In the first part of the course we'll consider systems of huge numbers of particles whose microscopic motions are random. The study of such systems is known as statistical mechanics. We'll find that by ignoring the details of the random motions of individual particles and looking at such systems in terms of only macroscopic observables we can come to very general and powerful conclusions about their behaviors. Ultimately we will be able to describe essentially all macroscopic thermal phenomena (thermodynamics), and we will be able to answer the question of why time appears to flow forward.
The latter part of the course is devoted to the study of linear waves, a class of coherent motions commonly present in systems of large numbers of particles (e.g. a big metal rod). Our approach to this problem will be to neglect the discreteness of the system and instead approximate the many particles by a continuous field, analogous to the electromagnetic field. As a result, we will be able to study both many sorts of waves in various media and waves, including electromagnetic waves. The same principles will apply in quantum mechanics as well, where matter itself is treated as a wave (although you'll have to wait until you take Physics 35 to see this).
The general characteristics of wave motion will be approached through the wave equation and the solution to the boundary value problem. Subsequent discussion cover energy relationships in waves, diffraction, interference, reflection, refraction and polarization. We'll also see how geometrical (ray) optics can be obtained from wave optics in the short-wavelength limit. If time permits, we'll broach the subject of nonlinear waves and solitons.
In the laboratory portion of the course we will get our hands on some thermal and wave phenomena and geometrical optics. We'll also continue our study of electronics, begun in Physics 33, with an introduction to operational amplifiers.
Throughout the course, new mathematical techniques including probability and combinatorics, Fourier analysis and matrix algebra and Green's functions will be introduced as needed. Mathematica may be introduced to broaden the range of problems you'll be able to tackle.
It's your affair how you allocate your time and I won't explicitly penalize you for being absent, but if you're going to be absent please do me the courtesy of letting me know in advance, if possible. It's a small class--I'll notice if you're missing. If you arrive late, please enter quietly at the back of the classroom. Obviously it will be your responsibility to find out what happened in the class you missed. Also, you must attend all of the labs.
Why do the homework?
I can't emphasize enough the importance of working the problems. In some of your classes homework is primarily evaluative; the point is for you to demonstrate what you've learned from the readings and lectures. In physics the homeworks are primarily instructional; you learn physics primarily by working problems. You must work the problems, think about the results, and understand any mistakes you've made if you wish to attain the type of understanding of the subject required of a working physicist. In at nutshell: If you can't work problems you don't know physics. I (or a grader) will grade the problems, and I'll post solutions to this web page as quickly as I can. I encourage you to read the solutions and understand any mistakes immediately. If it doesn't make sense, ask me about it right away---don't wait until right before an exam. Pragmatically, year after year I observe a strong correlation between performance on homework and overall performance in the courses I teach. While missing a homework or two does not have a profound impact on your grade, remember that the combined weight of the homework is that of a midterm exam, so skipping them all is not advised. A more serious consequence of skipping homework is missed opportunity to sharpen your understanding.
I instruct the grader to give full credit only to a clear, correct and complete solution. I always encourage the grader for the course to be picky in grading the homework. The point is not to frustrate you, but to bring to your attention problems (from sign errors to sloppy reasoning) so that you can clarify your thinking and improve your problem-solving skills. If you lose points, be grateful that someone is paying sufficiently close attention to your work to give you valuable feedback. Of course, the grader is a student like you, and has only a limited amount of time to expend on grading. I've told him/her that if s/he doesn't see your logic or can't read your solution, don't give credit. If s/he makes a mistake in the grading, simply bring it to me and when you explain it I'll correct the grade.
If you've got a compelling reason why you need an extension, come talk to me in advance. I will not grant a homework extension without penalty if you ask for it on the day the homework is due, so don't ask for one. [If you need such a last-minute or post-facto extension due to extenuating circumstances (e.g. death in the family, sudden illness, travel problem), you'll need to have the Dean of Students or your Class Dean formally make such a request to me and suggest a rescheduled due date. You should also take this route if you need an extension but you don't want to tell me why (say, it's for personal or legal reasons). If you explain your reason to a Dean and the Dean tells me it's OK, that's good enough for me.] In general, though, life will be easier for you and for me if you get into the habit of doing your best to finish the problem set on time and handing in as much as you've been able to complete by the deadline. Even if you finish your homework after the three-day deadline, you can turn it in. I won't award you credit, but I will record that you did the problem set.
The formal lab writeups are due one week from the day the lab was performed, unless otherwise noted. Late labs (and rewrites) will be penalized 10% per day late (including weekends). In any case, each lab not performed or lab report not submitted will automatically cost at least a full letter grade in the final course grade.
Unlike Physics 33, I will grade the lab reports myself. The standard for clarity and completeness will be somewhat higher than in Physics 33, and in particular I'll take care to note that you've done the error analysis. The book by D. C. Baird (listed below) continues to be a good guide to style in lab reports.
Each faculty member has a style of exam they prefer. Exams in this course will be challenging, and you should not be discouraged if you don't receive perfect scores. I try to construct problems so that if you've got an idea what the problem is about and you've got a moderate working knowledge of the physics needed to solve it, you should be able to get at least 2/3 of the points. If you get stuck during an exam, don't panic! You should formulate a question and come ask me. I may not answer your question, but I certainly won't fault you for asking. The purpose of the exams is for to demonstrate your knowledge, and it doesn't serve that purpose for you to turn in blank pages just because you didn't see the mental picture I had when I wrote the problem.
Unlike in Physics 33, in this course I do have some exams from previous years which you can use as a study aid. Just ask.
The midterms will be scheduled during lab time, dates listed in the table below. In the unlikely event you have a conflict with these days, let me know immediately.
The College requires that all written work for a course except for a final be submitted by 5 pm on the last day of classes. The physics department takes this deadline seriously. After that day/time, no homework or lab reports will be accepted, nor will I conduct exit interviews for labs.
The roles of lectures and textbooks
Lecture will not be a regurgitation of the text, a summary of all you need to know for the course, or a how-to guide for the homework. Rather, I'll try go deeper into selected points. In lecture I'll cover material and do demonstrations related to the readings, but I won't feel obliged to be comprehensive in those places where I feel the text is adequate and I may focus only on a few points that I feel are particularly interesting or subtle. You shouldn't expect to understand what's going on without close study of the readings, and you should come to class with questions you have on the readings. Further, after we settle into the semester a bit, I expect the classes will become less lecture-oriented and more participatory; it will be difficult to reap the maximum benefit from that format if you're not sufficiently prepared to fully participate.
For the problems you can't solve, talk to classmates, attend the problem sessions, or ask me. When you ask me, either try to give you just enough of a hint to get you through, or I'll guide you through the problem with a series of leading questions. I'll never just tell you how to do it. If you run out of time and don't finish the set, start earlier next week. When the solutions come out, look over them right away, before you've forgotten all of the points you were confused about. You think you'll just get clear on it before the next exam, but there's never as much time as you think.
On the other hand, if you find the class too slow for your liking, if you have questions that you aren't getting answers to, if you'd like more detail, if you are frustrated that we aren't digging deeply enough, if you crave more applications, come talk to me. I'm very happy to provide you with additional materials or explanations that will will stimulate you and challenge you at whatever level you can handle.
One word of warning: Amherst College students tend to have lots of extracurriculars of all types. I support this (enthusiastically), and I am occasionally willing to be flexible to facilitate your participation in range of activities, but don't let your extracurriculars overshadow your academics. If you become concerned that your courses are getting in the way of your extracurriculars, you definitely have the wrong mindset. Remember why you're here.
If circumstances in your life beyond the class are the problem, you can come talk to me, but also talk to your class Dean.
Required (should be available at the Jeffrey Amherst bookstore):
|1. January 27||Thermodynamics: Zeroth and First Laws
Jan 27:Introduction to the course
Administrative stuff / course overview / the big picture of thermal physics. Why does time flow forward? [Handouts: web page, student info sheet [source], Lab 1, Notes on Uncertainty analysis [source], Thermometers and Thermostats handout]
Jan 29: What is temperature?
Paradigmatic thermal process. Phenomenological approach to temperature: temperature, thermoscopes, zeroth law of thermodynamics. Temperature standard: constant volume gas thermoscope and T=CP.
[Handouts: Romer article on temperature]
Jan 30: Thermal Energy and Ideal Gases
Thermometers. Temperature related to energy via the specific heat. How is energy stored in matter? Ideal gases: Ideal gas law, kinetic theory model of ideal gas.
Jan 31: Kinetic theory model of ideal gas
Kinetic energy related to temperature. Internal energy stored as microscopic mechanical energy. How well does the model work? Comparison of prediction of specific heat to experiment. Equipartition theorem.
|Read: Schroeder, Chap. 1
Problems: Chap. 1: 1.8, 1.16, 1.20, 1.22, 1.27, 1.31, 1.34, 1.38, 1.39, 1.40
Deadline extended to Wed. 11:59 pm
Ideal gas thermometry
|2. February 3||Thermodynamics: First and Second Laws|
Feb 3: Ideal Gases / First Law
Quantum mechanics determines how degrees of freedom are counted for equipartition theorem. Typical energy scales for quanta. Einstein solid.
Heat, work defined. First law is conservation of energy. Expansion-compression work.
[Handouts: Lab 2]
Feb 5: PV diagrams and constrained processes
Quasistatic processes. PV diagrams. Calculating work and heat transfers for quasistatic isothermal, isobaric, isochoric, and adiabatic processes. Closed cycle processes.
[Handouts: Additional notes on specific heats, Notes on formal lab writeups [MSWord source] and uncertainty analysis [MSWord source], more notes on formal lab writeups [MSWord source], article on Liedenfrost effect]
Feb 5 Lab: Heat capacity and Latent Heat
Cv, Cp, in general and for ideal gases. Latent heat.
Feb 6: Approaching the Second Law: Two-state model
Microstates, macrostates, multiplicity. Two-state system: model, counting microstates, probabilities, multiplicities of macrostates.
Feb 7: Two-state model continued
Stirling's approximation. Simplifying the multiplicity function for large N. Sharpness of the multiplicity function and probability distribution. Sharp probability distribution implies well-defined physical properties.
|Read: Schroeder, Chap. 2
Problems: Chap. 1: 43, 46, 55; Chap 2: 4, 8, 18, 19, 22, 24
Latent heat of liquid nitrogen
|3. February 10||Entropy and the Second Law
Feb 10: Generalizing lessons of the two-state model
Fundamental assumption combined with sharp multiplicity function ensures macroscopic regularity for general systems.
Einstein model: model, equipartition theorem results. Quantum mechanics of harmonic oscillators. Microstates, macrostates.
[Handouts: Lab 3, notes on multiplicity in Einstein solids using generating functions]
Feb 12: Einstein Model / Systems in Thermal Contact
Calculating the multiplicity. Multiplicity in the classical limit. Multiplicity for general systems in thermal contact. Multiplicity for a pair of Einstein solids in thermal contact.
Feb 13: Irreversibility of Heat Flow / Ideal gases
Irreversibility: Multiplicity sharply peaked about a particular macrostate (equal number of quanta per oscillator). How sharp is the peak? Sharp peak in probability implies irreversibility.
Ideal Gases: Basic quantum mechanics of an ideal gas: particle-in-a-box. In the classical limit, multiplicity proportional to area of a shell in 3N-dimensional space.
Feb 14: Multiplicity of an ideal gas
Relation between multiplicity and area of 3N dimensional sphere. Area of d-dimensional sphere. Discussion of "thickness factor". Correction for indistinguishability of identical particles. Observe relation between exponent of U and number of d.o.f.
[Handouts: Entropy, Information and computation article]
|Read: Schroeder, Chap. 2
Problems: Chap 2: 2.25, 2.26, 2.30, 2.32, 2.34, 2.38, 2.40, 2.42, Appendix B: B.14
Measurement of Cp/Cv
|4. February 17||Entropy and Ideal Gases
Feb 17: Ideal gases in thermal and mechanical equilibrium
Multiplicity for gases in thermal contact are sharply peaked: thermal equilibrium. Multiplicity for gases in thermal contact when volume can fluctuate sharply peaked: thermal and mechanical equilibrium. Second law: multiplicity tends to increase, given the opportunity. Entropy defined via log of multiplicity. Entropy is extensive.
[Handouts: Complex numbers and the meaning of Exp[ix] [source], Complex numbers and AC circuits [source], Input and Output Impedance and Thevenin's Theorem [source], Solutions to problem set 2, Notes on use of the oscilloscope [source], Lab 4: operational amplifiers, Feedback operational amplifiers (Horowitz and Hill), Operational amplifiers (Diefenderfer) ]
Feb 19: Ideal gases (with a side of Op Amps)
Ideal gases: Sackur-Tetrode equation. Processes with gas in a piston: isothermal expansion, sudden expansion. Mixing ideal gases: Gibbs paradox. Reversibility and irreversibility.
Op Amps: What they are, what they're used for. Golden rules of op amps. Inverting amplifier.
[Handout: op amp data sheet]
Feb 19 lab: Op amps
Inverting amplifier. Noninverting amplifier. Thevenin's theorem, input and output impedance. Rule of 1/2.
Feb 20: Consequences of the Second Law: Thermodynamics
Microscopic understanding of thermal equilibrium. Thermal equilibrium condition: dS/dU same for subsystems. Defining temperature from entropy using the ideal gas. Temperature is a measure of a system's propensity to give up energy (but NOT necessarily a direct measure of energy). Temperature in Einstein solids. Three possible S vs. U curves.
Feb 21:Connection between Entropy and Heat / Two-state model
Five-step process to go from multiplicity to specific heat. Demonstrate for Einstein solid. Properties of Cv from shape of S vs. U curves. Obtaining entropy from the Cv(T). Cv(T) vanishes as T->0 (third law). Heat is energy that carries entropy.
Two-state model: energy of magnetic dipoles in a B field. Macrostates: energy or total magnetic moment.
|Read: Schroeder, through sec 3.4.
Problems: From Schroeder, Chap. 3: 3.5, 3.7, 3.8, 3.10, 3.14, 3.15, 3.16, 3.23, 3.25, 3.26
|5. February 24||Thermodynamics and Equilibrium
Feb 24:Two-state model / Mechanical equilibrium
Two-state model: Entropy (exact and Stirling approx.). Shape of S vs. U curve implies positive, negative and infinite temperature. Negative temperature "hotter" than positive temperature: 2-state system in contact with Einstein solid will equilibrate at positive temperature. Calculate T(U), invert to get U(T) and M(T).
Mechanical equilibrium and Pressure: Set up system to analyze.
[Handout: Solutions to problem set 3]
Feb 26: Complex Numbers, AC Circuits, and Op Amps
Review of complex number, Euler's formula, and the use of complex numbers in solving AC circuits. Start analysis of frequency-dependent op-amp circuits: high-pass inverting amplifier.
Feb 26 lab: Op Amps continued
Integrator. Differentiator. High pass inverting amplifier (cont'd). Three decibel point.
Feb 27: Generalized equilibrium
Generalize ideas from study of thermal equilibrium to mechanical and diffusive. Expression for pressure from condition of diffusive and thermal equilibrium. Verify in ideal gas. Thermodynamic identity. Chemical potential from diffusive equilibrium condition. Demonstrate that systems with larger chemical potential tend to give up particles. Extend thermodynamic identity.
Feb 28: Heat engines
Second law limits the amount of heat that can be turned into work in cyclic processes. Abstract model of heat engine. Efficiency defined. Second law limits maximum value of efficiency. Carnot cycle as a realization of reversible engine.
|Read: Finish Schroeder, Chap 3. Also Schroeder 4.1, 4.2, 4.3; 6.1,6.2.
Problems from Schroeder: 3.34, 3.36, 3.39, 4.1, 4.3, 4.14, 4.18, 6.5, 6.6
|Lab 4: Operational amplifiers (cont'd)||EI|
|6. March 3||Wrapup of thermal physics / start waves|
Mar 3: Carnot engines / Boltzmann factor
Carnot engines: Heat and work at each stage of a Carnot engine (with ideal gas working medium). Efficiency of Carnot engine.
Boltzmann statistics: Idea: characterize systems at fixed temperature. Derive Boltzmann factor.
[Handout: 2002 Midterm 1 and solutions]
Mar 5: Boltzmann factor finale
[Demos: Compression igniter, Two steam engines]
Quick recap of Boltzmann factor derivation. Partition function. Significance of kT as scale of typical energy fluctuations. Example: Two-state paramagnet. Example: simple harmonic oscillator.
Mar 6: Op amps and Optics
Op amps: Low pass inverting amplifier. Band pass amplifier.
Optics: Position of geometrical optics in the "grand scheme". Some facts basic geometrical optics facts: speed of light, index of refraction.
[Demos: Laser pointers, refraction in a tank of water]
Mar 7:Simple facts about geometrical optics
Snell's law, law of reflection, total internal reflection, chromatic dispersion. Applications: rainbows, optical fibers, fake diamonds. Fermat's principle: deriving Snell's law.
[Demos:Reflection, refraction, and total internal reflection in fishtank.]
[Handout: geometrical optics handout]
| Read Halliday, Resnick & Walker, Chap. 34.7, 34.8, start Chap. 35
Halliday Resnick & Walker, Chap 34: 49,51,56,57; Chap 35: 3,8,11
|7. March 10||Mirrors |
Mar 10: Mirrors (mostly plane)
Real and virtual images. Plane mirrors: Image distance vs. object distance, orientation and magnification, front-back reversal. some thought questions. Spherical mirrors: concave and convex-- summary. focal point and focal distance.
[Demos: plane mirror, big and small spherical mirrors]
Mar 12: Mirrors (mostly spherical), part I
Reflections are not equivalent to rotations. Focal points: single focus for paraxial rays only. Spherical aberrations. qualitative for convex and concave mirrors. f=R/2. sign conventions for f and R. proof that f=R/2 for paraxial rays. Images: state lens formula. ray tracing. example: concave mirror, object inside and outside focal point.
[Demos: blackboard optics -- concave spherical mirror]
Mar 13:Mirrors (mostly spherical), part II
Proof of spherical mirror formula. Proof of magnification formula. Summary of sign conventions for mirrors. Fermat's principle: recap. imaging rays are isochronous. mirrors that are conic sections make perfect images.
Mar 14:Fermat's principle in action
early version of Fermat's principle: rays are paths of least time. modern version: rays are paths of stationary time. Derive mirror equation and magnification formula using Fermat for general optical delay D(h). Relate D''(h) to focus. Demonstrate that D(h) for spherical mirror gives correct expession for focal length.
| Read Halliday, Resnick & Walker, Chap. 35
Halliday Resnick & Walker, Chap 35: 13,15,20,26,29,31,34,35,37
|Lab 4: Operational amplifiers (cont'd)||Formal|
|March 15-23||Spring Break!||---||woo-hoo!|
|8. March 24||
Mar 24: Spherical refracting surfaces
Location and nature of images (real or virtual) via ray tracing. Spherical refracting surface formula. Sign conventions. Focal length.
[Demos: blackboard optics -- spherical refracting surface]
Mar 26: Spherical refracting surfaces / Thin lenses
Derive spherical refracting surface formula. Converging and diverging lenses (qualitative). Thin lens formula and lensmakers formula. Sign conventions.
[Demos: blackboard optics -- converging lenses; lasers and lenses]
Mar 27: Thin Lenses
Derivation of thin lens formula. Finding images with ray tracing. Magnification. Check image distance and magnification for converging and diverging lenses.
[Demos: blackboard optics -- diverging lenses; lasers and lenses; a few big old lenses]
Mar 28: Thin Lenses from Fermat / Systems of Lenses
Deriving thin lens formula and lensmaker's equation from Fermat's principle. Systems of lenses.
|Read: Skim Pain, Chaps. 1-3 as a review of harmonic oscillators.
You can skip all of the starred sections.
|9. March 31||
Waves: 1D transverse waves on a string
Mar 31: Deriving the wave equation
Assumptions. Use Newton's second law to derive the waves equation in the small-slopes approximation. Verify the travelling wave solution.
Apr 2: Solving the wave equation
Small-slopes approx. means tranverse speed << wave speed. D'Alembert's solution to the wave equation: left-movers and right-movers. Qualitative kinematics.
Apr 3: Sinusoidal waves
Importance of linearity of eqn. Basic properties of sinusoidal waves. Aside: Finding travelling waves from initial conditions.
Apr 4: Superposition (Interference) of sinusoidal waves
Two examples of travelling waves from initial conditions. Causality and finite strings.
Interference Interference of two out-of-phase (+) waves. Interference in time: beating.
[Demos: Beats with tuning forks. Beats applet 1 , Beats applet 2 . ]
|Read: Pain, Chap.5, p. 113-122 and 135-142, and Pain, Chap. 10.
|10. April 7||
Fourier Synthesis: superposition gone wild
Apr 7:Standing waves and Fourier Synthesis
Fourier Synthesis: main idea. Fourier's theorem. Trig and exponential expansions. How to find the coefficients? Orthogonality relations.
[Demo: standing waves applet ]
[Handouts: lab, problem solutions]
Apr 9:Fourier analysis: example
Finding the Fourier coeffients using orthogonality relations. Derivation of an orthogonality relation. Trig Fourier expansion of square wave.
[Demo: one Fourier synthesizer , another Fourier synthesizer , a third Fourier synthesizer , a nice Fourier movie . If you have a microphone on your computer, you may be find this SILS speech analysis software interesting. ]
Apr 10:Qualitative features of Fourier series
Exponential series Fourier coefficients of square wave. Qualitative significance of large Fourier coefficients. Dirichlet conditions. General observations about coefficients. Vector space interpretation of Fourier analysis.
Apr 11:Pulses and Power spectra
Power spectrum. Fourier series for a periodic pulse. Bandwidth-pulse duration relation.
|Reading: Pain Chap. 10
Speed of Waves
|11. April 14||Acoustic Waves
Apr 14:Fourier transforms / Longitudinal waves
Fourier transform, coefficient equation, and orthogonality relation. Dirac delta function. Example: square pulse. Bandwidth-pulse duration relation again.
Longitudinal waves Qualitative features. Ideal fluid medium. Force on a fluid element is proportional to pressure gradient.
[Demos: transverse waves on a spring, longitudinal waves in a slinky]
Apr 16: Acoustic waves
Simplifying assumptions. Notation in the Lagrangian formulation.
Derivation of the wave equation: Conservation of mass, Newton's second law, equation of state. Combine and linearize to obtain 1-D wave equation and expression for speed of sound.
Longitudinal waves (II)
Longitudinal waves (III)
Longitudinal waves (IV)
Longitudinal waves (V)
Longitudinal waves (VI) ]
Apr 17: Speed of Sound
Speed of sound in isothermal approximation gives the wrong value. Adiabatic approximation is much better. Expression for speed of sound in terms of elastic bulk modulus. Keeping the nonlinearities gives amplitude-dependent speed of sound.
Apr 18: More on acoustic waves and wave equation
More on the propagation of shock waves. Quickie derivation of wave equation by linearizing early. Introduce convenient variables. Useful features of solutions to the wave equation.
|12. April 21||Transmission and Reflection of Waves
Impedance: Define acoustic wave impedance as ratio of driving force to particle velocity. Analog for transverse waves on a string.
Phase relations between physical quantities in acoustic waves.
Qualitative transmission and reflection: Reflected waves inverted upon "hard" reflection, not inverted upon "soft" reflection. Transmitted wave not inverted.
Reflections from boundaries
Reflection of a pulse
Reflection of a sine wave ]
Apr 23: Reflection and Transmission: Boundary Value Problems
Example: Reflection of transverse waves on a string from a rigid surface. Incident sinusoidal waves give standing waves in this case.
Example: Reflection of acoustic waves from a rigid surface.
Example: Acoustic waves produced by a moving boundary.
Set up scenario for transmission and reflection at an interface with impedance mismatch.
Apr 24: Transmission and Reflection at an Interface
Procedure: Write general form of wave equation in each region. Choose boundary conditions at infinity (e.g. sending in a wave from the left but not the right). Determine and apply boundary conditions at interface.
Example: Longitudinal waves in fluids
Boundary conditions for longitudinal waves at interface: displacement/velocity continuous, pressure continuous. Apply boundary conditions to obtain solution transmitted and reflected velocity waves in fluid. Transmission and reflection coefficients for velocity defined and obtained. Observations about qualitative features of velocity transmission and reflection coefficients.
Apr 25: Transmission, reflection and a plate
Finish qualitative observations of transmission and reflection at a massless interface.
Example: Transmission and reflection of acoustic waves through a thin massive plate.
Deduce boundary conditions. Note that arbitrary wave packets will be deformed, so we use exponential (sinusoidal) incident waves. Apply boundary conditions, obtain (complex) velocity transmission and reflection coefficients. Observe that plate acts as a low-pass filter.
Measuring the wavelength of light with a ruler
|13. April 28||Interference of spherical waves
Apr 28: Finish transmission/reflection / Start interference
Define pressure transmission/reflection coefficients, make general observations about them. Consider limits of extreme impedance mismatch and impedance matching.
Interference from point sources in 2D and 3D:
General form of wave from point sources in 2D and 3D. Discussion of r-dependence of amplitude. Sinusoidal waves. Scenario: interference from two point sources of equal frequency, viewed at a distance large in comparison to their separation (far-field approximation). Find expression for total wavefunction at observation point, note that it has nodes and antinodes reminiscent of standing waves in one dimension. Claim: nodes and antinodes lie on hyperbola.
[Demo: Ealing film loop: Interference of waves]
Apr 30: Two-source inteference
Curves of constant phase are hyperbola, since phase difference is proportional to difference in distance from observation point to source. Simple approximation for difference in distance in far-field limit (assume rays parallel); curves of constant phase become lines. Two-source interference from phasor perspective.
[Demo: ripple tank: 2D interference ,
Ealing film loop: Interference of waves]
May 1: Two-source interference in various limits
Time-averaged intensity. Small source spacing: single source of double amplitude. Large source spacing: closely-spaced fringes. Determining intensity maxima and minima by comparing difference distances in source- observation point distances to wavelength. Intermediate spacing: antenna patter. Consider each case in using (1) phasors, (2) exact expression, (3) distance difference vs. wavelength.
[Demo: antenna/interference ]
May 2: N-source inteference
Formulation of the problem and the far-field approximation. Maximal constructive interference occurs when path-length differences are integer numbers of wavelengths. Simple expression for phase difference in far-field approx. Exact expression for total wave at observation point: summing the series. Total time-averaged intensity. Examine resulting expression in large-spacing limit: rapidly oscillating numerator, denominator that blows up. Denominator gives the maxima corresponding to path-difference reasoning.
|Reading: Pain, Chap 12 (especially the latter part, starting at
Division of the Wavefront on p. 355 (5th edition))
Problems: Pain, Chap 12: 3,4,5,6,8,10,13,14,15,16
|14. May 5||Diffraction and Polarization
May 5: N-source inteference leads into diffraction
N-source interference in the large-source-spacing, small-spacing, and intermediate-spacing regimes. The latter was the case for the CD in the lab. Phasor perspective. N-source inteference in the limit N goes to infinity leads conceptually to diffraction: interference from extended, continuous sources. Archetypical problem: diffraction of waves from a slit.
N-source phasor applet
diffraction applet (I)
diffraction applet (II)
diffraction applet (III)
diffraction applet (III)
diffraction applet (IV)
May 7: Diffraction through a slit
Huygens principle. Expression for total wave as sum of point sources along the slit: N-source sum goes into an integral over the aperture. Fraunhofer (far-field) approximation. Expression for total wave at observation point in this limit. Evaluate integral, find locations of minima. Time-averaged intensity. Connection between diffraction through a slit and Fourier transform of a finite pulse.
May 8: Diffraction
Short-wavelength (geometrical optics) and long wavelength (point source) limits of diffraction through a slit. Obtaining diffraction time-averaged intensity from N-source interference time-averaged intensity. Finding the nodes of diffraction pattern by cancelling waves pairwise. Diffraction formalism can be extended to more complicated geometries (e.g. rectangle).
Ealing film loops: diffraction around obstacles, single slit diffraction.
ripple tank: diffraction
3D wave box: diffraction
May 9: Diffraction from arrays of slits / Polarization
Diffraction through a pair of slitshas an intensity pattern that is a product of that of a single slit times that of two point sources. This generalizes to N slits.
Rederive the wave equation from Maxwell's equations. General solution for a plane wave now has a left-mover and a right-mover associated with each component of the electric field vector, all of which are potentially different. Polarization is the direction of the E-field. Linear polarization. Circular polarization. Elliptical polarization. Natural light as randomly polarized. Linear polarizer sheets absorb one component of the incident E-field, allow the other to pass. Series of polarizers.
Sets of crossed polarizers on overhead projector. Glare reflected from the wax on the floor is polarized.]
|No problem set.||Finish interference||EI|
|May 12-16||Final exam period||Final: TBA|
Area Seminars and colloquia