Topic Name Description
Course Introduction Page Course Syllabus
Vector Algebra File Vectors in Euclidean Space

Read the first four sections of chapter 1 on pages 1–30. Work through the examples and try the odd-numbered exercises after each section. You can find the answers on page 189.

Double and Triple Integrals in Cartesian Coordinates Page Double Integrals

Watch this lecture series. Double integrals are typically used to calculate volumes.

File Multiple Integrals

Read the first two sections of chapter 3 on pages 101–109. Then, work through the examples and try the odd-numbered exercises 1–13 at the end of section 3.1 on page 104, and the odd-numbered exercises 1–11 at the end of section 3.2 on page 109. You can find the answers on page 191.

Page Triple Integrals

Watch these videos, which demonstrate how triple integrals could be used to calculate mass and find the center of mass of three-dimensional objects.

File Triple Integrals

Read section 3.3 on pages 110–112. Work through the examples and try the odd-numbered exercises 1–9 at the end of the section on page 112. You can find the answers on page 191.

Integrals in Curvilinear Coordinates File Vectors in Euclidean Space

Read section 1.7 on pages 47–50. Work through the examples and try the odd-numbered exercises 1–9 on page 50. You can find the answers on page 189. Some integrals are much easier to take when they are expressed in terms of coordinates that are not Cartesian (that is, cylindrical or spherical). This section explains how to convert curvilinear coordinates to Cartesian and vice versa.

Page James Sousa's "Triple Integrals Using Cylindrical Coordinates" and "Triple Integral and Volume Using Cylindrical Coordinates"

Watch these two videos on using triple integrals in cylindrical coordinate systems.

File Change of Variables

Read pages 121–123 of section 3.5. Work through the examples and try exercises 1 and 3 at the end of the section. The answers are on page 191.

Line and Surface Integrals Page Line Integrals and Vector Fields

Watch these videos, which introduce line integrals. Line integrals are needed to calculate the physical quantities of work and circulation.

File Line Integrals

Read section 4.1 on pages 135–142. Work through the examples and try the odd-numbered exercises 1–13 at the end of the section on page 142. The answers are on page 191.

Page Parameterizing a Surface

Watch this lecture series, which introduces and gives examples of surface integrals.

Page Surface Integrals

Watch this lecture series, which covers surface integrals in detail.

Page Flux in 3D and Constructing Unit Normal Vectors to Surface

Watch this lecture series. The flux of a field, electric or magnetic, is used to determine how strong the field is. Gauss' and Faraday's laws of electromagnetism involve the flux of electric and magnetic field, respectively.

File Surface Integrals and Divergence Theorem

Read section 4.4 on pages 156–164. Work through the examples and try the odd-numbered exercises 1–9 at the end of the section. The answers are on page 191.

Differential Equations Page Introduction to Differential Equations

Watch this lecture series, which introduces differential equations.

Page Separable Differential Equations

Watch this lecture series, which introduces separable differential equations.

File First-Order Differential Equations

Work through the six examples on the page. Be sure to solve them on your own before looking at the solutions.

1.1: Periodic Motion and Simple Harmonic Oscillators Page Springs, Hooke's Law, and Harmonic Motion

Watch these videos.

URL A Non-Harmonic Oscillator

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 nonlinear behavior. A damping term has also 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 on the motion of the oscillator. Check out the special cases suggested in the "Details" section of the demonstration: motion in the absence of damping (set the damping constant to 0), motion of the balloon when the string has no mass (the force of gravity no longer increases as the balloon goes up), and motion when the mass of the string is large and the damping constant is large. What happens to the balloon eventually in this case?

URL Oscillatory Motion

Read this chapter and try each of the six problems 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 solve these kinds of problems on the final exam.

URL Simple Harmonic Motion of a Spring

This 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. Use this worksheet as your guide in working with this demonstration.

1.2: Vibrations URL Vibrations

Download this book. This is a large file, but we will use it throughout the course. This version contains the solutions to the Self-Check questions. Read sections 1 and 2 of "Chapter 17: Vibrations" on pages 445–459.

1.3: Wave Motion Page Introduction to Waves

Watch this lecture series.

URL Free Waves

Read "Chapter 19: Free Waves" on pages 481–499. Answer the self-check questions. You can find the answers on page 553. Think about the discussion questions and solve problems 1–4 on pages 507–508. You can check some of the answers here.

URL Superposition of Waves

This demonstration illustrates the superposition of two waves traveling in opposite directions. First, try setting the frequencies of the two waves to be equal. Notice that as the time passes, the superposition of two waves goes from "double" the wave (the wave with the same frequency and twice the amplitude), when the waves are in the same phase, to "no wave" (the waves cancel each other out completely) when they are in the opposite phase. Then, explore what happens when the frequencies of the waves are close to each other, but a little bit different. How does the superimposed wave look like? This effect is easier to see when the frequencies are large. Try clicking on the "plus" icon in the top right corner of the demonstration, and selecting "autorun". Notice that the superimposed wave contains an oscillation within an oscillation, one with the sum, and another one with the difference of the original frequencies.

URL Waves on a String with a Mass in the Middle

Waves are partially reflected by local changes in the medium through which they propagate. This is illustrated here by the introduction of a point mass on a vibrating string. The transmitted wave becomes smaller in amplitude as the mass becomes larger. Why? Is there a phase shift associated with reflection/transmission? Why?

2.1: Introduction to Electricity Page Charge and Electric Force (Coulomb's Law)

Watch this lecture series.

URL Electricity and Circuits

Read "Chapter 21: Electricity and Circuits" on pages 561–565. Complete the self-checks (answers on page 1010). Think about the discussion questions and examples, and solve problems 1–9 on pages 601–602. You can check some of the answers here.

URL The Atom and E=mc2

Read "Chapter 26: The Atom and E=mc2" on pages 731–742. Answer the self-check questions (answers on page 1011). Think carefully about the Millikan's Fraud discussion, which illuminates the basis of science and how it is eventually self-correcting. Think about the discussion questions and examples, and solve problems 2 and 4 on page 787. You can check your answer to problem 4 here.

URL Electricity

Read this chapter. Try each of the problems in it before looking at the solutions. Make sure you understand how to approach solving the problem so that you can obtain the solution yourself.

URL Repulsion of Charged Objects

This demonstration illustrates the relationship between the charge on the two balls and the separation between them. Use this worksheet to work out this relationship by considering the balance of the forces acting on the charged balls.

URL Van de Graaff Generator

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.

2.2: Electric Field and Gauss' Law URL The Nonmechanical Universe

Read "Chapter 22: The Nonmechanical Universe" on pages 618–630. Answer the self-check questions (answers on page 1010). Think about the discussion questions and examples, and solve problems 1–7 and 10 on pages 644–645. You can check some of your answers here.

Page Electric Fields

Watch this lecture series.

URL Gauss' Law and Applications to Conductors and Insulators

Watch this lecture. Check your understanding by attempting this problem set. Solutions can be found here.

URL Gauss' Law

Read this chapter and try example 4.1 before looking at the solution.

URL Motion of a Charge in an Electric Field

This demonstration illustrates the effect that a uniform electric field has on the motion of an electric charge. Why do the charges follow a parabolic path? Think of an analogy that describes the effects of gravity on a projectile.

2.3: Electric Potential and Electric Potential Energy Page Electric Potential Energy, Electric Potential, and Voltage

Watch this lecture.

URL University of Texas: Richard Fitzpatrick's "Electromagnetism and Optics: Electric Potential"

Please click on the link above, and read this chapter after viewing the 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.

URL Electric Dipole Potential

This demonstration illustrates what you would find in a lab experiment where two electrically charged 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.

URL Lines of Force for Two Point Charges

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 an electric field. The lines of an 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 two charges' positions and magnitude 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.

2.4: Capacitors and Capacitance – Storage of Electric Energy Page Circuits with Capacitors

Watch this lecture series.

URL University of Texas: Richard Fitzpatrick's "Electromagnetism and Optics: Capacitance"

Please read this chapter. 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.

URL Parallel-Plate Capacitors

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. First, keep the voltage fixed, and set the values for the area of the plates, plate separation, and the dielectric constant. Calculate the capacitance using the formula given in the demonstration or the reading resources, and confirm that you get the same result as the program (pay attention to the units of all physical quantities!) Then, use the values of the voltage and the capacitance to calculate the charge on the capacitor. Again, confirm that your result is the same as that in the demonstration.

URL Partially-Filled Capacitors

This demonstration, again, can be used as an interactive homework problem. 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.) Use the formula for the capacitance in terms of the area of the plates, the distance between the plates, and the dielectric constant to write down the capacitance of the "filled" and "empty" capacitors. Notice that the distance between the plates of each capacitor depends on the percentage k of the dielectric material, as shown in the demonstration. Then, use the formula for the equivalent capacitance of the two capacitors connected in series to derive the result in the "Details" section of the demonstration. The values of A and d is fixed and given in the "Details" section of the demonstration. You can select a value of k and a dielectric material (you will have to look up the value of the dielectric constant), and calculate the capacitance of the partially filled capacitor. Again, confirm that your result agrees with the demonstration. 

URL Electric Field Energy in Capacitors

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 energy driven by applied voltage from the demonstration. First, calculate the charge on the capacitor's plates, and the resultant electric field between the plates. Use the formulas given in the description in the demonstration, or the readings above, and confirm your result. Then, calculate the electric energy stored in the capacitor, and the electric energy density. Again, confirm your result. 

3.1: Electric Current, Voltage and Resistance Page Circuits and Ohm's Law

Watch this lecture series.

URL Electricity and Circuits

Read sections 3–8 of "Chapter 21: Electricity and Circuits" on pages 566–608. Complete the self-checks (answers on page 981). Think about the discussion questions and examples, and solve problems 10–12, 16–17, and 24–38. You can check some of the answers here.

URL Electric Currents

Read the sections "Electric Circuits", "Ohm's Law", "Resistance and Resistivity", "EMF and Internal Resistance", and "Resistors in Series and in Parallel". Try to solve examples 7.1 and 7.2 before looking at the solutions.

URL Resistors in Series

Treat this demonstration as a laboratory experiment. Measure the voltage, resistance, and current levels for various conditions and use Ohm's Law to explain the results. Think how the equivalent resistance of the circuit is compared to that of individual resistors' (less or greater?) and how this affects the resultant current.

URL Resistors in Parallel

Treat this demonstration as a laboratory experiment. Measure the voltage, resistance, and current levels for various conditions and use Ohm's Law to explain the results. Think how the equivalent resistance of the circuit is compared to that of individual resistors' (less or greater?) and how this affects the resultant current. Which connection – series or parallel – should be used to produce the maximum possible current?

3.2: Electric Circuits Page Analyzing More Complex Resistor Circuits

Watch these lectures.

URL Electric Currents

Read the sections "Kirchhoff's Rules", "Capacitors in DC Circuits", "Energy in DC Circuits", and "Power and Internal Resistance". Try examples 7.3 and 7.4 before looking at the solutions.

URL Effective Resistance

Treat this demonstration as a laboratory project in which you calculate the resistance expected for a given network before "measuring" the actual resistance using the demonstration. Use this worksheet as a guide to derive the general formula for the equivalent resistance of the given network.

URL Galvanometer as a DC Multimeter

This demonstration illustrates how a current indicating meter (the galvanometer) can be used in simple circuits to measure voltage, current, or resistance over a wide range of values. Despite the text appended to the demonstration, there is no way to vary the circuit elements of the multimeter, and we are not supplied with the actual values used in the demonstration. However, the three selectable modes of operation all make a simple application of Ohm's Law, so see if you can understand the operation of the multimeter based on Ohm's Law and the indicated circuits.

4.1: Magnetic Field URL Circuits and DC Instruments

Read sections 22.1, 22.2, and 22.3. Use the conceptual questions to assess your understanding of these sections.

URL Observing Magnetic Fields with Iron Filings

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.

4.2: Magnetic Force on Moving Electric Charges Page Magnets and Magnetic Force

Watch this lecture series. The force on a charge in a magnetic field is related to the cross-product of the charge's velocity and the field vector. Two of these lectures review the cross-product of two vectors and how to calculate it. 

URL Magnetism

Read "The Lorentz Force", "Charged Particle in a Magnetic Field", and "The Hall Effect". Also, solve examples 8.1 and 8.2 before looking at the solutions.

URL Motion of a Charge in a Magnetic Field

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. This demonstration does not provide any measurements of the charge's trajectory, but you can observe how it changes qualitatively with the change of the parameters. As usual, change each parameter in turn while keeping the others constant to make conclusions. What affects the radius of the spiral path? Compare your observation with the theoretical results in the reading resources. How does the angle between the magnetic field vector and initial velocity affect the shape of the spiral? What happens if their direction is the same, and how can this be explained using Lorenz's force formula?

URL Motion of a Charge in Electric and Magnetic Fields

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?

4.3: Magnetic Field of a Current-Carrying Wire Page Magnetic Field Created by a Current

Watch this lecture series.

URL Magnetism

Read "Ampere Experiments", "Ampere's Law", "Ampère's Circuital Law", "Magnetic Field of a Solenoid", and "Gauss' Law for Magnetic Fields". Solve example 8.3 before looking at the solution.

URL Magnetic Field of an Electric Current

This is a simulation of another classic classroom demonstration, in which you trace out the magnetic lines of force surrounding a current-carrying wire. The magnetic force lines are circles surrounding the wire, and the direction of the magnetic field reverses when the current reverses. Practice using the right-hand rule to determine the direction of the magnetic field, and make sure you get the same result as in the demonstration.

URL Magnetic Field of a Current Loop

This demonstration illustrates the magnetic field surrounding a current loop. The magnetic field has a cylindrical axis of symmetry on the axis of the current loop.

4.4: Magnetic Materials URL Magnetism

Read "Origin of Permanent Magnetism".

URL Magnetization

This demonstration illustrates the process of magnetization of magnetic material in an external field. As the magnetic domains (small bar magnets) become more highly aligned, the magnetic field produced by the material increases. Note that when all (or most) domains are aligned, the material cannot become more highly magnetized. This is the phenomenon of magnetic saturation.

5.1: Faraday's Law URL Lenz's and Faraday's Laws

Watch this lecture. Check your understanding by attempting problems 3, 11, and 12 from this problem set. Check your solutions here.

URL Magnetic Induction

Read "Faraday's Law", "Lenz's Law", "Magnetic Induction", "Motional EMF", and "Eddy Currents". Solve examples 9.1 through 9.3 before looking at the solutions.

URL Magnetic Flux through a Wire Loop

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. Pick a value for the loop's radius and calculate the magnetic flux through the loop in the two cases when the field is uniform: (0,0, 1) and (1, 1, 1). Do your results agree with the demonstration?

URL Magnetic Braking via Eddy Currents

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?

URL EMF Induced in a Wire Loop

Use this worksheet as a guide to explore the demonstration.

URL Electromagnetic Ring Toss

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 that 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?

5.2: Inductance URL Inductance

Read "Mutual Inductance", "Self-Inductance", and "Energy Stored in an Inductor". Solve examples 10.1 and 10.2 before looking at the solutions.

URL Magnetic Field Energy in Inductors

This demonstration provides examples for both capacitors and inductors, but work only with the inductors for now. Treat this demonstration as a laboratory experiment in which you measure the inductance of various geometries and confirm from the theory that the values are correct. Get the electromagnetic field energy generated by applied current from the demonstration, and confirm them by direct calculation based on the theoretical relations listed here and in the lectures and readings above. First, calculate the inductance of the coil, using the formula given in the description in the demonstration, or the readings above. Then, calculate the current through the inductor using Ohm’s Law, and the magnetic field created by this current. Finally, calculate the magnetic energy stored in the inductor, and the magnetic energy density. Again, confirm your result.

You can also explore this demonstration conceptually, to get a feel for how the magnetic field inside the coil and magnetic energy density are affected by the geometry of the inductor, resistance, and applied voltage. Vary each of the parameters in turn and observe the changes in B and ρ. Why does not the magnetic field depend on the radius of the coil?

5.3: RC, RL and RCL Circuits URL Capacitance and Inductance

Read chapter 25 on pages 713–727. Answer the self-check questions (answers on page 1011). Think about the discussion questions and examples, and solve problems 1–8 on page 729.

URL Series RLC circuits

This demonstration shows the graphs of the voltage and current in the circuit containing a resistor, and inductor, and a capacitor connected in series, driven by the alternating voltage. The frequency of the alternating voltage supplied by the battery will determine the phase shift φ between the voltage and the current. Pick the values of the parameters and calculate the phase shift using the formula in the demonstration. Then, try to vary the frequency until you make the phase shift 0 (this might be difficult to accomplish, but try to make it as small as you can) and confirm that this frequency is close to the circuit's resonant frequency.

5.4: Electromagnetic Generators and Motors Page Electric Motors

Watch this lecture series.

URL Magnetic Induction

Read "The Alternating Current Generator", "The Direct Current Generator", "The Alternating Current Motor", and "The Direct Current Motor". Solve examples 9.4 and 9.5 before looking at the solutions.

URL Barlow's Wheel – A Primitive Electric Motor

This is a demonstration of one of the simplest electromagnetic motors ever invented. Before looking it up, figure out just how it works. (Hint: Lorentz force.)

URL Faraday Disk Generator

The Faraday Disk is a simple electric generator. Before looking it up, figure out how it works.

6.1: Maxwell's Equations URL Introduction to Maxwell's Equations

Most of this chapter reviews Maxwell's time-independent equations. Note of the definition of the electric flux density vector on page 7: D = ε E, where E is the electric field vector and ε is the dielectric constant times the free space permittivity εo. On page 12, the magnetic flux density vector is defined as 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). These equations 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.

URL Maxwell's Displacement Current

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 that 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 the magnetic field associated with a changing electric field as the displacement current.

6.2: Electromagnetic Waves URL Maxwell Equations and Electromagnetic Waves

Watch this lecture. Check your understanding by attempting problem 6 from this problem set. Check your solution here.

URL Why Things Radiate

This chapter develops the time-varying version of Maxwell's Equations and uses them to examine the properties of EM radiation and why this radiation is emitted in the first place. Stick with the math until you can see the physics.

URL Propagation of a Plane EM Wave

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.)

URL Measuring the Speed of Light with Marshmallows and a Microwave Oven

Try this experiment using your microwave oven. Why do the melted spots appear in a regular pattern?

6.3: Energy and Intensity of Electromagnetic Waves URL Maxwell Equations and Electromagnetic Waves

Watch this lecture. Check your understanding by attempting this problem set. Check your solutions here.

6.4: Spectrum of Electromagnetic Radiation URL Electromagnetic Waves

Read chapter 24, "Electromagnetic Waves". Think about the accompanying conceptual questions to assess your understanding of the chapter.

7.1: Geometric Optics URL The Ray Model of Light

Read chapter 28 on pages 814–826. It introduces the phenomena of reflection and refraction. Answer the self-check question (answer on page 1012). Think about the discussion questions and examples, and solve the problems on pages 829–830.

Page Reflection and Refraction

Watch this lecture series.

URL Geometric Optics

Read these sections. Work through examples 12.1 and 12.2 before looking at the solutions.

URL Specular and Diffuse Reflection

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.

URL Snell's Law of Refraction

This demonstration illustrates the way in which light bends at an interface between the two media. Use this worksheet as a guide when exploring this demonstration.

URL Total Internal Reflection

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 in 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 metalized 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.

URL Rainbows

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.

7.2: Paraxial Optics Page Mirrors and Lenses

Watch this lecture series.

URL Paraxial Optics

Read these sections. Work through examples 13.1–13.4 before looking at the solutions.

URL Ray Diagrams for Lenses

This demonstration illustrates the dynamics between focal length, object distance, and real/virtual focal points of a simple lens. Explore this demonstration for both convergent and divergent lenses. Drag the object closer to the lens and observe how the magnification changes with distance. Pay attention to the path of the rays. Notice that for the convergent lens, at some point, the image changes from real to virtual. How far is the object from the lens at this point? Change the height of the object and repeat the procedure. Does the same thing happen again?

7.3: Wave Optics URL Wave Optics

Read the sections "Introduction", "Huygens' Principle", and "Young's Double-Split Experiment". Work through example 14.1 before looking at the solution.

Page Interference of Light Waves

Watch this lecture series.

URL Two-Slit Constructive and Destructive Interference

Interference is a nearly ubiquitous effect in wave optics. This demonstration illustrates how light waves interfere both constructively and destructively. The electric vector of the electromagnetic radiation is shown waves moving from the slits. When you examine the diffraction image, the points of constructive interference (where the two electric fields add) appear red, while the points of destructive interference (where the two electric fields cancel) appear white.

8.1: Introduction to Relativity URL Introduction to Relativity

Watch this lecture.

URL Special Relativity

This article introduces special relativity. You do not have to memorize every single fact. Strive to make sure that it makes sense. When you finish, briefly summarize the main points.

Page Relativity in 5 Minutes

Watch this video to review the essential physics of relativity.

URL Two Swimmers

This applet demonstrates the expectation of Michelson and Morley that if the speed of light was constant through the ether (the speed of the swimmers is constant through the water) and the detection device is moving through the ether (the platform is moving through the water), then the travel time for the perpendicular light beams (the travel time for the two swimmers) would be different. Why does special relativity address the fact that the travel times in the Michelson-Morley were actually the same?

Page Albert Einstein's Life and Career

Watch this video, which discusses some of Einstein's work and interests outside of physics.

Page Alfred Einstein and the Theory of Relativity

Watch this video and make sure that you understand the logic of its two conclusions.

URL Russell's Thought Experiment

This demonstration illustrates what happens when you add a series of velocities that, in Newtonian mechanics, would reach speeds faster than the speed of light. The demonstration illustrates why relative speeds can add up to nearly the speed of light but cannot exceed that speed.

8.2: Simultaneity, Time Dilation, and Length Contraction URL Lorenz Transformations

Watch this lecture. Check your understanding by attempting problems 1, 3, and 5–7 from this problem set. Check your solutions here

URL Time Dilation: A Worked Example

This page shows how time dilation and length contraction can explain certain apparent paradoxes in relativity. Be sure that you understand the logic used by Jack and by Jill to account for the observations.

URL More Relativity: The Train and the Twins

This page shows two worked examples: a train moving through a tunnel and the twin paradox. The last example uses the Doppler effect to show a clever way to explain the difference in ages.

URL Relativistic Length Contraction

Explore this applet, which illustrates the paradox of a muon originating in the upper atmosphere striking the surface of the earth.

URL Spacetime Diagram

This demonstration illustrates how a set of events appears in two inertial frames moving with respect to one another.

URL Reviewing Relativity

Read "The Michelson-Morley Experiment" through "History of Special Relativity". These give a solid review of relativity.

8.3: The General Theory of Relativity URL General Relativity and Black Holes

Read this tutorial.

Study Guide Book PHYS102 Study Guide
Course Feedback Survey URL Course Feedback Survey