### Course Introduction

The physics of the universe appears to be dominated by the effects of four fundamental forces: gravity, electromagnetism, weak nuclear forces, and strong nuclear forces. These forces control how matter, energy, space, and time interact to produce our physical world. All other forces, such as the force you exert in standing up, are ultimately derived from these fundamental forces.

We have direct daily experience with two of these forces: gravity and electromagnetism. Consider, for example, the everyday sight of a person sitting on a chair. The force holding the person on the chair is gravitational, and that gravitational force balances with material forces that "push up" to keep the individual in place. These forces are the direct result of electromagnetic forces on the nano-scale. On a larger stage, gravity holds the celestial bodies in their orbits, while we see the universe by the electromagnetic radiation (light, for example) with which it is filled. The electromagnetic force also makes possible the advanced technology that forms much of the basis for our civilization. Televisions, computers, smartphones, microwave ovens, and even the humble light bulb are made possible by control of electromagnetism. The average physics major will spend more time understanding and applying the concept of electromagnetic force than he or she will spend studying any other type of force.

The classical (i.e., non-quantum) theory of electromagnetism was first published by James Clerk Maxwell in his 1873 textbook A Treatise on Electricity and Magnetism. A host of scientists during the nineteenth century carried out the work that ultimately led to Maxwell's electromagnetism equations, which is still considered one of the triumphs of classical physics. Maxwell's description of electromagnetism, which demonstrates that electricity and magnetism are different aspects of a unified electromagnetic field, holds true today. In fact, Maxwell's equations are consistent with relativity, which was not theorized until 30 years after Maxwell completed his equations.

In this course, we will first learn about waves and oscillations in extended objects using the classical mechanics that we learned about in PHYS101. We will also establish the sources and laws that govern static electricity and magnetism. A brief look at electrical measurements and circuits will help us understand how electromagnetic effects are observed, measured, and applied. We will then see how Maxwell's equations unify electric and magnetic effects and how the solutions to Maxwell's equations describe electromagnetic radiation, which will serve as the basis for understanding all electromagnetic radiation, from very low frequency, long wavelength radio waves to the most powerful astrophysical gamma rays. We will briefly study optics, using practical models largely consistent with the predictions of Maxwell's equations but that are easier to use. Finally, this course provides a brief overview of Einstein's theory of special relativity. We will assume that you have a basic knowledge of calculus. You may decide to refresh your knowledge with Saylor Academy's MA101 course, but the most important concepts from vector calculus and differential equations will be covered in Unit 1 of this course.

This course will require you to complete a number of problems. Unlike mechanics, most of the phenomena encountered in the field of electromagnetism are not found in everyday experience - at least, not in a form that makes the actual nature of the phenomena clear. As a result, learning electromagnetism involves developing intuition about a rather unintuitive area of physics. In the end, developing physical intuition is less about getting a right answer than it is about getting a wrong answer and then understanding why it is wrong. In an ideal situation, this course would require you to both work out problems concerning the phenomena and observe various important phenomena in the laboratory. However, because this is an online course, we do not have the luxury of lab sessions. We have included a number of interactive demonstrations to compensate for this. When you approach a problem, try to work out the size of those quantities that clarify the basic nature of the question proposed. Thinking of these numbers as data from an ideal laboratory will help you develop a sense of how electromagnetism works – a sense that most people do not get from the mathematical description of the physics.

### Unit 1: Selected Topics in Vector Calculus and Differential Equations

This unit contains mathematical background necessary to solve many problems in the study of electromagnetism. The reading in subunit 1.1 provides a formal summary of the topics in vector algebra that you should already know from the previous study of geometry, trigonometry, and mechanics. Use this subunit as a review, or refer to it as necessary as you work through the rest of the course. The resources in subunit 1.2 explain how to set up and take the integrals in two and three dimensions, including the line and surface integrals. These resources will help you to understand the physical concepts involving these kinds of integrals, such as flux and circulation. Finally, subunit 1.3 introduces basic differential equations, which you will come across frequently throughout the course.

**Completing this unit should take you approximately 13 hours.**### Unit 2: Mechanical Vibrations and Waves in Extended Objects

In PHYS101, we learned how to describe the motion of particle-like masses using classical mechanics. We will start PHYS102 by examining how objects of size – length, width, depth – behave. We will focus on vibrating systems and the propagation of mechanical waves through media; think of ripples traveling outward from a stone dropped into water. This course will also lay the basic foundation for the development of a classical theory of mechanics for extended solids.

**Completing this unit should take you approximately 7 hours.**### Unit 3: Electrostatics

We are now beginning our study of electricity and magnetism. We will discover that electricity and magnetism are two different aspects of the same phenomenon, which is usually referred to as electromagnetism. Our starting place will be electrostatics or, more simply, the rules governing the behavior of static charges. The first experiments on electrical phenomena were carried out by our friend from PHYS101, Thales of Miletus. He observed that one could generate a static charge on amber by rubbing it with wool.

**Completing this unit should take you approximately 20 hours.**### Unit 4: Electronic Circuit Theory

Although the study of electric and magnetic fields is interesting in and of itself, it may not seem directly useful in the real world. However, the interplay between these phenomena is responsible for much of the technology you see in your everyday life. For example, all electronics apply various features of electromagnetism, so that computers, HDTV, iMacs and iPads, smartphones, motors, fans, lights, and so on are applied electromagnetic devices. In this unit, we will take a quick look at the foundations of electronics, while at the same time adding to our understanding of electromagnetism.

**Completing this unit should take you approximately 8 hours.**### Unit 5: Magnetism

In Unit 3, we studied electric charges, potentials, and fields. We will now take a look at an important effect of moving charges: magnetism. Thales of Miletus set the stage for the scientific exploration of magnetism back in Ancient Greek times, when magnetism could only be observed via the behavior of natural magnets, called lodestones. Hans Christian Oersted first noted the relationship between moving electric charges and magnetism much later, when he accidentally discovered that an electric current could deflect a nearby compass needle in 1820. Forty-five years after Oersted made this observation, James Clerk Maxwell united electrical and magnetic phenomena into four reasonably simple equations known since as Maxwell's Equations.

**Completing this unit should take you approximately 7 hours.**### Unit 6: Electromagnetic Induction

In Units 3 and 5, you learned that stationary electric charges produce electric field, and moving electric charges (that is, electric current) produce magnetic field. In this unit, you will find out that the reverse is also true: changing magnetic flux produces electric field, or induces electric current. This is the phenomenon of the electromagnetic induction, which is a basic principles in such devices as generators of electric power, electric motors, and transformers.

**Completing this unit should take you approximately 13 hours.**### Unit 7: Maxwell's Equations

At this point in the course, we have developed the mathematical structure for and a general understanding of all of Maxwell's Equations. Now we want to sit back and summarize our findings by identifying what they are, what they mean, and how we can use them.

There are four Maxwell equations that describe all classical electromagnetism. Maxwell's equations take on a particularly simple form when describing the behavior of electric and magnetic fields in regions devoid of matter; that is, in a vacuum. (Note that for most purposes, air is close enough to being a vacuum that the presence of an atmosphere can be ignored.) These are Maxwell's free space equations.

There are four Maxwell free space equations. These include the two flux equations - the electric and magnetic forms of Gauss' law. These state that the electric or magnetic flux through a closed surface is proportional to the electric or magnetic charge enclosed within that surface. Note that in the magnetic case, there are no magnetic charges (also called magnetic monopoles), so that the magnetic flux through and closed surface is zero.

The other two free space Maxwell's equations are Faraday's Law of Induction and a modified version of Ampere's Circuital Law. Once again, these electric and magnetic equations have similar formalisms, thereby emphasizing the close relationship of the electric and magnetic fields. Faraday's Law of Induction states that the induced EMF in any closed circuit is proportional to the time rate of change of the magnetic flux through the circuit, while Ampere's Law states that the integrated magnetic field around a closed curve is proportional to the currents passing through a surface bounded by the curve. Maxwell's main contribution (beyond realizing that these four equations provided a complete theory of electromagnetism) was the discovery and description of the displacement current, which is a source of the magnetic field associated with the rate of change of the electric displacement field in a region.

Inside materials, Maxwell's Equations are modified by the electric permittivity and magnetic permeability of the materials, but they remain the basis for the classical model of electromagnetism. In this unit, we will concentrate on Maxwell's Equations as a single theory that unites the half-century of previous work on electromagnetism.

**Completing this unit should take you approximately 8 hours.**### Unit 8: Optics

An optical phenomenon involves the interaction between electromagnetic waves and matter. We will focus on visible, infrared, and ultraviolet light, but much of the study of optics will apply to some extent to radio waves and x-rays. The complete study of optics involves enormously complex mathematics, a thorough understanding of both classical and quantum optical effects, and a great deal of ingenuity for success.

For the purposes of this course, optics will be limited to the classical description of electromagnetism provided by Maxwell's equations: the full wave optics. Even this level of description is quite complicated for most optical phenomena, so we will apply simplified models to develop a basic understanding of how optics works. In geometric optics, we assume that all light travels in straight lines. In paraxial optics, we assume that all optical systems handle light rays near a symmetry axis of the optical system, which allows us to largely ignore aberration, a vast array of terribly complex optical effects. In theory, the full wave optics provides the most complete picture of optics possible with a classical description, but the most fascinating optical effects are (arguably) intrinsically quantum mechanical in nature. (Patience is a virtue.)

**Completing this unit should take you approximately 11 hours.**### Unit 9: Special Relativity

The physical descriptions we have studied to this point were based on a notion of absolute space and time. A model for this point of view was that space is filled everywhere by a continuous medium called the ether. Light and other forms of electromagnetic radiation were waves in this ether, analogous to sound waves in air. All other phenomena were to be understood as various manifestations of Maxwell's electromagnetism, which was originally based on a mechanical model of ether. It seemed reasonable that the 19th Century 'theory of everything' could be tied down by measuring the 'elastic' properties of the ether.

Toward the end of the 1800s, however, this model became associated with more and more hastily patched cracks. The detailed history of the gradual realization that ether models were not quite right is complex and technical. However, there is one rather clear indication of trouble. In 1887, Albert Michelson and Edmund Morley of the Case Institute (now Case Western University) performed an experiment using an optical interferometer in which they compared the speed of light in two beams traveling at right angles to each other. If the speed of light relative to the ether was always the same, the measured speed of light would be larger or smaller depending on the direction the experiment was traveling through the ether. The motion of the Michelson-Morley experiment was provided by the rotation of the Earth on its axis and the orbital motion of the Earth around the Sun, as well as the absolute velocity (if any) of the Sun relative to the ether.

They expected to see both diurnal changes and yearly changes in the relative velocities of light in the two paths. True, the changes expected by classical ether theory were small (on the order of 0.01% of the velocity of light), but the Michelson-Morley interferometer was able to detect velocity changes about 6-7 times smaller. To the surprise of all, there were no changes whatever observed. This experiment was widely repeated, using constantly improving equipment - a new version of the experiment carried out in 2002 established that the velocity of light is constant to better than 1 part in 1015 - one of the most precise physical measurements ever accomplished.

The explanation of the Michelson-Morley null result was length contraction, as developed by Hendrik Lorentz and George Francis FitzGerald. Length contraction explained the Michelson-Morley result, the idea being that matter is held together by electromagnetic forces (true), and so the actual size of objects will change with motion through the ether (false). In the end, it was Albert Einstein's formulation of the theory of Special Relativity that gave us a consistent explanation of all such phenomena. His primary postulate was to accept that the speed of light and the laws of physics are constant in all reference frames – including reference frames that are in motion. Oddly, despite the fact that Einstein's theory completely explained the Michelson-Morley result, he took no motivation for his theory from that experiment.

**Completing this unit should take you approximately 10 hours.**### Final Exam