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Introduction to Electromagnetism

Purpose of Course  showclose

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 nanoscale.  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 Physics 101.  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 (MA101), and you will need to know the principles covered in MA102 (a co-requisite for this course) as well.

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.

Course Information  showclose

Welcome to PHYS102.  Below, please find general information on this course and its requirements.

Primary Resources: This course is comprised of a range of different free, online materials.  However, the course makes primary use of the following resources:
In addition, there will be interactive laboratory demonstrations provided by the Wolfram Demonstrations Project.

The mathematics required for this course is somewhat more advanced than for Physics 101.  This will be particularly true for Unit 5 on Maxwell’s equations.  To prepare you for this level of mathematics, several resources are provided, including a lecture on vector notation by Professor Lewin, a reading by Professor Michael Corral on vector calculus, and one by Professor Richard Fitzpatrick on vectors.

Requirements for Completion: In order to complete this course, you will need to work through each unit and all of its readings and lectures.  You will also need to complete the Final Exam.

Note that you will only receive an official grade on your Final Exam.  However, in order to adequately prepare for this exam, you will need to work through the problems assigned throughout the course.

In order to “pass” this course, you will need to earn a 70% or higher on the Final Exam.  Your score on the exam will be tabulated as soon as you complete it.  If you do not pass the exam, you may take it again.

Time Commitment: This course should take a total of 142.25 hours to complete.  Each unit includes a “time advisory” that lists the amount of time you are expected to spend on it and each of its subunits.  These should help you plan your time accordingly.  It may be useful to take a look at these time advisories and to determine how much time you have over the next few weeks to complete each unit, and then to set goals for yourself.  For example, Unit 1 should take 9.25 hours to complete.  Perhaps you can sit down with your calendar and decide to complete subunit 1.1 (a total of 3.5 hours) on Monday night; subunit 1.2 (a total of 2.25 hours) on Tuesday night; subunit 1.3 (a total of 3.5 hours) on Wednesday night; etc.  Note that the course begins with a Math Review.  This review should be completed as you work through Units 1-5 of the course; you may refer back to the resources in this review as a reference as you study subsequent units.

Tips/Suggestions: Your studyof electromagnetism will require a level of mathematics beyond that required for your previous physics courses.  It is very important that you study the pertinent math before trying to work on the problems.  Otherwise, you will find the problems very frustrating.  The primary mathematical resource is a reading by Professor Michael Corral on vector calculus, from which 98 problems are assigned.  For maximum benefit, you should work on these gradually throughout the course, making sure that you are almost through by the time you get to Unit 5 on Maxwell’s equations.  Good luck to you in this undertaking you are about to embark on.

Learning Outcomes  showclose

Upon successful completion of this course, the student will be able to:
  • Determine the magnitude and direction of a vector from its components.
  • Determine the components of a vector from its components.
  • Add and subtract vectors.
  • Calculate dot and cross products of vectors.
  • Transform Cartesian coordinates into curvilinear coordinates.
  • Evaluate line, surface, and volume integrals.
  • Describe the properties of simple harmonic motion.
  • State the laws of electromagnetism, and identify the units of the physical quantities contained in the laws.
  • Define and apply Gauss’ law.
  • Solve point charge problems involving forces, electric fields, and electric potentials.
  • Compare and contrast the electric potential and the electric field.
  • Describe the magnetic field associated with the following: a moving charge, a magnetic dipole, and a long straight current carrying wire.
  • Identify and apply the force exerted by a magnetic field on a moving charged particle.
  • Define and apply Faraday’s law of induction.
  • Define and apply Ampere’s law.
  • Define and apply Ohm’s law.
  • Use the junction and loop rules to analyze basic circuits.
  • Analyze RC, Rl, and RCL circuits.
  • Describe how Maxwell’s equations resulted in the prediction of electromagnetic wave and the realization that light was an electromagnetic wave.
  • Draw diagrams that represent applications of geometric optics to various configurations of mirrors and lenses.
  • Determine the nature, location, and magnification of images formed by lenses and mirrors.
  • Identify the postulates upon which the Special Theory of Relative is based.
  • Solve problems involving time dilation and length contraction.
  • Compare and contrast the special and general theories of relativity.
  • List the most significant consequence of Einstein’s special and general theories of relativity.

Course Requirements  showclose

In order to take this course, you must:

√    Have access to a computer.

√    Have continuous broadband Internet access.

√    Have the ability/permission to install plug-ins or software (e.g., Adobe Reader or Mathematica Viewer).

√    Have the ability to download and save files and documents to a computer.

√    Have the ability to open Microsoft files and documents (.doc, .ppt, .xls, etc.).

√    Be competent in the English language.

√    Have read the Saylor Student Handbook.

√    Have completed PHYS101 as a pre-requisite.

√    Have basic knowledge of calculus or have completed MA101 and MA102.

Preliminary Information

  • Math Review Before and During Course

     

    Time Advisory   show close
    • Lecture: MIT: Professor Walter Lewin’s Physics 8.01 – Classical Mechanics: “Lecture 3: Vector Notation”

      Link: MIT: Professor Walter Lewin’s Physics 8.01 – Classical Mechanics: “Lecture 3: Vector Notation” (Adobe Flash)

      Also available in:
      YouTube
      iTunes U

      Instructions: Please click on the link above, and view the entire lecture, pausing to take notes, before moving on to the readings below.

      You should spend approximately 1 hour and 15 minutes on this lecture.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

    • Reading: Schoolcraft College: Professor Michael Corral’s “Vector Calculus”

      Link: Schoolcraft College: Professor Michael Corral’s “Vector Calculus” (PDF)

      Instructions: Please click on the link above and then click “calc3book.pdf” to download the PDF. Read “Chapter 1: Vectors in Euclidean Space,” Sections 1.1 through 1.8, on pages 1-58.  Studying these pages should take approximately 3 hours to complete.

      You should also complete the following problems for each section:

      Section 1.1, Introduction: Work on problems 1-3 on page 8.

      You should spend approximately 1 hour and 30 minutes on these problems.

      Section 1.2, Vector Algebra: Work on problems 1-3 and 6 on page 14.

      You should spend approximately 1 hour and 30 minutes on these problems.

      Section 1.3, Dot Product: Work on problems 1-5, 9-14, and 24-25 on pages 18-19.

      You should spend approximately 3 hours on these problems.

      Section 1.4, Cross Product: Work on problems 1-16 and 24-28 on pages 29-30.

      You should spend approximately 3 hours on these problems.

      Section 1.5, Lines and Planes: Work on problems 1-21 on page 39.

      You should spend approximately 3 hours on these problems.

      Section 1.6, Surfaces: Work on problems 1, 3, 5-7, 9-10, and 12-13 on page 46.

      You should spend approximately 3 hours on these problems.

      Section 1.7, Curvilinear Coordinates: Work on problems 1-13 on page 50.

      You should spend approximately 3 hours on these problems.

      Section 1.8, Vector Valued Functions: Work on problems 1-2, 5-6, and 6-11 on pages 57-58.

      You should spend approximately 3 hours on these problems.

      Answers to most of these problems begin on page 189 of the text.  Make sure you understand not only the solutions but how to approach solving these math problems so that you can obtain the solution yourself.  You will be responsible for being able to solve similar math problems on the Final Exam.

      Do not try to do this math review all at once.  It will be too frustrating, and you will lose track of the physics.  Try not to spend more than a few hours at a time working on the math.  The important thing is to finish by the time you are working on Unit 5.

      In addition to reading pages 1-58, scanning portions of this book that are not assigned above is worthwhile.  There are several advanced mathematical tools you must know in order to study electromagnetism.  Maxwell’s equations are often written as a set of vector partial differential equations.  Accordingly, it will be a good idea to learn a bit about vector calculus and partial derivatives.  However, do not let the math get in the way of understanding the physics.  Most calculus as well as the study of electromagnetism can be broken down (approximately) into algebraic equations to gain understanding.  This reading will help you gain familiarity with some of the mathematical ideas you will need, and this resource can be used as a general reference during the course.

      You should spend approximately 1 hour exploring other sections of the text.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Reading: University of Texas: Professor Richard Fitzpatrick’s Classical Mechanics: “Vectors”

      Link: University of Texas: Professor Richard Fitzpatrick’s Classical Mechanics: “Vectors” (HTML)

      Instructions: This reading is a review of vectors and vector calculus.  This reading also includes discussions on line, surface, and volume integrals, which will be important throughout the course.  Like Professor Michael Corral’s “Vector Calculus,” you should not try to complete this all at once.  Your will first use vector calculus in subunit 2.2 on Gauss’ law.

      You should spend approximately 3 hours on this review.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

Unit Outline show close


Expand All Resources Collapse All Resources
  • Unit 1: 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.

    Time Advisory   show close
    Learning Outcomes   show close
  • 1.1 Periodic Motion and Simple Harmonic Oscillators  
    • Interactive Lab: Wolfram Demonstrations Project: “Torsion Pendula”

      Link: Wolfram Demonstrations Project: “Torsion Pendula” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  Torsional Pendula are twisting analogs of the conventional linear spring oscillator.  Investigate their properties using this demonstration.

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Lecture: MIT: Professor Walter Lewin’s Physics 8.01 – Classical Mechanics: “Lecture 10: Hooke’s Law: Simple Harmonic Oscillators”

      Link: MIT: Professor Walter Lewin’s Physics 8.01 – Classical Mechanics: “Lecture 10: Hooke’s Law: Simple Harmonic Oscillators” (JWPlayer)

      Also available in:
      YouTube
      iTunes U

      Instructions: Please click on the link above, and view the entire lecture, pausing to take notes, before moving on to the readings below.

      You should spend approximately 1 hour and 15 minutes on this lecture.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

    • Interactive Lab: Wolfram Demonstrations Project: “A Non-Harmonic Oscillator”

      Link: Wolfram Demonstrations Project: “A Non-Harmonic Oscillator” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  This demonstration illustrates a very simple example of a non-harmonic oscillator – a helium balloon on a string.  There are two sources of non-linearity.  First, as the balloon rises, it lifts more string.  Therefore, the mass of the oscillator is a function of the position of the oscillating mass, which leads to non-linear behavior.  In addition, a damping term has been included which mimics the effect of air resistance.  Adjust the various control parameters to gain a feel for which parameters have a larger effect of the motion of the oscillator.

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Reading: University of Texas: Professor Richard Fitzpatrick’s Classical Mechanics: “Oscillatory Motion”

      Link: University of Texas: Professor Richard Fitzpatrick’s Classical Mechanics: “Oscillatory Motion” (HTML)

      Instructions: Please click on the link above, and read this chapter after viewing Professor Lewin’s lecture above.  There are 6 worked examples in the chapter.  Try each of these problems before looking at the solutions.  Make sure you understand not only the solutions but how to approach solving the problem so that you can obtain the solution yourself.  You will be responsible for being able to solve problems of this type on the final exam.

      You should spend approximately 1 hour and 30 minutes on this reading.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Interactive Lab: Wolfram Demonstrations Project: “Simple Harmonic Motion”

      Link: Wolfram Demonstrations Project: “Simple Harmonic Motion” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  The demonstration illustrates the motion of a mass on a spring.  When the mass is pulled down, the spring exerts a restoring force described by Hooke’s Law that pulls the mass upwards.  The result is that the mass travels up and down in simple harmonic motion, where the displacement of the mass is described by a sinusoidal curve.  Think of this demonstration as an experiment to verify (or not) the effect of Hooke’s Law on the period of oscillation.

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

  • 1.2 Vibrations  
  • 1.3 Wave Motion  
  • Unit 2: 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.

    Time Advisory   show close
    Learning Outcomes   show close
    • Reading: Fullerton College: Professor Benjamin Crowell’s Light and Matter: “Chapter 26: The Atom and E=mc2”

      Link: Fullerton College: Professor Benjamin Crowell’s Light and Matter: “Chapter 26: The Atom and E=mc2 (HTML)

      Instructions: Please read sections 1 through 4 and section 8 of “Chapter 26: The Atom and E=mc2” carefully on pages 705-766 of the PDF.  Answer the Self-Check questions in the text (answers on page 982).  Think about the Discussion Questions and Examples, and work out some numerical examples.  Think carefully about the Millikan’s Fraud discussion, which illuminates the basis of science and how it is eventually self-correcting.

      You should spend approximately 1 hour and 30 minutes on this reading.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

  • 2.1 Introduction to Electricity  
    • Reading: MIT: Professor Walter Lewin’s Physics 8.02 – Electricity and Magnetism: “Lecture 1: What Holds Our World Together?”

      Link: MIT: Professor Walter Lewin’s Physics 8.02 – Electricity and Magnetism: “Lecture 1: What Holds Our World Together?” (JWPlayer)

      Also available in:
      YouTube
      iTunes U

      Instructions: Please click on the link above, and view the entire lecture, pausing to take notes, before moving on to the readings below.

      You should spend approximately 1 hour and 15 minutes on this lecture.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

    • Reading: University of Texas: Professor Richard Fitzpatrick’s Electromagnetism and Optics: “Electricity”

      Link: University of Texas: Professor Richard Fitzpatrick’s Electromagnetism and Optics: “Electricity” (HTML)

      Instructions: Read this chapter after viewing Professor Lewin’s lecture above.  There are 3 worked examples in the chapter.  Try each of these problems before looking at the solutions.  Make sure you understand not only the solutions but how to approach solving the problem so that you can obtain the solution yourself.  You will be responsible for being able to solve problems of this type on the final exam.

      You should spend approximately 1 hour and 30 minutes on this reading.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Interactive Lab: Wolfram Demonstrations Project: “The Electroscope”

      Link: Wolfram Demonstrations Project: “The Electroscope” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  The electroscope (also called a gold-leaf electroscope) was invented in the late eighteenth century to measure the size of small static charges.  A charge is transferred to a pair of neighboring gold leafs.  Acquiring charges of the same sign, the gold leafs repel each other, taking on an angle which depends on the mass of the gold leafs and the size of the charge.

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Interactive Lab: Wolfram Demonstrations Project: “Repulsion of Charged Objects”

      Link: Wolfram Demonstrations Project: “Repulsion of Charged Objects” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  This demonstration illustrates the relationship between the charge on a gold-leaf electrometer and the separation of the leafs.  Make measurements on the demonstration, and try to work out the force balance equations which describe the behavior of the electrometer.

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Reading: Wolfram Demonstrations Project: “Van de Graaff Generator”

      Link: Wolfram Demonstrations Project: “Van de Graaff Generator” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  This demonstration illustrates how a Van de Graaff generator generates static charges and collects the charges on a metal sphere.  The voltage on the sphere is proportional to the amount of charge collected.  Though it appears that the collected charge on the sphere would increase indefinitely, in reality, paths for loss of the collected charge exist and typically limit the static voltage on the sphere to a fraction of a megavolt, although Van de Graaff generators specialized for use in nuclear accelerators can generate 10 megavolts or more.

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

  • 2.2 Electric Field and Gauss’ Law  
  • 2.3 Electric Potential and Electric Field  
    • Interactive Lab: Wolfram Demonstrations Project: “Electric Dipole Potential”

      Link: Wolfram Demonstrations Project: “Electric Dipole Potential” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  This demonstration illustrates what you would find in a lab experiment where two electrically charges bodies are placed on a table, and you measure the electric potential (roughly speaking, the voltage relative to a reference point) as a function of position on the table.  The electric potential is displayed as a series of equipotential curves, or curves along which the electric potential is constant.  Vary the position and strength of the charges, and then view the results with both the 3D and the contour plot.  Turn on the field direction, and notice that the electric field is everywhere perpendicular to the equipotential curves.  This is because the electric field is proportional to the gradient of the electric potential.

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Reading: University of Texas: Professor Richard Fitzpatrick’s Electromagnetism and Optics: “Electric Potential”

      Link: University of Texas: Professor Richard Fitzpatrick’s Electromagnetism and Optics: “Electric Potential” (HTML)

      Instructions: Please click on the link above, and read this chapter after viewing Professor Lewin’s lectures above.  Try all four worked examples before looking at the solutions.  Make sure you understand not only the solutions but also how to approach solving the problem so that you can obtain the solution yourself.  You will be responsible for being able to solve problems of this type on the final exam.

      You should spend approximately 1 hour and 30 minutes on this reading.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Lecture: MIT: Professor Walter Lewin’s Physics 8.02 – Electricity and Magnetism: “Lecture 4: Electrostatic Potential and Electric Energy”

      Link: MIT: Professor Walter Lewin’s Physics 8.02 – Electricity and Magnetism: “Lecture 4: Electrostatic Potential and Electric Energy” (JWPlayer)

      Also available in:
      YouTube
      iTunes U

      Instructions: Please click on the link above, and view the entire lecture, pausing to take notes, before moving on to the readings below.

      You should spend approximately 1 hour and 15 minutes on this lecture.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

    • Interactive Lab: Wolfram Demonstrations Project: “Lines of Force for Two Point Charges”

      Link: Wolfram Demonstrations Project: “Lines of Force for Two Point Charges” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  This demonstration is an extension of the previous demonstration on Electric Dipole Potential.  Shown here again is an electrostatic dipole where the strengths of the electric charges can be varied.  The graph shows the lines of electric field.  The lines of electric field are everywhere perpendicular to the equipotential curves.  Note that this does not mean that the MAGNITUDE of the electric field is constant along an electric field line; it only means that the magnitude of the electric field points along that line.  Vary the positions and magnitude of the two charges to gain some feel for how the electric field behaves.  The calculations required by the demonstration are complex, so wait between changes for the graph to once again become smooth.

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Interactive Lab: Wolfram Demonstrations Project: “Method of Images”

      Link: Wolfram Demonstrations Project: “Method of Images” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  This demonstration is valuable for gaining an understanding of the method of images, which is one of the more powerful elementary methods for calculating the electric potential and field resulting from a combination of charges and conductors.  In this case, the conductors take the form of a plane or a sphere.  The point is that a perfect conductor is always an equipotential surface and that metals are close enough to being perfect conductors for many purposes.

      The basis for the method of images is that the electric potential of a charge near a metal surface will be the same as that of a system of charges that produces the same equipotential surface as would the metal surface.  In the case of a first charge near a plane conductor, we form the simplest system of charges that produces a planar equipotential surface at the position of the conducting surface by placing a second charge behind the conducting surface.  The second charge is located behind the conducting surface on a line through the first charge positioned normal to the conducting surface.  The second charge is placed as far behind the surface as the first charge is in front of the surface.  Then, the electric potential of this pair of point charges has the same electric field as the charge near the metal surface but is much simpler to solve for.

      The logic behind the method of images applies when a charge is near a spherical conducting surface, save that the equipotential surface forced by addition of the second charge must reproduce the shape of the spherical conductor.  Note that although the demonstration states that the conductors must be grounded, this is not true; the method of images still works with ungrounded conductors.  The only difference is that the potentials are uniformly larger or smaller depending on just the electric potential on the conducting surface.

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

  • Unit 3: Magnetism  

    In Unit 2, 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.

    Time Advisory   show close
    Learning Outcomes   show close
  • 3.1 Magnetic Field  
    • Reading: MIT: Professor Walter Lewin’s Physics 8.02 – Electricity and Magnetism: “Lectures 11: Magnetic Field and Lorentz Force”

      Link: MIT: Professor Walter Lewin, Physics 8.02 – Electricity and Magnetism: “Lectures 11: Magnetic Field and Lorentz Force” (JWPlayer)

      Also available in:
      YouTube
      iTunes U

      Instructions: Please click on the link above, and view the entire lecture, pausing to take notes, before moving on to the readings below.

      You should spend approximately 1 hour and 15 minutes on this lecture.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

    • Reading: University of Texas: Professor Richard Fitzpatrick’s Electromagnetism and Optics: “Magnetism”

      Link: University of Texas: Professor Richard Fitzpatrick’s Electromagnetism and Optics: “Magnetism” (HTML)

      Instructions: Please click on the link above, select the links to the following subsections, and read these webpages in their entirety: “Historical Introduction,” “Ampère’s Experiments,” and “Ampère’s Laws.”

      You should spend approximately 45 minutes on this reading.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Interactive Lab: Wolfram Demonstrations Project: “Observing Magnetic Fields with Iron Filings”

      Link: Wolfram Demonstrations Project: “Observing Magnetic Fields with Iron Filings” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  This demonstration is designed to remind you of one of the most common elementary school demonstrations of magnetism, where fine iron filings decorate the lines of magnetic force, showing, as in the demonstration, the dipole-like magnetic field of a permanent magnet.  If we place a ferromagnetic material, such as iron, in a magnetic field, it will induce a magnetic field in the iron that opposes the external field.  As usual, one of Nature’s rules is to arrange matters so that the total energy of the system is as small as possible.  In this case, inducing an opposing field in the iron reduces the total magnetic energy.  Because iron filings tend to be long and skinny, the induced field turns them into tiny bar magnets, with north and south poles aligned such that the north pole of the iron filing points along the local direction of the magnetic field and orients away from the north pole of the external magnet.  Accordingly, when you place a few hundred iron filings on a surface over a magnet, you are able to visualize the magnetic lines of force.

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

  • 3.2 Magnetic Force on Moving Electric Charges  
    • Lecture: MIT: Professor Walter Lewin’s Physics 8.02 – Electricity and Magnetism: “Lectures 13: Moving Charges in B-fields”

      Link: MIT: Professor Walter Lewin’s Physics 8.02 – Electricity and Magnetism: “Lecture 13: Moving Charges in B-fields” (JWPlayer)

      Also available in:
      YouTube
      iTunes U

      Instructions: Please click on the link above, and view the entire lecture, pausing to take notes, before moving on to the readings below.

      You should spend approximately 1 hour and 15 minutes on this lecture.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

    • Reading: University of Texas: Professor Richard Fitzpatrick’s Electromagnetism and Optics: “Magnetism”

      Link: University of Texas: Professor Richard Fitzpatrick’s Electromagnetism and Optics: “Magnetism” (HTML)

      Instructions: Please click on the link above, select the links to the following subsections, and read these webpages in their entirety: “The Lorentz Force,” “Charged Particle in a Magnetic Field,” and “The Hall Effect.”  In addition, select the links for Examples 8.1 and 8.2, and work through these examples before looking at the solutions in the text.  Make sure you understand not only the solutions but also how to approach solving the problem so that you can obtain the solution yourself.  You will be responsible for being able to solve problems of this type on the final exam.

      You should spend approximately 1 hour and 15 minutes on this reading.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Interactive Lab: Wolfram Demonstrations Project: “Motion of a Charge in a Magnetic Field”

      Link: Wolfram Demonstrations Project: “Motion of a Charge in a Magnetic Field” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  This is an idealized version of a classic laboratory experiment carried out with a cathode-ray tube.  In this demonstration, you can see the entire path of the moving charge, rather than just its position at a screen (as you would with a cathode-ray tube).  The initial velocity and magnetic field vectors are indicated, allowing you to determine the direction and strength of the Lorentz force on the moving charge.  You should estimate the effect that the various parameters on the path of the electric charge have and compare your results with those predicted by the theoretical treatment in the lecture and text.

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Interactive Lab: Wolfram Demonstrations Project: “Motion of a Charge in a Magnetic Dipole Field”

      Link: Wolfram Demonstrations Project: “Motion of a Charge in a Magnetic Dipole Field” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  The actual title of this demonstration is “Dipole Fields Are Complicated.”  More accurately, the motion of electric charges in a magnetic dipole field is complicated.  Follow the suggestions for changing parameters listed within the demonstration.  Notice that very small changes in the system parameters sometimes result in large changes in the resulting trajectories.  (For example, by typing in the dynamic controls, set the magnetic field strength at 0.2, time of solution at 5000, X0 at 1.5, Y0 at 0.0, z0 at 0.4, Vx0 at Vy0 at 0.0, and plot quality at 50.  You will now find very different trajectories for Vz0 = 0.0186 and 0.0187.)  Extreme sensitivity to initial conditions is the most characteristic property of chaotic motion.  Try to find other parameters that suggest chaotic motion while keeping an eye on how trajectories change with the parameters.

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Interactive Lab: Wolfram Demonstrations Project: “Motion of a Charge in Electric and Magnetic Fields”

      Link: Wolfram Demonstrations Project: “Motion of a Charge in Electric and Magnetic Fields” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  This demonstration shows the motion of an electric charge in uniform electric and magnetic fields.  The charge, E field, and B field magnitudes are all controllable, as are the field orientations and the initial velocity vector of the charge.  Note that for nearly all combinations of parameters, the result is that the charge spirals toward a position, comes to a stop save for circular motion, and then reflects back in roughly the original direction.  Why?

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

  • 3.3 Magnetic Field of a Current-Carrying Wire  
  • 3.4 Electromagnetic Induction  
    • Lecture: MIT: Professor Walter Lewin’s Physics 8.02 – Electricity and Magnetism: “Lecture 16: Electromagnetic Induction”

      Link: MIT: Professor Walter Lewin’s Physics 8.02 – Electricity and Magnetism: “Lecture 16: Electromagnetic Induction” (JWPlayer)

      Also available in:
      YouTube
      iTunes U

      Instructions: Please click on the link above, and view the entire lecture, pausing to take notes, before moving on to the readings below.

      You should spend approximately 1 hour and 15 minutes on this lecture.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

    • Lecture: MIT: Professor Walter Lewin’s Physics 8.02 – Electricity and Magnetism: “Lecture 17: Motional EMF and Dynamos”

      Link: MIT: Professor Walter Lewin’s Physics 8.02 – Electricity and Magnetism: “Lecture 17: Motional EMF and Dynamos”  (JWPlayer)

      Also available in:
      YouTube
      iTunes U

      Instructions: Please click on the link above, and view the entire lecture, pausing to take notes, before moving on to the readings below.

      You should spend approximately 1 hour and 15 minutes on this lecture.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

    • Reading: Fullerton College: Professor Benjamin Crowell’s Light and Matter: “Chapter 25: Capacitance and Inductance”

      Link: Fullerton College: Professor Benjamin Crowell’s Light and Matter: “Chapter 25: Capacitance and Inductance” (HTML)

      Instructions: Please read “Chapter 25: Capacitance and Inductance” carefully (PDF pages 687-704).  Answer the Self-Check questions in the text (answers on page 982).Think about the Discussion Questions and Examples, and work out some numerical examples.

      You should spend approximately 1 hour and 15 minutes on this reading.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Reading: University of Texas: Professor Richard Fitzpatrick’s Electromagnetism and Optics: “Magnetic Induction”

      Link: University of Texas: Professor Richard Fitzpatrick’s Electromagnetism and Optics: “Magnetic Induction” (HTML)

      Instructions: Please click on the link above, select the links for the following subsections, and read these webpages in their entirety: “Faraday’s Law,” “Lenz’s Law,” “Magnetic Induction,” “Motional EMF,” and “Eddy Currents.”  Also, work through Examples 9.1-9.3 before looking at the solutions.  Make sure you understand not only the solutions but also how to approach solving the problems so that you can obtain the solutions yourself.  You will be responsible for being able to solve problems of this type on the final exam.

      You should spend approximately 1 hour and 30 minutes on this reading.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Interactive Lab: Wolfram Demonstrations Project: “Magnetic Flux through a Wire Loop”

      Link: Wolfram Demonstrations Project: “Magnetic Flux through a Wire Loop” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  This demonstration illustrates the relationship between magnetic flux and size and orientation of the surface through which the magnetic field is passing.  Note that the relationship is particularly easy to observe when constant magnetic fields parallel to the field lines can be seen.  Here, a constant flux is simply a constant number of field lines penetrating the wire loop.  Although the relationship is more difficult to observe when variable magnetic fields are used in the demonstration, the relationship is still the same: magnetic flux is the number of field lines penetrating the wire loop.

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Interactive Lab: Wolfram Demonstrations Project: “Magnetic Braking via Eddy Currents”

      Link: Wolfram Demonstrations Project: “Magnetic Braking via Eddy Currents” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  Many machines use magnetic brakes  from the small to the gigantic.  The analysis of the effect is simple in some ways and quite tricky in others.  There are two ways of looking at magnetic braking, both of which are mentioned in the demonstration write-up.  The basic idea is that when a conductor moves through a magnetic field, currents are induced that resist the motion.  These are called Eddy currents, and the reduction in the kinetic energy of the conductor is equal to the resistive heating caused in the conductor by the induced Eddy currents.  Another approach to explaining magnetic braking is that the Lorentz force acting on the electrons in the moving conductor acts to move the electrons outward, and the Lorentz force associated with that outward motion in the applied magnetic field serves to slow down the moving conductor.  These two ways of thinking about magnetic braking are equivalent; that is, they make the same predictions.  A couple of questions for reflection: If the conductor is a perfect conductor (no resistance, but not a superconductor), is there any braking effect?  Also, can a magnetic brake by itself bring a moving conductor to a complete stop?  Why, or why not?

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Interactive Lab: Wolfram Demonstrations Project: “EMF Induced in a Wire Loop”

      Link: Wolfram Demonstrations Project: “EMF Induced in a Wire Loop” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  Treat this demonstration as a laboratory experiment, measuring various combinations of parameters and results and then comparing these with the theory presented in the lectures and readings above.

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Interactive Lab: Wolfram Demonstrations Project: “Electromagnetic Ring Toss”

      Link: Wolfram Demonstrations Project: “Electromagnetic Ring Toss” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  This demonstration is modeled after a classic classroom demonstration.  A conducting ring is placed atop an electromagnet, and a large pulse of current passes through the electromagnet.  As seen in the lectures, readings, and demonstrations above, a current is induced in the conducting ring in a direction which opposes the formation of the magnetic field.  As usual, this is to minimize the total energy of the system.  The result is that the current in the ring generates a magnetic field with the opposite sign as that of the electromagnet.  Opposed magnetic fields repel, so the ring launches into the air.  Why does nothing happen when we use the split ring?

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

  • 3.5 Magnetic Materials  
  • 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.

    Time Advisory   show close
    Learning Outcomes   show close
  • 4.1 Electric Current, Voltage, and Resistance  
  • 4.2 Electric Circuits  
  • 4.3 Capacitance – Storage of Electric Energy  
    • Lecture: MIT: Professor Walter Lewin’s Physics 8.02 – Electricity and Magnetism: “Lecture 7: Capacitance and Field Energy”

      Link: MIT: Professor Walter Lewin’s Physics 8.02 – Electricity and Magnetism: “Lecture 7: Capacitance and Field Energy” (JWPlayer)

      Also available in:
      YouTube
      iTunes U

      Instructions: Please click on the link above, and view the entire lecture, pausing to take notes, before moving on to the readings below.

      You should spend approximately 1 hour and 15 minutes on this lecture.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

    • Reading: Fullerton College: Professor Benjamin Crowell’s Light and Matter: “Chapter 24: Electromagnetism”

      Link: Fullerton College: Professor Benjamin Crowell’s Light and Matter: “Chapter 24: Electromagnetism” (HTML)

      Instructions: Please read sections 1 through 5 and 8 in “Chapter 24: Electromagnetism” carefully (PDF pages 655-686).  Complete the Self Checks throughout the chapter (answers on page 982).  Think about the Discussion Questions and Examples, and work out some numerical examples to help decide on a position.

      You should spend approximately 1 hour and 15 minutes on this reading.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Reading: University of Texas: Professor Richard Fitzpatrick’s Electromagnetism and Optics: “Capacitance”

      Link: University of Texas: Professor Richard Fitzpatrick’s Electromagnetism and Optics: “Capacitance” (HTML)

      Instructions: Please click on the link above, select the links to each subsection, and read this chapter after viewing Professor Lewin’s lecture above.  There are 4 worked examples in the chapter.  Try each of these problems before looking at the solutions.  Make sure you understand not only the solutions but how to approach solving the problems so that you can obtain the solutions yourself.  You will be responsible for being able to solve problems of this type on the final exam.

      You should spend approximately 1 hour and 30 minutes on this reading.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Interactive Lab: Wolfram Demonstrations Project: “Energy Density of the Electrostatic Field of an Electric Dipole”

      Link: Wolfram Demonstrations Project: “Energy Density of the Electrostatic Field of an Electric Dipole” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  This demonstration illustrates the relationship between the density of electrostatic equipotential curves and the electrostatic energy density.  This is seen most easily as the charges come together.  Note that there is a minor bug that causes the demonstration to crash at the smallest charge separation.

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Interactive Lab: Wolfram Demonstrations Project: “Parallel-Plate Capacitors”

      Link: Wolfram Demonstrations Project: “Parallel-Plate Capacitors” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  This demonstration can be treated as a combination laboratory project and homework problem.  The capacitance of a parallel plate capacitor depends on the area and separation of the plates and the dielectric constant of the material between them.  In this demonstration, you will control the geometry and materials of the capacitor, and you will measure the charge resident on the capacitor as a function of applied voltage.  Use the theoretical description here and in the lectures and readings above to confirm that the workings of the demonstration agree with physical reality.

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Interactive Lab: Wolfram Demonstrations Project: “Partially Filled Capacitors”

      Link: Wolfram Demonstrations Project: “Partially Filled Capacitors” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  A partially-filled capacitor can be viewed as a pair of capacitors, one filled and the other unfilled.  (Note that this is only true for geometries where the dielectric interface is approximately on an equipotential surface, as nothing then changes when the extra pair of metal plates is inserted.  This same technique could be used on a partially-filled cylindrical or spherical capacitor, for example, provided the dielectric surface was cylindrical or spherical, respectively.)  Treat this as a laboratory experiment, and confirm that the demonstration is accurate.

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Interactive Lab: Wolfram Demonstrations Project: “Electric Field Energy in Capacitors”

      Link: Wolfram Demonstrations Project: “Electric Field Energy in Capacitors” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  This demonstration provides examples for both capacitors and inductors, but for now, work only with the capacitors.  Treat this demonstration as a laboratory experiment in which you measure the capacitance of various geometries and use theory to confirm that the capacitances are correct.  Then, determine the electromagnetic field energies driven by applied voltage from the demonstration and confirm those results by direct calculation based on the theoretical relations listed here and in the lectures and readings above.

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

  • 4.4 RC circuits, Inductance, and RL Circuits  
  • 4.5 RLC Circuits and Resonance  
  • 4.6 Electromagnetic Motors and Generators  
  • Unit 5: 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. 

    Time Advisory   show close
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  • 5.1 Maxwell’s Equations  
    • Lecture: MIT: Professor Walter Lewin’s Physics 8.02 – Electricity and Magnetism: “Lecture 27: Resonance and Destructive Resonance”

      Link: MIT: Professor Walter Lewin’s Physics 8.02 – Electricity and Magnetism: “Lecture 27: Resonance and Destructive Resonance” (JWPlayer)

      Also available in:
      YouTube
      iTunes U

      Instructions: Please click on the link above, and view the entire lecture, pausing to take notes, before moving on to the readings below.

      You should spend approximately 1 hour and 15 minutes on this lecture.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

    • Reading: Ampyx LLC: Dr. Glen Dash’s A Dash of Maxwell’s: A Maxwell’s Equation Primer: “Chapter 1: Introduction”

      Link: Ampyx LLC: Dr. Glen Dash’s A Dash of Maxwell’s: A Maxwell’s Equation Primer:Chapter 1: Introduction” (PDF)

      Instructions: Please click on the link above, and then click on part 1 to download the PDF.  Read this chapter after viewing Professor Lewin’s lecture above.  Most of this chapter reviews Maxwell’s time-independent equations but often from a different viewpoint that acts synergistically with our previously covered material.  Take particular note of the definition on page 7 of the electric flux density vector D = ε E, where E is the electric field vector and ε is the dielectric constant times the free space permittivity εo.  Similarly, on page 12 the magnetic flux density vector B = μ H, where H is the magnetic field vector and μ is the magnetic permeability, sometimes described as the relative permeability times the permeability of free space.  For this class, these equations that allow the use of Maxwell’s Equations in a material are assumed to be scalar functions of position.  The general case is that they are tensors, but the scalar approximation simplifies gaining an initial understanding of the behavior of electromagnetism systems.  Work through the examples until you understand how to approach solving similar problems.

      You should spend approximately 1 hour on this reading.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Interactive Lab: Wolfram Demonstrations Project: “Maxwell’s Displacement Current”

      Link: Wolfram Demonstrations Project: “Maxwell’s Displacement Current” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  This is an illustrative explanation of displacement current, which is arguably the linchpin of Maxwell’s full theory of electromagnetism, as well as the single most confusing concept in that theory.

      There is no electric current between the plates of a capacitor.  However, Ampere’s Circuital Law tells us that the integral of the magnetic field B around a closed loop C is proportional to the flux of the current density through a surface S attached to the loop.  This is independent of the shape of S.  In the demonstration, consider a closed loop positioned around one of the wires carrying electric current into the charging capacitor.  If S is chosen so that the wire penetrates the surface, the flux of the current density through S is simply the electric current.

      Now draw another surface S’ so that it passes between the capacitor plates, thereby making no contact with the current carrying wire.  There is no electric current between the capacitor plates, so it would appear that the flux of the current density through S’ is zero.  However, Ampere’s Circuital Law tells us that the magnetic field integral around loop C is still the same non-zero value.  We appear to meet a contradiction.

      What the apparent contradiction is actually telling us is that Ampere’s Circuital Law is incomplete.  The electric field between the plates of a charging capacitor changes  with time, so it would appear that a time-varying electric field must generate a magnetic field which is consistent with the current charging the capacitor.  The way Maxwell chose to think about this by identifying a fictitious current between the capacitor plates called the displacement current such that the total displacement current flux between the plates was equal to the current charging the capacitor.  Although there is no actual electrical current between the plates, we still refer to the source of magnetic field associated with a changing electric field as the displacement current.

      You should spend approximately 30 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

  • 5.2 Electromagnetic Waves  
    • Reading: MIT: Professor Walter Lewin’s Physics 8.02 – Electricity and Magnetism: “Lecture 28: Index of Refraction and Poynting Vector”

      Link: MIT: Professor Walter Lewin’s Physics 8.02 – Electricity and Magnetism: “Lecture 28: Index of Refraction and Poynting Vector” (JWPlayer)

      Also available in:
      YouTube
      iTunes U

      Instructions: Please click on the link above, and view the entire lecture, pausing to take notes, before moving on to the readings below.

      You should spend approximately 1 hour and 15 minutes on this lecture.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

    • Reading: Ampyx LLC: Dr. Glen Dash’s A Dash of Maxwell’s: A Maxwell’s Equation Primer: “Chapter 2: Why Things Radiate”

      Link: Ampyx LLC: Dr. Glen Dash’s A Dash of Maxwell’s: A Maxwell’s Equation Primer:Chapter 2: Why Things Radiate” (PDF)

      Instructions: Please click on the link above, and then click “Dash-of-Maxwells-Chapter-2.pdf” to download the PDF.  Read this chapter after viewing Professor Lewin’s lecture above.  This chapter develops the time-varying version of Maxwell’s Equations and uses them to examine not only the properties of EM radiation, but also why anything emits this radiation in the first place.  Again, stick with the math until you can see the physics.

      You should spend approximately 1 hour on this reading.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Interactive Lab: Wolfram Demonstrations Project: “Propagation of a Plane EM Wave”

      Link: Wolfram Demonstrations Project: “Propagation of a Plane EM Wave” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  This demonstration schematically indicates the time-dependent electric and magnetic fields associated with an electromagnetic wave.  Note that the electric and magnetic fields are mutually perpendicular to one another and to the path the wave follows.

      As the electric field changes in time, a magnetic field is generated as described by Maxwell’s generalization of Ampere’s Circuital Law.  As the magnetic field changes in time, an electric field is produced as described by Faraday’s Law of Induction.  Given this word picture, why are the electric field strength and the maximum magnetic field strength proportional at all times?  Which direction does the wave velocity point, toward positive or negative x?  (Hint: Look at Poynting’s Theorem.)

      You should spend approximately 30 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Interactive Lab: Wolfram Demonstrations Project: “Rømer’s Measurement of the Speed of Light”

      Link: Wolfram Demonstrations Project: “Rømer’s Measurement of the Speed of Light” (Wolfram NBP)
       
      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  Despite numerous experiments, no one before Rømer had been able to establish that the speed of light was not infinite.  For example, in the early seventeenth century, Galileo attempted to measure the speed of light by relaying signals between a pair of lanterns, but he could only establish that light traveled faster than a few kilometers per second.  Throughout the period 1668-1674, Rømer made and recorded accurate measurements of the eclipses of the moons of Jupiter.  He found that the times would vary by about 16.5 minutes during the course of a year, and he explained this as the time it takes light to traverse the diameter of Earth’s orbit around the Sun.  Using an inaccurate value for the size of Earth’s orbit, he obtained a value of about 214,000 km/sec for the speed of light – about 20% smaller than the actual value.

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Interactive Lab: Wolfram Demonstrations Project: “Measuring the Speed of Light with Marshmallows and a Microwave Oven”

      Link: Wolfram Demonstrations Project: “Measuring the Speed of Light with Marshmallows and a Microwave Oven” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  Try this experiment using your microwave oven.  Why do the melted spots appear in a regular pattern?

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

  • 5.3 Maxwell’s Culture  
    • Reading: Ampyx LLC: Dr. Glen Dash’s A Dash of Maxwell’s: A Maxwell’s Equation Primer: “Chapter 3: The Difference a Del Makes”

      Link: Ampyx LLC: Dr. Glen Dash’s A Dash of Maxwell’s: A Maxwell’s Equation Primer:Chapter 3: The Difference a Del Makes” (PDF)
        
      Instructions: Click on the link above, and then click “Dash-of-Maxwells-Chapter-3.pdf” to download the PDF.  Up to this point in our course, the integral formulation of Maxwell’s Equations has been presented.  However, Maxwell’s Equations are more often used in a different, although completely consistent, version – one expressed in terms of partial differential equations.  As this formulation requires junior-level math, we are not going to attempt to solve any problems.  However, it is worth reading this chapter to see that there is a totally different approach to describing electromagnetism.  The 2nd and 4th chapters of Corral’s Vector Calculus can help you try to make sense of Unit 5.3.  There is a deep connection between the integral and differential versions of Maxwell’s equations, but demonstrating that relationship is beyond the scope of this course.
       
      You should spend approximately 1 hour on this reading.
       
      Terms of Use: Please respect the copyright and terms of use displayed at this link.

  • 5.4 Electromagnetism Problems  
  • 5.5 Examples of Electromagnetic Phenomena  
  • Unit 6: 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.)

    Time Advisory   show close
    Learning Outcomes   show close
  • 6.1 Geometric Optics  
    • Lecture: UC College Prep’s Advanced Placement Physics B: “Lesson 48: Reflection and Refraction”

      Link: UC College Prep’s Advanced Placement Physics B: “Lesson 48: Reflection and Refraction” (Flash)

      Instructions: Please view the entire lecture, pausing to take notes, before moving on to the readings below.

      You should spend approximately 1 hour and 15 minutes on this lecture.

      Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0.  It is attributed to The Regents of the University of California and the original version can be found here.

    • Reading: University of Texas: Professor Richard Fitzpatrick’s Electromagnetism and Optics: “Geometric Optics”

      Link: University of Texas: Professor Richard Fitzpatrick’s Electromagnetism and Optics: “Geometric Optics” (HTML)

      Instructions: Please click on the link above and read it in its entirety.  Select the links for Examples 12.1 and 12.2, and work through these examples before looking at the solutions.  Make sure you understand not only the solutions but how to approach solving the problems so that you can obtain the solutions yourself.  Make sure you understand not only the solutions but how to approach solving the problems so that you can obtain the solutions yourself.  You will be responsible for being able to solve problems of this type on the Final Exam.

      You should spend approximately 1 hour and 30 minutes on this reading.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Interactive Lab: Wolfram Demonstrations Project: “Specular and Diffuse Reflection”

      Link: Wolfram Demonstrations Project: “Specular and Diffuse Reflection” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  This demonstration illustrates the difference between specular reflection (like a mirror) and diffuse reflection (like a piece of paper).  There is a continuum of behaviors between specular and diffuse reflection, and these are well-illustrated in this demonstration.  Note that the key is not the amount of incident light reflected, but rather the extent to which information about the original direction of the light is lost in the reflection.  The demonstration may run slowly on older computers.

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Interactive Lab: Wolfram Demonstrations Project: “Snell’s Law of Refraction”

      Link: Wolfram Demonstrations Project: “Snell’s Law of Refraction” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  This demonstration illustrates the way in which light bends at a tilted interface.  Treat the demonstration as a lab project, taking notes about the physical parameters for several angles and indices of refraction, and verifying that Snell’s Law is obeyed.

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Interactive Lab: Wolfram Demonstrations Project: “Total Internal Reflection”

      Link: Wolfram Demonstrations Project: “Total Internal Reflection” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  When a light ray is within a medium having a refractive index n1 and is incident on an interface between that medium and a second medium having a smaller refractive index n2, Snell’s Law tells you that the angle at which the light is refracted in the second medium is given by sin θ2 = (n1/n2) sin θ1.

      What happens if (n1/n2) sin θ1 is greater than 1?  Because sin θ2 cannot be greater than 1, the light ray cannot be refracted into the second medium.  As a result, the ray is reflected from the interface.  The reflection is total (neglecting possible processes of absorption which might occur right at the interface, such as dye molecules or the like), because there is no mechanism whereby any of the light can penetrate into the second medium.  (This is actually only the case for infinitely thick media, as light can penetrate a distance related to the skin depth.  However, for most practical purposes the reflection is complete.)

      Total internal reflection is unlike reflection from a metallized mirror, in which the metal absorbs some of the light incident on the surface.  This difference explains why the reflecting face of a prism is usually left unmetallized whenever that is consistent with its optical function; more light passes through the optical system than does when a mirror is used.

      You should spend approximately 30 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Interactive Lab: Wolfram Demonstrations Project: “Rainbows”

      Link: Wolfram Demonstrations Project: “Rainbows” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  The color of a rainbow results from variable dispersion of different wavelengths of light, but this demonstration goes further in illustrating why the rainbow appears in a circular bow in the sky.

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

  • 6.2 Paraxial Optics  
    • Lecture: UC College Prep’s Advanced Placement Physics B: “Lesson 49: Mirrors”

      Link: UC College Prep’s Advanced Placement Physics B: “Lesson 49: Mirrors” (Flash)

      Instructions: Please click on the link above, and view the entire lecture, pausing to take notes, before moving on to the readings below.

      You should spend approximately 1 hour and 15 minutes on this lecture.

      Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0.  It is attributed to The Regents of the University of California and the original version can be found here.

    • Lecture: UC College Prep’s Advanced Placement Physics B: “Lesson 50: Lenses”

      Link: UC College Prep’s Advanced Placement Physics B: “Lesson 50: Lenses” (Flash)

      Instructions: Please click on the link above, and view the entire lecture, pausing to take notes, before moving on to the readings below.

      You should spend approximately 1 hour and 15 minutes on this lecture.

      Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0.  It is attributed to The Regents of the University of California and the original version can be found here.

    • Reading: University of Texas: Professor Richard Fitzpatrick’s Electromagnetism and Optics: “Paraxial Optics”

      Link: University of Texas: Professor Richard Fitzpatrick’s Electromagnetism and Optics: “Paraxial Optics” (HTML)

      Instructions: Please click on the link above and read it in its entirety.  Select the links for Examples 13.1 through 13.4, and work through examples before looking at the solutions.  Make sure you understand not only the solutions but how to approach solving the problems so that you can obtain the solutions yourself.  Make sure you understand not only the solutions but how to approach solving the problems so that you can obtain the solutions yourself.  You will be responsible for being able to solve problems of this type on the Final Exam.

      You should spend approximately 1 hour and 30 minutes on this reading.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Lecture: California State University at Northridge: Norman Herr’s “Optics Lecture Slides”

      Link: California State University at Northridge: Norman Herr’s “Optics Lecture Slides” (PPT)
       
      Instructions: Please click on the link above, and then click “optics.ppt” to access the PowerPoint.  Work through these lecture viewgraphs to understand.  Pay special attention to the various animations of image formation.

      You should spend approximately 1 hour on this lecture.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Interactive Lab: Wolfram Demonstrations Project: “Spherical Aberration in Concave Mirrors”

      Link: Wolfram Demonstrations Project: “Spherical Aberration in Concave Mirrors” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  This demonstration illustrates the phenomenon of spherical aberration.  As the center of a sphere lays on the perpendicular ray through every point on the mirror surface, a spherical mirror produces a perfect focus of a point light source located at its radius of curvature.  However, this is not the case as such a mirror is used to form an image of a point source placed a long distance away.

      As the light moves away from the mirror, the focal point of the mirror moves toward the mirror.  The point of light and the focal point define an ellipsoid which is the ideal mirror shape to give a perfect focus.  As you may remember from high school geometry, when one of the foci of an ellipsoid is moved infinitely far away, the shape becomes a paraboloid.  This is why a parabolic mirror gives a perfect focus of a distant source of light.

      In the demonstration, you can experiment with the control parameters to evaluate how imperfect the focus of a spherical mirror is for a distant source.  You will discover that the imperfection, called spherical aberration, is a major influence when the ratio of the mirror focal length to its diameter (called the f/ratio) is small.

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

    • Interactive Lab: Wolfram Demonstrations Project: “Ray Diagrams for Lenses”

      Link: Wolfram Demonstrations Project: “Ray Diagrams for Lenses” (Wolfram NBP)

      Instructions: To use this demonstration, you must download and install the Mathematica Viewer from the Wolfram Demonstrations Project.  This demonstration illustrates the dynamics between focal length, object distance, and real/virtual focal points of a simple lens.

      You should spend approximately 15 minutes on completing this lab.

      Terms of Use: Please respect the copyright and terms of use displayed at this link.

  • 6.3 Optical Aberrations  
  • 6.4 Wave Optics  
  • 6.5 Optical Phenomena  
  • Unit 7: 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 which 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.

    Time Advisory   show close
    Learning Outcomes   show close
  • 7.1 Introduction to Relativity  
  • 7.2 Spacetime  
    • Reading: Wikibooks’ Special Relativity

      Link: Wikibooks’ Special Relativity (HTML)

      Instructions: Please click on the link above, and read the following sections: Introduction, Principle of Relativity, and Spacetime.  Pay special attention to the postulates of special relativity, light cones, and the Lorentz transformations.

      You should spend approximately 1 hour and 30 minutes on this reading.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

    • Reading: Nobelprize.org’s “Relativity”

      Link: Nobelprize.org’s “Relativity” (HTML)

      Instructions: Please click on the link above, read the introductory text on the webpage, select the links to “The Michelson-Morley Experiment” through History of Special Relativity,” and read all of these sections.  This reading is a solid review of relativity.

      You should spend approximately 1 hour on this reading.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • 7.3 Simultaneity, Time Dilation, and Length Contraction  
  • 7.4 The General Theory of Relativity  
  • Final Exam  

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