Essential Trig-based Physics Study Guide Workbook: Electricity and Magnetism

419 Pages·2017·13.782 MB·English

by  Chris McMullen

Essential Trig-based Physics Study Guide Workbook
Volume 2: Electricity and Magnetism
Learn Physics Step-by-Step

Chris McMullen, Ph.D.
Physics Instructor
Northwestern State University of Louisiana
Copyright © 2017 Chris McMullen, Ph.D.
www.monkeyphysicsblog.wordpress.com
www.improveyourmathfluency.com
www.chrismcmullen.wordpress.com
Zishka Publishing
All rights reserved.

ISBN: 978-1-941691-10-6

Textbooks > Science > Physics
Study Guides > Workbooks> Science
CONTENTS
Introduction 5
Chapter 1 – Coulomb’s Law 7
Chapter 2 – Electric Field 15
Chapter 3 – Superposition of Electric Fields 23
Chapter 4 – Electric Field Mapping 41
Chapter 5 – Electrostatic Equilibrium 55
Chapter 6 – Gauss’s Law 65
Chapter 7 – Electric Potential 89
Chapter 8 – Motion of a Charged Particle in a Uniform Electric Field 93
Chapter 9 – Equivalent Capacitance 99
Chapter 10 – Parallel-plate Capacitors 113
Chapter 11 – Equivalent Resistance 119
Chapter 12 – Circuits with Symmetry 133
Chapter 13 – Kirchhoff’s Rules 143
Chapter 14 – More Resistance Equations 155
Chapter 15 – Logarithms and Exponentials 165
Chapter 16 – RC Circuits 171
Chapter 17 – Bar Magnets 179
Chapter 18 – Right-hand Rule for Magnetic Force 183
Chapter 19 – Right-hand Rule for Magnetic Field 193
Chapter 20 – Combining the Two Right-hand Rules 201
Chapter 21 – Magnetic Force 207
Chapter 22 – Magnetic Field 219
Chapter 23 – Ampère’s Law 235
Chapter 24 – Lenz’s Law 251
Chapter 25 – Faraday’s Law 265
Chapter 26 – Inductance 283
Chapter 27 – AC Circuits 295
Hints, Intermediate Answers, and Explanations 325
EXPECTATIONS
•
Prerequisites:
As this is the second volume of the series, the student should already have studied
the material from first-semester physics, including uniform acceleration, vector
addition, applications of Newton’s second law, conservation of energy, Hooke’s law,
•
and rotation.
The student should know some basic algebra skills, including how to combine like
terms, how to isolate an unknown, how to solve the quadratic equation, and how to
•
apply the method of substitution. Needed algebra skills were reviewed in Volume 1.
The student should have prior exposure to trigonometry. Essential trigonometry
skills were reviewed in Volume 1.
•
Use:
This book is intended to serve as a supplement for students who are attending
•
physics lectures, reading a physics textbook, or reviewing physics fundamentals.
The goal is to help students quickly find the most essential material.
•
Concepts:
Each chapter reviews relevant definitions, concepts, laws, or equations needed to
•
understand how to solve the problems.
This book does not provide a comprehensive review of every concept from physics,
but does cover most physics concepts that are involved in solving problems.
•
Strategies:
Each chapter describes the problem-solving strategy needed to solve the problems
•
at the end of the chapter.
This book covers the kinds of fundamental problems which are commonly found in
standard physics textbooks.
•
Help:
Every chapter includes representative examples with step-by-step solutions and
explanations. These examples should serve as a guide to help students solve similar
•
problems at the end of each chapter.
Each problem includes the main answer(s) on the same page as the question. At the
back of the book, you can find hints, intermediate answers, directions to help walk
you through the steps of each solution, and explanations regarding common issues
that students encounter when solving the problems. It’s very much like having your
own physics tutor at the back of the book to help you solve each problem.
INTRODUCTION
The goal of this study guide workbook is to provide practice and help carrying out essential
problem-solving strategies that are standard in electricity and magnetism. The aim here is
not to overwhelm the student with comprehensive coverage of every type of problem, but
to focus on the main strategies and techniques with which most physics students struggle.
This workbook is not intended to serve as a substitute for lectures or for a textbook,
but is rather intended to serve as a valuable supplement. Each chapter includes a concise
review of the essential information, a handy outline of the problem-solving strategies, and
examples which show step-by-step how to carry out the procedure. This is not intended to
teach the material, but is designed to serve as a time-saving review for students who have
already been exposed to the material in class or in a textbook. Students who would like
more examples or a more thorough introduction to the material should review their lecture
notes or read their textbooks.
Every exercise in this study guide workbook applies the same strategy which is solved
step-by-step in at least one example within the chapter. Study the examples and then
follow them closely in order to complete the exercises. Many of the exercises are broken
down into parts to help guide the student through the exercises. Each exercise tabulates
the corresponding answers on the same page. Students can find additional help in the hints
section at the back of the book, which provides hints, answers to intermediate steps,
directions to walk students through every solution, and explanations regarding issues that
students commonly ask about.
Every problem in this book can be solved without the aid of a calculator. You may use
a calculator if you wish, though it is a valuable skill to be able to perform basic math
without relying on a calculator.
The mathematics and physics concepts are not two
completely separate entities. The equations speak the
concepts. Let the equations guide your reasoning.

— Chris McMullen, Ph.D.
Essential Trig-based Physics Study Guide Workbook
1 COULOMB’S LAW

Relevant Terminology
Electric charge – a fundamental property of a particle that causes the particle to experience
a force in the presence of an electric field. (An electrically neutral particle has no charge
and thus experiences no force in the presence of an electric field.)
Electric force – the push or pull that one charged particle exerts on another. Oppositely
charged particles attract, whereas like charges (both positive or both negative) repel.
Coulomb’s constant – the constant of proportionality in Coulomb’s law (see below).
Coulomb’s Law

According to Coulomb’s law, any two objects with charge attract or repel one another with
an electrical force that is directly proportional to each charge and inversely proportional to
the square of the separation between the two charges:
|𝑞𝑞1||𝑞𝑞2|
𝐹𝐹𝑒𝑒 = 𝑘𝑘 2
𝑅𝑅
The absolute values around each charge indicate that the magnitude of the force is positive.
Note the subscript on 𝐹𝐹𝑒𝑒: It’s 𝐹𝐹 sub 𝑒𝑒 (not 𝐹𝐹 times 𝑒𝑒). The subscript serves to distinguish
electric force (𝐹𝐹𝑒𝑒) from other kinds of forces, such as gravitational force (𝐹𝐹𝑔𝑔). The
proportionality constant in Coulomb’s law is called Coulomb’s constant (𝑘𝑘):
2 2
9 N∙m 9 N∙m
𝑘𝑘 = 8.99×10 2 ≈ 9.0×10 2
C C 2
9 N∙m
2
In this book, we will round Coulomb’s constant to 9.0×10 C such that the problems
may be solved without using a calculator. (This rounding is good to 1 part in 900.)
Symbols and SI Units

Symbol Name SI Units
𝐹𝐹𝑒𝑒 electric force N
𝑞𝑞 charge C
𝑅𝑅 separation m
2 3
N∙m kg∙m
𝑘𝑘 Coulomb’s constant C2 or C2∙s2
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Chapter 1 – Coulomb’s Law
Notes Regarding Units

The SI units of Coulomb’s constant (𝑘𝑘) follow by solving for 𝑘𝑘 in Coulomb’s law:
2
𝐹𝐹𝑒𝑒𝑅𝑅
𝑘𝑘 =
𝑞𝑞1𝑞𝑞2
2 2
N∙m 𝐹𝐹𝑒𝑒𝑅𝑅
2
The SI units of 𝑘𝑘 equal C because these are the SI units of 𝑞𝑞1𝑞𝑞2. This follows since the SI
unit of electric force (𝐹𝐹𝑒𝑒) is the Newton (N), the SI unit of charge (𝑞𝑞) is the Coulomb (C),
and the SI unit of separation (𝑅𝑅) is the meter (m). Recall from first-semester physics that a
Newton is equivalent to:
kg∙m
1 N = 1 2
2 s 3
N∙m kg∙m
2 2 2
Plugging this into C , the SI units of 𝑘𝑘 can alternatively be expressed as C ∙s .
Essential Concepts

The matter around us is composed of different types of atoms. Each atom consists of
•
protons and neutrons in its nucleus, surrounded by electrons.
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Protons have positive electric charge.
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Neutrons are electrically neutral.
Electrons have negative electric charge.
•
Whether two charges attract or repel depends on their relative signs:
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Opposite charges attract. For example, electrons are attracted to protons.
Like charges repel. For example, two electrons repel. Similarly, two protons repel.
•
The charge of an object depends on how many protons and electrons it has:
If the object has more protons than electrons (meaning that the object has lost
•
electrons), the object has positive charge.
If the object has more electrons than protons (meaning that the object has gained
•
electrons), the object has negative charge.
If the object has the same number of protons as electrons, the object is electrically
neutral. Its net charge is zero.
(Atoms tend to gain or lose valence electrons from their outer shells. It’s not easy to gain
or lose protons since they are tightly bound inside the nucleus of the atom. One way for
objects to become electrically charged is through rubbing, such as rubbing glass with fur.)
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Some materials tend to be good conductors of electricity; others are good insulators.
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Charges flow readily through a conductor. Most metals are good conductors.
Charges tend not to flow through an insulator. Glass and wood are good insulators.
When two charged objects touch (or are connected by a conductor), charge can be
transferred from one object to the other. See the second example in this chapter.
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Essential Trig-based Physics Study Guide Workbook
e

n
pppp ppppppppp

nnnnn e
e n nnnn

pp ppppppp
n

e

Metric Prefixes
Since a Coulomb (C) is a very large amount of charge, we often use the following metric
prefixes when working with electric charge.
Prefix Name Power of 10
−3
m milli 10
−6
µ micro 10
−9
n nano 10
−12
p pico 10
Note: The symbol µ is the lowercase Greek letter mu. When it is used as a metric prefix, it
is called micro. For example, 32 µC is called 32 microCoulombs.
Algebra with Powers
It may be helpful to recall the following rules of algebra relating to powers:
𝑎𝑎
𝑎𝑎 𝑏𝑏 𝑎𝑎+𝑏𝑏 𝑥𝑥 𝑎𝑎−𝑏𝑏 −𝑎𝑎 1 1 𝑎𝑎
𝑥𝑥 𝑥𝑥 = 𝑥𝑥 , 𝑏𝑏 = 𝑥𝑥 , 𝑥𝑥 = 𝑎𝑎 , −𝑎𝑎 = 𝑥𝑥
𝑥𝑥 𝑥𝑥 𝑥𝑥
0 𝑎𝑎 𝑏𝑏 𝑎𝑎𝑏𝑏 𝑏𝑏 𝑏𝑏 𝑏𝑏
𝑥𝑥 = 1 , (𝑥𝑥 ) = 𝑥𝑥 , (𝑎𝑎𝑥𝑥) = 𝑎𝑎 𝑥𝑥
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Chapter 1 – Coulomb’s Law
Coulomb’s Law Strategy
•
How you solve a problem involving Coulomb’s depends on which kind of problem it is:
In this chapter, we will focus on the simplest problems, which involve two charged
objects attracting or repelling one another. For simple problems like these, plug the
known values into the following equation and solve for the unknown quantity.
|𝑞𝑞1||𝑞𝑞2|
𝐹𝐹𝑒𝑒 = 𝑘𝑘 2
𝑅𝑅
𝑞𝑞1 𝑞𝑞2
𝑅𝑅

o
Look at the units and wording to determine which symbols you know.

o
A value in Coulombs (C) is electric charge, 𝑞𝑞.

o
A value in meters (m) is likely related to the separation, 𝑅𝑅.
o A value in N is a force, such as electric force, 𝐹𝐹𝑒𝑒.
2
9 N∙m
2
You should know Coulomb’s constant: 𝑘𝑘 = 9.0×10 C .
If two charged objects touch or are connected by a conductor, in a fraction of a
second the excess charge will redistribute and the system will attain static
equilibrium. The two charges will then be equal: The new charge, 𝑞𝑞, will equal
2
𝑞𝑞1+𝑞𝑞2 𝑞𝑞
• 𝑞𝑞 = 2 . Coulomb’s law then reduces to 𝐹𝐹𝑒𝑒 = 𝑘𝑘𝑅𝑅2.
If a problem gives you three or more charges, apply the technique of vector addition,
•
as illustrated in Chapter 3.
If a problem involves other forces, like tension in a cord, apply Newton’s second law,
•
as illustrated in Chapter 5.
If a problem involves an electric field, 𝐸𝐸, (not to be confused with electric force, 𝐹𝐹𝑒𝑒,
or electric charge, 𝑞𝑞), see Chapter 2, 3, or 5, depending on the nature of the problem.
Inverse-square Laws
Coulomb’s law and Newton’s law of gravity are examples of inverse-square laws: Each of
1
2
these laws features a factor of 𝑅𝑅 . (Newton’s law of gravity was discussed in Volume 1.)
|𝑞𝑞1||𝑞𝑞2| 𝑚𝑚1𝑚𝑚2
𝐹𝐹𝑒𝑒 = 𝑘𝑘 2 , 𝐹𝐹𝑔𝑔 = 𝐺𝐺 2
𝑅𝑅 𝑅𝑅
Coulomb’s law has a similar structure to Newton’s law of gravity: Both force laws involve a
1
proportionality constant, a product of sources (|𝑞𝑞1||𝑞𝑞2| or 𝑚𝑚1𝑚𝑚2), and 𝑅𝑅2.
10