Physics Scholarship

I calculate the electric field and potential at a point between two grounded spheres diametrically across from a point charge. I also find the surface charges induced on the two spheres.

First I consider an N-way switching circuit for a light bulb. Understanding this circuit could help troubleshoot wiring problems in your house. Second I present a sneaky series circuit involving two switches and two bulbs where each switch independently controls one of the bulbs. The magic behind this circuit was demonstrated to me by Hans Pfister at Dickinson College.

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.

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 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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!

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.

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.

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.

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.

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.

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.

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.

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?

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.

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.

Approximate and exact values of the angular positions, peak intensities, and integrated areas of the secondary maxima for single-slit Fraunhofer diffraction are compared.

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.

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.

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.

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.

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.

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.

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 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.

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.

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.

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.

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.

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.

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?

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.

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 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.

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.

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.

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.

I review a way to introduce the Legendre transform via partial derivatives. Examples from thermodynamics and classical mechanics are included to illustrate the method.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

If you know the numerical values (possibly complex) of the sum of two numbers and the sum of their cubes, what is the sum of their squares? When is there not a unique solution?

On average how many rolls of a die will it take to get a 4, if you are limited to some maximum number of rolls?

The probability distribution for the first digit of a list of measured values is derived. For example, there are about two-and-a-half times more books in your public library that have a page count between 300 and 399 than than are books that are between 800 and 899 pages long (assuming there are a negligible number of books shorter than 90 pages or longer than 2999 pages).

If sugar cubes are arranged into the shape of a rectangular box (with no vacancies), how many cubes can you see if your eye points "radially toward" one of the vertices of the box (so that three of its faces are visible)?

Student 1 walks by a set of initially closed lockers and opens all of them. Student 2 then closes lockers 2, 4, 6, .... Next student 3 closes locker 3, opens locker 6, closes locker 9, .... After student N toggles the states of the lockers in this way, which lockers between 1 and N will remain open?

Two methods, one algebraic and one graphical, are presented to obtain the formulas for the trig functions of 15 and 75 degrees from scratch.

A derivation of the sums from minus to positive infinity of the reciprocal of the sum or difference of integer squares or quartics and a constant.

Find the smallest integer that when divided by any L in a given list of integers gives a remainder of L-1.

Find a quartic polynomial in n for the sum of the first n integers cubed.

A penny and a nickel are each flipped a different number of times. What is the probability we get the same number of heads for the two coins?

A circle and square intersecting at 3 points define a 3-4-5 triangle which enables one to compare the properties of the circle and square.

How many squares are there on a chessboard? How many of them contain an equal number of white and black spaces?

You draw two balls numbered in order starting from 1 out of a bowl randomly. What is the probability the two numbers differ by some specified integer?

In how many different ways can you write a positive integer as a sum of two or more consecutive positive integers?

A method for multiplying two integers is proposed if you only know how to multiply by 2 (and its inverse, which is dividing even integers by 2).

We seek integer solutions for the ratio of two reduced integer polynomials.

There are in general two real solutions when an exponential function equals a linear polynomial.

A brief primer on how to convert numbers in arbitrary bases to stacked scientific notation.

A frog can hop forward with equal probability to any of n lily pads in a line ahead of him. How many lily pads will he visit on average before arriving at the final pad?

Any four-digit number formed by tracing out a parallelogram on a numeric keyboard (such as 5621) is divisible by 11.

Show that any prime number (greater than or equal to 5) can be written as the square root of 1 plus some integer times 24.

Divide a square into four isosceles triangles, none of which are identical to each other.

A challenging logic puzzle from the Wall Street Journal involving two numbers whose sum and product are each seen by only one person.

What four-digit number when multiplied by four is equal to the original number with its digits written in reverse order? How about any even-digit number with six or more digits?

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?

If you know the integral of a forward function, you can find the integral of its inverse function by integrating by parts.

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.

A derivation of the relationships between the side lengths and angles if one angle of a triangle is bisected by a line.

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.

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.

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.

Cardano's four-step method for finding the algebraic solution of a cubic in terms of its coefficients is compactly outlined.

A simple noncalculus derivation of the area of an ellipse is reviewed.

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?

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.

A quick sketch is presented of how the Jacobian enters into integrals when one transforms multiple variables (such as from rectangular to polar coordinates).

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!

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.

Four methods are presented for expanding the reciprocal of a polynomial in x as a power series for small x.

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.

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.

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.

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.

A summary of how to find the locations of all of the minima, maxima, and saddle points for a function of two independent variables.

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.

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.

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?

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.

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?

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?

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.

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.

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.

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.

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.

A derivation of Bessel functions of the first and second kind, together with some student exercises.

A one-page reference guide to common ODEs.

Do you use a power series or a Frobenius series? How do you handle various possibilities for the roots of the indicial equation?

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.

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.

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.

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.

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.

A few choice selections from this "golden oldie" series.

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.