PHYS 1004 Course Outline 1 Calendar description and prerequisites

Carleton University Physics Department -1- PHYS 1004 Winter 2011 

PHYS 1004 

Introductory Electromagnetism and Wave Motion 

Winter 2011 

Course Outline 

This course is designed to provide an introduction to electricity, magnetism, circuits, electromagnetic induction, 

and wave motion from a physics perspective for students in engineering programs. The associated 

laboratory and tutorial sessions meet during alternate weeks starting in the second week of term. Student 

evaluations will be based on labs, tutorial tests given during each tutorial session, weekly WebCT-based 

quizzes, and a final exam. 

1 Calendar description and prerequisites 

1.1 Calendar description 

This calculus-based course introduces electricity, magnetism, oscillations, waves and optics. The laboratory 

is an essential and autonomous part of the course. 

Precludes additional credit for PHYS 1002 and PHYS 1008. 

1.2 Prerequisites 

You must have successfully completed 

(i) 

or MATH 1004 Calculus for Engineering or Physics 

or MATH 1007 Elementary Calculus I 

PLUS (ii) either be concurrently registered in 

or ECOR 1101 Mechanics I 

or else have passed 

or PHYS 1003 Introductory Mechanics and Thermodynamics 

or PHYS 1001 Foundations of Physics I 

or PHYS 1007 Elementary University Physics I (with a grade of at least B−). 

If you do not have both of these requirements you must check with your instructor to obtain permission of 

the Physics Department to take this course. This could apply, for example, to students who have completed 

the equivalent of MATH 1004 or PHYS 1003 at another university. 

If you withdraw from ECOR 1101 during the term, you will be required to also withdraw from PHYS 1004.


2 Who teaches the course, when and where 

2.1 Lecture Timetable 

CRN Section Times Room Lecturer Lecturer’s office & contact info 

10923 A Wed 10:00–11:30 AT 102 Bruce Campbell Herzberg 3378 

Fri 10:00–11:30 

campbell@physics.carleton.ca 

613-520-2600 x4322 

14859 B Tue 19:30–21:00 AT 102 Heather Logan Herzberg 2450 

Thu 19:30–21:00 (course logan@physics.carleton.ca 

coordinator) 613-520-2600 x4319 

10927 C Mon 16:00–17:30 AT 102 Igor Ivanovic Herzberg 3346 

Wed 16:00–17:30 

igor@physics.carleton.ca 

613-520-2600 x5796 

10930 M Mon 19:30–21:00 AT 102 Igor Ivanovic Herzberg 3346 

Wed 19:30–21:00 

igor@physics.carleton.ca 

613-520-2600 x5796 

2.2 Laboratory and Tutorial Timetable 

CRN Section Time 

10924 A1 Mon 11:30–14:30 

10925 A3 Wed 11:30–14:30 

10926 A5 Mon 8:30–11:30 

14860 B6 Fri 11:30–14:30 

14861 B7 Wed 8:30–11:30 

14863 B8 Wed 14:30–17:30 

10933 C2 Fri 14:30–17:30 

10931 C4 Thu 13:30–16:30 

10928 C9 Tue 13:30–16:30 

10932 M11 ∗ Thu 18:00–21:00 

10929 M12 Fri 8:30–11:30 

11703 M13 ∗ Tue 18:00–21:00 

Room: 

Lab supervisor: 

Herzberg 4130 (all sections) 

Andrew Robinson 

( ∗ except Igor Ivanovic 

for sections M11 and M13) 

Supervisor’s office Herzberg 3368 

& contact info: robinson@physics.carleton.ca 

613-520-2600 x8922


3 Learning outcomes 

In the process of successfully completing this course, you will learn how to: 

• solve multi-step problems; 

• present written solutions to problems in a clear and logical way; 

• apply calculus techniques to real physical problems; 

• understand the concepts of electric and magnetic fields and the associated forces; 

• relate the motion of objects in electric fields to familiar concepts of force, work, and kinetic and 

potential energy; 

• determine the electric field and electric potential due to a known charge distribution using Coulomb’s 

law, Gauss’s law, and the relationship between electric field and potential; 

• use the electric field and potential formalism to understand capacitors; 

• understand electric current and the rules for modeling electric circuits from a physics perspective; 

• determine the magnetic field due to a known current distribution using the Biot-Savart law and 

Ampère’s law; 

• determine the forces on moving charges and current-carrying wires due to an external magnetic field, 

and understand the technological applications of these forces, e.g., in the electric motor and electric 

generator; 

• understand electromagnetic induction, determine the resulting induced currents, and understand 

their consequences; 

• understand oscillatory motion using the simple mass-on-a-spring model; 

• analyze alternating-current (AC) circuits from a physics perspective; 

• understand travelling waves with special emphasis on electromagnetic waves; and 

• analyze the effect of polarizing filters on electromagnetic waves. 

In the laboratory portion of this course you will learn how to: 

• present the results of laboratory experiments in a clear and well-organized written form; 

• quantify experimental uncertainties; 

• construct circuits from electronic components; 

• measure currents and voltages in circuits using common measuring devices; 

• use an oscilloscope for analysis of electronic signals; and 

• understand the behaviour of direct-current (DC) circuits, resistor-capacitor (RC) circuits, and the 

use of optical lenses.


4 How to succeed in this course 

4.1 Tips for success 

• Study effectively. The only way to learn physics is by working the problems. Work and rework 

all the examples in the text and the assigned homework problems. If you can do all the homework 

problems by yourself with only the attached formula sheet as an aid, then you’re on the road to a 

good mark in the course. 

• Do your homework. Aim to work through all the assigned homework problems before your 

tutorial. That way you will know what areas you need to focus on during the tutorial when working 

in groups and getting help from the TAs. Each tutorial test will include long-answer problem(s) 

based on the homework problems, so be prepared! 

• Take advantage of class time. Attend all the lectures, participate in class discussions and exercises, 

take notes, and don’t be afraid to ask questions. When taking notes, concentrate on the things 

that your instructor emphasizes on the chalkboard or slides. 

• Use your textbook. Read the assigned sections of the textbook before each class. Review them 

again after class and annotate your class notes as needed. At the end of each chapter of the textbook 

is a one- or two-page review of the important concepts covered in each chapter. 

• Get all the term marks you can. Attend all the tutorials and be prepared for the tutorial tests. 

Attend all the labs and turn in all your lab reports on time. Do all the WebCT quizzes on your own 

and before the deadline. (Note that it is easy to work in groups on the WebCT quizzes and get all 

the answers from your friends, but this defeats the purpose of learning by doing these problems and 

it will not prepare you for the final exam. The final exam will have many similar problems which are 

worth twice as many marks as all the WebCT quizzes put together.) 

• Get help if you need it. If you’re having trouble with a concept or homework problem, go to your 

instructor’s office hours or the Drop-In Centre for help. We are here to help you learn! 

4.2 Getting help 

• Office hours: Each of the course instructors will hold office hours a couple times per week. Times 

and locations will be posted in WebCT and announced in class. The instructors can also be reached 

by email. 

• Drop-In Centre: The Physics Department normally runs a Drop-In Centre staffed by TAs for help 

in first-year Physics courses. The times and location will be posted in WebCT and announced in 

class.


5 Course components and marking scheme 

5.1 Marks and passing conditions 

The marking scheme is as follows: 

Theory (total 70%) Weekly WebCT quizzes (best 9 of 11) 10% 

Tutorial Tests (best 4 of 5) 25% 

Final Exam 35% 

Lab Experiments 30% 

Course total 100% 

In order to pass the course, your overall mark must be greater than 50% AND you must achieve 40% or 

above on BOTH the Theory AND the Lab Experiments components of the course. 

Students with an overall course mark above 50% but who achieve between 40% and 49% on either Lab 

Experiments or Theory will be given a grade of D−, no matter how good their overall mark is. 

5.2 Weekly WebCT quizzes 

Each week you will have a WebCT-based quiz consisting mostly of conceptual questions or questions 

requiring a brief calculation. The purpose of these quizzes is to help you learn the material and verify your 

basic understanding before you move on to the more complicated homework problems associated with the 

Tutorials. Each week there will be a Self-Quiz that you can do for practice as many times as you like, and 

a time-limited Quiz that will be graded. You should read the relevant textbook sections before you start 

the time-limited quiz. The quizzes are open-book. Have lots of scratch paper handy while doing the quiz: 

even simple calculations are much easier when done on paper than in your head. 

There are many ways to subvert the purpose of these quizzes (e.g., working in groups, copying answers). 

Note though that 60% of the final exam will be made up of similar problems and it is in your interest to 

put in the effort to learn the material and do the quizzes well by yourself. 

WebCT quizzes are under the “Assessments” section of the WebCT page and will be due at 16:00 on 

Fridays. It is your responsibility to complete the quiz by the deadline. There is no way to make up a 

missed quiz once the deadline has passed. If you have a legitimate technical problem (browser crash, 

internet failure) and you contact us by email before the deadline, we can reset your quiz for you to start 

fresh. 

Solutions to the WebCT quizzes will be posted shortly after the deadline. You should review the solutions 

to be sure you understand the answers. 

Your best 9 out of the 11 WebCT quizzes will make up 10% of your overall course mark. 

5.3 Tutorials 

There are 5 Tutorials in the term. They take place in your regular lab time slot in Herzberg 4130, 

alternating weeks with the labs. The Tutorial schedule is given in Sec. 8. During the first two hours of 

the Tutorial you will work on problems and have the opportunity to ask questions of the TAs. Working 

in groups is encouraged. During the last hour of the Tutorial there will be a 45-minute Test which you


do on your own and hand in for marking. The Test will consist of 2 to 5 multiple-choice or short-answer 

questions plus one or two questions requiring written solutions. Your best 4 out of the 5 Tutorial Tests 

will make up 25% of your overall course mark. 

The Tutorial Tests (and the final exam) will be based on the homework problems given in Sec. 7. It is 

thus important that you prepare the homework problems prior to the Tutorial and use the Tutorial time 

to understand these problems well. 

Solutions to the Tutorial Tests will be posted in the notice board outside the lab (Herzberg 4130) at the 

end of each Tutorial week. 

What to bring: Bring your student ID card, writing instruments, and a calculator, plus a ruler if you 

want. No other aids are allowed for the Tests. The formula sheets that make up the last two pages of this 

course outline will be provided to you with the Test. 

Attend your own Tutorial section only. To be able to write the Test in a different section, you must 

obtain written permission from your lecturer. Such permission will usually be granted only for emergencies 

or medical reasons, or official activities such as Engineers Without Borders. 

Tutorial Test make-ups: If you miss a Tutorial Test, immediately contact your lecturer and explain 

why. If the reason is illness, a doctor’s note is required. Students with valid reasons will be given written 

permission to write the Test in a different section later the same week if possible. If this is not possible, you 

must obtain permission to write a make-up test at the end of term. These will all be written on Wednesday 

April 6; the time will be announced toward the end of term. Note that you need to get permission at the 

time you miss the test or as soon as you are back at school after an illness or accident. Retroactive 

permission will not be given at the end of term. 

5.4 Labs 

Labs start during the second week of term (January 10-14). Bring your copy of the lab manual and a beige 

lab booklet with you to the first lab. For each lab you will be writing a lab report. Some of these will 

be Short Reports while others will be Formal Reports. Instructions on writing lab reports will be given 

during the lab and the format can be found in the Lab Manual. You have one week to complete the Lab 

report, which will be due in at the start of the next tutorial. The exceptions to this are the last two labs, 

for which the report will be written up and turned in before the end of the lab period. Some of the lab 

report is done before you attend the lab session. This is referred to as the “Prelab”. This must be written 

in your lab booklet. It will be checked and marked by your TA during the lab session and marks will be 

deducted if it is not done in advance. See page 2 of the Lab Manual. 

For students repeating this course, you may request to be exempt from the lab (and have your lab mark 

carried forward from before) if you have completed all the lab experiments with an overall lab mark of at 

least 60%. You must contact your lab supervisor and obtain explicit permission to be exempt from the 

lab. Note that you will not be exempted from the Tutorials, which meet in alternate weeks during your 

lab period. 

5.5 Final Exam 

The Final Exam will be held during the exam period, April 7–21 (including Saturdays)—the date will be 

announced by the university by February 18. All four lecture sections will write the Final Exam together.


You may bring only writing implements, a calculator, and a ruler to the Final Exam. The formula sheets 

that make up the last two pages of this course outline will be provided. The Final Exam will include four 

problems requiring written solutions, of which you will choose three to do. These problems will be based 

on the assigned Homework Problems. The remainder of the exam will consist of multiple-choice questions. 

The exam will include a few questions relevant to the laboratory. 

If you miss the Final Exam for a good reason such as illness, you may apply for a Deferred Exam through 

the registrar’s office. A Deferred Exam replaces only the Final Exam portion of your mark. Deferred 

Exams for Winter 2011 will be scheduled during June 13–23. In order to be eligible for a Deferred Exam 

you must have earned at least 7 out of the possible 35 marks on term work in the theory component of 

the course (i.e., the Tutorial Tests and WebCT quizzes) and at least 12 out of the possible 30 lab marks. 

6 Required textbook and materials 

Textbook: Fishbane, Gasiorowicz & Thornton, Physics for Scientists and Engineers, Custom Edition for 

Carleton University PHYS 1004 (Pearson, 2005), ISBN 0-558-87457-6. 

[Available at the Bookstore (Uni Centre) and at Haven Books (at Sunnyside and Seneca St.)] 

This is a custom printing specifically for Carleton University containing the chapters we will cover. Alternatively, 

you may use the full text: Fishbane, Gasiorowicz & Thornton, Physics for Scientists and 

Engineers, Third Edition (Pearson Prentice Hall, 2005), ISBN 0-13-142094-1. 

Lab manual: Laboratory Manual for PHYS 1004, Winter 2004 edition or more recent. 

[Available at Science Stores, 118 Steacie Building.] 

You must bring this with you to each Lab session. 

Lab booklets: Five (5) beige booklets, Carleton University Laboratory Report. 

[Available at the Bookstore (Uni Centre).] 

You must bring one with you to each Lab session.


7 Homework problems 

The following problems, from Fishbane, Gasiorowicz & Thornton, are assigned as homework. You should 

attempt to complete them all before the corresponding Tutorial. They will not be collected; instead, the 

Tutorial Test during the last hour of each Tutorial will be based on these assigned problems. During the 

first 2 hours of each Tutorial you will work through any of these problems you had difficulty with and be 

able to ask questions of your peers and the tutorial TAs. This is your opportunity to clear up any remaining 

questions on the material before the Tutorial Test. The last set of problems, on material covered during 

the last 2.5 weeks of term, will not be on a Tutorial Test but is fair game for the final exam. 

Tutorial 1: 

Ch. 1 P# 56, 57, 60, 83, 84 

Ch. 6 P# 11, 16, 37, 42, 43, 46, 72 

Ch. 7 P# 2, 6, 7, 14, 20, 28 

Tutorial 2: 

Ch. 21 P# 31, 44, 46, 52, 55 

Ch. 22 P# 26, 28, 30, 31, 48, 51 

Ch. 23 P# 5, 8 

Tutorial 4: 

Ch. 26 P# 11, 19, 40, 55, 58 

Ch. 27 P# 4, 15, 18, 27, 28, 52, 56, 57 

Tutorial 5: 

Ch. 28 P# 5, 9, 19, 43, 48, 50, 53, 80 

Ch. 29 P# 3, 8, 9, 15, 21, 30, 32, 39, 46 

Ch. 30 P# 3, 7, 12, 13, 17, 20, 37, 48 

Ch. 32 P# 2, 13, 22, 24, 38, 45, 48 

Tutorial 3: 

Ch. 23 P# 24, 26, 36, 37, 40, 41, 47 

Ch. 24 P# 22, 26, 29, 40, 45, 51, 53, 64 

Ch. 25 P# 7, 9, 19, 26, 33, 36, 54 

Remaining course material: 

Ch. 13 P# 1, 5, 12, 19, 38 

Ch. 32 (sec. 32.5) P# 53, 58 

Ch. 33 P# 12, 15, 18, 24, 35 

Ch. 14 P# 2, 4, 14, 15 

Ch. 34 P# 1, 5, 6, 15, 16, 22, 28, 51, 56, 59 

Note: P = Problems given at the end of the chapter. You should also try to do all of the “Understanding 

the Concepts” questions at the end of each chapter. 

Problem-solving is a skill. The only way to get good at it is to practice on problems for which you have not 

already seen the solution.


8 Schedule of lectures, labs, and tutorials 

8.1 Schedule 

Week 

Activities 

Week 1 (Jan 3–7) Lectures 1 and 2 

No lab or tutorial 

WebCT Quiz 1 due at 16:00 on Friday 


Lab: Intro to the Lab; Error Analysis; 

Lab: Introduction to DC Circuits (15% of your lab mark; 

Lab: Short Report due at start of next tutorial) 


Fri Jan 14 is the last day to change courses or sections 


Tutorial 1 

DC Circuits lab report due at start of tutorial 



Lab: Kirchhoff’s Rules (15% of your lab mark; 

Lab: Short Report due at start of next tutorial) 


Week 5 (Jan 31–Feb 4) Lectures 9 and 10 

Tutorial 2 

Kirchhoff’s Rules lab report due at start of tutorial 


Week 6 (Feb 7–11) Lectures 11 and 12 

Lab: RC Time Constant (25% of your lab mark; 

Lab: Formal Report due at start of next tutorial) 


Week 7 (Feb 14–18) Lectures 13 and 14 

Tutorial 3 

RC Time Constant lab report due at start of tutorial 


The final exam schedule will be available by Fri Feb 18 

Reading Week (Feb 21–25) 

No lectures, labs, or tutorials


Week 

Activities 

Week 8 (Feb 28–Mar 4) Lectures 15 and 16 

Lab: Oscilloscope part 1 


Week 9 (Mar 7–11) Lectures 17 and 18 

Tutorial 4 



Lab: Oscilloscope part 2 (25% of your lab mark; 

Lab: Short Report due at the end of this lab period) 



Tutorial 5 


Week 12 (Mar 28–Apr 1) Lectures 23 and 24 

Lab: Simple Lens (20% of your lab mark; 

Lab: Short Report due at the end of this lab period) 

Review days (Apr 4–5) 

Lecture review sessions 

No labs or tutorials 

Tutorial Test make-ups (Apr 6; for those given written permission during term) 

Lab make-ups (for those given permission during term)


8.2 Material to be covered in each lecture 

To make best use of limited lecture time, lectures will concentrate on the more difficult material, while the 

simpler introductory material will be left for you to read and understand by yourself. This material is still 

examinable, even though it will be covered in less detail in lectures. 

You should always read all the sections of the textbook to be covered before coming to class. 

Textbook sections 

Lecture Topic Covered today Self study 

1 Introduction to the course; vectors Ch. 1.6 – 

2 Work; kinetic energy; power Ch. 6.1–6.3, 6.5 – 

3 Conservative forces; potential energy Ch. 6.4, 7.1–7.2 7.2 

4 Electric charge; Coulomb’s Law Ch. 21.1–21.3 – 

5 Coulomb’s Law, electric field, field lines Ch. 21.4, – 

Ch. 22.1–22.2 

6 Electric field of continuous charge distributions; Ch. 22.3–22.5 – 

motion of charged particles in an electric field 

7 Gauss’ Law and electric flux Ch. 23.1–23.3 – 

8 Determining electric fields; Gauss’ Law and conductors Ch. 23.3–23.5 – 

9 Electrical potential energy; electric potential; Ch. 24.1–24.4 – 

determining electric fields from electric potentials 

10 Electric potentials of charge distributions; Ch. 24.5–24.6 24.7 

electric potentials of conductors 

11 Capacitance; capacitors; energy in electric fields; Ch. 25.1–25.5 25.6 

capacitors in parallel and series; dielectrics 

12 Electric currents, current density in materials; Ch. 26.1–26.4, 26.5–26.6 

resistivity and resistance; Ohm’s Law; resistors Ch. 26.7 

in series and parallel; power 

13 DC circuits; emf, real batteries; Kirchhoff’s rules; Ch. 27.1–27.3, 27.4 

energy; RC circuits Ch. 27.5 

14 Intro to magnetism, force on a moving charge and on Ch. 28.1–28.5 28.1, 28.6 

current-carrying wires; torque on a current loop, 

electric motor


Textbook sections 

Lecture Topic Covered today Self study 

15 Magnetic fields, Ampère’s law, field due to a long Ch. 29.1–29.5 – 

straight conductor; magnetic flux; solenoids; 

Biot-Savart law, magnetic dipole; Maxwell’s 

correction to Ampère’s law 

16, 17 Magnetic induction, Faraday’s and Lenz’s laws; Ch. 30.1–30.5 – 

motional emf; eddy currents; induced electric 

fields; generators 

18, 19 Inductance and inductors; self-inductance of a Ch. 32.1–32.4 – 

solenoid; energy in inductors & magnetic fields; 

RL circuits 

20 Simple harmonic motion (SHM), terminology; Ch. 13.1–13.4, 13.2 

spring-block system; energy in SHM; LC oscillator Ch. 32.5, 32.7 

21, 22 Alternating currents; AC source with separate Ch. 33.2–33.4, 33.1 

R, L, C elements, phasors; power, rms quantities 33.5 (filters only) 

23 Maxwell’s equations, Ch. 34.1 – 

introduction to travelling waves Ch. 14.1–14.4 

24 Travelling electromagnetic waves, speed of light; Ch. 34.2–34.3 – 

energy carried; polarization Ch. 34.5


9 University policies 

9.1 Students with disabilities 

Students with disabilities requiring academic accommodations in this course must register with the Paul 

Menton Centre for Students with Disabilities (PMC) for a formal evaluation of disability-related needs. 

Documented disabilities include but are not limited to mobility/physical impairments, specific Learning 

Disabilities (LD), psychiatric/psychological disabilities, sensory disabilities, Attention Deficit Hyperactivity 

Disorder (ADHD), and chronic medical conditions. Registered PMC students are required to contact 

the PMC every term to have a Letter of Accommodation sent to the Instructor by their Coordinator. In 

addition, students are expected to confirm their need for accommodation with the Instructor no later than 

two weeks before the first assignment is due or the first in-class test/midterm. If you require accommodations 

only for formally scheduled exam(s) in this course, you must request accommodations by March 

11. 

PMC contact information: 500 University Centre, (613) 520-6608, email pmc@carleton.ca, TTY: (613) 

520-3937, web http://www2.carleton.ca/pmc. 

9.2 Other accommodations 

You may need special arrangements to meet your academic obligations during the term because of pregnancy 

or religious obligations. Please review the course outline promptly and contact your instructor with 

any requests for academic accommodation during the first two weeks of class, or as soon as possible after 

the need for accommodation is known to exist. 

9.3 Copying, plagiarism, and other forms of cheating 

You should read and be familiar with the university’s policies on academic integrity, given in Section E.14 

of the Academic Regulations of the University: 

http://www.carleton.ca/calendars/ugrad/0910/regulations/acadregsuniv14.html 

In this course, these rules are relevant mainly for lab reports (do not copy someone else’s) and tutorial 

tests and the final exam (do not attempt to use unauthorized materials or collaborate with other students). 

Such offences will normally result in a mark of zero for the lab report, tutorial test, or exam in question. 

In addition, a report will be sent to the Dean of your Faculty, for possible further disciplinary action.


Reference material for exam and tutorial tests 

N A = 6.022 × 10 23 mol −1 

G = 6.674 × 10 −11 N m 2 kg −2 

g = 9.81 m s −2 

m e = 9.109 × 10 −31 kg 

m p = 1.673 × 10 −27 kg 

e = 1.602 × 10 −19 C 

K = 1/(4πɛ 0 ) = 8.988 × 10 9 N m 2 C −2 

c = 2.998 × 10 8 m/s 

Permittivity, free space, 

ɛ 0 = 8.854 × 10 −12 C 2 N −1 m −2 

Permeability, free space, 

µ 0 =4π × 10 −7 T m A −1 = (H/m) 

1 J = 1 N m 1 eV = 1.602 × 10 −19 J 

1 C = 1 A s 1 V = 1 J/C 

1 Ω = 1 V/A 1 F = 1 C/V 

1 Wb = 1 T m 2 1 H = 1 T m 2 /A 

1 Hz = 1 s −1 1 T = 1 N/(A m) [10 4 Gauss] 

(1 + x) n = 1 + nx + n(n − 1)x 2 /2! + 

n(n − 1)(n − 2)x 3 /3! + ...if |x| < 1 

sin(α ± β) = sin α cos β ± cos α sin β 

cos(α ± β) = cos α cos β ∓ sin α sin β 

vf 2 = vi 2 + 2a∆s s f = s i + v∆t + 1 2 a(∆t)2 

v f = v i + a∆t 

Spring-block:F = −k∆s U sp = 1 2 k(∆s)2 

W > 0 ⇒ energy transferred to an object by force 

K = 1 2 mv2 , ∆K = K f − K i U g = mgh 

Ignoring dissipative energy losses: 

∆E sys = ∆K + ∆U = W ext or E f = E i + W ext 

Conservation of E mech in an isolated system 

(W ext = 0): 

∆E mech = ∆K + ∆U = 0 

Work-KE theorem: ∆K = W net [J] 

Work W force = ∫ f 

i 

⃗F · d⃗s [J] 

Power P = dW/dt = F ⃗ · ⃗v [W] 

∆U = U f − U i = −W force [J] 

If external agent does work against force: 

∆U = W ext 

A force is conservative if the work to move a mass 

or charge between two points is path independent. 

Coulomb’s Law: 

F 1on2 = F 2on1 = K|q 1||q 2 | 

r 2 

[N] 

Shell Theorems: A shell of uniform charge: 1) attracts 

or repels an external charge as if all of the 

shell’s charge were at its centre; and 2) exerts no 

net electrostatic force on a charge in its interior. 

⃗E = ⃗ F /q 0 

[N C −1 or V/m] 

Electric Dipole 

2 charges +q, −q separated by s; dipole 

moment |⃗p| = qs directed from −q to +q 

⃗τ = ⃗p × E ⃗ U = −⃗p · ⃗E 

Electric Fields: 

point charge: | E| ⃗ = 1 q 

4πɛ 0 r 2 

non-conducting ∞ sheet: | E| ⃗ = σ/(2ɛ 0 ) 

conducting ∞ sheet: | E| ⃗ = σ/(ɛ 0 ) 

Gauss’s Law: 

Φ e = ∮ E ⃗ · dA ⃗ = 

Q in 

ɛ 0 

Electric potential, V = U/q 0 [V] where U 

is the electrostatic potential energy 

∆V = V f − V i = − ∫ f 

i 

⃗E · d⃗s 

⃗E from V: E s = − ∂V 

∂s 

V of point charge: V = 1 q 

4πɛ 0 r 

System of 2 charges: U 12 = V 1 q 2 = 1 q 1 q 2 

4πɛ 0 r 

for several charges: U = U 12 + U 13 + U 23 + ... 

Capacitance: C = Q/∆V C [F] 

parallel-plate capacitor: C = ɛ 0A 

d 

in series: 1/C eq = ∑ i 1/C i 

in parallel: C eq = ∑ i C i 

U C = 1C(∆V 2 C) 2 

u E = 1ɛ 2 0E 2 

with dielectrics: ɛ 0 → ɛ = κɛ 0 , 

dielectric constant: κ = ɛ/ɛ 0 

Current: I = dQ/dt [A] 

Current density: J ⃗ : | J| ⃗ = I/A in direction of E. ⃗ 

v d is drift speed: n e is conduction e − /m 3 

J = n e ev d 

Resistance: R = ∆V R /I [Ω] 

Ohms law ⇒ R independent of ∆V R 

R = ρL/A units of resistivity ρ are [Ωm]. 

Conductivity σ = 1/ρ [Ω −1 m −1 ] 

J = σE J = n e ev d 

1 

R in parallel: 

R eq 

= ∑ 1 

i R i


R in series: R eq = ∑ i R i 

Power, P = iV (general case). 

If Ohm’s law holds: P = i 2 R = V 2 /R 

emf: E = dW 

dQ 

[V] 

Kirchhoff’s Rules: 

∑ 

Loop: i ∆V i = 0 

∑ 

Junction i I i = 0 

RC circuit: 

Discharging: Q(t) = Q 0 e −t/τ τ = RC [s] 

Charging: Q(t) = Q max (1 − e −t/τ ) Q max = CE 

Magnetic fields: Force from B ⃗ 

moving charge: Fon ⃗ q = q⃗v × B ⃗ 

wire current: Fwire ⃗ = I ⃗ l × B ⃗ 

2 parallel wires: F || wires = µ 0lI 1 I 2 

2πd 

(|| attract, anti|| repel) 

∮ 

Ampere’s Law: ⃗B · d⃗s = 

dΦ µ0 I through + µ 0 ɛ e 0 dt 

long straight wire: B ⃗ µ = 0 I 

2π d 

(tangent to circle,rhr) 

B solenoid = µ 0 nI where n = N/l 

Biot-Savart Law, current element I∆⃗s: 

⃗B = µ 0 I∆⃗s×ˆr 

4π r 2 

Magnetic dipoles: 

loop’s magnetic dipole moment: ⃗µ = NIA (rhr I) 

mag field on axis: Bloop ⃗ = µ 0 2⃗µ 

4π z 3 

Torque on current loop: ⃗τ = ⃗µ × B ⃗ 

Potential energy of magnetic dipole: U = −⃗µ · ⃗B 

Magnetic Flux through loop: 

Φ m = N ∫ ⃗B · dA 

⃗ [Wb] 

loop 

Faraday’s Law: E = ∣ ∣ ∣ 

dΦ m ∣∣ 

dt 

with direction of induced current such that 

induced B ⃗ will oppose the change in Φ m . 

Induced electric field: E = ∮ E ⃗ · d⃗s =− dΦ m 

dt 

Inductance: L [H] =[Wb/A] 

solenoid: L = Φ m 

I 

= µ 0N 2 A 

l 

E coil = L ∣ dI 

∣ direction from Lenz’s Law 

dt 

∆V L = −L dI 

dt 

U inductor = 1 2 LI2 u B = 1 

2µ 0 

B 2 

LR circuit: I = I 0 e −t/τ τ = L R 

frequency f = ω/2π (Hz) 

period T = 1/f [s] 

Spring-block system: 

U = 1 2 kx2 

x displacement, k spring constant ω = 

√ k 

m 

LC Circuit: ω = √ 1 

LC 

Q(t) = Q 0 cos ωt 

AC circuits: 

capacitive reactance: X C = 1/(ωC) [Ω] 

inductive reactance: X L = ωL [Ω] 

I rms = I max / √ 2 V rms = V max / √ 2 

E rms = E max / √ 2 P ave = IrmsR 

2 

Travelling waves: 

v = λf k = 2π/λ ω = vk 

D(x, t) = A sin(kx − ωt + φ 0 ) 

Electromagnetic waves: 

E = E max sin(kx − ωt), B = B max sin(kx − ωt) 

c = 1/ √ ɛ 0 µ 0 E = cB ⃗ E ⊥ ⃗ B 

Poynting vector: ⃗ S = 

1 

µ 0 

⃗ E × ⃗ B [W/m 2 ] 

I = S ave = E 2 rms/(cµ 0 ) 

index of refraction: n = c/v 

Malus’s Law 

initially unpolarized: I = 1I 2 0 

initially polarized: I = I 0 cos 2 θ 

Error Propagation Equations 

[ ( 

σ z = ∂z 

σ ) 2 ( 

∂x x + ∂z 

σ ) ] 1 2 2 

∂y y + ... 

for z = ax + by − cu + ... 

σ z = [ (aσ x ) 2 + (bσ y ) 2 + (cσ u ) 2 + ... ] 1 2 

for z = Ax n y m u −p 

σ z = z [ (nσ x /x) 2 + (mσ y /y) 2 + (pσ u /u) 2 + ... ] 1 2 

oscillatory motion: x(t) = A cos(ωt + φ 0 ) 

angular frequency ω [rad/s] 

Last edited: 2010-12-19 12:12:00

PHYS 1004 Course Outline 1 Calendar description and prerequisites

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