravitational Force due to a Spherical Shell
An elegant but subtle proof that the field outside a uniform shell is the same as that of a point mass at its center is summarized.
- Rolling Cylinders Race Down an Inclined Plane
A solid cylinder, a hollow cylinder, and an iron cylinder bonded to a cylindrical wooden sleeve (shaped like an optical fiber preform) race down an incline starting from rest. What is the order of arrival of the objects at the bottom?
- Oscillatory Motion with and without Damping and Driving
I review the familiar solutions for undriven undamped oscillations, undriven underdamped oscillations, and sinusoidally driven damped oscillations in steady state. Only basic differentiation and trigonometry is used to verify that these solutions satisfy Newton's second law. Graphs of the frequency-dependent amplitude and phase difference are included for the third (resonant) case.
- Model of a Viscoelastic Solid
A spring in series with a dashpot is modeled and shown to be in good agreement with experimental data for the bounce of a steel ball off a cylinder of silly putty.
- Review of the Brachistochrone Problem
I review the derivation of some key formulae for the brachistochrone problem, including the Beltrami identity, the parametric solution, the descent time and arclength, and the relation to the isochrone problem.
- Perfectly Inelastic Collision Between a Disk and a Stick
A disk strikes a stick at an acute angle (relative to its axis) between 0 and 90 degrees. If the collision is perfectly inelastic, a different fraction of the system's kinetic energy is lost if the disk adheres to the stick than if it does not. This example illustrates that one should NOT define "perfectly inelastic" to imply the colliding objects stick together.
- Shooting at a Constant-Velocity Target
As in a video game, you can fire constant-speed bullets at a constant-velocity target. Knowing the velocity and initial position of the target, and the speed of the bullets, at what angle should you fire and how long will it take until you get a hit? Assume the bullets move faster than the target, so as to guarantee a hit.
- Number of Galaxies in the Universe
A simple but accurate back-of-the-envelope estimate of the number of galaxies in the universe is found based on its age and on the typical number of stars in a galaxy. The key assumption is that the visible mass of the universe is just enough that it is flat.
- Derivation of Kepler's Third Law and the Energy Equation for an Elliptical Orbit
Introductory textbooks usually derive Kepler's third law and the total mechanical energy of a satellite orbiting a much heavier body only for the special case of circular orbits. However, using algebra alone, one can obtain the generalized expressions valid for an elliptical orbit.
- The Lagrangian Method in the Introductory Course
One can derive Newton's second law starting from conservation of mechanical energy. I call this the "Lagrangian method" because it is equivalent to solving the Lagrange equation. It is a nice method for solving problems that involve forces of constraint.
- A Semiclassical Derivation of Eigenenergies
Using de Broglie's relation and standing wave conditions, together with classical mechanics, one can easily derive the energy levels for a particle in a box, a hydrogen atom, and a simple harmonic oscillator.
- Transformation Equation for Center-of-Mass Work
The equation is derived that relates work computed in an inertial frame to that computed in some other frame that translationally accelerates relative to it. An important special case is the transformation between two different inertial frames, in which case the impulse is frame invariant but the work in general is not.
- Rolling Stack of Cans
Three cans are stacked in a horizontal triangular pile and released from rest. Ignoring friction, what is the speed of each can as the top one strikes the table?
- Maximum Bob Height of an Interrupted Pendulum
The swing of a simple plane pendulum is interrupted by a peg around which the string bends. What is the position of the highest point of the swing as a function of the height of release of the bob, for a given peg position? This is a nice review exercise combining concepts from 2D kinematics, centripetal force, and conservation of mechanical energy.
- Speed and Amplitude of a Tsunami
Some rough estimates using Newton's second law and the equation of continuity enable one to derive the well-known result that a shallow water wave has a speed equal to the square root of the acceleration of gravity and the water depth. One can use this to predict that the wave pulses will get narrower and taller as they approach shore.
- Coriolis Correction to Freefall
If a rock is dropped down a mineshaft on Earth, its flight path gets deflected from a plumb bob hanging from the initial location due to the Coriolis force. One can calculate exactly the easterly and southerly deflections (in the northern hemisphere).
- Wave Pulse on a Hanging Rope
A heavy hanging rope is given a brief shake. How long does it take for the resulting pulse to travel up and down the rope? An approximate solution is discussed for both traveling and standing waves set up on the rope.
- Perpendicular-Axis Theorem for Volumes
The perpendicular-axis theorem is generalized from laminae to volumes. The result has applications to the computation of moments of inertia of spherically symmetric objects and to the calculation of moments about rotated coordinate axes.
- Kepler's Equation
A satellite is in an elliptical orbit about a planet. Given the eccentricity and period of the orbit, find the relationship between the angular position and the transit time of the satellite, both measured starting from the position of closest approach between the satellite and planet.
- Jumping Frog
What is the minimum speed a frog needs to jump over a log of circular cross section, if it can leave the ground from any point it likes? (Hint: The frog will NOT brush the top of the log.)
- Evaluating a Common Integral
I discuss the pros and cons of two different methods of solving the second integral of the Euler-Lagrange equation that arises in the well-known soap-bubble problem. The curve of revolution is a catenary (hyperbolic cosine).
- Synchronous Orbit of a Satellite about a Binary Star
A lightweight satellite synchronously orbits an equal-mass binary star system. The satellite and the stars are in circular orbits about their common center of mass. What are the possible positions of the satellite whereby this can occur?
- Potential Energy of Stretched Springs
In this paper, I discuss the conditions under which the elastic potential energy of a set of stretched springs can be calculated from the displacements of the ends of each spring from their equilibrium (as opposed to their relaxed) positions. I consider both longitudinal and transverse displacements. Being able to confidently write down this potential energy is crucial to the solution of coupled oscillator problems.
- Power to Create a Water Jet
How much power is required to launch a vertical jet of water with given base radius to some specified maximum height? This is problem 8.91 in Giancoli and is a nice exercise in the application of the concepts of energy and force.
- Maximum Range of a Projectile Launched from a Height
It is well known that a surface-to-surface projectile has maximum range equal to the initial speed squared divided by the freefall acceleration when launched at 45 degrees. What are the maximum range and optimum launch angle when the projectile instead starts above the landing surface?
- Theory of Holonomic Constraints
This is a half-page summary of how to find the equations of motion and the generalized constraint forces for a system subject to holonomic constraints using the Lagrange equations.
- Box Pulled on a Rough Surface by a Winch
A box on a rough horizontal surface is being pulled from above by a winch being cranked at a constant angular speed. Find the acceleration of the box and the tension in the pulling cable as a function of the horizontal distance to the winch.
- Newton's Laws
This document is intended to guide a one-lecture introduction to Newton's three laws of motion in a first course for majors. I try to be a bit more careful than typical textbooks about the definitions of force, mass, and reference frame, yet without getting mired in conceptual quicksand.
- Formal Derivation of Centripetal Acceleration
This is in effect a simplified version of the document below entitled "Acceleration Components in 2D" for the special case of UCM. It is suitable as a derivation of the standard formula for centripetal acceleration in a calculus-based course early in the semester before rotational motion or polar unit vectors have been properly introduced.
- Constant Acceleration in Special Relativity
An object experiences a constant acceleration for an extended period of time. Decide what this statement means and then calculate its relativistic speed at the end of the time interval if it started from rest. Consider both the cases where the time interval is measured in the lab frame and in the object's proper frame.
- Rolling Friction of a Free Wheel
A free wheel which is perfectly round and rolls on a rigidly flat surface cannot experience any contact friction. In reality, both the wheel and road deform slightly. It is then possible to introduce rolling friction with its own coefficient to simultaneously account for both the translational and rotational deceleration. Such deformations are not needed for a driven or braked wheel, however.
- Rotating Space Station
If you drop a ball while standing in a space station which is rotating to provide artificial gravity, does the ball land at your feet? For that matter, if you drop a ball from a tall tower on Earth, does the ball land at the base of the tower?
- The Black Hole Shredder
An ordinary brick falls into a black hole. Will the large tidal stresses tear the brick apart before it winks out of view behind the event horizon?
- Multiple Strings and Pulleys
In order to apply Newton's second law to problems involving multiple pulleys, blocks, and strings, it is necessary to find equations relating the accelerations of the various masses to each other. I give an example of how to do this for a quasi-1D problem; the idea is to express the positions of the masses in terms of the fixed lengths of the strings and then differentiate twice.
- Acceleration Components in 2D
Four adjectives are commonly used when discussing acceleration in plane polar coordinates: centripetal, radial, tangential, and azimuthal. Most introductory textbooks only use some of these; others use all of them but without clearly explaining the differences. In fact, none of them are synonyms and each is packed with content. In this handout I explain all four terms both qualitatively and mathematically; a brief appendix considers two possible definitions of the word "tangential."
- Displacement and Pressure Amplitudes of Sound
The pressure oscillations in a longitudinal sound wave are in phase with the velocity oscillations of the fluid molecules. I summarize three derivations of this relation including the constants in the proportionality. This is a useful exercise in distinguishing the wave and molecular speeds.
- Kinetic Energy of a Rigid Body
Under what circumstances should the KE of a rigid body be calculated using the translational formula, the rotational formula, or the sum of the two? I remind you of how to prove the fact that, when properly applied, all three choices give the same answer. A couple of practice exercises from Serway are suggested to illuminate this.
- Formulae for Collision Problems
This handout briefly summarizes the formulae for the three standard kinds of collisions (elastic and perfectly or imperfectly inelastic) in both 1D and 2D. This is not intended to be memorized or carried into exams, but instead to help a student find the bottom line in the morass of equations thrown at them in the momentum chapter of typical intro textbooks.
- Local Vertical on Earth
I review the solution of the problem of finding the direction of a freely hanging plumb bob on a spherical rotating planet, correcting an apparent typo in Arnold Arons' book.
- Small-Angle Oscillations of an Arc
Here is a lovely problem for introductory mechanics. Find the period of SHM of the physical pendulum consisting of a uniform circular arc balanced at its midpoint on a knife edge. Remarkably, the answer is independent not only of its mass but also of what fraction of a complete circle it happens to be!
- Two Rotational Equilibrium Problems
1. A ladder of uniform mass density on a rough floor leans against a rough wall. What is the minimum angle the ladder can make with the floor and not slip? A number of textbooks state this problem is indeterminate. I leave it to the reader to decide if my solution is flawed.
2. Two balls (not necessarily identical) are stacked inside a hollow cylinder. Under certain conditions, the cylinder topples over. What are these conditions on the masses and radii of the balls?
- Falling Ball Puzzle
A ball is launched horizontally into a semi-cylindrical depression, makes a perfectly elastic collision with it, and is observed to rise straight up. How high will the ball rise above the lip of the depression? Your answer is to be expressed in terms of the radius of the depression only. This is a good test of how organized you are at problem-solving.
- Talking like Donald Duck
Why does your voice sound high pitched when you inhale helium gas? One sometimes hears it said that vocal chords are like tuning forks. Unfortunately the frequency of a tuning fork is independent of the gaseous medium it is in--think for example of the usual introductory lab where such a fork is struck over top of a column of variable length.
- Mass on a Vertical Spring
When simple harmonic motion is discussed in introductory textbooks, the mass connected to the spring usually slides on a frictionless horizontal surface. Unfortunately, it is pretty hard to demonstrate that in the real world, so we arrange things vertically. Then we assign some homework problems where the mass hangs vertically. But um professor sir, doesn't gravity mess up the force and potential energy considerations when you do this?
- Momentum Carried by Mechanical Waves
This is a thread from PHYS-L discussing whether mechanical waves carry net momentum. Five articles from AJP are cited which I strongly recommend. It is interesting to see how much disagreement there is on such a topic among physics educators and professionals.
- Classical Doppler Effect
I derive and summarize the classical Doppler shifts in the wavelength, frequency, and wave speed for three special cases: only the source moving, only the observer moving, and only the medium moving. In each case, one of the three wave parameters is unchanged. Finally, I put the three effects together to derive the shifts when everything is moving.
- Completely Inelastic Collisions
Prove that a maximum amount of kinetic energy is lost in a completely inelastic collision between two point masses. Use only high school algebra, conservation of linear momentum, and the definition of kinetic energy to do it in 1D. Then use partial differentiation to do it for the fully general case of 3D.
- Elastic Collisions in 1D
Many students have trouble remembering the quadratic formula. Hey, I often have the same problem and besides it usually creates a computational mess and having to choose between two solutions is a nuisance. Here is a neat trick for solving one-dimensional elastic collisions to get general algebraic expressions for the two final velocities.
- Number of Independent Kirchhoff Equations
I present a method for counting the number of independent Kirchhoff Current Junction and Voltage Loop Rule equations by inspection of a circuit. A trick is required if some wires cross over other wires without electrical contact.
- Equivalent Resistance of a Wheatstone Bridge
An excellent illustration of the method of nodal potentials for circuit analysis is to find the equivalent resistance of a Wheatstone bridge of resistors.
- A Surprising Circuit Symmetry
A problem in Halliday, Resnick, and Walker involves showing that an ammeter reading does not change when the ammeter and battery in a circuit are interchanged. I present three different solutions of this surprising situation.
- Charged Parallel Metal Plates
Arbitrary charges are placed on two large parallel metal plates. Find the electric fields everywhere in space. (For specificity, suppose +2 C is put on the left plate and -3 C on the right plate.) Spell out all physical assumptions needed in your solution.
- Electric Field of a Uniformly Charged Straight Rod
There is a simple identity that makes it easy to calculate the field of a uniform rod at an arbitrary point in space. After deriving the general result, I specialize it to various cases of interest, including semi-infinite and infinite rods, and a point on the axis of a finite rod beyond one of its ends.
- Surfaces of Zero Potential Around a Quadrupole
The two cones surrounding a zz quadrupole on which the electrostatic potential is zero are derived and graphed. The half-angle of the cones relative to the z axis is 55 degrees, but they deviate near the quadrupole so that they do not intersect, passing between the positive and negative charges of each opposing dipole pair making up the linear quadrupole.
- Inductance of a Flat Circular Coil
The self-inductance of a flat circular coil from PASCO is estimated using the Biot-Savart law, with the two integrals computed in Mathematica. The results are in fair agreement with an experimental measurement of the actual inductance.
- Induced Electric Field for a Solenoid of Uniformly Increasing Current
The nonconservative electric field induced everywhere in space for ideal solenoids of both circular and square cross section are calculated when the current in the windings is increased at a constant rate. Contour lines and field lines are instructively plotted.
- Relation between the Ampere-Maxwell and Biot-Savart Laws
For a charge density in uniform motion, the Ampere-Maxwell and Biot-Savart laws can be shown to be equivalent, in analogy to the fact that Gauss' and Coloumb's law are equivalent in electrostatics.
- Magnetic Field of a Current Loop in the Plane of the Loop
The magnetic field in the plane of a current loop is calculated by numerically integrating the Biot-Savart law. The results are graphically compared to the standard results at the center of the loop and at large distances from the loop.
- Four Derivations of Motional EMF
As most instructors of E&M know, the familiar example of a conductor moving in a magnetic field is a rich playground for exploring forces, electromagnetic fields, and energy. In this document, I have taken the various bits and pieces and tied them together into one continuous story suitable for a standard classroom lecture.
- Inductance Calculations
A recent paper in AJP gives a formula for computing the inductance of a device in terms of the self and mutual inductances of the elementary filamentary loops making up the circuit. I show for example how to apply the formula to calculate the inductances of a solenoid and of a coax cable.
- Further Thoughts on the Ideal Solenoid
Here is a second method (simpler than that presented below) of rigorously deriving the magnetic field inside and outside an ideal solenoid of arbitrary cross-sectional shape. I end with a list of AJP references on this topic.
- Time Constant for Charging a Pair of Capacitors
Giancoli P26.47 considers a resistor and capacitor in parallel connected to another parallel RC pair. What is the time constant of this circuit? A straightforward approach is to use Kirchhoff's rules.
- Rewired Capacitors
Tipler problem 25.59 asks one to wire together in series three previously charged capacitors and find the final charges on and voltages across each. This is a nice application of Kirchhoff's rules and a counter-example to the "memoroid" that capacitors in series must have the same charges!
- Double Coil Inductance
Tipler problem 30.54 asks one to calculate the inductance of a series pair of coaxial solenoids carrying the same current in opposite directions. The difficulty is that the two solenoids do not have the same length. In this note, I derive a particularly simple approximation to the solution by neglecting the end effects. For the actual numbers given in the problem, this is not very well justified and some interested party is invited to come up with a next order approximation to the solution.
- Magnetic Moment due to Electron Spin
The magnetic dipole moment of a circulating charge is equal to the product of the gyromagnetic ratio and the angular momentum. The standard introductory derivation of this ratio comes out too small for a spinning electron by a factor of 2 (neglecting radiative corrections due to virtual photons). In this paper, I summarize two approaches that have been previously proposed to account for this factor within the contexts of introductory E & M and QM courses.
- Magnetic Field Outside an Ideal Solenoid
The usual textbook arguments for why the field outside an ideal solenoid is zero are not very convincing. In this handout I directly integrate the Biot-Savart law by interpreting an ideal solenoid as a semi-infinite sheet of current rolled into a cylinder. That is, I solve Griffiths P5.44. The results provide a nice way to think about the issue, analogous to why the electric field inside a cylinder of uniform surface charge is zero.
- Resistivity of Copper
A microscopic model for the resistivity of a metal is developed in order to prove Ohm's law. A classical model using Maxwell-Boltzmann statistics and hard-ion-core collisions correctly gives the field independence and ballpark magnitude. However, more detailed agreement and the linear temperature dependence requires using the Fermi-Dirac distribution and scattering only from lattice imperfections. Amazingly, the only material parameter needed in the final result is the atomic number density. Good agreement for the resistivity and temperature coefficient of copper are obtained.
- Two Capacitance Formulae used in Lab
First an expression is derived relating the dielectric constant of some material to three capacitance measurements using an appropriate meter: the capacitance at minimum plate separation (which is nonzero because of the spacers used), the capacitance with the sample in place, and the capacitance with the sample removed but the plate separation unchanged. Secondly a formula is deduced in three different ways for the measured force between two plates as a function of the potential difference applied across them.
- Infinite Square Lattice of Resistors
A well-known problem is the equivalent resistance of a semi-infinite ladder of one-ohm resistors. What is the resistance between two adjacent nodes of a 2D lattice of such resistors? The answer can be easily found using superposition of potentials, although some care is warranted when this idea is examined closely.
- Electrostatic Equilibrium of Two Hanging Spheres
Two pith balls of equal mass but different charges (of the same sign) hang on equal-length threads from a common point. What is the relationship between the two angles they make with the vertical? Suppose we decide to solve this problem by drawing a free-body diagram and invoking Newton's third law and translational equilibrium. If you choose a coordinate system with horizontal and vertical axes, you get an equation which is very hard to solve. However, if you choose a different coordinate system for each sphere, with one axis oriented along the string, the two solutions are easy to find. This is an excellent example to throw out to students who insist on always using horizontal-vertical coordinate systems.
- Hand-Cranked Generator and Capacitor
A Pasco hand-cranked generator is connected to one of those amazingly compact 1-F capacitors. After charging it up briefly, you stop turning the crank and then release your grip on it. The generator now spins as a motor with the capacitor serving as a battery. However, even though the current in the circuit has reversed, the crank continues to turn in the same direction! Isn't this a violation of energy conservation: how does the crank "know" that you took your hand away?
- Charges on Series Capacitors
Introductory textbooks do not properly explain why the charges on (initially discharged) capacitors wired in series (to a low-frequency source) must all be equal. This is not exactly true for real capacitors; some unspecified idealizations have been assumed. In this thread from PHYS-L, the nature of these idealizations are clearly identified.
- Two-Slit Interference Using a Thermal Source of Light
I explain how it is possible to perform two-slit interference with light from the sun or an incandescent light bulb even without passing that radiation through a pinhole first. The key idea is that the difference in pathlengths from a point on the source to the two slits needs to be less than the coherence length, which is on the order of the peak wavelength for a blackbody. This condition is easily satisfied.
- Highway Mirages
A differential equation is obtained for the trajectory of a light ray descending diagonally through air whose index of refraction varies with height. It is applied to two examples in which the index decreases montonically with decreasing altitude above a hot road. In one case, that causes the ray to eventually hit the critical angle and turn around, giving rise to a mirage of water on the road ahead. In the second example, however, the ray asymptotically approaches the horizontal without ever reaching it, much less curving upward.
- Single-Slit Diffraction
Approximate and exact values of the angular positions, peak intensities, and integrated areas of the secondary maxima for single-slit Fraunhofer diffraction are compared.
- Location of a Position-Sensitive Detector on a Rotary Plate
A laser beam is incident on a PSD on a rotating platform. By measuring the position of the beam spot on the array detector as a function of the angle of rotation of the plate, one can determine the absolute location of the detector relative to the plate's axis.
- Summary of Thin-Film Interference
I summarize the standard introductory discussion of thin-film interference with a single master formula. This contrasts with the variety of special-case formulae that typical physics textbooks present instead, leading students to misapply them in other situations.
- Phase Change upon Reflection
The phase changes for reflection at normal incidence from an interface between nonmagnetic, nonconducting media are needed in the discussion of thin-film interference. Many introductory textbooks "justify" these phase changes using an analogy to free vs fixed boundary conditions for a mechanical wave reflecting off the end of a string. This is a bit strange considering that Maxwell's equations were usually discussed just a couple of chapters previous. Given these equations, it only takes a simple diagram and a few lines of algebra to formally derive the phase changes. As a bonus, index matching naturally arises as the third possibility when comparing the refractive indices of the two media. The reflectance and transmittance can also be easily introduced in this context.
- Optical Length of a Day
Because of atmospheric refraction, the length of a day is a bit longer than the geometric duration it would have in the absence of this effect. I make a crude estimate of the extra daylight we would get by modeling the atmosphere as a homogeneous spherical shell of air and compare it to the actual value of about 4 minutes.
- Hamiltonian Formulation of Geometric Optics
I review a recent Hamiltonian formulation of optics starting from Snell's law and photon concepts which naturally leads to the concept of group index of refraction. I briefly contrast this with an alternative formulation starting from Fermat's principle.
- Optical Molasses
Some simple numerical values describing laser cooling of atoms are easily calculated, making a nice homework problem in a modern physics course. Specifically, one can estimate the slowing per photon, the net radiative force, the atomic stopping time, the minimum temperature, and the required laser intensity.
- Cone Artist
Place a prism on a sheet of paper and trace its base. Lift off the prism and shade in the resulting rectangle. Call that the object. Now replace the prism but flip it over vertically so that its apex is in contact with and bisects the object. Look through the base. The question is: What is the width of the resulting image? (Hint: There is both an inverted and an uninverted image, depending on the angle at which you look into the base. What is the minimum width of the uninverted image?)
- Light Output of a Three-Way Bulb
Light bulb packages usually state not only the electrical wattage but also the useful light output in lumens. How is the latter related to the former? In this paper, I consider the example of a GE soft white luminescent 50-100-150 W bulb.
- Angle of Minimum Deviation through a Prism
Introductory texts often allude to or draw a diagram of a ray refracting symmetrically through a prism but seldom prove that this occurs at the angle of minimum deviation. In this note, I show this two ways. First by brute force, and secondly using an elegant argument based on optical reversibility. I end with a reminder of how a measurement of the angle of minimum deviation can be used to find the index of refraction of a prism.
- Cat's Eye Retroreflector
A lens with a mirror at the focal plane (such as in the eye of a cat) retroreflects light. In this note, the retroreflection efficiency is computed as a function of angle of incidence and numerical aperture for a simple model. The results could be used to estimate how bad "red eye" might be.
- Moon Tans
The ratio of the luminosities of the sun and full moon at zenith are calculated. I take the opportunity presented by this delightful problem to review the basic concepts of radiometry: the bidirectional reflectance distribution function, irradiance and radiance, albedo, and the properties of Lambertian surfaces. Actual lunar data from the Global Ozone Monitoring Experiment are used. I discuss the poorly known fact that while the sun is a Lambertian emitter, the Moon is not a Lambertian scatterer. The final result is in excellent agreement with the known luminosities and imply that to get a good moon tan, you would have to bask about a million times longer in its light than in that of the sun.
This is a quick summary of how to derive the angles at which the primary and secondary rainbows for water droplets in air are seen. The key point is that they arise at the angles of minimum deviation. A negative angle for the secondary bow explains its inverted color spectrum. Successive orders of rainbows dominate for droplets whose truncated index of refraction is the successive integers.
- Parabolic Mirror
Introductory textbooks often mention that non-paraxial rays result in spherical aberrations and that this can be avoided by using a parabolic mirror. It is very simple to prove the validity of this claim using a sketch, as well as to quantify the size of the blur spot for a spherical mirror. The following concepts arise naturally in the discussion: the relation between the focal length and geometrical shape for both a parabolic and a spherical mirror, the directrix for a parabola, and where the non-paraxial rays strike the focal plane for a spherical mirror.
- Radiative Coupling between an Object and its Surroundings
An object is placed inside an opaque cavity. What is the radiative heat load on the sample? Most textbooks incorrectly state that the answer is independent of the nature of the cavity. In this four-page paper, I review the correct analysis of the situation and apply it to two specific geometries: coaxial cylinders and infinite parallel planes. I end with two practical considerations. First, the fact that Kirchhoff's law for the equality of the absorptivity and emissivity holds not just for the integrated values but also wavelength by wavelength and angle by angle implies that the thermal radiation emitted by a sample is polarized. Second, I derive an expression for the exponential relaxation of a sample to its final temperature when it is optically heated or cooled.
- Faraday Isolators and Kirchhoff's Law
Want a challenging puzzle to chew on? Consider a Faraday isolator: two linear polarizers whose transmission axes are oriented 45 degrees relative to each other, between which is located a magnetic rotator which rotates the plane of polarization of a beam by 45 degrees in the same direction regardless of the direction of propagation of the light. This constitutes a one-way light valve, used to protect lasers from harmful back-reflections. Now place a sample inside a cavity whose walls are made of this stuff. Light gets out but not back in, right? IF SO, THE SAMPLE WILL RADIATE AWAY ALL ITS ENERGY AND COOL DOWN TO ABSOLUTE ZERO! Save thermodynamics (and the principle of optical reversibility) for us, will you?
- Lensmaker Formula
Any two points on a plane wavefront travel the same optical pathlength when brought to a focal point by a lens. (For example, this arrangement is typically used to attain the Fraunhofer regime for single or multiple slit diffraction.) Using this concept alone, one can derive the laws of lenses such as the lensmaker formula for a thin lens in the paraxial approximation.
- Fresnel Boundary Conditions
Continuity of the tangential components of the electric and magnetic fields (expressed in complex form) across an interface are easily shown to lead to three results: the law of reflection, Snell's law, and continuity of the tangential components of the field amplitudes (from which the Fresnel equations for the reflection and transmission coefficients are derived).
- Additively Combining Two Samples of Helium Gas
If we open a stopcock separating two flasks of a monatomic ideal gas, must the final equilibrated value of any extensive state variable be equal to the sum of the initial equilibrated values for the two samples?
- Computing a Partial Derivative for a Van der Waals Gas
Computing partial derivatives is a common problem in thermodynamics and can be difficult for novices. Here I discuss an interesting example from the literature and present a simple solution to it.
- Thermodynamics of an Open System
For an open system, one cannot simply identify TdS with dQ and μdN with the energy transfer due to particle exchange. A simple example involving a monatomic ideal gas makes the issues clear.
- Entropy of a Classical Ideal Gas of Distinguishable Atoms
The usual textbook formula for the entropy of a classical ideal gas of distinguishable particles is wrong because it violates the second law of thermodynamics. In actuality, the standard Sackur-Tetrode expression applies to either distinguishable or indistinguishable particles.
- Brightness Temperature of a Laser
The brightness temperature of a helium-neon laser is calculated to be on the order of 10 billion kelvin by finding the temperature of a blackbody of the same emitting area that has been filtered spectrally and angularly to match the laser beam.
- Legendre Transforms for Dummies
I review a way to introduce the Legendre transform via partial derivatives. Examples from thermodynamics and classical mechanics are included to illustrate the method.
- Thermal de Broglie Wavelength
I show that the reciprocal cube root of the quantum concentration (appearing in the partition function of an ideal gas) is equal to half of the de Broglie wavelength thermally averaged over the Maxwell distribution of molecular speeds.
- Connecting the Work-Energy Theorem to the First Law of Thermodynamics
Conventional texts develop topics in the order: forces, work, energy, thermodynamics. Within the context of this particular sequence, how can one bridge from the work-energy theorem in mechanics to the first law in thermodynamics? I compactly summarize one logical approach in a single page, deferring some of the subtleties to footnotes.
- Model for the Atmosphere
A simple atmospheric model is summarized which makes realistic predictions for the temperature, density, and pressure variations with altitude up to about 15 km, without making ad hoc assumptions common in typical intro physics textbooks.
- Density of States of a Particle in a Box
By differentiating the volume of a hypersphere (as calculated in the Mathematical Physics section), I derive the general formula for the DOS of a particle in a hyperbox of arbitrary dimensionality.
- Density of Air down a Bore Hole
Gases are usually distinguished from condensed media by a factor of about 1000 difference in density. But what if we lived in deep underground caverns rather than at the earth's surface: At what depth would an ideal gas have unit specific gravity? I answer this question both with and without accounting for the change in ambient temperature with depth.
- Adiabatic Expansion of Soda Pop
A small amount of water is placed in the bottom of a corked bottle which is pressurized to the point that it blows its top. Under appropriate circumstances, a mist forms in the bottle, as presented for example in Demo 15-04 of the Video Encyclopedia of Physics Demonstrations. In this note, I analyze the pressure and temperature changes to explain why condensation occurs.
- Van der Waals Equation of State
Introductory thermodynamics and chemistry textbooks often "justify" the Van der Waals equation of state with a few sentences of nonsense. Here I try to do a slightly better job of motivating the formula by relating the two constants in the equation in an approximate manner to molecular constants of the gas in question.
- Thermally Induced Agitation of a Simple Pendulum
The equipartition theorem is generalized to handle cases where the energy is proportional to the generalized coordinates to any positive power (i.e., not necessarily two). As an example, I show that different choices of coordinates for a simple pendulum (to wit, the angle and angular momentum, the horizontal displacement and linear momentum, or the vertical displacement and linear momentum) all give the average energy to be kT. Another nice case is a photon, for which the energy is linearly proportional to the momentum.
- Thermodynamics of a Classical Ideal Gas
The purpose of this paper is to remind you of how to calculate the entropy and chemical potential of a classical ideal gas. That is, it is assumed the gas obeys classical statistics for the translations and rotations. (The vibrations are presumed to be frozen out.) For example, the entropy in the monatomic case is known as the Sackur-Tetrode equation. The chemical potential is related to the average occupancy of available states.
- Internal Energy of a Nuclear Gas
Consider a gas of nucleons in an infinite-square-well potential whose radius is proportional to the cube root of the mass number. An expression for the internal energy per nucleon is found as a function of temperature, by first deriving a formula for the chemical potential. The results clearly highlight the limited validity of this common model for the nuclear potential.
- Important Corrections to Stowe
This is a list of serious flaws in Keith Stowe's otherwise very readable textbook, "Introduction to Statistical Mechanics and Thermodynamics." The list was prepared by Dan Schroeder (who has himself written the introductory book "Thermal Physics") and divides into three categories: chemical potential, multiplicity of a classical system, and counting polarization states.
- Density of States via the Heisenberg Uncertainty Principle
Stowe gives two different expressions for how the density of states of a system should vary with its internal energy and number of degrees of freedom. Both are wrong. I derive the correct expression by appealing to Heisenberg's Uncertainty Principle. The examples of a particle in a 3D box, and of a simple harmonic oscillator in 1D, 2D, and 3D are explicitly considered.
- Thermal Processes
A one-page chart summarizing the work, heat, change in internal energy, and pressure-volume graph for the standard four thermal processes applied to ideal gases, as derived in a typical introductory physics course. No calculus is invoked.
- Algebraic Proofs of Two Equations
In this one-page handout I present proofs of two important equations related to thermal processes: the work done during the isothermal expansion of an ideal gas, and the relationship between pressure and volume during the adiabatic expansion of an ideal gas. Both are presented without proof in Sec. 15.5 of Cutnell & Johnson. I use College Algebra alone, so that bright non-calculus students can follow it. This is a nice reminder of the power of logarithms and exponentials.
- Glossary of Thermodynamic Terms
A glossary of terms for the typical introductory physics course. This is intended to help students navigate the minefield of familiar words used in very precise ways. Both macroscopic and microscopic definitions are included.
- Tile a Square With Four Nonidentical Isosceles Triangles
Divide a square into four isosceles triangles, none of which are identical to each other.
- My Number is 136
A challenging logic puzzle from the Wall Street Journal involving two numbers whose sum and product are each seen by only one person.
- Multiply by Four and Reverse
What four-digit number when multiplied by four is equal to the original number with its digits written in reverse order?
- Postage Stamps Problem
You have two different denominations of stamps. Using various numbers of each kind of stamp, what possible total amounts can you make? In particular, under what circumstances is there an upper limit on the amount you cannot make, and what is a formula for that limiting amount?
- Finding the Integral of an Inverse Function
If you know the integral of a forward function, you can find the integral of its inverse function by integrating by parts.
- Law of Universal Procrastination
NSF reports of when submissions to grant requests are made during the open window of acceptance leads to a universal hyperbolic law of procrastination. Notably, about half of all submissions will be on the due day, be it for grant proposals, tax returns, student papers, or what have you.
- Triangle Bisector Theorem
A derivation of the relationships between the side lengths and angles if one angle of a triangle is bisected by a line.
- Repeating Decimals
A quick proof that any repeating decimal fraction can be expressed in rational form by dividing the repetend by that number all of whose digits are 9 and having as many digits as the repetend has.
- Quant Quiz
Wall Street Journal published 5 sample questions from a recent Math Competition on page C13 of the Wednesday 4 March 2015 issue of their newspaper.
- The Problem of a Boy Born on a Tuesday
If you know that at least one of the children in a two-child family is a boy, what is the probability the family has two boys? How about if you know at least one child is a boy born on a Tuesday? How about if you know the older child is a boy? Remarkably the answers to these three questions are all different. You may wish to compare it to the Problem of Two Aces listed below.
- Solution of a Cubic Equation
Cardano's four-step method for finding the algebraic solution of a cubic in terms of its coefficients is compactly outlined.
- Area of an Ellipse
A simple noncalculus derivation of the area of an ellipse is reviewed.
- Probability of Being Chosen in Repeated Independent Trials
There are a bunch of numbered objects in a group. Suppose that the probability that an object's number is selected is a constant p during repeated trials. What are the odds that that number will be selected at least once over the course of n trials?
- Solving M Equations in N Unknowns by Gaussian Elimination
I review the number of solutions that arise for a system of linear equations. The idea is to row reduce the augmented matrix and then inspect for the presence of rows where every entry except possibly the last is zero.
- Why the Jacobian Transforms Variables in an Integral
A quick sketch is presented of how the Jacobian enters into integrals when one transforms multiple variables (such as from rectangular to polar coordinates).
- Problem of Two Aces
You are dealt two cards from a shuffled deck. What is the probability of getting two aces? If you know that one is an ace, what is the probability that the other is an ace? If you know that one is the ace of spades, what is the probability that the other is an ace? Each of these questions has a different answer, clearly demonstrating that the more information you have, the more your odds go up. If you start an argument at a party by asking these questions, DON'T BLAME ME!
- Riemann Zeta of 4 as Needed for Stefan-Boltzmann Law
A quick review of the standard method of evaluating ζ(4) using Parseval's relation for a triangular wave. Knowledge of this result enables one to find the value of the Stefan-Boltzmann constant.
- Small-Argument Expansion of a Polynomial in a Denominator
Four methods are presented for expanding the reciprocal of a polynomial in x as a power series for small x.
- Names of the Trigonometric Functions
I summarize some mnemonics for remembering the values of the trig functions, along with an abbreviated etymology of their names. This article is a short but useful handout for students who cannot remember which function is which.
- Cosines of Common Angles
There is a nice pattern to the values of the cosines of common angles in the first quadrant. A pattern involving square roots and the integers 0 through 4 is found.
- Center of Mass of a Uniform Triangular Plate
One can find the center of mass of a triangular plate using an algebraic scaling argument. It is easiest to first do a right triangle. Then an oblique triangle can be split into two right triangles, both having positive mass for an obtuse triangle and one having negative mass for an acute triangle.
- Griffiths Form of the Potential for a Semi-Infinite Slot
I derive the summation of the series for the solution of the Laplace equation inside a rectangular slot of height b between two grounded plates and with a constant potential along the edge on the y-axis.
- Extrema & Saddle Points for a Function of Two Variables
A summary of how to find the locations of all of the minima, maxima, and saddle points for a function of two independent variables.
- Modified Taylor Series
Standard 1D Taylor series can be modified to expand f(x+h) or f(g) where h(x) and g(x) are functions of x. As an example, a series expansion for sin(sin(x)) is obtained.
- Chords of a Circle
Draw a circle and choose any point inside the circle and call it P. Then draw a chord that passes through point P. Point P divides the chord into two lengths. Prove that the product of those two lengths would be the same for any chord you drew through point P. As a corollary, derive the equation of a circle, or equivalently the Pythagoras theorem. As another corollary, show that the locus of points describing the perpendicular intersection of two lines passing through two fixed points is a circle.
- Matrix Coefficients for a System of Linear Equations
Consider the matrix equation Ax=d. A standard problem is to find x given A and d. But what is the general solution instead for A, given x and d?
- Coins Puzzle
While blindfolded, you are handed a tray of identical coins. You are told how many of them are heads up. How can you divide them into two groups such that each group has an identical number of heads? You may manipulate the coins any way you like, but have no means of determining which are heads and which are tails. Hint: There is a simple algorithm which works even if the initial number of heads is odd.
- Looped String Puzzle
You hang a loop of string over a bunch of parallel rods. No matter which rod you pull out of the bunch, the loop falls to the ground. How was the string wound around the rods?
- Shaking Hands at a Party
A bunch of couples at a party shake hands. No one shakes his spouse's hand and every person except one shakes a different number of hands. How many hands did the odd person out shake hands with?
- An Induction Proof for Prime Numbers
Prove that the difference between n raised to the power p and n, where n is any whole number and p is any prime number, is divisible by p. This establishes a famous theorem about primes due to Fermat.
- Equilateral Triangle on the Surface of a Sphere
In this document I derive formulas for the interior angle and area of a triangle with sides of equal length on the surface of a unit sphere. This is a nice exercise showing that Euclidean geometry changes in curved space.
- Integral Representation of the Riemann Zeta Function
In texts such as Arfken, the integral representation of the Riemann Zeta function (needed for example in the derivation of the Stefan-Boltzmann law) is obtained by contour integration. Here is a much simpler derivation, accessible to a student who is familiar with the definition of the gamma function.
- Volume of a Hypersphere
The volume of a hypersphere in n dimensions is derived. This is a wonderful exercise in the use of the gamma function and gives an amazingly compact form for the final result.
- Irrationality of Square Roots
Can you prove that the square root of 3 is irrational? How about of 4.1? The proof is amazingly simple, by extension of a well-known proof that the square root of 2 is irrational.
- Frobenius Series Solutions of Bessel's Equation
A derivation of Bessel functions of the first and second kind, together with some student exercises.
- Solutions to Elementary First- and Second-Order Differential Equations
A one-page reference guide to common ODEs.
- Flowchart for Series Solutions to Differential Equations
Do you use a power series or a Frobenius series? How do you handle various possibilities for the roots of the indicial equation?
- Functions expressed in terms of Hypergeometric Series
A variety of functions encountered in Boas are expressed in terms of hypergeometric and confluent hypergeometric series. No attempt is made to derive these expressions; the purpose of this one-page handout is simply to whet interest in this series.
- Uniformly Charged Wire Outside a Grounded Cylinder
An infinite wire carrying a uniform linear charge density runs parallel to an infinitely long, grounded, conducting cylinder. The potential everywhere outside the cylinder is derived using series expansions. The paper ends with some student exercises.
- Solving Newton's Second Law in One Variable in the Absence of Dissipation
This one-page note gives an algorithm for finding the time needed by a particle moving along a specified curve to get from a given point to another, assuming that no net nonconservative force acts on the particle. This is essentially a disguised version of the work-energy theorem.
- Derivations of Stirling's Approximation
Stirling's Approximation is derived two different ways. First, a quick proof uses Taylor's theorem, as is appropriate for the thermodynamics teacher who needs a 10-minute class derivation. Second, a much longer proof due to Mermin is presented; in his usual style, it meanders through many flowery meadows, including the "compound interest" formula for e and Wallis' formula for pi.
- Multipole Expansion of the Electrostatic Potential
The electrostatic potential of an arbitrary finite charge distribution is expanded in powers of Legendre polynomials. The monopole, dipole, and quadrupole terms are written out explicitly and compared to analogous quantities in mechanics. The handout ends with six student exercises.
- An Easy Method for Partial Fraction Decomposition
If you know how to find residues of simple poles (which is usually the first example one learns about in connection with residues), then you can easily decompose partial fractions without having to laboriously solve simultaneous equations. This brief handout gives an example outlining the idea.
- Physics Cinema Classics
A few choice selections from this "golden oldie" series.
- Mechanical Universe and Beyond
If you have seen this excellent series of 52 videos by David Goodstein, the following set of keywords may suffice to jog your memory. A web address where you can watch the series online is included.
- Video Encyclopedia of Physics Demonstrations
I watched this entire 25-laserdisc collection of physics demonstrations. I summarize my favorites and tie them to specific courses and textbooks in a typical physics curriculum. Select videos can be watched online.