Download a method for teaching algegra based advanced placement physics

Teresa A. Walker
1532 Blackbird RD NE
Rio Rancho, NM 87144
(505) 892-8041
[email protected]
April 17, 2008
Dave Westpfahl, Ph. D
W. D. Stone, Ph. D
Julie Ford, Ph. D.
New Mexico Tech
Master’s of Science Teaching Program
801 Leroy Place
Socorro, NM 87801
Dear Drs. Westpfahl, Stone, and Ford:
You will find attached my final Independent Study Project entitled “A Model for Teaching
Algebra-based Advanced Placement® Physics and Calculus-based Advanced Placement®
Physics in the Same Classroom”. This will complete my Master’s of Science Teaching degree.
Hours of research and writing have gone into producing this project to support AP Physics
teachers and students. In an effort to provide improved education for high school students,
Advanced Placement ® (AP) Physics provides students with advanced physics concepts,
prepares them for college, and makes them more attractive to competitive colleges. Providing
this report to physics educators will provide them support in teaching both algebra-based AP
Physics and calculus-based AP Physics in the same classroom. Providing this program at their
school will allow physics instructors with a smaller population of students to offer the class by
increasing enrollment.
I would like to thank you for your time in reading and reviewing this project. Please contact me
if you have additional concerns or questions.
Sincerely,
Teresa A. Walker
A MODEL FOR TEACHING ALGEBRA-BASED ADVANCED PLACEMENT®
PHYSICS AND CALCULUS-BASED ADVANCED PLACEMENT® PHYSICS IN THE
SAME CLASSROOM
Prepared for Physics Educators
And
New Mexico Tech
Masters of Science Teaching (MST) Program
Particularly
David J. Westpfahl
Professor, Astrophysics
W. D. Stone
Professor, Mathematics
Julie Ford
Professor, Technical Communication
Prepared by
Teresa Walker
[email protected]
http://www.orgsites.com/nm/walker/
Instructor
Rio Rancho High School, Rio Rancho Public Schools
Rio Rancho, New Mexico
Submitted April 17, 2008
ABSTRACT
This project provides a plan for teaching algebra-based Advanced Placement® Physics (AP
Physics B) and calculus-based Advanced Placement® Physics (AP Physics C) in the same
classroom. This project provides a general timeline for presenting all the material required by
both classes and in particular provides self-guided lesson plans for students when the material of
each of the courses is entirely different. The material covered in the self-guided lesson plans for
AP Physics B includes waves and optics, thermal physics and fluid mechanics, and nuclear
physics. The material covered in the self-guided lesson plans for AP Physic C includes
rotational motional, calculus-based electricity including Gauss’s Law, and calculus-based
magnetism including Ampere’s Law.
The purpose of this project is to provide high school physics educators with support and to
provide physics students with quality lessons on the specific topics described. The goal for
teaching these two classes concurrently in the same classroom is to improve enrollment in AP
Physics. Literature research was done on the specific topics using college physics textbooks and
the internet. The author’s experience teaching the subjects this way for six years also contributed
to the lesson plans presented.
iii
TABLE OF CONTENTS
ABSTRACT ……………………………………………………………………………………. iii
LIST OF TABLES AND FIGURES …………………………………………………………..... v
INTRODUCTION ……………………………………………………………………………… 1
OVERVIEW OF COURSE TOPICS AND ORDER………………………………………….... 3
PRACTICAL THOUGHTS ON MANAGING THE COURSES TOGETHER ……………….. 6
RECOMMENDED BOOKS AND INTERNET RESOURCES ………………………………... 7
UNIT 3, WAVES AND OPTICS (PHYSICS B) AND ROTATIONAL MOTION (PHYSICS C)
……………………………………………………………………………….…… 9
Physics B Unit 3, Lesson 1, Wave Beginnings (B31Wave) …………………………………….. 9
Physics B Unit 3, Lesson 2, Reflection and Refraction (B32Reflect) …………………………. 14
Physics B Unit 3, Lesson 3, Lenses and Mirrors (B33Lenses) …………………………...……. 15
Physics B Unit 3, Lesson 4, Diffraction and Thin Films (B34Diffract) ……………………….. 20
Physics C Unit 3, Lesson 1, Calculus-based Rotation (C31Rotate) …………………………… 23
Physics C Unit 3, Lesson 2, Rotational Inertia (C32Inertia)………………………...…………. 25
Physics C Unit 3, Lesson 3, Incline and Pulley (C33Incline) ………………………….…….… 27
Physics C Unit 3, Lesson 4, Other Rotational Situations (C34Other) ……………………….… 28
UNIT 5, FLUID MECHANICS AND THERMAL PHYSICS (PHYSICS B) AND
ELECTRICITY IN DEPTH (PHYSICS C) ……………………………………………………. 30
Physics B, Unit 5, Lesson 1, Pressure and Buoyant Force (B51Pressure) …………………..… 30
Physics B, Unit 5, Lesson 2, Bernoulli and Torricelli (B52Bernoulli) …………………...……. 33
Physics B, Unit 5, Lesson 3, Gas Laws (B53Gas) ……………………………………………... 34
Physics B, Unit 5, Lesson 4, Thermodynamics (B54Thermo) ………………………………… 36
Physics C Unit 5, Lesson 1, Gauss’s Law for Spheres (C51Gauss) …………………………… 39
Physics C Unit 5, Lesson 2, Continuous Charge Distributions (C52Continuous) …………….. 41
Physics C Unit 5, Lesson 3, RC Circuits (C53RC) ……………………………………………. 43
Physics C Unit 5, Lesson 4, Electric Potential from Electric Field (C53Potential) ……………. 45
Physics C Unit 5, Lesson 5, Other Field, Potential and Capacitance Problems (C54Other) ...… 48
UNIT 7, PHYSICS B NUCLEAR AND QUANTUM PHYSICS AND PHYSICS C
MAGNETISM IN DEPTH ………………………………………………………………..…… 50
Physics B Unit 7, Lesson 1, Nuclear Physics and Radioactivity (B71Nuclear) ……………….. 50
Physics B Unit 7, Lesson 2, Quantum Physics and Wave Particle Duality (B72Quantum) …… 52
Physics B Unit 7, Lesson 3, Bohr and Energy Level Diagrams (B73Bohr) …………………… 53
Physics C Unit 7, Lesson 1, Ampere’s Law (C71Ampere) ……………………………………. 55
Physics C Unit 7, Lesson 2, LR Circuits (C72LR) …………………………………………….. 56
Physics C Unit 7, Lesson 3, Other Magnetism Problems (C73Other) …………………………. 58
CONCLUSION…………………………………………………………………………………. 59
BIBLIOGRAPHY AND WORKS CITED…………………………………………………...… 61
iv
LIST OF TABLES AND FIGURES
Table 1. Side by Side Comparison of Order and Topics Covered for Physic B and Physic C
………………………………………………………………………………. 3 to 5
Table 2. Additional Book and Internet Resources for Physics B and Physics C
………………………………………………………………………...…….. 7 to 8
Table 3. Unit 3 Overview, Physics B Waves & Optics and Physics C Rotational Motion
………………………………………………………………………………….... 9
Figure 1. An ambulance experiencing the Doppler Effect…………………………………….. 10
Table 4. Table of harmonics and overtones……………………………………………………. 13
Table 5. Synopsis of mirrors and lenses………………………………………………………. 16
Table 6. Answer chart for the practice problems for ray tracing…………………..…….. 19 to 20
Figure 2. Free body diagrams for an Atwood Machine ……………………………………….. 26
Table 7. Unit 5 Overview, Physics B Fluid Mechanics & Thermal Physics and Physics C
Electricity in Depth ………………………………………………………………………….…. 30
Figure 3. Schematic of energy input and output for a heat engine…………………………….. 38
Table 8. Unit 7 Overview, Physics B Nuclear and Quantum Physics and Physics C Magnetism
in Depth ……………………………………………………………………………………….... 50
v
INTRODUCTION
The purpose of my project is to provide support materials and guidance for physics teachers who
wish to teach both types of Advanced Placement® (AP) Physics concurrently in the same
classroom setting. The two courses are algebra-based Physics B and calculus-based Physics C.
Physics C consists of two parts, (1) Mechanics and (2) Electricity and Magnetism.
This year, and for the past 6 years, I have taught the courses concurrently in the same classroom.
Approximately 30 students each year at my high school took the courses, 60% taking the single
course of Physics B and 40% taking the two parts of the Physics C courses. The single course of
Physics B or the two courses of Physics C are essentially equal to two semesters of freshman
level college physics. I am often asked, “Why do you teach two AP classes at the same time?”
This introduction answers the “Why” question. It also provides information to answer the other
important question, “How?”
What is the value of AP classes to students? College professors, high school teachers, and
students may all have different views of these courses, both pro and con. Research also gives a
variety of findings, also both pro and con. College Board administers all AP exams and for the
school year beginning in the fall semester of 2007 has audited the content of every course that
carries the designation AP, which is a registered trademark of the College Board. Two studies
conducted at the University of Texas and funded by College Board show that students who take
AP classes and pass the tests have higher college grade point averages and receive a bachelor’s
degree in fewer years (Chute, 2007). College Board itself reports two different studies, one from
University of California at Berkeley and one from the National Center for Educational
Accountability, which find that an AP exam grade, and particularly an exam grade of 3 (which is
considered passing), are strong predictors of ability to complete a college degree (College Board,
2007). However, there have been reports of the erosion of the value of AP classes and exams
(Marklein, 2006). Other independent research studies showed little or no relationship between
AP classes and college success (Chute, 2007). Selective schools such as Harvard, Yale, and
MIT require students to take additional courses even if those students have passed the AP exam
in that subject (Marklein, 2006). College Board president Gaston Caperton has stated that
College Board has accelerated its drive to reach underserved students of differing ethnic, racial
and economic backgrounds (Caperton, 2005). His hope is that AP classes can help to increase
rigor in schools (Marklein, 2006).
Regarding the value of AP classes to students, my educational philosophy is to give all students,
the best chance at success in their lives. I teach students of differing academic levels, some who
are absolute geniuses. I also teach those who are learning English for the first time or have
significant learning disabilities. Not everyone can be a brain surgeon, the epitome of all career
choices according to my grandmother. Some students can be brain surgeons or engineers or
computer systems analysts. They just don’t know it. College physics is required for many
math, science, and technology-related degrees. I don’t want students’ possibilities limited by an
inability to pass college physics classes. One of the best ways I can prepare students for college
physics and open the most avenues for success is to teach AP Physics. Whether they scrape by
or perform brilliantly, they have gained greater ability. They will be better able to deal with the
1
difficulty, disappointment and failure they will encounter because they experienced these things
in AP Physics.
What value does teaching this class have for me? I hear a lot of talk about teacher burnout.
Colleagues complain about the stress of our job. Young teachers leave the profession within a
year or two. Even I get tired of explaining the solution to a common misunderstanding for the
100th or the 1000th time. AP Physics has its common misunderstandings, but it also has new and
exciting ones every single year. I am fond of telling my students that I have completed 30 years
worth of AP Physics problems and yet, every year, there is at least one new one that makes me
think. The problem encourages me to be flexible, inventive, and to look at things in a new way.
Also, I get to teach wonderful students in AP Physics. They are not all at the top of their class,
but they are willing to take risks, try something new and difficult, and to work hard. They have
their own unique personalities, and they are never dull. The variety inherent in the class and the
students helps me avoid teacher burnout. This variety keeps me loving my job, which benefits
me and all my students, not just one classroom.
Job satisfaction isn’t the only factor to consider when discussing concurrent physics courses.
Money is a significant issue. It is the subject of a question on one of my final exams. “What
question do I always ask you? What makes the world go round?” The answer is money. Most
teachers know that administrators require a minimum number of students in an elective class for
the course to be offered. In my school they talk about money and numbers of teaching positions
in terms of full-time equivalents (FTE’s). I can’t tell you everything about FTE’s, but I do know
that in public education, just like in the corporate world, money is important. To keep within my
school district’s budget, an elective class must have at least 20 students. AP Physics is an
elective class and I want to teach it so I will do all that is reasonable to get 20 students enrolled
in the class. One of those reasonable things is to offer Physics B and Physics C in the same
classroom.
I have provided you with the reasons why I teach AP Physics as I do. Perhaps you are just
teaching one of the AP Physics classes. Or maybe you already teach concurrent courses or are
considering doing so. I have developed a step-by-step plan for teaching this material. This plan
includes self-guided lesson plans for students taking either calculus or algebra-based physics
when the studied topics don’t overlap. This provides students with the greatest benefit when
time is limited. If you would like to contact me or need additional information, please visit my
website at http://www.orgsites.com/nm/walker/.
2
OVERVIEW OF COURSE TOPICS AND ORDER
At this point, if you are reading this I am going to assume that you are actually considering
teaching some course of AP Physics, either Physics B, Physics C, both courses separately, or
both concurrently in the same classroom. There are a few items of general information that I
think would be helpful and I am going to go over those in this section.
College Board oversees all AP classes and administers AP exams. The courses that would
actually be taught simultaneously are algebra-based Physics B and calculus-based Physics C.
Physics C consists of 2 parts, (1) Mechanics and (2) Electricity and Magnetism. Physics B is
one course designed to cover the first 2 semesters of introductory college-level algebra-based
physics. Physics C, Mechanics and Physics C, Electricity and Magnetism are designed to cover
1 semester each of introductory college-level calculus-based physics. I teach 2 AP Physics
courses at Rio Rancho High School (RRHS). They are titled AP Physics B and AP Physics C.
The curriculum of both C courses is covered in AP Physics C at RRHS. Topics overlap in the 2
different courses about half of the time. The other half of the time, totally different topics are
covered. A comparison of the topics covered can be found on the College Board web site at:
http://apcentral.collegeboard.com/apc/members/repository/ap05_phys_objectives_45859.pdf.
The goal of teaching either of the classes is for a student to understand the concepts of 2
semesters of introductory college-level physics, comparable with the first 2 semesters of physics
offered at a college or university. In order to do this I have chosen to structure the scheduling of
the classes as shown in the table below.
Table 1. Side by Side Comparison of Order and Topics Covered for Physic B and Physic C.
Physics B
Physics C, Mechanics AND
Electricity and Magnetism
Course Unit Outlines
Course Unit Outlines
1) Newtonian Mechanics
1) Newtonian Mechanics
a) Kinematics including vectors,
a) Kinematics including vectors,
displacement, velocity, and acceleration
displacement, velocity, and acceleration
for 1 dimensional motion and 2
for 1 dimensional motion and 2
dimensional/projectile motion
dimensional/projectile motion
b) Newton’s Laws including static
b) Newton’s Laws including static
equilibrium, dynamic equilibrium of a
equilibrium, dynamic equilibrium of a
single particle and systems of two or
single particle and systems of two or
more objects
more objects
c) Work, energy and power including the
c) Work, energy and power including the
Work-Energy Theorem, conservative
Work-Energy Theorem, conservative
forces, potential energy and
forces, potential energy and
conservation of energy
conservation of energy
d) Conservation of linear momentum for a
d) Conservation of linear momentum for a
system of particles including impulse,
system of particles including center of
collisions, and explosions
mass, impulse, collisions, and
explosions
3
2) Centripetal Motion, Gravitation, and
Oscillations Basic Rotational Dynamics
and Torque
a) Centripetal motion and gravitation
including uniform centripetal motion,
Newton’s Law of Universal
Gravitation, and circular orbits of
satellites and planets
b) Oscillations including pendulums, mass
on a spring, and simple harmonic
motion
c) Basic Rotational Dynamics and Torque
3) Waves and Optics
a) Properties of traveling waves
b) Wave motion including properties of
standing waves, Doppler effect, and
superposition
c) Physical optics including interference
and diffraction, dispersion of light and
the electromagnetic spectrum
d) Geometric optics including reflection
and refraction, mirrors and lenses
4) Electricity
a) Electrostatics including charges, field,
potential, force from Coulomb’s Law,
field and potential of point charges, and
field and potential of a planar system of
charges
b) Electrostatics of conductors and parallel
plate capacitors
c) Electric circuits including current,
resistance, and power, steady state
direct current circuits with battery and
resistors only and steady state
capacitors in circuits
5) Fluid Mechanics and Thermal Physics
a) Fluid mechanics including hydrostatic
pressure, buoyancy, fluid flow
continuity, and Bernoulli’s equation
b) Temperature and heat including the
mechanical equivalent of heat, heat
transfer, and thermal expansion
c) Kinetic theory and thermodynamics
including the kinetic theory of ideal
gases, the ideal gas law, the first of law
of thermodynamics and PV
2) Centripetal Motion, Gravitation, and
Oscillations, Basic Rotational Dynamics
and Torque
a) Centripetal motion and gravitation
including uniform centripetal motion,
Newton’s Law of Universal
Gravitation, and circular and general
orbits of satellites and planets
b) Oscillations including pendulums, mass
on a spring, and simple harmonic
motion
c) Basic Rotational Dynamics and Torque
3) Rotational Motion
a) Torque and rotational statics
b) Rotational kinematics and dynamics
c) Angular momentum and its
conservation for point particles and for
extended bodies including rotational
inertia
4) Electricity Basics
a) Electrostatics including charges, field,
potential, force from Coulomb’s Law,
field and potential of point charges, and
field and potential of a planar system of
charges
b) Electrostatics of conductors and parallel
plate capacitors including dielectrics
c) Electric circuits including current,
resistance, and power, steady state
direct current circuits with battery and
resistors only and steady state
capacitors in circuits
5) Electricity in Depth
a) Gauss’s Law
b) Electric field and potential due to
charge distributions of
spherical/cylindrical symmetry
c) Spherical and cylindrical capacitors
d) Capacitors in circuits specifically
transients in RC circuits
4
relationships, and the second law of
thermodynamics and heat engines
6) Magnetism
a) Magnetostatics including forces on
charged particles moving in magnetic
fields, forces on current-carrying wires
in magnetic fields, and magnetic fields
due to long current-carrying wires
b) Electromagnetism including
electromagnetic induction, Faraday’s
Law, and Lenz’s Law
7) Atomic and Nuclear Physics
a) Atomic physics and quantum effects
including photons and the photoelectric
effect, atomic energy levels, and waveparticle duality
b) Nuclear physics including nuclear
reactions, conservation of mass number
and charge, and mass-energy
equivalence
6) Magnetism Basics
a) Magnetostatics including forces on
charged particles moving in magnetic
fields, forces on current-carrying wires
in magnetic fields, and magnetic fields
due to long current-carrying wires
b) Electromagnetism including
electromagnetic induction, Faraday’s
Law, and Lenz’s Law
7) Magnetism in Depth
a) Biot-Savart and Ampere’s Law
b) Inductance including LR and LC
circuits
c) Maxwell’s equations
Out of the seven units that I teach for each class, four of the units are essentially the same and
three units are completely different. Daily handouts for each of the units that are completely
different are the purpose of this project.
5
PRACTICAL THOUGHTS ON DAILY MANAGEMENT OF THE COURSES
I would like to provide you with some practical ideas for day-to-day management when teaching
Physics B and Physics C in the same classroom. I think it breaks down to a few useful
suggestions. My most important suggestion is to keep a firm grip when scheduling time, both
throughout the semester and the class period. Secondly, have high expectations for time
management and initiative from your students and encourage and motivate them. Finally, I think
you should find the best resources and support system. The goal of these suggestions is to allow
you to provide the most beneficial experience for your students in the most efficient way.
Managing time is absolutely the most important aspect of teaching AP Physics and teaching
Physics B and Physics C in the same classroom requires even more dedicated time management.
Do as much as you can together with both classes for your own convenience and the students’
ease of understanding. In the Overview of Course Topics and Order, I tried to give you my best
thoughts on how I schedule topics throughout the semester. Of course, it is important to make
sure there is time to cover all topics. When you are covering topics that are different, time
management through the class period is important. I typically divide the class time in half and
teach each separate topic for half the class. I switch which part of the class is taught first every
day so that no group claims favoritism. At the times when the topics are different, it is so
important for the students to be responsible, both for their time and their learning which leads me
to the next thing I’d like to discuss.
If you are teaching Physics B and Physics C concurrently in the same classroom, you are
probably doing this to provide as many students the opportunity to take AP Physics as possible.
Make sure students are aware of this from the beginning of the class and that because of this
opportunity that they need to flex their college muscles, be independent, and take on more
responsibility for their own learning. Students may need to be reminded of this over the
semester, particularly later in the semester as the exams loom closer and stress is high. I try to
keep motivation as high as possible by reminding students of what they stand to gain from the
class. I particularly emphasize the college experience they are gaining before even going to
college. I also remind them of the benefit of having a less strenuous freshman year of college. I
will also say that the occasional granola bar or slice of pizza as positive reinforcement (feel free
to read this as “bribe”) is very useful.
Resources and support are so important. There are numerous books, websites, and people that
can be resources for teaching AP Physics. You will not be able to use them all so find and
choose the ones that will be most helpful to you. In the next section, I have provided a section
on the books and websites that have been most helpful to me. I also suggest that all teachers
teaching any class of AP Physics seek out the AP Summer Institute class nearest to them if they
have never taken it and even consider taking a refresher if it has been a while. You will receive
valuable materials and practical advice from the instructor as well as a chance to meet and
network with other AP Physics teachers in your geographic area. I also suggest joining the AP
Electronic Discussion Group. When you join this group, be prepared to receive a flood of mail
and figure out how to put it into its own folder. You will not be able to read all e-mails, but it
will allow you to know what teachers across the country are discussing and ask for their help if
needed. This has been very helpful to me on several occasions.
6
RECOMMENDED BOOK AND INTERNET RESOURCES
Sometimes I have students complain that they don’t understand the textbook. I understand that.
It has happened to me many times. However, I will not accept that as an excuse for why they
don’t understand the material. My resource mantra chanted over and over again to my students
and to myself is “If you can’t figure something out from your book, then find another resource
and keep doing that until you find one that is best for you.” I believe that this is an absolutely
necessary skill. Other resources include additional textbooks, paperback review books, the
internet, and other students and instructors. Below you will find some possible resources listed.
I have used three different textbooks in teaching Physics B over the years and two different
textbooks teaching Physics C. There are also other books I use for reference and several internet
websites that I check first for further information. This not a comprehensive list of helpful
materials, just a list of the ones I use.
Table 2. Additional Book and Internet Resources for Physics B and Physics C.
Type of
Resource
Physics B
Textbooks
List of Resources



Physics C
Textbooks



Other Book
Resources


Advanced

Placement®
Physics

Websites


College Physics by Raymond A. Serway and Jerry S. Faughn. 7th Edition.
Published by Thomson Brooks Cole.
Physics by John D. Cutnell and Kenneth W. Johnson. 6th Edition. Published by
Wiley & Sons, Inc.
Physics: Principles with Applications by Douglas C. Giancoli. 6th Edition.
Published by Prentice Hall.
Fundamentals of Physics by David Halliday, Robert Resnick, and Jearl Walker.
7th Edition. Published by John Wiley & Sons, Inc.
Physics for Scientist and Engineers by Paul A Tipler and Gene Mosca. 5th
Edition. Published by W. H. Freeman and Company.
Principles of Physics A Calculuus-Based Text by Raymond A. Serway and John
W. Jewett, Jr. 4th Edition. Published by Thomson Brooks Cole.
Conceptual Physics by Paul G. Hewitt. 10th Edition. Published by Addison
Wesley.
Cracking the AP Physics B & C Exams by Stephen A. Leduc. 2006-2007
Edition. Published by Random House.
AP Physics B Course Home Page.
http://apcentral.collegeboard.com/apc/public/courses/teachers_corner/2262.html.
AP Physics C: Electricity and Magnetism Course Home Page.
http://apcentral.collegeboard.com/apc/public/courses/teachers_corner/2263.html
AP Physics C: Mechanics Course Home Page.
http://apcentral.collegeboard.com/apc/public/courses/teachers_corner/2264.html
Table of Information and Equations Tables for AP® Physics Exams.
http://apcentral.collegeboard.com/apc/public/repository/physics_equation_tables
_2008_09.pdf
7
Table 2. (continued) Additional Book and Internet Resources for Physics B and Physics C.
Other
Internet
Resources







AP Physics B by Dolores Gende. http://apphysicsb.homestead.com/.
How Stuff Works. http://www.howstuffworks.com/.
HyperPhysics by Carl R. Nave. http://hyperphysics.phyastr.gsu.edu/hbase/hph.html.
The Physics Classroom Tutorial by Tom Henderson.
http://www.glenbrook.k12.il.us/gbssci/phys/Class/BBoard.html.
The Physics Factbook by Glenn Elert. http://hypertextbook.com/facts/indextopics.shtml.
The Physics Hypertextbook by Glenn Elert. http://hypertextbook.com/physics/.
Wikipedia. http://www.wikipedia.org/.
8
UNIT 3, PHYSICS B WAVES & OPTICS AND PHYSICS C ROTATIONAL MOTION
Table 3. Unit 3 Overview, Physics B Waves & Optics and Physics C Rotational Motion.
Physics B Waves & Optics
1) Wave Beginnings - Understand and
make calculations for waves including
the wave equation, superposition, the
Doppler Effect, beats, standing waves,
waves on a stretched string,
fundamental frequencies and
resonance.
2) Reflection and Refraction- Make
calculations for and have a conceptual
understanding of Snell’s Law, ray
tracing for interfaces such as air-water
and air-prism, and apparent depth.
3) Lenses and Mirrors - Understand
mathematically and conceptually the
lens/mirror equation for diverging and
converging lenses and concave and
convex mirrors and ray trace for any of
these and describe the image.
4) Interference, Diffraction and Thin
Films- Understand and make
calculations for double slit diffraction,
single slit diffraction, diffraction
gratings and thin films.
Physics C Rotational Motion
1) Rotation Beginnings – Contrast and
compare linear quantities and rotational
quantities and make calculations for
equilibrium requiring τnet =0.
2) Moment of Inertia and Pulleys – Gain a
deeper conceptual understanding of
moment of inertia, use calculus to
determine the moment of inertia of a
uniform rod and be able to take moment of
inertia of a pulley into account.
3) Rotational Motion for an Inclined Plane
– Compare and contrast the use of
Newton’s 2nd law and conservation of
energy to solve problems for objects of
varying moments of inertia rolling down an
inclined plane.
4) Other Rotational Situations – Understand
conceptually and be able to make
calculations for the rotational analogies to
the Work-Energy Theorem, power,
Newton’s 2nd Law, angular momentum, and
precessional motion.
Physics B Unit 3, Lesson 1, Wave Beginnings (B31Wave)
Wave Beginnings - Understand and make calculations for waves including the wave equation,
superposition, the Doppler Effect, beats, standing waves, waves on a stretched string,
fundamental frequencies and resonance.
Start your homework today by labeling the top of each paper with the date and the exciting code:
B31Wave and the page number. Make sure to label any additional pages with your exciting code
and the page number so neither one of us gets confused. Feel free to add anything else at the top
you need to help you remember what topics you are studying. Today you are going to make sure
you know some wave basics and specific information on several topics. Wave basics and their
calculations are a significant part of every AP Physics B test. I am covering a lot of topics in this
lesson, but it is most important to know the basics. Topics such as the Doppler Effect, beats,
standing waves, waves on a stretched string, and fundamental frequencies and resonance are part
of this class, of course, but you will have to choose to what depth you want to study this material.
According to the prerequisites for this class, you should have done some study of waves before
this time. I expect that you understand the essentials about the following terms and concepts:
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crest, trough, wavelength, transverse, longitudinal, period, frequency, superposition, nodes,
antinodes, standing waves, constructive and destructive interference, compression and
rarefactions of longitudinal waves, in phase waves, out of phase waves, decibels, nanometers, the
electromagnetic (EM) spectrum including radio waves, infrared, visible light, ultraviolet, x-rays
and -rays (gamma rays), the trend of wavelengths and frequencies for EM waves, the visible
spectrum of light along with its trends in wavelengths and frequencies and the ballpark limits for
the wavelengths at both ends of the visible spectrum, the speed of light, speed of sound,
interference, diffraction, polarization and the Doppler effect. Now that was a very long list.
1. Go back and read the long list again and write down any term you need to double check or if
you know everything about all these wave terms, write down as the answer to this homework
question #1 as “I know all this stuff”.
2. If you need some help on these terms two great places to look are your textbook and the
internet. Usually there are separate chapters or sections in your textbook for (A) waves, (B)
sound waves, and (C) light waves. Write down the chapters and page numbers or internet
resources for each of these 3 topics.
3. What is the wave equation? Tell what each variable is and its most common unit.
4. What is the speed of sound?
5. What is the speed of light?
6. Explain in some detail what the drawing below has to do with the Doppler Effect. Be sure to
describe the effect on wavelength, frequency and pitch.
v
Figure 1. An ambulance experiencing the Doppler Effect.
7. How are frequency and pitch related to each other?
8. If you are standing still and a car/ambulance is coming toward you, what happens to the pitch
of the sound you observe?
9. If you are standing still and a car/ambulance passes you and moves way from you, what
happens to the pitch of the sound you observe?
This is the Doppler Effect. **It is important to know that the actual frequency/pitch of the
sound does not change, just the frequency/pitch observed!** Think about that vroom noise a car
10
makes as it passes you now in terms of frequency and pitch. Make the noise. See if your
mimicry follows the physics. Now there are calculations that can be made to find the specific
frequency for the observer. This can be done with a single equation if there is a plus/minus and
minus/plus symbol is used in the numerator and denominator of the equation (
). Or there
can be multiple equations for different situations. There are three different velocities in the
equation, the velocity of the sound, the velocity of the source, and the velocity of the observer.
Haul your textbook out or nimble up those internet searching fingers and find this equation in a
reference.
10. Write down a single equation or a multiple equations if needed to calculate the observed
frequency.
11. Look at the equation(s). Could you use it/them if you had to; if the equation and all the
information you needed to solve it were given to you? Be honest.
Now for the good news…students have been very, very rarely asked to find an actual frequency.
However, questions on the concept of the Doppler Effect abound. Make sure you can answer
them. My students always ask me what this has to do with Doppler Weather Radar, the most
common time they hear the word “Doppler”. If you want to know the answer to this question,
you might try this website:
http://premiuma.accuweather.com/phoenix2/help/adc/pr_radar.htm#how_doppler from
accuweather.com or a more basic one on radar from How Stuff Works.
http://www.howstuffworks.com/radar.htm.
Beat Frequency - Beat frequency or just “beats” is a specific sound and frequency pattern created
due to two objects that are making a sound at nearly the same frequency. There is alternating
constructive and destructive interference. My students in the band always seem to know exactly
what I am talking about when we discuss this, better than I do! Wait until the lab and you will
get the chance to simulate this. Woo hoo! There is a very characteristic frequency pattern that is
associated with beat frequency. This is typically found in all textbooks.
12. Using a book or the internet, find the picture and make a sketch. Here is a hint on a possible
web site: http://en.wikipedia.org/wiki/Beat_%28acoustics%29
Standing Waves and Waves on a Stretched String - Standing waves are commonly produced on
stretched strings and most easily understood if we discuss both at the same time. Those stretched
strings can be found on guitars, violins, violas, cellos, and basses! A standing wave on a string
vibrates so quickly that the trough and crest can be seen at the same time. Check out this picture
on the internet or find one in your textbook
http://www1.union.edu/newmanj/lasers/Light%20as%20a%20Wave/standing%20wave2.JPG.
Standing waves exhibit nodes, places where the string doesn’t really vibrate up and down at all,
where you supposedly could touch the string and not disturb the waves. Notice that standing
waves on a string are classic places to be able to see and understand the terms node, antinode,
trough, crest, wavelength, and amplitude. **Please note that the simultaneously seen crest and
trough are not part of the same wave. The crest goes with the next trough down the line not the
one directly underneath it and vice versa. You have to be very careful when counting the
number of standing waves on a string.** On the internet picture I referenced above, there is
actually only one entire wavelength and a small part of another. If you don’t understand this,
you need to discuss this with me in greater detail later.
13. Now make a quick sketch of several standing waves on a string. Trace one full wavelength
and label it correctly with the following terms: node, antinode, trough, crest, wavelength, and
amplitude.
,
11
The speed of a wave along a string is related to wavelength and frequency using the wave
equation (v =f ) , but it is also related to the tension in the string and the mass per unit length of
the string according to the following equation.
F
v
T

In this equation v is the speed (m/s), FT is the force of tension (N), and  is the mass per unit
length (kg/m). I am sorry that they are using  again. It is not my fault that the letters are used
over and over again. Don’t get mad at me. Also the symbol  is sometimes written as the
quantity m/L or mass divided by length which is actually just the definition of mass per unit
length or length density. So it is plausible that there could be a free response problem where
there is a mass hanging over a pulley connected to a string that there is a standing wave on. You
could be given a picture of these waves and the amount of the mass and the mass density per unit
length. Then you could calculate the weight of the mass, the tension in the string, the velocity of
the waves on the string, the number of waves from the picture, and the frequency of the waves.
This would be a great free response problem! I have not seen one exactly like it, but I have seen
all these elements in other problems. Also please note that the equation above for velocity as a
function of tension and length density is not found on your equation sheet. This does not mean it
has never been used on the AP exam. It has. They just provided the equation for you. I also
wonder if it isn’t possible to make up a multiple-choice question or two out of this equation
considering the following trends.
14. If the tension in the string increases what happens to the velocity, wavelength, and
frequency?
15. If the string is thicker or in other words has a greater length density, what happens to the
velocity, wavelength and frequency?
Fundamental Frequencies and Resonance - A resonant frequency is a natural frequency related to
the physical properties of the object that is vibrating and causing the sound. The lowest resonant
frequency is the fundamental frequency, also called the first harmonic and usually abbreviated f1.
Both standing waves on strings and columns of air create standing waves and have these
fundamental frequencies/harmonics. This part of sound wave discussion is significant to musical
instruments since a significant number of these make music due to strings (guitar, violin, etc.) or
columns of air (flute, trumpet, etc.) Luckily for us the standing waves for strings and air
columns are related. The following discussion will be much more understandable if you have
some pictures to look at. So please find the information about fundamental frequencies or
harmonics in your textbook or find a couple of good sites on the internet. You should have
separate pictures for strings and for columns of air.
16. Write the page numbers from your book or give the URL from the websites that you are
using and have the picture in front of you before reading further.
First let’s talk about strings. If you had trouble finding something about fundamental
frequencies/harmonics of standing waves, check out Hyperphysics, Vibrating Strings,
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/string.html. Scroll down to harmonics. Let’s
consider a string that has only two nodes, one at each end. This means that there is just one crest
and trough or one big “bump”. This corresponds to one half of a wavelength. Remember what I
said about counting standing waves carefully. So the length of the string, L = λ/2. Or we can
rewrite this as  = 2 L. We are going to keep the tension in the string and also it’s mass per unit
12
length the same so that speed of the wave remains the same. With only 2 nodes we have the
largest wavelength we can get and therefore the lowest frequency also known as the fundamental
frequency, f1. Also recall that v =  f, so v = 2L f1 and solving for this fundamental frequency
f1=v/2L. However, the velocity doesn’t change so if we want to consider different frequencies,
there will different wavelengths also. If you didn’t understand all that logic, read the previous
paragraph again, check out your book or internet resource, talk to your classmates and then go
on. If you still have questions, you will have an opportunity to discuss this with me later. This
lowest frequency is the fundamental frequency. It is also called the first harmonic.
Next let’s consider a standing wave on a string that has three nodes, two big “bumps”, or one full
wavelength. These are all the same. This is the second largest wavelength possible which leads
us to the second harmonic and the first overtone (f2). Please note that overtones can be
remembered to be over harmonics so that the first overtone is the second harmonic and the
second overtone is the third harmonic. Notice that the number of the harmonic and the number
of the frequency are the same and the number of the overtone is one less. Some books use f0 for
the fundamental frequency, but I have found that to be used less often. Remember I didn’t make
up the lettering system. You don’t get to shoot the messenger, ha ha! For this second harmonic
(f2) we know that  = L and using the wave equation we the equation we get v = L f2and the
second harmonic’s frequency, f2= v/L note that 2f1 = f2. Let’s see if we can see some patterns
here and understand the general equation found in the book.
17. Copy the table below into your notebook and fill in the table in a similar fashion to the table
was filled out for f1 and f 2 .
Table 4. Table of harmonics and overtones.
Frequency
Harmonic
Overtone
Nodes
Bumps
Wavelength
f
First
1
f
Not
Applicable
First
2
1
3
2
2
L  /2
 = 2L
L  2 / 2
=L
f
f
f
Second
Frequency
Equation
v
f 1 = 2L
v
f 1 =  2 f1
L
3
4
nth
n
Remember that columns of air also create standing waves and share some equations and
properties with standing waves on strings. Make sure to have the pictures you found earlier of
the waves in columns of air readily available. There are two possibilities with columns of air, a
column of air open at both ends and a column of air that is only open at one end. If you have a
pipe that is open at both ends, it uses the same equation as the standing waves on a string.
13
f n
n
v
 nf
2L
1
n  1,2,3...
If the column of air is closed at one of the ends, this restricts air movement at that end and we get
only odd multiples of the fundamental harmonic. The equation for this is
f n
n
v
 nf
4L
1
n  1,3,5...
Be sure to study the pictures that you have found to help make this clearer. With the two
equations directly above, you should be able to handle any standing waves on strings or in
columns of air that come your way.
18. Write down the equation for the frequency for the 7th harmonic for a column of air open at
both ends.
19. Write down the equation for f3 for a column of air open at only one end.
Notice that these equations are not given on your equation sheet. However, these types of
questions have appeared on the exam so at least a cursory knowledge of the properties of
standing waves is necessary.
20. Yahoo! We are done with basic waves. Just a note, we will not be studying speed as a
function of bulk modulus or temperature, intensity of waves or intensity level in decibels.
Now please write down the two most important things you learned from this lesson.
Physics B Unit 3, Lesson 2, Reflection and Refraction (B32Reflect)
Reflection and Refraction- Make calculations for and have a conceptual understanding of
Snell’s Law, ray tracing for interfaces such as air-water and air-prism, and apparent depth.
Label each page of today’s homework with today’s code B32Reflect, the page number, and the
date. You are going to learn all the concepts of this unit in terms of light although these concepts
are true of other types of waves. You are going to gain some understanding of reflection and
refraction.
1. Please find the sections of your book or websites that discuss reflection, refraction, Snell’s
Law and thin films and write down the references for this information.
2. Show your understanding of the following terms in the most concise way possible, either a
few words, an equation, or a picture. You may want to complete this as you do the rest of the
work for this section.
A) reflection
B) refraction
C) Snell’s Law
D) thin films
E) light ray
F) incident angle
G) reflected angle
H) normal
I) diffuse reflection J) index of refraction K) critical angle
L) prism
M) total internal reflection
N) prism
O) apparent depth
P) fiber optics
Reflection and Refraction - Please answer the following questions. Write down any additional
questions you have so you can ask me later.
3. What is the shape of the path that light follows as it moves?
4. How are the reflections from smooth surfaces different from bumpy surfaces?
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5. How does the measurement of angles in this section differ from the way we have most
commonly measured angles in the past.
6. What can you tell me about the relationship between the angle of incidence and the angle of
reflection?
Snell’s Law of Refraction and Geometric Optics - Answer the following questions
7. Does the speed of light ever change? How does the speed of light in a vacuum compare to its
speed in water?
8. I like to create a picture in my mind about index of refraction. Tell me how at least a couple
of these words relate to index of refraction: thick, thin, clear, opaque, sheer, or dense? How
do would these terms you have chosen relate to water and air?
9. Write down a mathematical definition of index of refraction.
10. As a light wave refracts which of these change and which do not: speed, frequency, and
wavelength? Write each word down with the word “yes” (it changes) or “no” (it doesn’t).
11. Draw two pictures, one showing light traveling from air to water and the other showing light
traveling from water to air. On both pictures be sure to include the normal, the incident ray,
the reflected ray, and the refracted ray. Below each picture write a “rule” that helps you
remember how the refracted ray bends based on the refractive index of the incident medium
and the refractive index of the refracting medium. The “rule” must include the word
“normal”.
12. Write down Snell’s Law and the name of each variable in it and how they are related to the
subscripts. Feel free to illustrate or use previous illustrations if that will help.
13. What is dispersion and what does it have to do with rainbows?
14. Make a drawing of a prism that is an equilateral triangle and show an incident beam of white
light that enter the first plane of the prism at an angle that is different than 90°. Then show
the ray tracing of how the prism refracts red, green, and blue light through the prism and on
the other side. Which bends more red or green? Green or blue?
15. Why couldn’t the incident beam of light in the last question enter at 90°?
16. What is total internal reflection and what does it have to do with fiber optics?
17. Give a mathematical equation for critical angle.
18. Write down a rule for total internal reflection and its relation to the index of refraction for the
two media involved that starts “Total internal reflection can only occur……”
Apparent Depth - Here is an interesting web site that shows a virtual lab on using apparent depth
to measure index of refraction. http://www.mathsphysics.com/Physics/RandA.htm
19. Write an equation for apparent depth.
20. Okay we made it through the very exciting work on reflection and refraction! Good job!
Write down the two most significant fundamentals that you learned in this lesson.
Physics B Unit 3, Lesson 3, Lenses and Mirrors (B33Lenses)
Lenses and Mirrors - Understand mathematically and conceptually the lens/mirror equation for
diverging and converging lenses and concave and convex mirrors and ray trace for any of these
and describe the image.
Today’s lesson will cover mirrors, lenses, ray tracing, magnification, and the lens/mirror
equation. Please label all homework papers with the exciting code B33Lenses and the date and
15
the page number. Please also locate references in either your textbook or on the internet that
discuss ray tracing for lenses and mirrors and write these references down.
Equations for this lesson are found below. The subscript o as in so refers to the object and the
subscript i as in si refers to the image.
1
1 1
 
Lens/Mirror equation
f
so si
R
s
h
m=  i
m= i
f=
2
so
ho
m is magnification (a unitless number)
so is length the object is from the mirror or the lens
si is distance the image is from the mirror or lens
f is focal length
R is the radius of curvature
hi is the image height
ho is the object height.
All variables except m are measured in length units. They are often given in centimeters. This is
just fine as long as all other numbers in the equation are in centimeters also. Any unit will work
as long as all units are the same for these equations. So you usually do not need to convert the
units. R the radius of curvature can be given for a mirror. The focal length of the mirror is half
this radius of curvature. Generally the rays of light for lenses and mirrors move from left to right
like you normally read. However, problems have been known to appear where the light moves
from right to left. After some practice you will already know exactly what to do and this will not
be a problem for you. Below are some illustrations of how to classify the mirrors and lenses.
This is very important because once you can classify the mirror or lens, it leads you into specific
ways that problem is done.
Table 5. Synopsis of mirrors and lenses.
Concave mirror
Skinny woman
Convex mirror
Pregnant woman
Converging lens
Fattest in the middle
Skinniest at the ends
The ends converge
At least one convex
face
This is a double
convex lens
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Diverging lens
Skinniest in the
middle
Fattest at the ends
The ends diverge
from the center
At least one concave
face
This is a double
concave lens
General information for mirrors includes:
concave ↔ f is positive (on left side of the mirror) Usually the left side is the positive side – but
positive is always the object side
convex ↔f is negative (Usually on the right side of mirror)
so is always positive
If si is positive then the image is real.
If si is negative then the image is virtual. (on the wrong side of the mirror from where we expect
it – opposite object side)
If m is positive then the image is upright. (right side up)
If m is negative then the image is inverted. (upside down)
General information for lenses includes:
converging ↔ f is positive–Usually left side –Always object side
diverging↔ f is negative (Usually right side)
so is always positive
If si is positive then the image is real.
If si is negative then the image is virtual. (on the wrong side of the lens from where we expect it
– object side, not other side)
If m is positive then the image is upright. (right side up)
If m is negative then the image is inverted. (upside down)
The following information gives the “Big Picture” of Ray Tracing. You will not be able to
understand this unless you practice! After you have done several different types of ray tracing
diagrams I hope you realize you are making the same rays over and over again using some
general rules. You must know where the focus is for each type of lens and mirror.
 Concave mirrors and converging lenses are extremely similar (BASICALLY THE SAME),
AND so are convex mirrors and diverging lenses (BASICALLY THE SAME).
 YOU MAY WANT TO MEMORIZE THAT CONVEX AND DIVERGING ARE THE
SAME BY REMEMBERING THAT YOU WANT TO GET AWAY (DIVERGE) FROM
SOMEONE WHO VEXES YOU (CONVEX)
 Concave Mirrors and Converging Lenses create real objects can when the object distance is
greater than the focal length. This is the only time real objects are created.
 Convex mirrors and diverging lenses ALWAYS create virtual objects AND objects located
between the focal point and a converging lens or concave mirror ALWAYS create virtual
objects.
 General information on the 3 principal rays – The AP Physics test usually only requires 2 out
of the 3. Did I mention that you will not be able to understand this unless you practice.
 Draw a ray parallel to the principal axis that touches the top of the object and then goes
through the focus
 Draw a ray that goes through the focus and the top of the object and then goes parallel to
the principal axis.
 For mirrors a line through the center of curvature (C or 2f) and the top of the object, the
reflection is along the same line. For lenses a line through the top of the object and the
horizontal and vertical center of the lens (optical center).
Note: If one of the basic rays passes through the object, then the image will be virtual and the
rays must be put on the opposite side of the lens or mirror than where they usually are. To locate
the object, draw an arrow from the principal axis to the place where the 3 principal rays meet. If
17
the object is virtual, it is located where the 3 dashed lines intersect. PLEASE NOTE, if the
curvature of the mirror isn’t just right, all 3 rays may not intersect perfectly. This is not a
problem. Remember this is a graphical method and those are never as perfect as the actual
mathematical calculations. You can make a best guess (I draw a really big dot) or just take a
point where 2 of the principal rays intersect. WEIRD HINTS: We expect to see an object
through a lens! The expected side for the image is the opposite side from the object!
We expect an object to be reflected from a mirror! The expected side for the image is the same
side as the object! In order to understand ray tracing, you must practice it several times!
Specific ray tracing directions for specific situations are below. Please skim through them and
then use then to complete the practice problems assigned on the next page.
RAY TRACING -CONCAVE MIRRORS FOR OBJECTS LOCATED MORE THAN 1
FOCAL LENGTH FROM THE MIRROR
 Draw a ray parallel to the principal axis that touches the top of the object and then reflects at
the surface of the mirror through the focus.
 Draw a ray that goes through the focus and the top of the object and then reflects at the
surface of the mirror parallel to the principal axis.
 Draw a ray that goes through the center of curvature (C = 2f) and reflects along the same line
RAY TRACING - CONVERGING LENSES FOR OBJECTS LOCATED MORE THAN 1
FOCAL LENGTH FROM THE LE NS
 Draw a ray parallel to the principal axis that touches the top of the object and then bends at
the center of the lens to pass through the focus on the other side of the object
 Draw a ray that goes through the focus on the object side of the lens and the top of the object
and then bends at the center of the lens to go parallel to the principal axis
 Draw a ray from the top of the object and passes through the center of the lens (the point at
which the vertical center of the lens intersects the principal axis)
RAY TRACING- CONCAVE MIRRORS FOR OBJECTS LOCATED BETWEEN THE
FOCAL POINT AND THE MIRROR
 Draw a ray parallel to the principal axis that touches the top of the object and then reflects at
the surface of the mirror through the focus, HOWEVER, since this ray passes through the
object the image is virtual. Draw a dashed line on the opposite side of the mirror that is an
extension of the reflection line.
 Draw a ray that goes through the focus and the top of the object and then reflects at the
surface of the mirror parallel to the principal axis, then extend the reflected line on the other
side of the mirror with a dashed line.
 Draw a ray that goes through the center of curvature and the top of the object (C = 2f) and
then extend this line on the other side of the mirror with a dashed line.
RAY TRACING -CONVERGING LENSES FOR OBJECTS LOCATED BETWEEN THE
FOCAL POINT AND THE LENS
 Draw a ray parallel to the principal axis that touches the top of the object and then bends at
the center of the mirror to pass through the focus on the other side of the object.
HOWEVER, because the rays drawn will never meet, the image is virtual. Now draw a
dashed line on the same side of the lens that is an extension of the line after the bend at the
center of the lens.
 Draw a ray that goes through the focus on the object side of the lens and the top of the object
and then bends at the center of vertical plane of the lens to go parallel to the principal axis,
THEN extend the line (dashed) on the same side of the lens
18

Draw a ray from the top of the object and passes through the center of the lens (the point at
which the vertical center of the lens intersects the principal axis), THEN extend this line with
a dashed line backward from the object
RAY TRACING FOR CONVEX MIRRORS FOR ALL OBJECTS – Remember this mirror only
results in virtual objects
 Draw a ray parallel to the principal axis that touches the top of the object and then bends as it
goes through the mirror to go through the focus on the other side of the mirror (remember f is
negative for this mirror). The line on the other side of the mirror should be dashed.
 Draw a ray that would go through the focus on the other side of the mirror and bend it at the
center of the mirror to reflect parallel to the principal axis. Extend the bent line on the other
side of the mirror with a dashed line
 Draw a ray that goes through the center of curvature as it would appear on the other side of
the mirror and the top of the object (C = 2f). This line should be dashed on the other side of
the mirror.
RAY TRACING –DIVERGING LENSES FOR ALL OBJECTS – Remember this lens only
results in virtual objects
 Draw a ray parallel to the principal axis that touches the top of the object and then reflects
back from the lens to go through the focus on the same side of the lens that the object is on
(remember f is negative for this lens). The line that reflects from the lens should be dashed
 Draw a ray from the top of the object that would go through the focus on the other side of the
lens and reflect it at the center of the lens parallel to the principal axis. Extend the line
parallel to the axis on the left side of the mirror with a dashed line
 Draw a ray that goes through the vertical and horizontal center (optical center) of the lens and
the top of the object. This line does not need to be dashed.
Practice: Make ray-tracing diagrams for problems 1 through 6. This is done most easily by using
graph paper!
1. Concave mirror, object distance 20 cm, object height 5 cm, radius of curvature 30 cm
2. Concave mirror, object distance 10 cm, object height 5 cm, radius of curvature 30 cm
3. Convex mirror, object distance 10 cm, radius of curvature 40 cm, object height 5 cm
4. Converging lens, object distance 10 cm, focal length 15 cm, object height 5 cm
5. Converging lens, object distance 20 cm, focal length 15 cm, object height 5 cm
6. Diverging lens, object distance 10 cm, focal length 20 cm, object height 5 cm
7. Fill in the chart shown below so, si, f, R (only for mirrors), ho, hi and m for problems 1
through 6 from your ray diagrams (always the first row you come to). Then solve for these
quantities mathematically using the equations you were given in this lesson and fill in those
rows on your chart. How did your graphical method (ray tracing) compare to the math?
What are the strengths and weaknesses of ray tracing?
Table 6. Answer chart for the practice problems for ray tracing.
Problem # and type
S0
Si
F
R
(mirrors
only)
1 concave mirror
diagram >f
1 concave mirror
math >f
19
H0
Hi
m
Table 6. (continued) Answer chart for the practice problems for ray tracing.
2 concave mirror
Diagram <f
2 concave mirror
math <f
3 convex mirror
diagram
3 convex mirror
math
4 converge lens
diagram < f
4 converge lens
math <f
5 converge lens
diagram >f
5 converge lens
math >f
6 diverge lens
diagram
6 diverge lens math
Physics B Unit 3, Lesson 4, Interference, Diffraction, and Thin Films (B34Interfere)
Interference, Diffraction and Thin Films- Understand and make calculations for double slit
diffraction, single slit diffraction, diffraction gratings and thin films.
Label each page of today’s homework with today’s code B34Interfere, the page number, and the
date. Again, you are going to learn all the concepts of this lesson in terms of light although these
concepts are true of other types of waves. I have specifically seen problems that involve sound
from stereo speakers.
1. Find the pages in your book or websites on the internet that discuss Young’s double slit
experiment, single slit diffraction, diffraction gratings and thin films and write down these
references.
Let’s start with interference. We have talked before about constructive and destructive
interference.
2. How are amplitude and intensity affected for two waves that undergo constructive
interference?
3. How are amplitude and intensity affected for two waves that undergo destructive
interference?
4. Constructive and destructive interference are a significant part of Young’s Double-Slit
Experiment. Skim through one of your resources and then in two sentences tell me what
constructive and destructive interference have to do with Young’s experiment.
20
Look on page 2 of your Table of Equations for AP Physics Exams and find all the equations that
are given for the topics we have studied in this unit in the section labeled WAVES AND
OPTICS.
5. There is a classic illustration that goes with Young’s experiment that illustrates the variables
for this situation in the two equations found on our equation sheet, d sin θ = m λ and xm ~
mλL/d. Find the classic illustration and make a quick, but understandable sketch of it.
Make sure you have labeled xm, d, θ, and L on the illustration. The letter y might be shown
on the illustration instead of xm. They represent the same thing.
Now I want you to gain a better understanding of the illustration. Young’s experiment yields
bright fringes (maxima) and dark fringes (minima). Make sure you know what these terms
mean. When a bright fringe occurs the light that travels there has to arrive exactly in sync.
When this happens, Ta Da!, there is constructive interference or light of greater
intensity/amplitude. The light from each slit has traveled one wavelength or an integer multiple
of a wavelength further or shorter in the path it takes. The only exception to this is the central
maximum where light has traveled the same distance from both slits. The difference in the
length the light traveled is typically referred to as “path difference” and this is often asked about
on the test.
6. If we aren’t considering the central maxima, what is the minimum path difference for the first
bright fringe?
For dark fringes, the light has to arrive completely out of sync. That means that a crest from the
light from one slit has to arrive exactly when a trough from the light from the other slit arrives.
Then, Aha!, there is destructive interference or darkness.
7. How does the path difference for dark fringes differ from that of bright fringes?
Okay now go back and look at that lovely illustration you drew for question 5. Now answer the
following questions.
8. What is L?
9. What is θ?
10. What is xm?
11. What is d?
12. Now even more irritating, what is m?
Books and resources typically give different equations for bright fringes than for dark fringes. I
am crazy and think you can use the same two equations for both. Or maybe I am not so crazy.
13. Give three examples of a numerical value for m for bright fringes using the two equations
given on the AP equation sheet.
14. Give three examples of a numerical value for m for dark fringes using the two equations
given on the AP equation sheet.
15. Now consider some other things. One of our equations has an equal sign, =, and the other
has an approximation sign, ~. So of course this leads me to a question. Why? The answer to
this question is a question. What is the small-angle approximation and how does that
approximation get us from the equation with the equal sign to the equation with the
approximation sign?
16. In 1 to 2 sentences tell how Young’s experiment could determine the wavelength of an
unknown source of light.
17. What does Young’s experiment have to say about the wave nature of light?
Now have a quick read in your resource about single slit diffraction and diffraction gratings.
Notice that the typical book or resource gives new equations for single slit diffraction and
21
another one for diffraction gratings. Now again your teacher is totally living the la vida loca and
thinks you can use the same two equations on the AP equation sheet for these two situations.
18. What is d for single slit diffraction?
19. How is there a path difference for single slit diffraction?
20. What is d for a diffraction grating?
21. Make a drawing of waves showing the intensity of light on the screen for (a) double slit
diffraction, (b) single slit diffraction and (c) a diffraction grating.
Okay, I think that is it for double slits, single slits, and diffraction gratings. But wait… There is
another significantly important type of interference that happens with thin films. Locate this
topic in one of your references because now I want you to think about thin films. I used to say
that I hated thin films. Truthfully I still don’t love them, but at least now I can do problems
involving them and get the problems right every time! That is because I have created Walker’s
Sandwich method of thin films. Woo hoo! You know what a sandwich is, don’t you? I am sure
you do, but I have to define the “Walker” thin film sandwich. A sandwich is a filling of
something placed between two pieces of bread. The two pieces of bread are the same, but the
sandwich filling is totally different. A piece of bread with a filling and then another filling
without that second piece of bread is not a “Walker” thin film sandwich. Wait until I give you
my totally exciting discussion of little old ladies and East European women body builders for
Lenz’s law. That will be even more fun than my sandwich discussion! And that leads us to my
thin film discussion.
Okay, you are familiar with thin films without ever having discussed them mathematically. You
have seen water puddles with a slightly oily surface on them where you can see a rainbow
haven’t you? That rainbow is caused by thin film interference. You can see this same rainbow
near the surface of a soap bubble can’t you? Thin films are also used on eyeglasses and other
types of glass as anti-glare coatings. They are also used in fiber optics. So first let’s contemplate
this whole stupid sandwich thing. Think of the soap bubble. The soapy water that forms the
bubble creates a thin film! And light traveling from air through the soapy water into the air
inside the bubble creates an interference pattern that allows us to see the rainbow! It is the
indices of refraction of the air-soapy water-air interfaces that create the sandwich! So if you
have a ray of light (think arrow) that goes from a medium with a low refractive index (air) into a
medium with a higher refractive index (soapy water) and then back into a medium with lower
refractive index (air), you have a thin film sandwich! If you are looking at a film of water on a
piece of glass, then you are going from a low index of refraction (air 1) to a higher index (water
1.33) to an even higher one (glass 1.5). This is NOT A SANDWICH. Thin film problems on the
AP Exam involve calculating the minimum thickness of the thin film and doing ray tracing and
saying where phase changes occur. It is important to note that it is not required that the two
layers on either side of the thin film have the exact same indices of refraction, they just need to
follow the same trend, either larger than the thin film or smaller than the thin film. If this is true,
then it is still a sandwich. Okay I know this isn’t easy, but read the thin film notes below, then
answer these questions.
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Thin Film Notes
A. When going from thinner to thicker there is a ½  phase change at the REFLECTION
SURFACE.
B. When going from thicker to thinner there is no phase change.
C. Recall that the wavelength of the light changes in the second medium. Remember the
velocity slows and since the FREQUENCY NEVER CHANGES then the wavelength
must also be smaller so divide the vacuum wavelength by the refractive index to get the
new wavelength.
D. So in going from thinner to thicker there is a /2 path difference, but the wave also has to
travel an additional distance of 2t where t is the thickness of the film. So if 2t = /2 then
the waves recombine in phase and there is constructive interference. A general equation
for constructive interference is:
2nt = (m + ½ ) OR
two times the thickness times the refractive index of the thin film = a multiple of /2
E. If the extra distance 2t traveled by the light is a multiple of , then the two waves
recombine out of phase then 2t =  and you get destructive interference. The general
equation for destructive interference is:
2nt = m OR
Two times the thickness times the refractive index of the thin film = a multiple of 
F. The above equations are valid when there is only one phase reversal, i.e. if it is a thin film
sandwich where the thin film is contained between two media that have a smaller index
of refraction or a larger index of refraction than the film itself (i.e. soap bubble which is
an air-soapy water-air sandwich!). If it is NOT a sandwich, i.e. you go thin, thick, thicker
or vice versa, THEN THE TWO EQUATIONS ARE REVERSED!
22. Let’s say that you have a soap bubble with a refractive index of 1.35. The refractive index of
air is 1. What is the minimum thickness of the bubble for constructive interference to occur?
At which interfaces is there a 180° phase change?
23. There is a puddle of water (n=1.33) with a thin film of vegetable oil on top of it (n=1.47) and
then air (n=1) on top of that. What is the minimum thickness of the oil for constructive
interference to occur? At which interfaces is there a 180° phase change?
24. Let’s consider a biology slide with a glass bottom (n=1.55) then water (n=1.33) and then a
plastic slide cover (n=1.46). Is this is sandwich or not?
25. That’s it for this lesson. Write the two most important things you learned in this lesson.
Physics C Unit 3, Lesson 1, Rotation Beginnings (C31Rotate)
Rotation Beginnings – Contrast and compare linear quantities and rotational quantities and
make calculations for equilibrium requiring τnet =0.
Start your homework today by labeling the top of each paper with the date and the exciting code,
C31Rotate, and the page number. Make sure to label any additional pages with your exciting
code and the page number so neither one of us gets confused. Feel free to add anything else at
the top you need to help you remember what topics you are studying.
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1. Find references in your textbook or on the internet that describe rotational kinematics and
situations for equilibrium where the net torque is zero. Write these references down.
2. Remember that rotational quantities are analogous to mechanics quantities you already know.
And they use the same equations. In most textbooks, there is a chart with the two sets of
kinematics equations (i.e. v = v0 + at and  = 0 + t) found side by side. Find this chart and
write the page number as the answer to this question.
Also some books use the word “angular” to refer to these quantities. I have also seen them
referred to as “rotational” and “radial”. I will use all three terms interchangeably AND you
should be familiar with all three terms. So let’s look at some basics. Read the ~ sign as “is
analogous to”.
 (rotational acceleration) ~ a, acceleration
 (rotational velocity) ~ v, velocity
 (rotational displacement) ~ x , displacement
 (torque) ~ F, force
I (moment of inertia) ~ m, mass
L (angular momentum) ~ p, momentum
K (angular kinetic energy) ~ K, linear kinetic energy -- Wait a minute. Those are both the same!
Rest assured they are supposed to be.
There are a few additional things to know. All angular components are equal to their linear
components divided by the radius. So  =x/r and  = v/r and =a/r. This makes sense to get rid
of the unit of length contained in linear quantities. So if you divide the linear quantity
acceleration, a, by length you really get 1/s2 or radians/s2. Because of this, I think of radians as
not really being units, but as filling in a space. Also remember 360 = 2 radians=1 revolution.
3. You also need to be aware of how centripetal acceleration relates to rotational quantities.
Derive or find an equation for centripetal acceleration in terms of r and ω.
4. What is the difference between tangential acceleration and total translational acceleration?
5. Write the equation for rotational kinetic energy.
6. Write down an equation that does not contain an integral for moment of inertia.
7. Write down the integral equation for moment of inertia. There will be more on this soon.
8. Write down the units for moment inertia.
9. (a) Write down the page for the Table with Moments of Inertia. (b) What is the moment of
inertia for a thin cylindrical shell? (c) What is the moment of inertia for a solid disk?
(d)What is the moment of inertia for a solid sphere?
10. The units for rotational kinetic energy are the exact same units as our linear ones. But wait,
the equations are totally different. For this part of the homework, prove that both equations
give the exact same units.
11. There are other analogous equations. Write the equation for torque that is not r x F. You
don’t have to look this up. It is analogous to FNET =ma.
12. What are the units for torque?
13. We have learned that the requirements for equilibrium in terms of force are that Fnet = 0
along both the x and y axes. Now with inclusion of torque, what are the requirements for
equilibrium?
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14. This is the end of today’s lesson. Now I suggest you read carefully through examples in your
book about basic rotational kinetic energy, calculations of moment of inertia that do not
require calculus, and equilibrium problems that include torque. Also please write as the
answer for this question the two most scintillating things you learned in this lesson.
Physics C Unit 3, Lesson 2, Moment of Inertia and Pulleys (C32Inertia)
Moment of Inertia and Pulleys – Gain a deeper conceptual understanding of moment of inertia,
use calculus to determine the moment of inertia of a uniform rod and be able to take moment of
inertia of a pulley into account.
Okay, okay, you need to do that labeling thing again. Today’s label is C32Inertia. Please put
this code, the date, and the page number as usual at the top of each page as you do the
homework. That is the first exciting thing. The next exciting thing is that we are actually going
to use calculus today! Yesterday you should have written down the integral form of the equation
for moment of inertia. We will start with that. First I want you to find either in your book or on
the internet or both a place where an example is done to find the moment of inertia of uniform
solid cylinder.
1. For your first homework problem, copy down the process step by step, to find the moment of
inertia of a uniform solid cylinder. Write the process down like you had to explain to
someone else how to do it. This means not just writing down mathematical steps, but short
phrases explaining the process. This has not ever been asked on the AP exam that I can find
a history of, but similar questions have been and it is important to be able to understand this
process. I apologize for the fact that both lower case “r” and upper case “R” are used, but the
basic rule is that lower case r is a variable and upper case R is a constant, in this case one of
the limits of integration.
2. Now see if you can achieve an analogous process to find the inertia of a uniform rod that is
pivoted about one end. Use a linear density for the rod instead of a volumetric one like you
did for the cylinder. By the way we will be using these densities over and over again for this
APC course, so let’s get the typical nomenclature down. ●Volumetric density, like the one
you studied in chemistry where density equals mass divided by volume, is given the Greek
letter lower case rho, ρ. The typical unit of ρ is kg/m3. ●Surface area density, the only
example I can think of is number of pixels on a computer screen, is given the Greek letter
lower case sigma, σ, typical unit kg/m2. ●Linear density, mass per unit length is given the
Greek letter lower case lambda, λ, typical unit kg/m. You will be using λ for this problem.
We will use these same Greek letters again when we discuss charge density in association
with Gauss’ law and then again with Ampere’s law …you should be hearing an evil witchlike laugh in your head right now. Typically this problem, the one about the moment of
inertia for a uniform rod, is done using an infinitesimally small sliver of the rod of width dx
which then has a mass of λ dx and your limits of integration are from 0 to L. Try this
integration and then see if your answer matches the one in the table that you found yesterday.
I have seen this very integration on the AP exam more than once.
Let’s consider a pulley with significant rotational inertia. An object rolling down an incline and
object(s) attached to a pulley where rotational inertia has to be taken into account are probably
the two most classic rotational physics problems. Since they are on the exam so often, you will
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go over them in some significant detail. Please note just because you are learning about an
object rolling down an incline and a pulley where moment of inertia is important, this does not
mean that you will not have to deal with the easier situations of an object just sliding down a
ramp or a pulley where moment of inertia is not important. You will be asked to do both on the
test and you must read carefully to see which kind of a problem you are being asked to do.
Remember, this is as much a reading test as it is a physics test. Reading is Fundamental!
Let’s consider the classic problem of an Atwood machine. This is a pulley with significant
moment of inertia. The pulley is considered to have a radius R and a rotational inertia I. There
are two masses looped over it, m1 and m2. Let’s say that m2 is heavier than m1 and that m2 is
accelerating down and m1 is accelerating upwards. You are typically asked to find the
acceleration of the masses and the tension in each side of the rope. So let’s look at this problem.
The free body diagram for each object is shown below.
FT1
m1
FT2
m2
I
FT1
Fg1 = m1g
FT2
Fg2 = m2g
Figure 2. Free body diagrams for an Atwood Machine.
The first thing I would like you to notice is that unlike previous problems where we have ignored
rotational inertia, the tension on each side of the rope is not the same. So I have labeled them as
FT1 and FT2. In a previous lesson you learned the definition of torque () = I. Torque is also a
cross product equal to r x F. I will take an arbitrary sign convention of clockwise is positive and
counterclockwise is negative. In other words, any force that acts to make the pulley rotate
clockwise we will consider as being positive while any force that acts to make the pulley rotate
counterclockwise we will consider as negative. Remember rotational mechanics have significant
analogies in linear mechanics. So we know that FNET = ma and we can also say that NET = I
and NET =  r x F so that I =  r x F. Notice that there are three unknowns, a, FT1, and FT2 and
you will need three equations. Here are the equations. First let’s sum FNET around m1. The
equation is FT1 – m1g = m1a. Now let’s sum FNET around m2. The equation is FT2 – m2g = - m2a.
Remember my sign convention is always that down is negative and m2 is accelerating down.
The third equation is provided by rotational motion and that is FT2R – FT1R = I. Recall also
a
that  = a/R. Substituting this in we get FT2R – FT1R = I . Here are our three equations in a
R
nice neat list:
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FT1 – m1g = m1a
FT2 – m2g = - m2a
a
FT2R – FT1R = I
R
Notice that the acceleration (a) in all three equations is exactly the same acceleration. This will
always be the case for the types of problems that we are going to consider. You will get the
excitement of considering other types of problems where this is not true once you are in higherlevel classes in college! This problem works out nicely if we consider the pulley to be a solid
cylinder (seems like we talked about this before) with a moment of inertia of ½ MR2.
3. Substitute the moment of inertia of the pulley into the third equation and solve the three
simultaneous equations for the acceleration in terms of the three masses, m1, m2, and M the
mass of the pulley.
4. Well, that seems like plenty to learn today to me. Write down the two stimulating and
essential things you learned from this lesson.
Physics C Unit 3, Lesson 3, Rotational Motion for an Inclined Plane (C33Incline)
Rotational Motion for an Inclined Plane – Compare and contrast the use of Newton’s 2nd law
and conservation of energy to solve problems for objects of varying moments of inertia rolling
down an inclined plane.
Please label all pages of this homework assignment with its code, C33Incline, the date and the
page number. I will just be boring and tell you to find the section in your book or an internet
website that describes the rolling motion of a rigid object.
1. Write down the pages in your book or the URL of a website that can be used as a resource for
rotational motion on an inclined plane. Most problems consider an object rolling down an
incline. So you will start with that. Let’s start by considering a sphere. Please start by
making a free body diagram; you know I always think that is valuable. If you are to create
the free body diagram of a sphere on an incline, the forces you need to consider are the
gravitational force, the normal force, and the frictional force. If there is no frictional force,
the object won’t roll; it will just slide down the incline! Wow! I have seen a big deal made
of these free body diagrams. Now ask yourself exactly where each force would originate on
the object? In a previous lesson we know that it does matter where the force of gravity and
other forces actually are due to torque. In this free body diagram, the force of gravity should
start at the center of the sphere (the center of mass for a uniform object) and point down. The
normal force should start at the exact point of contact between the sphere and the incline and
point perpendicularly upward from the incline. The frictional force should also start at the
point of contact of the sphere and the incline and point uphill and parallel to the ramp.
2. Draw a correct free body diagram of a sphere on an incline. Let’s consider an incline of a
given height, h, and a given angle, θ. You can solve for the velocity of the sphere at the
bottom of the incline by using conservation of energy. And you can solve for the
acceleration using the same principle. However, like other inclined plane problems you have
solved you can also use Fnet = ma. You need to be able to do both. Let’s start with
conservation of energy. There are two types of energy that you are considering as the ball
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rolls, its translational kinetic energy (1/2 mv2) and its rotational kinetic energy (1/2 I2). In
this case the velocity of the in the ½ mv2 equation is the velocity of the center of mass of the
sphere. Added together these are the sphere’s total energy and at the bottom of the ramp
should equal its gravitational potential energy at the top of the ramp according to the law of
conservation of energy.
3. Look up the moment of inertia of a solid uniform sphere of mass M and radius R.
4. Set the sum of both types of kinetic energy equal to the gravitational potential energy. Then
solve for v at the bottom of the incline assuming the sphere starts from rest. Remember that
 = v/R and substitute that in and magically (not really) all the R’s should cancel out.
5. Now that you know the v at the bottom and v0 = 0 and knowing θ and h you can find the
length of the ramp and find the acceleration of the sphere. Oh, by the way, you can probably
find an example on the internet that will show you this problem exactly for a sphere or a
cylinder.
6. Now let’s consider doing the same thing using Fnet= ma and of course our new and exciting
friend τ which equals both I and F x r! You will consider that the sphere has a coefficient
of friction . Go back to that free body diagram you drew and prove that the acceleration of
the object is a = g sin θ - g cos θ using Fnet = ma.
7. Now you can use the two equations for torque, τ = I and FR and set them equal to each
other. You already looked up moment of inertia for the sphere. The force F that cases the
rolling is friction and so τ = the force of friction times the radius. Or in other words I =
mg cos θ. Substitute the correct moment of inertia into this equation and solve for .
Remember that  = a/R. Substitute that in too or it won’t be right.
8. Put the  you solved for in part 7 into the equation that you solved for in part 6 and solve for
the acceleration. It should be the same answer you got for question 5. Say Aha!
9. Now use kinematics and solve for the velocity at the bottom of the incline. This answer
should be the same as question 4. Say Yahoo!
10. Okay that should give you the basics for dealing with these kinds of problems. There
however, is one more classic problem you need to know the answer to. A solid sphere, a
solid cylinder and a box are having a race down similar inclines. They all have the same
mass and the cylinder and sphere have the same radius. The box slides down a frictionless
incline and the sphere and cylinder roll down the same incline without slipping. Who wins
the race?
11. What if we compare two spheres rolling down an incline where one has some slipping and
the other has no slipping? Which one will win then?
12. What causes one sphere to have some slipping and the other to have no slipping?
13. Check out a website provided by Hyperphysics: http://hyperphysics.phyastr.gsu.edu/hbase/hoocyl.html#hc1 or check out any website that talks about spheres,
cylinders, and boxes racing.
Physics C Unit 3, Lesson 4, Other Rotational Situations (C34Other)
Other Rotational Situations – Understand conceptually and be able to make calculations for the
rotational analogies to the Work-Energy Theorem, power, Newton’s 2nd Law, angular
momentum, and precessional motion.
28
Okay, we have finally made it. This is the last new material for rotational situations. Yahoo!
Label this lesson C34Other, the date and the page number. There are a couple of other rotational
analogues we haven’t discussed. So let’s get these figured out.
1. There is a rotational analogue for the Work-Energy theorem. Find that equation and write it
down. Could you use it?
2. I have always told you that the centripetal force can do no work. Did I lie? Is there work
done during rotational motion?
3. What is the rotational analogue for power?
4. This has nothing to do with angular quantities, but I know it is shown in this chapter because
it is in one of those cute tables that show the angular quantities on one side and the rotational
quantities on the other side. There is a differential equation for force that uses translational
quantities of momentum and time and is Newton’s second law! Write it. Remember it is a
differential equation! Prove to me mathematically how this is equivalent to Fnet = ma.
5. Write the analogous rotational equation for the one you gave the answer to in number 4.
6. What is the equation for angular momentum that is a cross product?
7. Give simplified version of the previous equation that is in terms of the magnitudes of the
vectors only.
8. Write the equation for angular momentum in terms of rotational quantities only that is
analogous to our old friend p = mv.
9. Is angular momentum conserved? Is there an analogous equation for my poo = p equation?
10. Now let’s talk about the ice skaters. They are very important and often asked about. Think
about the last time you saw an ice skater doing a spin. When he wanted to spin faster and
faster he pulled his arms in. How does that change his moment of inertia? Write down a
couple of sentences that describe what is happening in terms of physics during the spin.
11. Read/skim something about precessional motion. Write down 1 to 2 sentences about what
precessional motion in terms of observation and physics.
12. Read through any example problems you can find about conservation of angular momentum
where there is a net torque.
13. You have completed the last rotational motion assignment! Yahoo! Make me happy by
writing down the two most important notions from today’s lesson.
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UNIT 5, PHYSICS B FLUID MECHANICS & THERMAL PHYSICS AND PHYSICS C
ELECTRICITY IN DEPTH
Table 7. Unit 5 Overview, Physics B Fluid Mechanics & Thermal Physics and Physics C
Electricity in Depth
Physics B Fluid Mechanics & Thermal
Physics
1) Pressure and Buoyant Force –
Understand both conceptually and
mathematically the variance of pressure
with fluid depth, pressure as a function of
force and area, and buoyant force.
2) Bernoulli and Torricelli – Understand
basic flow rate through pipes and be able to
describe and utilize Bernoulli’s equation
and related equations including Torricelli’s
Law, particularly understanding how these
concepts relate to the conservation of mass
and energy.
3) Gas Laws – Be able to interpret and make
calculations related to the following topics:
thermal expansion, the ideal gas law, the
kinetic theory of gases and rate of energy
transfer during conduction.
4) Thermodynamics – Be able to interpret
and make calculations related to the
following topics: thermodynamic work and
PV diagrams, the laws of thermodynamics,
and heat engines.
Physics C Electricity in Depth
1) Gauss’s Law for Spheres - Comprehend
and be able to analyze Gauss’s law for
spherical distributions including
conductors, nonconductors of uniform
charge density and nonconductors with
densities as a function of r.
2) Continuous Charge Distributions - Be
able to determine and recognize
appropriate problem solving techniques for
finding the electric field and the potential
from continuous charge distributions
particularly rings or arcs of charge.
3) RC Circuits - Grasp the numerical and
theoretical concepts associated with RC
circuits.
4) Electric Potential from Electric Field Using the integral relating electric field to
potential and the derivative relating
potential to electric field, be able to make
determinations both mathematically and
conceptually.
5) Other Field, Potential and Capacitance
Problems - Gain insight and awareness of
other Gaussian surfaces, more difficult
continuous charge distributions, and
capacitance for geometries different from
rectangular parallel plates.
Physics B, Unit 5, Lesson 1, Pressure and Buoyant Force (B51Pressure)
Pressure and Buoyant Force – Understand both conceptually and mathematically the variance
of pressure with fluid depth, pressure as a function of force and area, and buoyant force.
Let’s do first things first. First haul out your Table of Equations for AP Physics Exams. There is
a section on it labeled Fluid Mechanics and Thermal Physics. Look at the equations. The topics
relating to these equations are what we will be covering before the next test. Pressure is the one
of the unifying concepts of this unit so let’s start with a discussion of pressure and then move on
30
to a discussion of buoyant force. Please label all of today’s homework with B51 Pressure, the
date, and the page number.
1. Let me be blatantly obvious, please write down these 3 equations you will be using in this
lesson, (1) P = P0 + ρgh, (2) Fb = ρVg and (3) P = F/A. Write down your best guess of what
each letter represents.
2. Look in your book or on the internet and find a discussion of the following topics and write
down a reference or two for the study of these topics: fluids, density, specific gravity,
pressure, Pascal’s Principle, buoyancy, and Archimedes’ Principle. You might see Bernoulli
in there, but we will be studying Bernoulli tomorrow. We will not be studying the topics
related to deformation of solids. Don’t forget to write down some page numbers or URL’s.
3. I am pretty sure that you already know the equation for density, but just to be sure, write
down the formula for density.
4. This ain’t your chemistry teacher’s density. Or, in other words, there are no units of grams
per centimeter cubed or grams per milliliter here in physics. Write down the units used in
physics for density.
5. Write down the density of water at 4° C in correct physics units.
6. Write down the page number of the table in your textbook which lists the densities of
common substances. You may need to refer to this in order to do homework.
7. Let’s figure out what the heck specific gravity is. It really is an easy concept. What is the
difference between the density of glycerin which is 1260 kg/m3 and its specific gravity of
1.26? Why is specific gravity unitless? What is the specific gravity of copper which has a
density of 8920 kg/m3? What is the density of gold which has a specific gravity of 19.3?
Why is specific gravity useful?
8. Referring back to the last equation I asked you to write down, P = F/A. Write down the
name of each variable and its units.
9. Define the Pascal in terms of Newtons and meters. Define the Pascal in terms of the units of
kg, m and s and only.
10. Write a sentence that explains the concept of why a person can lay on a bed of nails.
11. Write a sentence that explains why a woman of the same weight will poke holes in the
football field when she is wearing high-heeled shoes when she doesn’t make any holes when
she is wearing soccer cleats.
12. Find that equation I gave at the beginning with an h in it which gives pressure as a function
of height. It is up next. If I have a bathtub with 3 inches of water in it and a milk carton with
3 inches of water in it and a glass with three inches of water in it, which one will have the
greatest pressure at the bottom? Why? Would the answer be different if there were milk in
the milk carton?
13. Explain in less than 3 sentences why there is atmospheric pressure on earth in terms of air,
fluid mechanics concepts, and height.
14. What is the difference between gauge pressure and absolute pressure?
15. Okay now a new equation the one for buoyant force. Write down the name of each variable
and tell me what it relates to, either the object or the fluid the object is in. I assume you
know the units of each variable, so you should make sure you do.
16. It is amazing that you can also write an equation for the force of gravity in terms of density
and volume. Please write this equation.
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17. Now pretty please draw a pretty picture of a rectangular piece of wood floating in tub of
water. Make sure to show a part of the wood below the water and part of it above the water.
Draw a free body diagram showing the two most significant forces acting on the wood.
Make sure both forces are expressed in terms of density and volume and that each density
and volume have subscripts on them indicating which density and volume they are! Using
this information derive an equation that will give you a fraction or percent of the wood that is
submerged. Give an equation that will give the fraction or percent of the wood that is above
the surface. This is a very, very classical question and is often found on the AP exam. Make
sure you read carefully to see whether the question asks for the volume submerged or volume
above the surface.
18. Why didn’t you include the buoyant force of the air in the previous problem?
19. Now for the very exciting and absolutely necessary Archimedes’ crown problem. I always
want to say that jewelers use this method for determining if an object is really gold or not,
but actually jewelers use chemical tests (Is It Real Gold, 2007). This problem is classic and
is shown in almost every physics textbook. A crown that is supposedly gold is weighed both
in air and in water. Let’s pretend that the weight in air is 10 N and the weight in water is 8
N and go through the problem. First things first of course, please draw two free body
diagrams: (1) the crown hanging on a spring scale in air and (2) the crown under the same
conditions except that it is in water.
20. You should have drawn three forces for the each of the situations to be technically correct.
There should be the force of the spring scale up and a buoyant force up and the force of
gravity down. Yes, it is true even the crown in the air is experiencing a buoyant force
because it is immersed in a buoyant fluid, air. Yes, in all those free body diagrams I have
shown of books and tables and lawnmowers and boxes there should always have been a
buoyant force due to air, but I never showed it. Why? Because the buoyant force of air is so
small it can generally be neglected. So if you forgot the buoyant force of air, I will forgive
you. However, the buoyant force of the water on the crown is not negligible. Now go back
to your free body diagram and write the equations for buoyant force and gravity in terms of
density. Make sure you use a subscript that distinguishes the two kinds of density for each.
They are very different.
21. Okay now I would like you to use Fnet = ma for each FBD and prove what you have is
equivalent to the equations I have listed below.
(1) Fs in air = ρcrownVcrowng = Fg
(2) Fs in water + ρfluidVcrowng = ρcrownVcrowng = Fg
Please note from the example I have started with Fs in air is 10 N and Fs in water is 8 N.
22. Notice that the volume of the fluid replaced is the same as the volume of the crown. Also I
hope that you can see that the difference between the two weights is equal to the buoyant
force which also equals ρfluidVcrowng. Now you should be able to solve for the volume of the
crown, the mass of the crown and the density of the crown. Solve for these things and since
we discussed the density of gold previously, discuss whether the crown is gold or not.
23. Write down the two most striking things that you learned about fluid mechanics in this
homework assignment.
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Physics B, Unit 5, Lesson 2, Bernoulli and Torricelli (B52Bernoulli)
Bernoulli and Torricelli – Understand basic flow rate through pipes and be able to describe and
utilize Bernoulli’s equation and related equations including Torricelli’s Law, particularly
understanding how these concepts relate to the conservation of mass and energy.
Today’s homework code is B52Bernoulli. Please make sure to put this on each page of your
homework along with the page number of the page you are writing on. Okay, let’s get started.
1. Find and write down some resources in your textbook or on the internet that discuss
Bernoulli’s equation and Torricelli’s equation and how they relate to the flow of water.
2. There are two main forms of Bernoulli’s equation. One is found on your AP equation sheet.
It is a little different in that it has an entire word in it, “constant”. Find and write this
equation down.
3. The other form of Bernoulli’s equation you can usually find on the internet or in your
textbook and it is very similar to an equation that you learned about while discussing
conservation of energy. Find this equation and write it down.
4. Write in a couple of sentences describing how Bernoulli’s equation relates to conservation of
energy.
5. Now before you start using Bernoulli’s equation, let’s do some thinking about flow rates.
Knowing what you do about basic physics units, give me two possible units for flow rate, one
for mass flow rate and one for volumetric flow rate. Now consider how these two flow rate
units are related…okay that’s enough of that. Hmm, aren’t kg and m3 the units of something
else? Hmm. Okay I am hoping with all this hmming that you see that you either multiply or
divide by density to change one unit into the other.
6. Now how else can you calculate volume? Last time I looked volume for a cylinder (and
what is a pipe, but a really long cylinder) it was the cross sectional area of the end multiplied
by the height of the cylinder. But we really want to talk about volumetric flow rate. How
can we get that time unit in there? Don’t we already have this very exciting quantity with
units of length per time that you learned about on practically the first day of physics? I think
it is called velocity. Yes that’s it. So you can also get volumetric flow rate by taking the
cross sectional area of the pipe times the velocity of the fluid. Now in terms of variables and
knowing that information, write an equation for mass flow rate in a pipe.
7. Okay now you need to find out what happens if you change the cross sectional area of the
pipe. Write an equation for our first (1) mass flow rate in a pipe of given cross sectional
area. Write another equation for a second (2) mass flow rate in a pipe of with a different
cross sectional area. What is the same in both places? The first thing is density. The
“assumption” of using these equations is that our fluid is incompressible. You can’t squish it
and change its density. What is also true of this situation is that mass is conserved. What
flows past one point in a certain time in the pipe also flows past any other point in the same
time. So mass flow rate is a constant and your two equations are equal to each other. Set
them equal to each and show that you get this equation A1v1 = A2v2.
8. Now for a very, very important concept. Using the equation you derived in the problem
before and the Bernoulli equation, tell me what happens to velocity of flow and pressure in a
pipe if you decrease the cross sectional area of a pipe while keeping the average height of the
pipe the same.
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9. Let’s derive Torricelli’s Law. Using Bernoulli’s equation, consider a situation where you
have a large tank of water. The top of the tank is open to the atmosphere and there is a hole
near the bottom a distance h down from the top. Water is spouting from the hole near the
bottom of the tank, but because the tank is so large we can consider the velocity of fluid flow
at the top of the tank to be zero. (a) Derive an equation for the velocity of the water exiting
the hole in terms of h. (b) What equation does it look like? (c) Knowing the hole at the
bottom of the tank has a radius of r, derive an equation for volumetric flow rate from this
hole. (d) Knowing that the tank is filled with a liquid of density ρ, derive an equation for
mass flow rate from this hole.
10. Let’s consider one more thing. There is something that can be used to measure pressure in a
pipe called a Venturi tube. Essentially it is a vertical tube set into a horizontally flowing
pipe. The height of fluid in the tube gives the pressure in the pipe related to the equation P =
P0 + ρgh. Draw a quick sketch of a pipe that has been “crimped” to a smaller cross sectional
area in the middle with the same, but larger cross sectional area at both ends. Add 3 Venturi
tubes, one each for beginning, middle, and end and sketch qualitatively the height you would
expect in each of the 3 tubes.
11. That’s it for Bernoulli. Tomorrow, thermodynamics. Write down the two most striking
ideas from this lesson.
Physics B, Unit 5, Lesson 3, Gas Laws (B53Gas)
Gas Laws – Be able to interpret and make calculations related to the following topics: thermal
expansion, the ideal gas law, the kinetic theory of gases and rate of energy transfer during
conduction.
Today you will be learning about the behavior of gases and along with that I will throw in a little
bit of thermal expansion and a small discussion of conduction. The code for this assignment is
B53Gas. Please label each page of your assignment with that code, the date and the page
number.
1. First of all I would like you to find some references. You will need to write down book page
numbers or an internet URL for each of the following topics: (a) thermal expansion, (b) ideal
gas law, (c) kinetic theory of gases, and (d) heat transfer by conduction.
2. Let’s start with thermal expansion. Find an equation for thermal expansion in your reference
and write it down. It should contain an α and I prefer the one with the Δ’s. The α is not
rotational acceleration, but is the coefficient of linear expansion and of course the Δ means
“change in”. Make sure the equation you have written down is either the same as the one on
your Table of Equations for AP Physics Exams or that you can understand how they are the
same. Also write down a reference for a table of coefficients of expansion for different
substances. You may need these values for homework. If you are a given a problem of this
nature on the AP exam, you will be given α or you will be given data and asked to solve for
α. The topic of thermal expansion is often one of the first thermal physics topics covered in
your textbook, usually right after a discussion of temperature. Since the equation requires
temperature in the calculations, you need to understand temperature also. During thermal
physics we will use two temperature scales, °C and K, the Celsius and Kelvin temperature
scales. The Kelvin temperature scale is an absolute temperature scale which means that it is
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3.
4.
5.
6.
7.
8.
referenced to absolute zero, the temperature at which all molecular motion stops. Please note
that temperatures in Kelvin don’t use the degree sign. Degrees of the K and °C scales are the
same size and temperatures can be converted from °C to K and vice versa with little trouble.
You should add 273.15 to temperatures in °C to get K and subtract 273.15 from temperatures
in K to get °C. Truthfully for our purposes, you really only need 3 significant figures and so
273 will be good enough. We will not use °F scale and so you don’t need to know how to
convert from or to this scale. Now after saying all this very important stuff there are two
rules that you need to remember for every single calculation involving temperature in
thermal physics: (a) When you use a ΔT in calculations you may use either °C or K in your
calculations; (b) When you use an absolute T in calculations (it is not ΔT or a subtraction, T
– T0), you must use temperature in K! If you mess up rule (b), you will have done all your
calculations incorrectly. Now write down in a sentence or two why it is okay to use either
scale when you are using ΔT or T-T0 in a calculation.
Okay after that long discussion of temperature, let’s get back to thermal expansion. You are
lucky enough in New Mexico to see the evidence of this process everywhere in buckling
sidewalks and cracks in asphalt. You are only given an equation for the coefficient of linear
expansion on your homework sheet. There are equations for area expansion and volumetric
expansion. These equations are analogous and use α also, but it the constant is replaced by β
where β=2α in the area equation. The constant is γ in the volumetric equation where γ = 3α
in the volumetric equation. Write down why all the constants are multiples of α.
Now on to the ideal gas law. Write it down. Write down the constant R from your AP
Physics equation sheet. Write down what the unit of pressure multiplied by volume is, or in
other words 1 Pa times 1 m3 equals what? Write down what the temperature units you
absolutely must put into this equation.
I love PV = nRT and you will definitely use it. This section will feel a bit “chemistry-like”,
but we will use our excellent physics skills. You need to know Avogadro’s number. Write it
down with a full sentence surrounding it so that I can tell you know what it means.
Chemistry books often define Boyle’s Law or Gay-Lussac’s Law, but not me. You only
need one equation for all these laws. Assume that you have a constant amount and therefore
number of moles, n, of a gas and remember that R is a constant, and then derive an equation
for an ideal gas that starts at one pressure, volume and temperature and then changes to a
different pressure, volume and/or temperature. Put some kind of subscripts on the variables
so you can tell them apart. You can derive any kind of constant pressure, constant volume,
or constant temperature equation directly from this one equation. Of course this is a very
useful equation. It is not found on your equation sheet. I expect you to be able to derive it at
a moment’s notice. You will probably have it memorized quite quickly.
By the way just for funsies define these terms: (a) isothermal, (b) isobaric, (c) isochoric, and
(d) adiabatic. I have seen multiple-choice questions that are just as easy as knowing the
definitions of these words!
Now on to the kinetic theory of gases. References usually give a list of postulates or
assumptions of this theory. I have seen anywhere from 4 to 12 assumptions/postulates of this
theory. At the risk of being helpful (you know how I hate that, ha ha!), here are two websites
where I looked it up (a) http://www.grc.nasa.gov/WWW/K-12/airplane/kinth.html and (b)
http://en.wikipedia.org/wiki/Kinetic_theory. You may hate my choice of internet resources,
but the point is to show how very differently the assumptions of this law can look. Make
sure you check out your textbook too and I actually love the way it is presented in the
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Chemistry in the Community textbook. You do not have to memorize the scientific language
in these postulates, but you should know the basics of this theory. So please -jot down, -in
bulleted form, -in just a few words each, the basics of this theory.
9. There are several equations on your equation sheet related to the kinetic theory of gases.
There is an equation for the ideal gas law in terms of Boltzmann’s constant (kB). There is an
equation for average kinetic energy in terms of kB. There are two equations for vrms which is
the root-mean-square velocity of the molecules of the gas. One of the vrms equations is for a
mole of an ideal gas and one is for a molecule of an ideal gas. Also please recall that we use
kilograms for mass and the periodic table of elements uses grams, so there are ~2 E -3 kg of
H2 in a mole as well as 2 grams in a mole. Now finally here are the points I want to make!
(a) Double the temperature in Kelvin (OF COURSE) and what happens to the average kinetic
energy? (b) Double the temperature in Kelvin and what happens to the root-mean-square
velocity of the molecules of an ideal gas? (c) The average energy of an ideal gas is related to
what variable? (d) How does this relate to the unit of pressure multiplied by volume?
10. Conduction, convection, and radiation! Define them and give an example of each in as few
of your own words as you can. There is no calorimetry on the APB exam any more. Did I
mention my years of research using high pressure, high temperature flow calorimeters?
Okay, I will stop whining and get over it.
11. Okay now back to conduction. There is an equation for the rate of energy transfer through a
slab of area A and length L due to conduction. Write it down from one of your resources.
Compare it to the one given on your Table of Equations for AP Physics Exams. Know what
the variables are and be prepared to use the equation.
12. That is it for our first pass through thermal physics. Please write down the most sensational
perceptions that you have gained from this lesson.
Physics B, Unit 5, Lesson 4, Thermodynamics (B54Thermo)
Thermodynamics – Be able to interpret and make calculations related to the following topics:
thermodynamic work and PV diagrams, the laws of thermodynamics, and heat engines.
Today’s lesson is about thermodynamics specifically PV diagrams and PV work, the laws of
thermodynamics and heat engines. For today’s assignment, please label each page with the code
B54Thermo, the date and the page number.
1. Check out references on today’s topics on the internet or in your textbook and write down
how you can find these references again.
2. Okay in the previous assignment I had you define the terms isothermal, isobaric, isochoric
and adiabatic. Make sure you remember these definitions. You can often visualize the topics
below in terms of an ideal gas being compressed in a syringe and then you can relate this to
engines. So having said these very exciting things, lets consider the definition of work W =
-PΔV. This is an equation for the work done on a gas. What was that underline for? You
will soon see (evil cackling wicked witch type laughter is heard in the background) my
pretties. So let’s visualize that syringe. Let’s consider taking the syringe full of ideal gas
and immersing it in a cold bath while leaving the plunger of the syringe free to move. We
can understand from the ideal gas law as the gas cools the volume will decrease. Leaving the
plunger free keeps the pressure at atmospheric pressure and constant. The change in volume
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will have a negative sign, so the work done on the gas will be positive. If the gas expands the
change in volume will be positive and the work done on the gas will be positive. Please
realize that there is work done on the gas by the environment and work done by the gas on
the environment. These are equal in magnitude, but opposite in sign and you must,
absolutely must keep the sign convention straight in your head. The equation W = -PΔV is
for the work done on a gas! Find a picture or mnemonic device or something to help you
keep this straight. For now, just write down a couple of phrases that will summarize the
signs of W for this very long paragraph. Also please note that if there is no change in volume
there ain’t no work done on the gas no how. Remember I only used bad grammar to make
really important points! How does this underlined quadruple negative sentence relate to
those vocabulary words at the beginning of this paragraph?
3. Let’s see if you actually understand what you just wrote down. I have a syringe of ideal gas
at 1 E 5 Pa. I add heat to the syringe and its volume changes from 1 m3 to 2 m3 (okay, okay,
it is a really, really, really big syringe…golly gee it is just an example). What is the work
done on the gas? What is the work done by the gas?
4. State the first law of thermodynamics in words only the fewest number of words possible.
5. Write an equation for the first law of thermodynamics found in a physics resource. If your
equation has a minus (-) sign in it, change it to a plus. Your equation should now match the
equation found on your AP equation sheet. Other resources have different ideas about the
sign convention and this has been discussed and debated by scientists, but I suggest that you
stick with the currently accepted system. I am not going to debate the merits of sign
conventions. Please also look at the very front page of your Table of Equations for AP
Physics Exams, the one with all the constants on it. At the bottom of the page there are a
couple of asterisks. Look at the asterisk next to IV. Read the sentence next to it. This tells
you that the current accepted sign convention is that W represents the work done on the gas.
Back to the syringe, this means if V gets smaller that ΔV is negative and W (W = -PΔV)
done on the gas is positive and W done by the gas is negative. If V gets larger, the ΔV is
positive and W (W = -PΔV) done on the gas is negative and W done by the gas is positive. I
must think all this sign convention stuff is important if I keep repeating it over and over
again, huh?
6. Now what is this Q in the equation form of the first law of thermodynamics? Write down
what it is. Please remember that Q is a type of energy so that you never forget that Q and W
have the same units.
7. What is the U in the equation form of the first law of thermodynamics? It is not potential
energy like that of a spring or gravity. What are the units of U?
8. Remember the end of the previous homework assignment where we discussed kinetic
molecular theory and Kavg and the fact that Pressure multiplied by Volume gives units of
energy. I am hoping that you can see by those equations that K = f(T) and PV = f(T) and that
energy of a gas is a function of temperature which leads me to my second quadruple negative
of this assignment. If there is no change in temperature there ain’t no change in the internal
energy of the gas no how. How does this relate to those vocabulary words I reminded you of
in question 2 of this assignment?
9. These problems typically are associated with PV diagrams. Let’s contemplate what certain
types of processes look like on this diagram. What is the shape of the line/curve that
represents an isobaric process on a PV diagram?
10. What is the shape of the line/curve that represents an isochoric process on a PV diagram?
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11. What is the shape of the line/curve that represents an isothermal process on a PV diagram?
What is the shape for an adiabatic process? If an isothermal line/curve and an adiabatic
line/curve were drawn on a PV diagram starting from a common point, how could you tell
one from the other?
12. If you have triangular shaped process (or indeed any closed process of any shape) drawn on a
PV diagram, it starts and ends at the same pressure and volume. What can you tell me about
the beginning internal energy and the ending internal energy of the gas?
13. How would you find the work for an isobaric process on a PV diagram?
14. How would you find the work for a triangular shaped process on a PV diagram?
15. Okay, please find a PV diagram for a Carnot cycle either in your textbook or on the internet.
Make a quick sketch of it. Mark on your sketch which parts of the 4 step process are
isothermal. Mark on your sketch which parts of the 4 step process are adiabatic. Put an
arrow to show where in the process heat is added and another arrow to show where in the
process heat is removed.
16. Write down the equation for the efficiency of a Carnot engine. Make sure it matches the one
on your AP equation sheet. If it doesn’t use the one on your equation sheet or manipulate the
equation you have until it matches. You will use the one on your equation sheet the most so it
is easiest to work with this equation all the time. No I am not just being irritating by not just
telling you to write down the one from your equation sheet in the first place. You have to be
able to see where these equations actually come from! What is TH? What is TC? What do
the units of these two quantities absolutely have to be in?
17. You can see my all time favorite picture to describe overall efficiency of a heat engine on the
below. The Carnot efficiency of a heat engine is considered the “ideal efficiency”. There is
another equation for efficiency on your equation sheet. Write it down. Yes, these two
equations are often set equal to each other so be prepared for that. Tell me in a few words
each what QH, QC and W are. Using the very attractive picture below that I drew all by
myself with the help of Microsoft word, write a conservation of energy equation for QH, QC
and W. Write in a sentence or two how the regular old efficiency equation (not the Carnot
one) relates to grades on homework or tests.
QH
W
QC
Figure 3. Schematic of energy input and output for a heat engine.
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18. Okay that is it for our second pass of thermodynamics. I know this is a lot of material. It
will become clearer as you practice what you learn. Good luck! Now please write down the
two most fantastic understandings you took from this lesson.
Physics C Unit 5, Lesson 1, Gauss’s Law for Spheres (C51Gauss)
Gauss’s Law for Spheres - Comprehend and be able to analyze Gauss’s law for spherical
distributions including conductors, nonconductors of uniform charge density and nonconductors
with densities as a function of r.
This most exciting day of days has arrived. Today’s lesson is about Gauss’s Law. Today you
will be learning about spherical Gaussian surfaces. There actually is some nice calculus
associated with Gauss’s law, but it probably won’t show up where you think it will. Please label
each page of your assignment with the code C51Gauss, the date and the page number.
1. Find references for Gauss’s law in your textbook and on the internet and write them down. It
is worth it to do some looking on the internet. These problems are classic problems and
getting information about them from more than one independent source will help you
understand this material better. An internet site that I think has pretty comprehensive
coverage of Gauss’s law is Hyperphysics. This is their link for a basic statement of Gauss’s
law, http://hyperphysics.phy-astr.gsu.edu/hbase/electric/gaulaw.html#c1 and here is a link
for some of the types of spherical distributions you will be looking at,
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elesph.html#c3.
2. In one of your references find an integral version of Gauss’s law. Write it down. Compare it
to the equation for Gauss’s law on your Table of Equations for AP Physics Exams. Let’s
make sure you understand the basics of this equation. So what does the circle on the integral
sign mean?
3. Which charge does Q refer to?
4. Name epsilon naught in words and give its value.
5. Find the symbol ФE somewhere in one of your resources. It is not on your equation sheet,
but it relates to Gauss’s law in a very significant way. Describe the meaning of this symbol
in a phrase or so.
6. There are a couple of common Gaussian surfaces. One of these is a closed cylinder. This
can be a short and fat cylinder (tuna fish can) that can be used for a continuous sheet of
charge. The cylinder can also be long and skinny (tennis ball can) that can be used for a long
line of charge. The most common Gaussian surface you will encounter is a sphere. You will
practice with this surface extensively, but you will need to know the cylindrical ones also and
I will assign those in the future.
7. What is the surface area of a sphere with a radius of r?
8. Now let’s look at the integral side of the equation of Gauss’s law. You might think you will
get to do a lot of calculus with this lovely integral sign, however, when you integrate E·dA
over the closed surface of a sphere, the limits of integration are essentially zero and the area
of the sphere, 4πr2, and the closed integral of E·dA turns out to be very simple for spheres is
always E(4πr2). So knowing what the integral side of the equation of Gauss’s law equals,
find an equation for the electric field a distance r from a point charge using a spherical
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Gaussian surface and knowing that our k of 9E9 is actually equal to 1 divided by the quantity
4 times pi times epsilon naught. Have you ever seen this equation before?
9. Let’s consider a metallic (AKA conducting) sphere, either a solid sphere or a hollow
spherical shell of a given radius with a charge Q. The answer is the same for both the solid
sphere and the hollow shell. Find out what the electric field is inside this shell a distance r
from the center of the shell, but still with r being less than the actual radius of the shell.
Please also recall that for conducting surfaces all the charge resides on the outside of the shell
spread as far apart as possible since like charges repel. What is Qenclosed? So then what is the
electric field inside a conductor?
10. Now a conducting sphere (solid or hollow – they give the same result) of a given radius and
charge Q. Using Gauss’s law, derive an equation for the electric field a distance r from the
center of the sphere, but this time r is larger than the actual radius of the sphere. Remember
that it doesn’t matter if you already know what the answer is; you still must show the work.
Have you seen this equation before? Doesn’t it look just like Coulomb’s law for a point
charge? And it doesn’t matter what the radius of the conducting sphere is, as long as the
equation is for the electric field outside the conducting sphere. As a matter of fact it reminds
me of when we find the acceleration due to gravity using the law of universal gravitation.
The distance used is r even though all the mass is not a point at the center of the earth.
11. So you can now handle showing that the electric field inside a conductor is zero and you can
derive Coulomb’s law using Gauss’s law for a distance r from a point charge or the center of
a conducting sphere. These have all actually been a part of AP exam in the past. Now let’s
look at a nonconducting (plastic) sphere of radius R with a uniform charge distribution and
total charge Q. Typically you are asked to find the electric field inside the sphere and outside
the sphere. Or in other words E for r<R and for r>R. These are the typical variables used for
the constant radius of the sphere, R, and the variable distance, r, at the point where you want
the value of the electric field. Once more with feeling: big R is a constant and little r is a
variable. Let’s consider the electric field inside the sphere. Imagine a Gaussian sphere inside
the plastic sphere. So now evaluating the integral side of the Gauss’s law equation you
should get your old friend E(4πr2). Remember this is the surface area of the Gaussian sphere
and this has the variable radius r. Now we have to figure out Qenclosed inside the sphere. We
can use a simple volumetric ratio. The total volume of the plastic sphere is (4/3)πR3. The
volume of our Gaussian sphere is (4/3)πr3. The total charged enclosed is equal to the volume
of the Gaussian sphere (4/3)πr3 divided by the total volume of the plastic sphere (4/3)πR3
times the total charge Q. Substitute this into Gauss’s law and solve for E, the electric field
inside the sphere. You should be able to check this from an example in your textbook or on
the internet. Also please note that while the electric field inside a conductor is zero, the
electric field inside a nonconductor is not zero. Go on to the next number to find the electric
field outside a nonconductor.
12. Well, if you are considering a nonconductor that has a total charge Q and our Gaussian
surface is a sphere of radius r that totally encloses the sphere (remember r > R), then Qenc is
Q! And what does the electric field equation look like? Feel free to use either Coulomb’s
law or Gauss’s law. I am pretty darn sure you have seen this equation once or twice or a
million times before.
13. Okay let’s get down to the exciting stuff now. Remember I told you there is some very nice
calculus involved in using Gauss’s law. Have you seen any yet? I agree with you. We
haven’t yet. So let’s see it now. I would like you to consider a nonconducting sphere of
40
nonuniform charge density. Let’s say that the charges varies radially outward with a
volumetric charge density of ρ = Br where r is a variable radial distance from the center of
our sphere and B is a constant. Now we have to find Qenc in this sphere and this is where you
get to do the nice calculus. Remember while talking about moment of inertia, I told you that
you would be using the symbols ρ, σ and λ for density again when we talked about Gauss’s
law? Well here is ρ again. Density is mass divided by volume, right? It is the same for
charge density so in this case ρ = charge divided by volume or ρ = Q/V or looking at a
differential form of this equation and rearranging a little bit, dq = ρ dV or a small amount of
charge equals charge density times a small amount of volume. However, you need
everything in terms of r which is actually what you will be integrating. So what is dV in
terms of r? Well, V = (4/3) πr3 and taking the derivative with respect to r, dV = (4/3) πr2
(1/3) dr or 4πr2 dr. Hey that’s just the area of a sphere times dr! You are, of course, totally
correct. So substituting our function for dV and our function for ρ into dq = ρ dV, you
should get dq = (Br) (4πr2) dr. Now you can integrate this to find Qenc both inside the sphere
and for the entire sphere. If you need to find the electric field and therefore the charge inside
the sphere your limits of integration are from 0 to some variable distance r. If you need to
find the electric field outside the sphere, then you need to find the total charge on the sphere
and your limits should be 0 to the radius of the sphere R. Knowing all this very exciting
information, find equations for the electric field both inside (r<R) and outside (r>R) of this
sphere. Then just to prove that it makes sense, substitute R in for r in each of the equations to
get the electric field at the limiting condition of the actual radius of the sphere and you
should get the exact same equation! Wow!
14. After all this discussion of Gaussian surfaces, let me ask you a very classic question. A
charge +Q is enclosed in a Gaussian cube, what is the electrical flux through the surface area
of the cube? Remember that flux is equal to the integral, but it is also equal to the other side
of the Gauss’s law equation too. This is an easy question to answer. Notice that I didn’t ask
for electric field.
15. I know our discussion of Gauss’s law was very, very fun. All that needs to be done to make
a new problem is to make the charge density some new function, perhaps ρ = Ar2 or ρ = A
cos (Br). Wouldn’t those problems be great? Okay, I think that is definitely enough for now.
Isn’t Gauss’s law fun? Write down the two most striking things you have learned in this
lesson today.
Physics C Unit 5, Lesson 2, Continuous Charge Distributions (C52Continuous)
Continuous Charge Distributions - Be able to determine and recognize appropriate problem
solving techniques for finding the electric field and the potential from continuous charge
distributions particularly rings or arcs of charge.
Good morning. Another day, another assignment. Today you will be learning about finding
electric potential (V measured in volts) and electric field for some other continuous charge
distributions. No Gauss’s law today, but you will be using some of the same skills you used in
the previous assignment. And there is a little of calculus today, so this is exciting! Please label
each page this assignment C52Continuous along with the date and the page number.
1. You are going to be looking at finding electric field and electric potential from continuous
41
2.
3.
4.
5.
6.
7.
8.
charge distributions. Typical possibilities include rings of charge or lines of charge. Find
references on these types of problems either in your textbook or on the internet. Usually
electric field and electric potential are found in different chapters, but the concepts are
similar and you are often asked to solve for both of these in the same problem. So I am
combining them! Write down the references.
You will also be considering calculus forms of equations you have already worked with. I
am going to write the calculus form and I want you to write down an analogous equation we
have seen before that is almost the same and tell me when you use this previous equation you
have seen. (a) E = k ∫ dq /r2; (b) V = k ∫dq/r; (c) E = -(dV/dr) and ΔV = -∫E dr. Notice that
these equations are not on your equation sheet, but they are differential forms of long used
equations and you will be expected to use them.
Remember again I told you those symbols for types of density would come up again. You
saw ρ yesterday. Today we have λ. Let me just remind you of what those symbols are
because σ is coming up again soon. ●Volumetric density is given the Greek letter lower case
rho, ρ. ●Surface area density is given the Greek letter lower case sigma, σ. ●Linear density
is given the Greek letter lower case lambda, λ. Sorry there is no answer to this number. It is
purely informational. I know you are devastated. Try to hang on.
The most common type of problem to consider is a ring of charge with a linear charge
density λ and a radius R. Let’s find the total charge of the ring. Let’s consider a small arc of
the ring ds, the length of the arc (using rotational variables) is r dθ, so dq = λ r dθ and
integrating Q = ∫ λ r dθ with the limits of integration being from 0 to 2π. Notice that r
doesn’t vary with θ. Show the work to prove Q = 2πRλ. Or we could have skipped all that
calculus stuff and said the total charge is just λ times length. The length of the ring is 2πR
(circumference of a circle) and so Q = 2πRλ.
Now let’s find the electric field at the center of the ring. REMEMBER that electric field is a
vector and since it is a complete ring every tiny arc of the circle ds has another tiny arc of
circle that is exactly opposite it and it doesn’t matter if the ring is positively charged or
negatively charged, every single small piece if electric field is cancelled out by a piece of
equal magnitude and opposite direction on the other side of the ring. So what is the electric
field at the center of the ring? I know it is an easy question.
Now to find the electric potential at the center of the ring. We have that equation V = ∫dq/r
and from number 4) we know that dq = λ r dθ so V =k ∫ λ (r dθ) / r using the limits from 0 to
2π, then what does V equal? You didn’t really need calculus for this either! DO NOT fall
into the trap of thinking well, E = kQ/r2 or E = k ∫ dq /r2and all the charge is an equal distance
from the center and the total charge of the ring is 2πRλ and telling me the electric field is
2πRλ/R2 or some such nonsense. You must keep in mind the vector summative nature of E
and the scalar summative nature of V. Those things haven’t changed since we first discussed
them for point charges and this is another very nice way to see if you understand vector and
scalar quantities for electricity.
Now doing 4, 5, & 6 might have been the beginning to a very nice free response problem.
What happens next? Let’s consider not a full ring of charge, but an arc of half a ring and
let’s orient it so that it looks like a bowl sitting on a table. Make a sketch of this arc. You
will definitely need it soon. The arc still has a radius of R and a charge density λ and it is
positively charged. What is the total charge of this arc?
What is the electric potential of this arc at a point that would be the center of the entire ring,
if there were an entire ring? Notice that when there was a whole ring I asked for E first and it
42
was definitely easier since it didn’t require any math really, only a knowledge of the nature
of vectors. Now I am asking for electric potential first. Hmmm.
9. Now let’s find the electric field of this arc at a point that would be the center of the entire
ring if there were an entire ring. Let’s consider the two top edges of the bowl. Mark off a
tiny part of the ring at the two top edges and let’s make two small arrows that show the
direction of the electric field and the center of the arc. One goes left and the other goes right
and they cancel out so this is going to be easy…Wait a minute, not so fast. Let’s now
consider two small pieces of the arc each about half way down the bowl. Mark these two
pieces at a symmetrical place on both sides. Now draw arrows for the direction of the
electric field at the center of the arc. The arrows are equal in magnitude and opposite in the x
direction, but not in the y direction. What is the direction of the electric field at the center for
this ENTIRE positively charged arc? Why? Now you will need to sum up (integrate) the
electric field due to all those small arcs. This is not as easy as the previous considerations.
10. Now let’s actually solve for the electric field. Consider one of the two small arcs at the top
of the bowl. The charge of that arc is dq = λ r dθ. You also know that E = k ∫ dq /r2 so in this
case E = k ∫λ r dθ /r2 and we need to consider only the y components of the field because all
the x components cancel out. So E = k ∫ (λ dθ /r) (sin θ). Since λ and r are both constants,
they can come outside of the integral and E = (kλ/r) ∫sin θ dθ with the limits of integration
being from π to 2π. Solve for the electric field at the center of the arc.
11. Try finding the electric field at the origin for a continuous line of charge of length L and a
distance A away from the origin. The line of charge has a continuous charge density of λ
also.
12. I think that is it for continuous charge distributions. Be sure to bring any questions you have
to the next time we meet. Please write down the two most remarkable ideas from this lesson.
Physics C Unit 5, Lesson 3, RC Circuits (C53RC)
RC Circuits - Grasp the numerical and theoretical concepts associated with RC circuits.
Please label each page of this assignment with the code C53RC, the date and the page number.
1. Find some resources either in your textbook or online dealing with circuits containing
resistors and capacitors, also called RC circuits. Write down these references.
2. The first thing I want to do is remind you, believe it or not, of something we learned in
mechanics when we talked about air resistance. I showed you these types of equations that
represented (1) exponential increase up to an asymptote and (2) exponential decrease. The
equations can be represented by these general forms (1) K = K0 (1 – e(-t/τ)) for exponential
increase up to an asymptote and (2) K = K0 e(-t/τ) for exponential decrease. You saw the
equation for velocity of an object that experienced air resistance where the resistive force was
equal to some constant times velocity, FAR = bv. Go back and remind yourself of that
concept and write down for me the equation for the velocity of that object that took the form
of one of the two equations above.
3. What is τ for the equation you just wrote?
4. Now let’s consider a resistor of resistance R connected in series with a capacitor of
capacitance C and a battery with an EMF or voltage of ε. Please draw this circuit and label
each piece with its letter and throw in a switch too!
43
5. Please recall what I kept saying about capacitors when we talked about circuits before. At
the instant the switch is closed (t=0) the capacitor acts like a ________ and after a long time
(as t → ∞) the capacitor acts like a_________ which is exactly what it looks like. Please fill
in the blanks. You should already know the calculations for these limiting conditions.
6. Now you want to consider the time in between t = 0 and t → ∞. During this time the current
of the capacitor, the voltage over the capacitor, the charge of the capacitor and the voltage
over the resistor all change from min to max values or vice versa. Write down Ic, Vc, Qc, and
VR and tell me whether they are a min or max at both t = 0 and t → ∞.
7. In order to do these problems you will be using Kirchoff’s loop rule, but you need to be
reminded of or find out some basics first. What is the differential equation to find current in
terms of Q and t?
8. Write down the equation for the voltage drop over a capacitor. You already know this
equation from our previous discussion of capacitors.
9. Now go around the circuit starting at the battery and show that the Kirchoff’s loop rule for
this simple RC circuit is ε – IR – Q/C = 0.
10. However, current I is also equal to dQ/dt. Substitute this into the previous equation. Divide
the entire equation by R and rearrange it so that you get dQ/dt by itself on the left hand side
of the equation and show that you get dQ/dt = (ε/R) – (Q/RC).
11. Then get a common denominator for the equation of RC and write down the equation you get
now.
12. Rearrange the equation again to get all the terms involving Q on the left and everything else
on the right and show that you get dQ/(Q – Cε) = (-1/RC) dt. Integrate this equation with the
limits from 0 to Q on the left and 0 to t on the right. Then get rid of the logarithm. Show that
you get the form Q = Cε (1 – e (-t/RC)) or Q = Qmax (1 – e (-t/RC)) since the capacitance times the
voltage (Cε ) is the maximum charge on the capacitor. Sketch a graph of Q versus t.
13. Now differentiate the equation you just found with respect to t and show that you get I =
(ε/R) e (-t/RC) or I = Imax e (-t/RC). Sketch a graph of I versus t.
14. Now consider the limiting conditions of t = 0 and t → ∞ and evaluate the last two equations
in terms of these limiting conditions. Do you get the values that you expect for Q and I?
15. Also consider VR and VC in terms of the equations you just got. Figure out analogous
equations for VR and VC using reasoning and sketch graphs of both VR and VC versus time.
16. What is the time constant τ for this equation?
17. Knowing that you cannot exponentiate a number with units, 1 Farad times 1 Ohm equals
what unit?
18. Those are the basics that you need to know for charging a capacitor. Now let’s consider
discharging the same capacitor, C, using the same resistor, R. Consider that the switch is
closed on the preceding circuit for a good long time and the capacitor completely charges and
current stops flowing in the circuit. What is the energy stored in the capacitor?
19. Then you are able to magically open the switch and remove the battery from the circuit
without discharging the capacitor and now you have a circuit with just a fully charged
capacitor and a resistor and an open switch. Draw this circuit.
20. What will happen when you close the switch now? To the current through the circuit? To
the charge of the capacitor? To the voltage over the capacitor? To the voltage over the
resistor? Write a phrase or two to answer each of these questions.
21. Write the Kirchoff’s loop rule for the voltage drops around this new circuit.
22. Substitute in dQ/dt for I and rearrange the equation getting all the Q terms on the left and
44
everything else on the right. Integrate both sides of the equation and get an equation for Q in
terms of t for a discharging capacitor. Is it what you expect?
23. Differentiate the equation that you got in the step before this to get current I as a function of
t. Is it what you expect?
24. Make quick graphs of QC, IC, VR and VC versus time.
25. What happened to the energy stored in the capacitor? Did it disappear?
26. How would the time constant change if you had two resistors in series with the capacitor? If
there were two resistors in parallel with each other and then in series with the capacitor? If
there were two capacitors in series with 1 resistor? Two capacitors in parallel with each
other and then in series with 1 resistor? What values should you use for C and R for these
situations?
27. You should now understand the basics of RC circuits. Please write down the two most
notable impressions from this lesson.
Physics C Unit 5, Lesson 4, Electric Potential from Electric Field (C54Potential)
Electric Potential from Electric Field - Using the integral relating electric field to potential and
the derivative relating potential to electric field, be able to make determinations both
mathematically and conceptually.
Today we will be discussing the relationship between electric potential and electric field. You
will use some of the equations I gave you in the previous assignment and you will look at the
same types of charged spheres that were discussed in the first Gauss’s Law assignment. Please
label this assignment C53Potential along with the date and page number.
1. You will need to find references about how electric potential and electric field are related
either in your textbook or on the internet. If you are looking on the internet, the best
resources I have found are college websites. Write down your references.
2. Let’s recall an equation I gave yesterday, ΔV = -∫E dr. I didn’t really use it, but it went along
with the other equations. I hope that you see that it is related to an equation we have for the
electric field inside a capacitor, E = -V/d. Doing a little rearranging E = -ΔV/Δd, -ΔV = E
Δd, and replacing d with r and moving the negative sign I get ΔV = -E Δr. This last one
looks more like the integral equation that I gave. Now recall that potential is not generally an
absolute value, it is a difference between two levels like gravitational potential energy.
Perhaps it is measured between 120 V and ground or it can be measured between any two
other points. Perhaps it also helps to visualize measuring voltage in a circuit where you have
to measure the voltage over (or between) two points. In more detail the equation ΔV = -∫E dr
is given as VB – VA = -∫E dr where the upper limit of integration is rB and the lower limit is rA
and rB and rA are only the radial distances away. Let’s look at this for a simple situation for a
point charge and substitute in the equation for E for an electric field VB – VA = -∫(k q/r2) dr.
Using the limits of integration rB and rA as being the radial distance from the point charge,
1 1
show that you get VB – VA = kq [  ].
rB rA
3. Now it is the usual convention to define the potential a long way away as zero. In other
words define rA = ∞ and VA= 0 and then simplify the above equation and take off the
subscripts. Write down what you get and write down where you have seen it before. Please
45
note that the equation you got for this homework question is the potential for a point charge.
If you are asked to find the potential difference between two different points in an electric
field, you will get something more like your answer in number 2.
4. Now let’s find the potential inside and outside a conducting sphere. Gee it seems like I am
asking all the questions I did for Gauss’s Law for spheres…and I am! And now I am going
to tell you the answer (I have lost it, gone totally round the bend giving out answers like this).
The potential inside a charged conducting sphere of total charge Q and radius R is kQ/R the
same as at a point on the surface of the sphere. The potential does not change inside a
conducting sphere, it is constant, always kQ/R. There are nice written explanations of why
this is in references such as your textbook and on the internet. Feel free to read about what
it says in one of my favorite internet references, HyperPhysics: http://hyperphysics.phyastr.gsu.edu/hbase/electric/potsph.html. Let me see if I can present you with a picture in your
mind to hold this there and then you will go through an example. Let’s find the electric
potential at the center of a charged sphere of radius R and total charge Q. Recall the previous
assignment about ring of charge. When we found the potential at the center of the ring, it
was equal to the total charge of the ring divided by the radius multiplied by k, of course.
This was because all the charge was equally distant from the center of the ring. This is the
same with the conducting sphere. Recall that for conducting spheres all the charge resides on
the outside of the sphere distributed as evenly as possible so that the individual charges can
get as far away from each other as possible because they are repulsed! So now I hope it is
okay with you that the potential at the center of a conducting sphere is kQ/R. But let’s
contemplate a point say halfway between the center and the surface. Now all the charge is
not equally distant. That is true, but I would like you to recall two things. The electric field
inside a conductor in equilibrium has to be zero and E = -dV/dr. So if the electric field is
zero inside a conductor and electric field is equal to the rate of change of electric potential,
then the electric potential inside the conductor cannot change. Consider my explanation,
read your book’s explanation and check the internet too, but let’s go on to a mathematical
example. I want to get the potential of a conducting sphere of total charge Q and radius R at
its surface and also its center. I am going to use the convention that potential is zero at r = ∞.

For the potential at the surface V∞ - VR = -  Edr and E outside the sphere equals kQ/r2 so V∞
R

Qdr
. Evaluate this integral and write down your answer. Don’t forget the
2
R r
limiting condition about r and V at infinity.
5. I hope you just wrote down, taking care of the multiple changes of sign, the answer that VR =
kQ/R, just the same thing we get for a point charge. Don’t just hand wave at the sign
changes. You will have to do problems that are more complex than this and if you can’t
figure out the sign changes now, you certainly won’t be able to then and it will matter more.
Now let’s consider the math showing us the potential inside the conducting sphere. A
method of solving these problems that I have used over and over is to start at infinity and
move in towards the point on the sphere that I want to find. And yes I do imagine myself in a
tiny spaceship traveling in from infinity with wonderful goggles on that allow me to view
how the potential changes as I get closer to and then inside the conducting sphere. The
equation is constantly flashed up on the viewing screen in my spaceship! Okay now
- VR = -k 
46

R
R
0
considering this situation, in general I would have V∞ - VR = -  Edr and VR – V0 = -  Edr .
If I add the left sides of both equations (V∞ - VR) + (VR – V0), I get V∞ - V0. And then if I

R
R
0
add the two right hand sides of the equation I get -  Edr + (-  Edr ) and I just evaluated the
first integral like 2 minutes ago and the electric field inside the conductor is 0 so the second
integral is just zero. So solve all this and show me that at for V0 you get kQ/R!
6. That was a lot of work just to prove what the potential inside a conducting sphere is just
kQ/R. However, the process will help us as we go along. Let’s now consider a
nonconducting sphere of radius R with a total charge Q and a uniform charge distribution.
We can consider the potential outside the sphere, but I hope that you have seen so many
situations where that reduces down to just what it is for a point charge that you will see that
the answer is slightly boring, V = kQ/r. Remember that big R is the constant radius and little
r is the variable radial distance! You will get asked this easy question on the AP exam, but
they know it is easy and it will only be worth 1 or 2 points. Just say it acts like a point charge
and write down the equation. The interesting question is to find the electric potential inside
the sphere! On number 11 of the C51Gauss assignment you should have found that the
equation for the electric field inside the sphere was E = kQr/R3. So lets find V for some
arbitrary distance r from the center of the sphere and of course r<R. Now it is convenient for
r
us to use the potential between the surface and our arbitrary point Vr - VR = -  Edr .
R
Substituting in with the correct electric field equation and taking some quick liberties with
r
kQ
showing steps I get Vr - VR = - 3  rdr . Show that by evaluating the integral you get Vr R R
kQ
kQ 2 2
VR = - 3 (r2 – R2) or Vr - VR =
(R – r ) . Now don’t forget that VR= kQ/R.
2R
2R 3
kQ
r2
(3  2 ) which is
Substitute this in and do the algebraic manipulation to show that Vr =
2R
R
the typical form found in most resources.
7. Now as if I haven’t already put you through enough pain, let’s go for one more. Recall in the
Gauss’s law assignment, I asked you to find the electric field for a nonconducting sphere of
nonuniform charge with a charge density ρ = Br and a radius R. I asked you to find the
electric field both inside and outside the sphere. This was number 13 on that assignment.
Here are some intermediate calculations I made and my final answers. Qenc for r<R is
r
 Br 4r
2
dr and equals πBr4. Evaluating the exact same integral with the different limits
0
from 0 to R (the actual radius of the sphere) I got Qenc for r>R (outside the sphere) to be
πBR4. This led me to find that E inside of the sphere was Br2/(4є0). E outside the sphere is
just good old kQ/r2. Substituting in E = kπBR4/r2 or if I want to get rid of k then E =
BR4/(4є0r2). Now we went through all that just so that we can look at the potential of the
sphere. What is the potential at r > R? You can just treat it as a point charge and V = kQ/r
using the entire charge of the sphere πBR4 and V = kπBR4/r. What is the potential at r=R? It
47
is also just kQ/r using the r = R so V = kπBR4/R = kπBR3. Again, these questions about
outside the sphere will be asked, but their solutions are rather simple. Again the difficult
question is to find V inside the sphere. However, it is analogous to what was done for the
r
nonconducting sphere of uniform charge. Again Vr - VR = -  Edr . Substituting in the
R
r
correct electric field equation (E = Br /(4є0) inside the sphere) I get Vr - VR = -B/(4є0)  r 2 dr .
2
R
Solve this for Vr.
8. Change the limits of integration on the last problem and show the potential at the center of
the sphere, V0, equals BR3/(3є0).
9. Now just to drive you absolutely crazy, there is another way to do this. Let’s solve for the
potential at the center of the sphere since it reduces to such a cozy fraction. Do you recall the
equation V = k ∫dq/r? If not, this is a good time to recall it. It is found in the lesson
R
dq
C52Continuous in number 2. Substituting = k  . We have an equation for Q inside the
r
0
4
sphere as πBr . I can either differentiate q in terms with respect to r and get dq = 4πBr3dr
which is exactly the same as dq = ρ dV or (Br) 4πr2 dr. I can substitute this equation into the
R
4Br 3 dr
integral and get V = k 
. Please simplify and evaluate this equation for the
r
0
potential at the center of the sphere.
10. I think that is enough for finding electric potential for now. There will be more excitement
soon! Please write down the two most sensational concepts you learned from this lesson.
Physics C Unit 5, Lesson 5, Other Field, Potential and Capacitance Problems (C55Other)
Other Field, Potential and Capacitance Problems - Gain insight and awareness of other
Gaussian surfaces, more difficult continuous charge distributions, and capacitance for geometries
different from rectangular parallel plates.
Now let’s just clear up some loose ends about Gauss’s law and some other ideas related to
symmetry of spheres. Label each page of your homework with the code C54Other, the date and
the page number.
1. Let’s start with references. Today we will be specifically discussing the electric field due to
an infinite line of charge of uniform density, the electric field due to an infinite
nonconducting sheet of charge, finding electric field from potential, finding the electric field
and potential at a distance x away from the center of a ring of charge along the x axis going
through its center and finding the capacitance of two conducting plates that are spherical
shells. These are typical examples found in textbooks and on the internet. It is worth your
while to find an example and follow along with these. Please find references for the types of
problems I have listed and write those references down.
2. Derive an equation for the electric field a distance r away from an infinite line of charge of
uniform charge density λ. This requires the use of a symmetrical cylindrical Gaussian
surface. This is usually a textbook example and can be found on the internet if you need
48
some help. There are a few questions below related to this problem.
3. Considering problem number 2, what is the direction of the area vector?
4. In the Gauss’s law equation,  E  dA = Q/є0, what does the symbol,  , between E and dA
mean?
5. Why don’t the end caps of the Gaussian cylinder in number 2 affect the answer?
6. Derive an equation for the electric field a distance r away from an infinite nonconducting
sheet with a uniform surface area charge density σ. This also requires the use of a cylindrical
Gaussian surface.
7. In number 6, only the end caps of the Gaussian surface matter and the outer cylindrical shell
doesn’t. Why is this so?
8. Consider a ring of charge of radius R with a total charge +Q and uniform linear charge
density. Derive an equation for the electric field a distance x away from the center of the
ring on the x-axis that runs through the center of the ring. Again this is a common textbook
example or internet example.
9. Consider a ring of charge of radius R with a total charge +Q and uniform linear charge
density. Derive an equation for the electric potential a distance x away from the center of the
ring on the x-axis that runs through the center of the ring. You should be able to find this
example in your textbook or on the internet.
10. In the last lesson, you considered multiple problems in which you found the electric potential
using an equation for electric field. Now you need to consider how a problem might progress
if you are given the potential and need to find the electric field. Let’s consider a conducting
sphere of radius R that is charged up to a potential V. Determine the charge Q on the shell in
terms of R and V.
11. Let’s consider that conducting sphere of radius R and potential V from the previous problem.
Two halves of a conducting spherical shell that have a radius larger than the inner conducting
sphere are carefully placed around the inner conducting sphere without touching it or
disturbing it. The charge on the outer shell is zero, but there is definite movement of charge.
Assuming the charge on the inner sphere is positive, sketch a picture showing the charge
distribution on the inner sphere and the outer shell.
12. If the outer shell is then connected to ground, what will happen? Again sketch a picture
showing the charge distribution on the inner sphere and the outer shell.
13. These two metal spheres actually form a capacitor! There are two metal surfaces a given
distance apart and both have a net charge on them, equal in magnitude and opposite in sign.
How do you find the capacitance of this kind of capacitor? First you will find the potential
between the outer shell and the inner sphere, then we can use the equation Q = C ΔV. You
will go over this in a practice problem.
14. That’s it for other capacitance, electric field and potential problems. Write down the two
most significant perceptions from this lesson, please.
49
UNIT 7, PHYSICS B NUCLEAR AND QUANTUM PHYSICS AND PHYSICS C
MAGNETISM IN DEPTH
Table 8. Unit 7 Overview, Physics B Nuclear and Quantum Physics and Physics C Magnetism
in Depth
Physics B Nuclear and Quantum Physics
1) Nuclear Physics and Radioactivity –
Understand the three types of radioactive
decay and the byproducts of each, be able
to balance nuclear equations and be able to
use the most famous physics equation
E=mc2 making calculations in both Joules
(J) and electron volts (eV).
2) Quantum Physics and Wave Particle
Duality – Gain an appreciation and
comprehension of the photoelectric effect,
de Broglie wavelength, Heisenberg
Uncertainty Principle and wave particle
duality.
3) Bohr and Energy Level Diagrams –
Achieve appropriate comprehension of the
Bohr model of the atom and how this
relates to energy level diagrams,
particularly understanding how to calculate
energy, frequency and wavelength of
emitted photons.
Physics C Magnetism in Depth
1) Ampere’s Law – Gain an awareness of
how Ampere’s law is used to calculate
magnetic field and be able to apply this to
both short answer questions and problems.
2) LR Circuits – Understand the ideas and
calculations associated with LR circuits.
3) Other Magnetism Problems – Consider
and evaluate additional calculus-based
magnetism problems such as the current
induced in a rectangular loop of wire a
distance away from a wire that carries
changing current and therefore a changing
magnetic field.
Physics B Unit 7, Lesson 1, Nuclear Physics and Radioactivity (B71Nuclear)
Nuclear Physics and Radioactivity – Understand the three types of radioactive decay and the
byproducts of each, be able to balance nuclear equations and be able to use the most famous
physics equation E=mc2 making calculations in both Joules (J) and electron volts (eV).
Today’s exciting homework code is B71Nuclear. Don’t forget to put this code, the date, and
page number at the top of each page of homework. Please find the pages in your book that
discuss nuclear physics. Some of the parts of this discussion might include these key words:
binding energy, alpha, beta, gamma, or nuclear reactions.
1. Write down the pages numbers in your book where information is found about radioactive
decay. As usual there are always great references on the Internet. Here is the URL for the
discussion of this information in Hyperphysics: http://hyperphysics.phyastr.gsu.edu/hbase/nuccon.html#c1.
2. Now let’s start with the basics. What are the three different types of radioactive decay?
3. What particle is associated with each type?
4. Which type is the most dangerous and why?
50
5. You also need to know how to balance nuclear equations, so let’s do a little “chemistry”
learning. If I talk to you about carbon-12, carbon-13, and carbon-14, what do the 12, 13, and
14 mean?
6. If I write the symbol for carbon-12 in this fashion, 126C , what do the 12 and the 6 mean?
7. Write similar symbols for carbon-13 and carbon-14.
8. Write similar symbols for the neutron, proton and electron.
9. Write similar symbols for the beta particle and the alpha particle.
10. Why didn’t I have you write a similar symbol for the gamma particle?
11. Write down any interesting thoughts you have about the way the symbols were written for
carbon-12, carbon-13, and carbon-14 and the way the symbols are written for the neutron,
proton, and electron.
12. Now we are going to do a little bit of nuclear equation balancing. Consider this equation,
238
234
A
92 U  90 Th  Z X , what are the values for A, Z, and X? You need to look at a periodic
table. What is the name of the particle?
14
A
13. Next equation, 14
6 C  7 N  Z X . What are the values for A, Z, and X? What is the name of
the particle?
12
A
14. Next equation, 12
7 N  6 C  Z X . What are the values for A, Z, and X? What is the name of
the particle?
15. Write a chemical equation for gamma decay.
16. What is the difference between fission and fusion?
17. Now for the true excitement, during both fission and fusion there is tiny loss of mass and this
entire mass is converted to energy according to Einstein’s famous equation, E=(Δm)c2. Give
c for this equation with 5 significant figures. The delta is important, as you will see in a
minute.
18. Please notice that quantities of the same units as K = ½ mv2 are found in the equation E =
mc2. If m is given in kilograms and c in m/s, what is the unit of E?
19. I know that last one was a dorky question, but I just wanted to make sure. It used to bother
me that you lost energy both when small molecules were put together to make larger
molecules (fusion) or when large molecules are split into smaller molecules (fission), but I
have a nice car analogy. If I had a car and I took it completely apart separating every single
piece that could be, I would certainly lose a tiny piece of mass, no matter how careful I was,
even if it was only a tiny piece. And if I had all the parts to make a car starting from scratch
and putting each and every piece together, I would certainly have a tiny piece of mass left
over when I was done. So now it makes more sense to me.
20. Let’s consider the mass of just a single proton. Find the energy released in Joules by the
annihilation of just one single proton. The mass of the proton in kilograms is on the
constants sheet as is the value of c in m/s.
21. Wow, that is a lot of energy! Physicists like to be as efficient as the next person and using
the mass in kilograms of something as small as a proton is really just inconvenient to write,
so there is another mass unit, the atomic mass unit or amu. The unit itself is just abbreviated
u. What is the mass of a proton in atomic mass units? Give at least 7 significant figures.
22. Why do I want so many significant figures? Why is it possible to have so many significant
figures?
23. What is the conversion between the unit u and kg?
51
24. What is mass defect? What is mass deficit? What is binding energy? What is disintegration
energy?
25. Of course since there is a more convenient unit of mass, there is also one for energy. The
Joule even though I consider it to be small unit, is just too large. The new energy unit is the
electron volt, abbreviated eV. What is the conversion between electron volts and Joules?
26. On the Table of Information for AP Physics, the following line is found: 1 u = 1.66 X 10-27
kg = 931 MeV/c2. Explain the 931 MeV/c2.
27. Let’s consider a typical problem now. A fusion reaction of deuterium (Hydrogen-2) plus
tritium (Hydrogen-3) yields Helium-4 and one neutron. What energy is released by this
reaction considering only one molecule of deuterium and one molecule of tritium? I am
giving the following values for masses: mass of deuterium is 2.014102 u, mass of tritium is
3.016049 u, mass of helium 4 is 4.002603 u and the mass of a neutron is 1.008665 u. Give
the energy released by the reaction both in Joules and mega-electron volts.
28. How many molecules of deuterium are needed to provide the energy needs of the entire
United States for one year if the U. S. uses 6.25 E 34 MeV per year?
29. How many kilograms of ocean water would be needed to provide the U. S. energy use for
one year? Assume one drop of water is 0.1 grams and there is 1 deuterium molecule for
every 6000 H2O molecules.
30. I think that’s it for beginning nuclear physics! Woohoo! Please write down the two most
valuable ideas from this lesson.
Physics B Unit 7, Lesson 2, Quantum Physics and Wave Particle Duality (B72Quantum)
Quantum Physics and Wave Particle Duality – Gain an appreciation and comprehension of
the photoelectric effect, de Broglie wavelength, Heisenberg Uncertainty Principle and wave
particle duality.
Today’s exciting homework code is B72Quantum. Don’t forget to put this code, the date, and
page number at the top of each page of homework.
1. Please find references in your book or on the Internet that discuss today’s topics including
Planck, Einstein and the photoelectric effect, photons, de Broglie wavelength, Heisenberg
Uncertainty Principle and wave particle duality. Please write these references down.
2. Give a sentence or two in synopsis of Max Planck and his contributions to this equation
found on your AP Equation Sheet, E = hf.
3. Give the name of each variable in the equation, E = hf, its units and two values if it is a
constant.
4. What is the photoelectric effect and who discovered it?
5. What did Einstein win the Nobel Prize in Physics for?
6. What is a photoelectron?
7. What is stopping potential?
8. What is a photon?
9. What is the work function? Which has a larger work function sodium or iron? Does this
make sense to you
10. Write the equation from your Table of Equations for AP Physics Exams that includes the
work function.
52
11. Sketch the graph of Kmax versus frequency. Give the work interpretation of the x-axis
intercept on this graph.
12. True or False and Why? Increasing the intensity of a light with a frequency below the
threshold frequency will cause photoelectrons to be ejected from a metal.
13. Why the heck is the word “quantum” so important in the phrase quantum physics?
14. What is wave particle duality? Describe an experiment that illustrates the wave nature of
light. Describe an experiment that demonstrates that wave is a particle.
15. What is the equation for the de Broglie wavelength? Summarize de Broglie’s discovery in a
sentence or two.
16. Consider a physics teacher of mass 80 kg moving at her top speed walking speed of 19
minutes per mile. What is her wavelength?
17. What is the mass of a photon?
18. There is another equal sign on your Table of Equations for AP Physics Exams right after E =
hf. Write the rest of the equation. How is this related to the de Broglie equation?
19. Can a photon have momentum?
20. Do the answers to 17 and 19 bother you? Answer yes or no and then write one additional
sentence of explanation.
21. Give 1 to 3 sentences of synopsis for the Heisenberg Uncertainty Principle.
22. Give 1 to 3 sentences of synopsis for the Compton Effect and draw a quick sketch that
represents Compton scattering, a combination of nuclear physics and the conservation of
momentum.
23. I guess that is it for quantum physics for today. Please write down the two most outstanding
perceptions from today’s lesson.
Physics B Unit 7, Lesson 3, Bohr and Energy Level Diagrams (B73Bohr)
Bohr and Energy Level Diagrams – Achieve appropriate comprehension of the Bohr model of
the atom and how this relates to energy level diagrams, particularly understanding how to
calculate energy, frequency and wavelength of emitted photons.
Do not freak out or anything, but this is the last typed lesson you will get in this class! The
homework code for today is B73Bohr. Please write this code, the date, and page number at the
top of each page of homework.
1. Find references in your textbook or on the Internet relating to atomic physics particularly the
Bohr model of the hydrogen atom and energy level diagrams.
2. Make a quick sketch of the Rutherford Gold Foil Experiment and describe it in 1 to 3
sentences.
3. Read through the page or two of narration and equations concerning the Bohr model of the
hydrogen atom. There are associations between this and several other topics you have
studied. Write one sentence for each explaining how the Bohr model of the hydrogen atom
relates to the following topics: (a) centripetal motion; (b) kinetic energy; (c) Coulomb’s law;
(d) electric potential energy; (e) electric potential. Just a note that you do not need to know
every iota of narrative and mathematical information of Bohr’s model of the hydrogen atom.
However, having some knowledge of this makes it so much easier to know and do what truly
is required.
53
4. A nice equation when considering atomic spectroscopic data is this one,
1
n
 RH (
1
1
 2 ),
2
n f ni
although I don’t think you will use this equation mathematically very often in this unit. It
gives a nice reference point for evaluating the specific wavelengths produced in the hydrogen
spectrum. It was known that only very specific wavelengths were emitted for the hydrogen
spectrum 50 years before it was known why. You get to evaluate the why pretty darn soon!
1
Lucky you. Other classic equations you should consider are E n  E1 2 for hydrogen and a
nf
more general form of this equation that can be used for one-electron atoms such as ionized
Z2
helium or doubly ionized lithium, E n  (13.6eV ) 2 . These two equations lead us to a
n
discussion of energy level diagrams. Find a copy of the energy level diagram for hydrogen.
Make a quick sketch of it. Where is zero on this diagram? Where is the largest magnitude of
E? Is it a positive or negative value? What are the units of E?
5. Okay, if an electron absorbs a certain amount of energy, it is excited to a higher orbit. Then
after the electron hangs around at this orbit for a while it returns to its lower orbit and emits a
photon of energy. The energy level of the photon emitted is the difference between the
energy of the two levels. You can find the energy of the photon emitted by subtracting the
energy of the two levels. Since you know the exciting equation E = hf, then you can find the
frequency of the photon and then you can find the wavelength of the photon using the
equation c = f. Okay now derive an equation for  specifically in terms of E and the known
constant c. If you know the wavelength you should have some idea whether or not the
emitted photon wavelength is in the visible light spectrum or higher than that or lower than
that. And this is a nice way to tie in a question involving light waves with nuclear physics!
6. Arrows can be used to show the transition from one energy level to another. On your sketch
from #3, draw an arrow in a new color that you didn’t use on the diagram that shows the
transition from energy level 1 to 3. Draw another arrow that shows a transition from level 2
to level 3. Draw another arrow that shows a transition from the ground state to the being
completely removed from the atom. Which way do the arrows point?
7. Okay now there are supposed to be a couple of big Aha!’s here. Let me be totally obvious
and list them out. (a) Wavelengths of emitted photons are only possible if they coordinate to
the change in energy from one level to another. The absolute values of the energy levels
aren’t as important to the problems that you will be doing as the difference in the energy
levels. (b) Given an energy level diagram you should be able to draw an arrow that
corresponds to a change in energy level and draw it in the correct direction. (c) Given the
diagram you should be able to find the frequency, wavelength and momentum (remember E
= pc) of the emitted photon. (d) Given numbers for the absolute values for the energy levels
for a real or fictitious atom, you should be able to construct an energy level diagram and then
do all of the above.
8. Just for clean up, describe the Michelson-Morley Experiment in 2 to 4 sentences.
9. Same as the last question, describe the Davisson-Germer Experiment in 2 to 4 sentences.
10. That is it for nuclear physics. Make my day and yours by listing the 2 most significant
insights from today’s lesson.
54
Physics C Unit 7, Lesson 1, Ampere’s Law (C71Ampere)
Ampere’s Law – Gain an awareness of how Ampere’s law is used to calculate magnetic field
and be able to apply this to both short answer questions and problems.
Please label the pages of this assignment with the code C71Ampere, the date and a page number.
1. The beauty of Ampere’s Law is that once you have already covered Gauss’s Law, Ampere’s
Law is not that big of a deal. Find references that deal with Ampere’s Law in your textbook
or on the Internet. Write down those references.
2. Write down Ampere’s Law from one of your references. Check it against the equation that
appears on your Table of Equations for AP Physics Exams. Make sure they match or that
you understand any differences between them.
3. What is the meaning of the circle on the left hand side of the integral? It is not the same as
for Gauss’s Law.
4. Define the current that I refers to. How is this analogous to the Q in Gauss’s Law?
5. Name mu naught in words and give its value.
6. What is the most common shape of Amperian Loop’s you will be using?
7. Analogously from what you know of the exciting calculus of the left hand side of Gauss’s
law, solve the integration for the left hand side of Ampere’s Law.
8. Let’s solve Ampere’s law for the magnetic field a distance r away and outside a wire carrying
a current of I. Where have you seen this equation before? Very exciting isn’t it?
9. Now consider the magnetic field inside a wire of radius R carrying current I where the
current is uniformly distributed over the cross sectional area of the wire. I hope that it is
obvious to you that the magnetic field outside the wire is still that same old equation we have
always used. This is why I am only asking you to consider the more interesting question of
the magnetic field inside the current carrying wire. What is Ienc in this situation? The
magnetic field that I want you to consider is inside the wire a distance r from the center of the
wire, with r<R. Since the current is uniformly distributed write steps to show that Ienc = I
(r2/R2).
10. Using your answer to the previous question and steps analogous to those you have used for
Gauss’s law and find a function for the magnetic field inside the wire.
11. What would the shape of the equation of B versus r be for r<R? Go ahead and sketch a graph
of B versus r, making sure to note where R occurs.
12. That’s almost all we need to know for Ampere’s Law. There could be a lovely question
where the current is given a density function, σ. Remember it is an area symmetry since it is
a 2-dimensional Amperian loop instead of a 3-dimensional Gaussian sphere. Remind me to
see if there is a known problem like this. I cannot remember one at this time.
13. There are other significant derivations using Ampere’s Law. It can be used to find the
magnetic field inside a solenoid. For this derivation, a rectangular loop is used. Draw a
sketch of the cutaway view of a solenoid and draw a rectangular loop around only the
incoming or outgoing current. Derive an expression for the magnetic field inside a solenoid.
Use your knowledge of RHR’s, dot products, and calculus to evaluate the integral for each
segment of the rectangle. Feel free to use your textbook, other textbooks, or the internet to
help you.
14. Another significant question that can be solved using Ampere’s law is to find the magnetic
55
field is for a coaxial cable. Find out the basics of the coaxial cable. If you drew an
Amperian Loop that went outside the coaxial cable, what would Ienc be? Knowing this, give
a reason or two to use coaxial cables.
15. Let’s consider that the coaxial cable is made from an inner conducting wire and an outer
conducting sleeve and that the space between the inner conducting wire and the outer
conducting sleeve is filled with silicon oil that has a dielectric constant  = 2.5. Derive an
expression for the capacitance of this cylindrical capacitor.
16. One final thing, there is a lovely law called the Biot-Savart Law. Find this law from a
reference source and write it down. Tell me whether or not it appears on your equation sheet.
Using your textbook, other books or the internet as aides, use the Biot-Savart Law to derive
the expression for the magnetic field outside a wire carrying a current I, a distance r from the
center of the wire. Tell me which you like better, Ampere’s Law or Biot-Savart. Give
supporting reasons.
17. That’s all folks, for Ampere’s Law. Please write down the two most noteworthy insights
from today’s lesson.
Physics C Unit 7, Lesson 2, LR Circuits (C72LR)
LR Circuits – Understand the ideas and calculations associated with LR circuits.
Please label each page of this assignment with the code C72LR, the date and the page number.
1. Find some resources either in your textbook or online dealing with circuits containing
resistors and inductors, also called LR circuits. Write down these references.
2. What is an inductor? What does it look like in a circuit diagram?
3. What is the symbol for inductance? What is the unit of inductance?
4. Please recall what I kept saying about capacitors when we talked about circuits before. At
the instant the switch is closed (t=0) the capacitor acts like a wire of zero resistance and after
a long time (as t → ∞) the capacitor acts like a broken wire because this is what it is really
and what it really looks like in a circuit! Of course I have a similar sentence for inductors.
Remember how the laws for capacitors and resistors in regular circuits just seemed to be
switched around? For resistors in series, they add normally, but for capacitors in series, they
add upside down? Well in these RC and LR circuits there is an analogous situation and the
laws seem to be switched around again. So here is the sentence for inductors. At the instant
the switch is closed (t=0) the inductor acts like ________ and after a long time (as t → ∞) the
inductor acts like a ___________ because this is what it is really and what it really looks like
in the circuit diagram! Please fill in the blanks. I also have a very important sentence for
inductors. They hate change like an old grandma or grandpa person. More about that later.
5. Again you will end up with the classic “look” equations for (1) exponential increase up to an
asymptote and (2) exponential decrease. Again, these equations can be represented by (1) K
= K0 (1 – e(-t/τ)) for exponential increase up to an asymptote and (2) K = K0 e(-t/τ) for
exponential decrease. If you have forgotten the basics of these equations, please remind
yourself.
6. There is an equation for the voltage drop across an inductor. It is a differential equation.
Find it and write it down.
7. Now let’s consider a resistor of resistance R connected in series with an inductor of
56
inductance L and a battery of with an EMF of ε. Please draw this circuit and label each part
of the circuit with its variable letter and throw in a switch too!
8. Now we want to consider the time in between t = 0 when you will shut the switch and t → ∞.
During this time the current through the inductor, the voltage over the inductor, and the
voltage over the resistor all change from min to max values or vice versa. Write down IL, VL,
and VR and tell me whether they are a min or max at both t = 0 and t → ∞.
9. In order to do these problems we will be using Kirchoff’s loop rule. Let’s go around the
circuit starting at the battery and show that the Kirchoff’s loop rule for this simple LR circuit
is ε – IR – L(dI/dt) = 0.
10. Before you integrate this equation, first divide the entire equation by R, (ε/R) – I –
(L/R)(dI/dt) In integrating this equation, it is easiest to let (ε/R) – I equal U and then dU = dI. Make these substitutions into the differential equation. Rearrange the equation to get dU
and U on one side and dt and everything else on the other side. Then integrate both sides,
using the limits of integration from U0 to U on the dU side and from 0 to t on the dt side.
Then substitute back in for U. Rearrange until you get a nice equation with our classic form!
Of course you should be showing all steps.
11. Does current exponentially increase to an asymptote or exponentially decrease? How does
this make sense with previous thoughts about how an inductor acts at t=0 and as t → ∞? And
what about all that stuff about old grandma and grandpa people, how does that make sense?
12. Make a sketch of I versus t for the circuit and for VL and VR versus t also.
13. What is the time constant τ for this equation in terms of L and R?
14. Knowing that you cannot exponentiate a number with units, 1 Henry divided by 1 Ohm
equals what unit?
15. The equation you found in number 10 is for the current when “charging” an inductor. But
there is no stored charge. However, there is definitely stored energy. Write the equation for
the energy stored in an inductor. Consider that I is always changing and you get a changing
amount of energy stored in the inductor until the switch has been closed for a long time.
16. Those are the basics that you need to know for storing energy in an inductor. Consider that
the switch is closed on the preceding circuit for a good long time and the inductor stores all
the energy it can. Then you are able to open the switch and magically remove the battery
from the circuit without disturbing the inductor and now you have a circuit with just a fully
energized inductor and a resistor and an open switch. Draw this circuit.
17. What will happen when you close the switch now? To the current through the circuit? To
the voltage over the inductor? To the voltage over the resistor? Write a phrase or two to
answer each of these questions.
18. Now I would really like to have you write the Kirchoff’s loop rule for the voltage drops
around this new circuit without a battery and then make you solve the differential equation
and get an equation for the current I as a function of time, but I am going to let you just write
down the equation without any calculus! Oh…the guilt. The equation should be familiar!
19. Make quick graphs of I, VR and VL versus time.
20. What happened to the energy stored in the inductor? Did it just disappear?
21. Now consider the following. How would the time constant change if you had two resistors in
series with the inductor? If there were two resistors in parallel with each other and then in
series with the inductor? Etc. etc.?
22. You should now understand the basics of LR circuits. Please put pen to paper and note the
two most remarkable ideas from this lesson.
57
Physics C Unit 7, Lesson 3, Other Magnetism Problems (C73Other)
Other Magnetism Problems – Consider and evaluate additional calculus-based magnetism
problems such as the current induced in a rectangular loop of wire a distance away from a wire
that carries changing current and therefore a changing magnetic field.
I do not want you to be too relieved or happy, but I do want you to know that this is the final
lesson of the semester. Please label today’s homework with the code C73Other, along with the
date and the page number.
1. Find references in your textbook for the chapters on magnetism skim the chapters checking
for example problems that you have not covered yet in this class. Also find the section in
your book or a reference from the Internet about Maxwell’s equations. Write these
references down.
2. There are several classic calculus-based magnetism problems that do not use Ampere’s law.
Let’s consider a few of these. First, find the integral equation for F = BIL and write it down.
3. Consider a closed 2-dimensional semicircle of wire carrying a current I where there is a
magnetic field that is in the same plane as the semicircle and is perpendicular to the straight
side of the wire. Find the total magnetic force acting on the wire. Try it on your own, see if
it is an example in your textbook or try the Internet. Here is a possible URL to give you a
hint: http://www.phys.uri.edu/~gerhard/PHY204/tsl185.pdf.
4. Consider a 2-dimensional closed rectangular loop of wire where there is a magnetic field that
is in the same place as the rectangle and perpendicular to two sides of the rectangle. Sketch
this situation. What are the forces on the two sides of the rectangle that are perpendicular to
the magnetic field? What does it cause the rectangle to do? Give me your best mathematical
treatment of this problem. Feel free to use these resources: you, your peers, others sentient
beings knowledgeable about physics, books, and the Internet.
5. What about a circular loop of wire with a radius of R carrying a current of I. What is the
magnetic field at a point a distance a away from the loop along the axis that goes through
center of the loop? Feel free consult good resources that have been mentioned before.
6. Consider a rectangular loop of wire with a resistance R and a length L and a width W that is a
distance c away from a current carrying wire where the current is a function of time
according to the equation I = mt +b. What is the current induced in the wire? Again, you
should consult good resources.
7. Find a reference for this equation, I = NevdA, and tell what all letters in the equation are
named and what the units of each letter are and what the equation is most often used for.
8. List the 4 Maxwell equations in derivative form (not the gradient form which uses a symbol
like an upside down Δ). Give a sentence that gives your best conceptual interpretation of
each law.
9. That’s it. Please write down the final two most salient thoughts related to this lesson.
58
CONCLUSION
Why do students take Advanced Placement Physics? Why do instructors teach the class?
Should an instructor teach the algebra-based Physics B course or the calculus-based Physics C
courses? What is the difference between Physics B and Physics C? How can enough students be
recruited so that the class can be taught? How can you guide students to know and understand
some difficult concepts on their own? These are the questions that I have answered with this
project. Some of them I deliberately set out to answer. One of these questions appeared of its
own volition as the project progressed. The purpose of this project is to provide support
materials for high school physics instructors and students. These materials were specifically
written to provide physics instructors with self-guided study for students when they were
teaching algebra-based Physics B and calculus-based Physic C concurrently in the same
classroom. The topics covered are those where the two courses are completely different. This
project is my guide of “How” to accomplish teaching these two courses in the same classroom.
What is the benefit to students taking Advanced Placement Physics? There are mixed feelings
about this among educators, students, and reports in the media. Some universities do not give
college credit for AP classes. Some studies show that students are greatly benefited in college
from AP classes. Other studies show that taking these classes in high school makes no significant
difference in their secondary education. I think willing students benefit from these courses. I do
not think it matters whether or not the college of their choice gives them credit for their passing
AP score or not. These students have been exposed to college-level material and have coped
with the difficulties that are inherent in this kind of material. They are more prepared for all
college-level material. I always tell my students that if they take the same course again in
college, they will be smiling and getting high grades while other students in the class are crying!
What is the benefit to instructors teaching the class and what is the benefit to offering the two
classes concurrently in the same classroom? Teachers get to interact with some of the brightest
and most motivated students. They get to teach material that still challenges them. This helps
keep instructor morale high. Offering both classes at the same time allows for a greater pool of
students to take the class. In smaller schools or larger schools where competition for top
students is fierce, this will enable more students to take the class at a mathematical level that they
are prepared for. Schools, students and educators benefit from having AP Physics as part of the
curriculum.
My original vision for these lessons for my students to complete independently was to provide
them with detailed information during the time what they were learning was totally different
from some of their classmates. I started out creating detailed notes for the topics. Then I
realized the students would gain a greater understanding if I didn’t just provide detailed notes,
but if I asked them good questions that would lead them to the information they need to know.
Then I could discuss the results with them. At this point in my project I changed the emphasis
from just setting down information to a combination of giving some information and asking
students questions that would lead them to the information they need to know and then
discussing that information with them.
59
The purpose of this project is to provide high school physics instructors and students with
support material for AP Physics B and AP Physics C when the material of the courses is totally
different, particularly if the two courses are taught concurrently in the same classroom. A
timeline overview of when the topics are taught was given. Specific self-guided student lessons
for these topics were provided. More rigorous high school physics courses lead to improved
college physics performance. Being successful in physics in college opens the door to more
career choices for students. Teaching more rigorous high school physics courses benefits all
physics students and the educators who teach them.
60
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