This is the second post in my series on self studying physics. The first post of the series on introductory mechanics can be found here.
In the prior post in this series I described how I decided to self study physics after completing my math degree. Discovering the excellent guide “So You Want to Learn Physics” (which my titles are a reference to) by Susan Rigetti gave me a roadmap with which to fulfill my aspirations. The material covered in this post represents the second unit in her guide: electrostatics. Quite honestly I am as surprised as I am happy to have completed the first two units of the guide. Many of my prior self study experiences have been either two week sprints or on and off again affairs. This is my first project that has involved daily effort over a period of months. I am at once proud of my accomplishments and humbled by the depth and beauty of physics.
Writing these posts gives me the opportunity to document and share my experience with the material. Seeing content, in written or video form, from others as they learned material has always comforted and inspired me. In any case, by describing at length the material covered in each unit I hope to provide guidance about which sections are important, where the trouble spots are, and what the culminating takeaways should be. Naturally my perspectives will continue to evolve long after completing this material as I see how it is applied elsewhere, so I may return to make updates. In the future, when I have completed additional material and gained a more holistic understanding of physics, I will integrate all these posts into a polished and cohesive whole.
Without further ado, here we go.
Prerequisites
To succeed in this material you must have a solid grasp of the basic mechanics material. This includes concepts such as force, energy, motion in three dimensions, and rotational motion. If you have completed the material in the previous post or have taken a course in university physics then you will have the necessary physics knowledge. Mathematically, you must be comfortable with manipulating algebraic expressions and using trigonometry. While you could likely make it through the mechanics material with only a cursory understanding of calculus, to succeed here you must be comfortable with differential and integral calculus (normally calculus 1 and 2 in the US), and ideally up to the level of multivariable and vector calculus (typically called Calculus 3 in US schools). While you could get through without having to compute any nasty integrals yourself (unfortunately by avoiding many of the more interesting problems), many of the concepts rely on line and surface integrals for explanation.
Content
The book I used for this course was again “University Physics with Modern Physics” by Young and Freedman, 14th edition. There are other editions of this book which would likely work just as well so use whichever edition you can get your hands on. The material is covered in chapters 21 through 32. It may feel odd skipping over chapters 15 to 20 (for now) but I did not notice any loss of continuity. Chapter 31 on alternating current and chapter 32 on electromagnetic waves could benefit from a deeper understanding of waves (chapters 15 and 16) but the material on periodic motion in chapter 14 covers the basics you need. If you are following the material in the same order I am, then you will be doing a deep dive into vibrations and waves in the next course.
A course in electromagnetism involves the study of electricity and magnetism. These phenomena are like two sides of the same coin since both are caused by the electromagnetic force, which is one of the four fundamental forces of our universe. Electromagnetism plays a role in everything from the smallest atoms to the functioning of stars. It also critical to our modern technological civilization. Students studying physics will normally cover electromagnetism twice in their studies (or three times if they go to graduate school). The first time is a lower division course often called physics 2 or electrostatics which is almost always the second course physics students take right after introductory mechanics. The second time is an upper division one or two course sequence in electrodynamics. Electrostatics refers to situations where there are no moving charges while electrodynamics refers to situations involving moving charges. Calling this course electrostatics is not quite accurate because it does cover some situations that involve moving charges; however, these simple situations mostly involving just one non-stationary charge.
This material can be roughly divided into three sections. Chapters 21 through 26 focus exclusively on electric forces, ignoring magnetic forces. Chapters 27 through 30 focus on magnetism but also involve electric forces since there is an intimate link between moving charges and magnetic forces. The final two chapters, 31 and 32, bring all of the previous material together to discuss alternating current and electromagnetic waves. Starting mostly with chapter 24 and running throughout there is coverage of circuits and technology.
In the course of studying this material, I came across the wonderful 8.02x MIT lectures by Walter Lewin. The playlist for these videos can be found here. The lectures do not line up perfectly with the order of material in this book, but that is okay. His lectures involve a great number of demonstrations which are informative and entertaining.
I will now give a brief description of each chapter and what it covered. Afterwards, I will discuss how I studied and my final thoughts on the course.
Chapter 21: The first chapter covers the basics of electric charge and electric force. This material is reminiscent of the coverage of gravity in chapter 13, since both gravity and the electric force are inverse square laws. This means the strength of the force drops off with the square of the distance. However, while gravity is an attractive force, the electric force can either repel or attract depending on whether the charges are the same or different. This relationship is governed by Coulomb’s Law which is almost identical in form to Newton’s Law of gravitation. An important concept introduced here is that of electric fields, which provides a nice way of visualizing and thinking about electric force.
Chapter 22: The second chapter introduces the concept of electric flux and Gauss’s Law. Electric flux is useful for calculations related to electric fields but it also becomes extremely important later once magnetism is introduced. A good analogy for flux is to imagine fluid flowing through a surface. The ‘fluid’ in this analogy has to do with the electric field lines that were introduced in the previous chapter. Gauss’s law relates the electric flux through a closed surface to the charge enclosed by that surface. This is the first of Maxwell’s four equations, but you will get to that point later on.
Chapter 23: In this chapter, you will learn the concept of electric potential energy. Electric potential energy has many similarities to gravitational potential energy. This has several uses. First and foremost here it allows for situations to be analyzed in terms of their energy in addition to the forces acting within the system. Later on (as in, starting next chapter) electric potential plays a particularly important role as the voltage in electric circuits. You will notice in this chapter a number of examples and problems relating to oppositely charged parallel plates. You should do these examples and problems, since this exact scenario will be crucial in the next chapter on capacitors.
Chapter 24: Here we are introduced to the first of many circuit elements which will be covered: the capacitor. A capacitor is a way of storing electric potential energy. In its most basic form a capacitor consists of two oppositely charged parallel plates. Since the plates are oppositely charged, there is a potential difference between them. This is why I mentioned that doing examples and problems relating to parallel plates in the previous chapter would be so important. The chapter explores how different ways of constructing a capacitor or altering its properties will affect its energy storage capacity. It also covers what happens when capacitors are connected in series or parallel configurations. Examples, asides, and problems for the chapter highlight various applications of capacitors in microphones, touchscreens, fuel tanks, and more. This chapter (and all the other ones too) turned me into an applications ‘fun fact’ machine that likely annoyed my friends and family.
Chapter 25: In this chapter, we are introduced to the concept of electric current and its behavior in circuits. While talking about capacitors, the notion of potential differences causing charge to move from one location to another hung in the background. This is current. Unfortunately, due to some rather poor choices for notation by historical scientists (like choosing the sign of the electron to be negative) some things related to current feel ‘backward’. For example, the direction of current is opposite of the direction in which the electrons move. Talking about how current moves through materials leads to resistance, where some materials are better conductors than others. Finally, the chapter describes electromotive force (emf), which is what provides the potential difference in a circuit which causes the current. A source of emf would be a battery or a generator. This lets us put together our first complete circuits near the end of the chapter. A key tool for analyzing circuits is the circuit diagram, so when the chapter describes the symbols for circuit diagrams, pay attention! I found the exercises in this chapter to be quite enjoyable.
Chapter 26: This chapter on direct-current circuits is all about applying what has been learned so far. First, you will learn how to combine resistors in series and in parallel, much like what was done with capacitors in chapter 24. However, resistors behave completely differently when hooked up in this way, so you will want to pay attention to the differences. Next, you will learn about Kirchhoff’s Rules, which are simple rules that, when combined with the series and parallel rules, will allow you to analyze any circuit. For example, you may have a network of emf sources, resistors, capacitors, or other components and want to know how much current is flowing through each of the components, or the voltage drops over the components. Essentially, the way this works is by converting the circuit into a system of linear equations. Then you can use (basic) linear algebra to solve the system and get the information you desire. Naturally, this technique is incredibly important for any real world usage of circuits.
The middle of the chapter concerns the use of several measuring devices: ammeters, voltmeters, ohmmeters, and potentiometers. These measure current, potential, resistance, and emf, respectively. Next, you will learn about R-C circuits, which are circuits that contain a resistor and a capacitor, normally in series. What stands out about this section (and makes it a bit more challenging) is that it involves currents that vary with time as the capacitor is charged and discharged. The final section of this chapter is on power distribution systems. This includes a basic description of some of the safety features in place to prevent shorts or overloads. The exercises and problems for this chapter feature ‘complicated’ circuits. It can be intimidating to approach some of these problems (I know I was intimidated). Take it one step at a time and remember all of the various rules which can be used to simplify circuits. This is about as complicated as the circuits get until chapter 30.
This is the final chapter solely dealing with the electric force. All of the remaining chapters involve a mixture of electric and magnetic force.
Chapter 27: Here we get our first proper introduction to the magnetic force. In many ways, the first half of the chapter parallels both chapter 21 and chapter 22. Magnetism is introduced as a phenomenon and then as a field. Then we see magnetic flux. The later half of the chapter deals with a variety of applications. There are some large differences between electric and magnetic interactions. First of all, magnetic force depends on velocity: a motionless charge experiences no magnetic force while a fast moving charge experiences a large magnetic force. Second, the magnetic force depends on the direction of the velocity. The direction of the magnetic force is perpendicular to both the magnetic field and the velocity. For this reason, it is calculated with a vector cross product. Some applications include mass spectrometers, magnetic bottles, speakers, and electric motors.
Chapter 28: In the previous chapter, the magnetic fields were mostly present without explanation. In this chapter we learn where magnetic fields come from: moving charges. The bulk of the chapter is spent working out the magnetic field generated by various current configurations. I am not going to lie, this was one of the more tedious chapters to get through. But persevere and you will be rewarded with a solid understanding of how current can be used to generate magnetic fields. A helpful tool called Ampere’s law is introduced in the chapter which is able to take advantage of symmetries to simplify certain calculations. Example 28.1 is a particularly important example to work through. Attempt the problem without looking at the solution first and then read the solution carefully. The evaluation section reveals one of the tidbits which led to special relativity.
Chapter 29: While we previously saw how a moving charge (such as a current) can create a magnetic field, we now see how a magnetic field can create a current. The key is that the magnetic field must be time varying. This was initially a surprise to me and it took some time to adjust to the concept. While at first the need for a time-varying magnetic field in order to generate a current feels like a limitation it turns out to have many marvelous applications such as generators and metal detectors. The chapter ends with Maxwell’s equations and a discussion of superconductivity. Rather than represent a new topic, Maxwell’s equations essentially encapsulate everything which has been covered so far in one neat package. The four equations that make up Maxwell’s equations have already been introduced in previous chapters, but now they are brought together to provide the basis for everything the course has covered.
Chapter 30: This chapter serves as a continuation of the previous chapter. It also marks the return of more complex circuits. Chapter 30 is all about inductance and its applications. While the mutual induction between circuits can be a nuisance, mutual inductance between coils is the basis for transformers which are an integral part of the modern power grid. Coils can also experience self-inductance. This gives rise to circuit elements called inductors, the main purpose of which is to oppose rapid variations in current in a circuit. Much like the previous chapter unified the math of the course, this chapter unifies the circuit of the chapter by exploring how resistors, capacitors, and inductors interact with each other.
Chapter 31: So far, most of the material regarding circuits has involved direct current. However, most of the power systems in the world involve alternating current. This chapter covers alternating current and related topics such as resonance and radio. This is a challenging chapter and I recommend reviewing chapter 14 of this book, and possibly skimming chapter 16 as well. (If you are going in the order of material that I am, you will have read up through chapter 14 before skipping to chapter 21 to start this course.) Don’t panic is the material in this chapter (and in the next one, too) does not ‘click’ right away. It will all make more sense after a more detailed study of vibrations and waves. While direct current is (mostly) constant, alternating current is in constant flux. Since many magnetic effects are caused by a time-varying current, this makes alternating current ideal for taking advantage of these effects. This material really inspired me to want to learn more about circuits and how they are designed. At some future point I will do a course on circuits and post a link to it here.
Chapter 32: In this final chapter, the book turns from applications back to theory and we get an introduction to electromagnetic waves, aka light. Using all of the material that came before, the book shows how time-varying electric and magnetic fields can propagate themselves in the form of a wave even through empty space. This material is of particular importance for (among other things) astronomy since the main way we see the universe outside our solar system is through the light that reaches us. It is also important for things like x-rays and microwaves. Once you have completed this chapter, congratulations! You have finished ‘electrostatics’.
How I Studied
All in all, the way I studied this material was almost exactly the same as the way I studied the introductory mechanics material. I recommend reviewing what I wrote there if you have not already read it.
The main way I studied was by using the book and supplementing it with Walter Lewin’s 8.02 MIT lectures, which can be found on YouTube. I linked to the playlist earlier in this post. When watching the lectures I did not take notes and instead focused on gaining an understanding. My process for working through each chapter in the book went like this: First, gain an overall sense of the chapter by reading the section titles, looking at the review page, and skimming some of the problems. Second, give the chapter a quick read through like reading a novel, identifying some of the key points. Avoid looking at the solutions to the example problems during this step. Third, do a detailed read through while taking notes. During this stage I pay more attention to each step, rereading derivations until I understand it well. I also stop and do every example problem when I get to it, without looking at the solution first. Then I can compare my solution with the detailed solution and explanation given in the book. This is one of the most important steps in the learning process. Finally, I complete a number of the end of chapter exercises and problems. I discussed this before but the amount of exercises you do is up to you. Just remember that the more you do the better your understanding will be.
It took me roughly one hundred hours of study over six weeks to complete this material. I took over a hundred pages of notes and completed a couple hundred problems. Overall, the effort was comparable (on the same order of magnitude) as my effort in introductory mechanics.
Final Thoughts
My experience in this course was quite different from my experience with the first course. Part of it was external. When I started studying physics my methods were rather undisciplined. I was unsure if I would actually make any meaningful progress. There was a long (six month) time gap between completing the first half of the course and returning to finish the second half. However, when I picked it up again I made significant strides in organization and discipline. What I am getting at is that I had all the benefits of lessons learned in the previous course when I started this one.
Other factors were internal to the material itself. Many of the topics covered in introductory mechanics are familiar from everyday experience. We all have an intuitive sense for how (everyday) things move, fall, or slide. Despite the fact that we use electricity everyday (you are using electricity to read this post, in more ways than one), most people do not have the same level of intuition for how electricity itself works. I know I did not. My smattering of random bits of electrical knowledge had to be completely replaced as I learned the material.
The area I felt this difference the most was in the units. Basic units like seconds, meters, and kilograms are familiar and even units built from these like newtons, joules, and pascals evoke a tangible image in the mind. The units involved in electromagnetism were, at least for me, much less intuitive. Coulombs, volts and amperes are straightforward, but once you throw in farads, telsas, henries, ohms, and webers it gets a bit confusing. Of course, the difficulty of relying on intuition alone is why physics is all about using math to describe the world. So for me this course involved a lot more of letting the math be my intuition. This course used a slightly higher level of mathematics, which was expected. It was nice to dust of some calculus knowledge which I hadn’t needed in a while. Calculations for problems were longer so the time spent on solving problems made up a larger proportion of my overall time.
Overall, I loved this course and walked away from it feeling empowered to better understand the world around me. In particular, I feel better equipped to understand how technology works and policy issues relating to energy and technology.
The next course in my sequence is Vibrations and Waves. I will place a link to that post here when it is written.
Happy studying, and thanks for reading!