PHYS102: Introduction to Electromagnetism

Before we begin, watch this video that explains why it is important to study electromagnetism. There are two types of charges, and like charges repel, while unlike charges attract. When equal amounts of both charge types are combined in an object, it becomes electrically neutral. The forces from balanced charges of opposite types inside a neutral object cancel out when viewed from the outside. This explains why we do not often notice the effects of electromagnetism directly, which creates the impression that it must be a weak kind of force. In reality, electric forces are much stronger than gravity! We will come back to this in later lectures.

Read this introduction to Hooke's Law that covers the concept of energy. Using the language of energy, deformed springs store potential energy, which can be converted into kinetic energy when the spring is released. In this process, the spring force causes acceleration by doing mechanical work.

Watch this video to see some additional examples of how to apply Hooke's Law.

When a spring causes acceleration, the motion often overshoots the equilibrium that it wants to return to because the oscillating mass has inertia. After it overshoots, the restoring force reverses direction and causes an opposite acceleration. This is how oscillations are created: the motion keeps reversing and overshoots its equilibrium every time.

Read this text to learn how we characterize oscillations quantitatively. The standard unit of frequency is called hertz (abbreviated Hz). It is no coincidence that you find the same unit labeling your radio dial: later, we will see that radio stations transmit electromagnetic waves of specific frequencies.

This text describes simple periodic motion by making graphs of the spring deformation over time. The graphs show a universal shape no matter what the specific oscillator looks like, and it is described mathematically by a sine or cosine function.

In the description of harmonic motion, the choice between sine and cosine functions depends on how the oscillation was launched at the initial time. Since a harmonic oscillation repeats exactly after one period, different launch conditions only have two possible effects: they determine the amplitude (the maximal deformation during the oscillation) and also the times at which that maximum is reached. In a graph, changing the time of maximum deformation corresponds to shifting the plot left or right along the time axis without changing its overall shape.

A sine function can be mathematically thought of as a cosine function that is shifted by a specific amount.

This video discusses this idea mathematically, but allows the shift along the time axis to have an arbitrary value to account for arbitrary launch conditions at a chosen time (usually called t = 0). This introduces the concept of the phase of oscillation.

Simple harmonic motion does not just happen with springs. For example, an old-fashioned grandfather clock makes use of simple harmonic motion to keep accurate time. A simple pendulum is defined as having an object with a mass of small size (the pendulum bob) that is suspended from a light wire or string. Examples include the pendulums that guide the movement of time on a clock, a child's swing, a wrecking ball, or even a sinker or weight at the end of a fishing line.

This text explores the conditions where a pendulum performs simple harmonic motion and derives an interesting expression for its period. For small displacements, a pendulum is a simple harmonic oscillator. As with all harmonic motion, the period of a pendulum is independent of the amplitude. This robustness of the period is what makes pendulum clocks work with great accuracy.

If you are familiar with rotational motion, you will recognize that the motion of a pendulum is really a back-and-forth rotation around the point of suspension. For a discussion of the pendulum from this point of view, watch this optional video.

Read this optional text to dive deeper into the relationship between rotational motion and harmonic oscillations in general.

In terms of energy, damped oscillations cannot maintain a constant balance of kinetic and potential energy because some of the mechanical energy is drained away in the form of thermal energy. This eventually causes the mechanical motion of the oscillator to return to equilibrium and stay there. At that point, all the energy of the oscillation has been converted to microscopic, invisible motion at the molecular level in the surrounding air or the oscillator itself.

Read this text which explains how a dramatic increase in the oscillation amplitude as the driving period is adjusted can lead to resonance. Notice the frequency where the driven oscillation occurs. Is it the natural frequency of the oscillator, or is it the frequency with which the external force is applied? If you try the experiment in Figure 16.26, you will see that you control the frequency – not the paddle ball. However, the response of the oscillation amplitude depends dramatically on which frequency you choose.

Waves come in various forms, such as sound, light, or ripples on the surface of a body of water. They all have the same underlying properties. The main difference between waves and the movement of matter particles is that waves can pass through each other without affecting each other's motion.

When more than one wave is present in the same place, they combine. This is known as superposition, and it creates wave patterns that are called interference. Waves also demonstrate diffraction, which is the ability to bend around an obstacle. When a wave encounters a boundary between two media, it undergoes reflection (traveling backward) and transmission (which goes along with refraction).

Are waves a type of motion? Yes, if they exist in a medium – such as sound waves in the air or ripples on a pond. The building blocks that the medium is made of (for example, air molecules) perform a coordinated motion when a wave passes over them. However, they return to their original positions when the wave is gone. This is something waves have in common with oscillators: for all the building blocks of the medium, there is an equilibrium position that can be perturbed (for example, the flat surface of a still pond), and there is a restoring force similar to that of a spring. Just as in a simple harmonic oscillator, mass gives the medium inertia, so it reacts to its own internal restoring forces with some delay.

But waves are not simply collections of oscillators, all doing their own little dances. The spring-type forces inside the wave medium have one additional function, which is to couple neighboring oscillators to each other. This is what creates the highly coordinated motion that we identify as a wave. This is something waves have in common with forced oscillations. Each oscillating part of the wave medium feels forces from its neighbors and can receive and transmit energy. As a result, waves can transport energy across a medium, even if the medium as a whole stays put!

Pay attention to the difference between transverse and longitudinal waves. The oscillations of a transverse wave are perpendicular to the direction of the wave's advance. A longitudinal wave travels in the direction of its oscillations.

When you play this demonstration, you will see that the frequency you use to jiggle the end will also determine the frequency of the wave. After trying this manually, click "Oscillate" at the top right to see what happens if you excite a wave by making the end follow a simple harmonic motion described mathematically by the sine function. That is called a periodic wave.

Periodic waves are characterized by their wavelength, which is the distance the wave travels during one period – the time it takes for the oscillation to go through a complete cycle. Watch this video to see how the wavelength and period are related to the speed of the wave propagation.

To illustrate wave speed in the simulator, it was essential to use a very long string available so that there are no reflections at the other end. They would obscure the pattern that we are trying to describe, which is called a traveling wave. As with the harmonic oscillator, we can also write down a mathematical description of the wave patterns shown above. You can see this in this video.

Watch this video, which shows how to put this mathematical function into the context of the physical property of the wave we are trying to describe (its wavelength).

Watch this video to review how the concepts fit together. It demonstrates another graphical representation of a traveling wave.

Review this video, which combines the concepts you have encountered so far.

Read this text to learn more about standing waves and musical beats and see an interactive demonstration of wave interference on a water surface.

Watch this video to see how constructive and destructive interference is created by adding two waves while taking the sign of the wave's displacement into account at every point.

The superposition principle for waves may still seem a little mysterious. To avoid misconceptions, be aware that not all waves obey this principle! When you see two boats leaving wakes on a still lake, you can observe the wakes passing through each other to form interference patterns, just as the superposition principle predicts. But for a ship in stormy seas, the wake will be obliterated by the ocean's crashing waves, which is because waves of large amplitude usually fail to form superpositions. Waves obey the superposition principle only if the restoring forces in the medium obey Hooke's Law!

Hooke's Law is a proportionality between force and displacement that breaks down when the displacement from equilibrium is too large. A simple example of this breakdown is the pendulum we studied earlier. The component of the gravitational pull that acts to restore the pendulum to its vertical equilibrium ceases to be proportional to the angle with the vertical when that angle becomes too large. When we talk about wave interference, we assume that the wave amplitudes are small enough to make Hooke's Law valid. Watch this video, which summarizes the relationship between interference and beats.

Read this text to learn more about energy in waves. The amount of energy in a wave is related to its amplitude. When we talk about the amplitude of a wave, we refer to deviations from the equilibrium of the medium carrying the wave. Large-amplitude earthquakes produce large ground displacements. Loud sounds have higher pressure amplitudes and come from larger amplitude source vibrations than soft sounds. Large ocean breakers churn up the shore more than small ones.

At the level of the wave medium, a wave is a displacement that is resisted by forces that have the dual tendencies of restoring equilibrium and coupling neighboring regions. The speed of the wave also depends on the mass contained in the medium because this implies inertia. Inertia governs the delay with which the energy put in at one end reaches the other end. The medium stores the energy of the wave as the potential energy of the coupling forces and as the kinetic energy of the oscillating mass. Since the mass only oscillates in place, a wave can transport energy over large distances without any mass being transported overall.

Static electricity refers to a constant imbalance of electric charges within or on the surface of a material. The charge remains until it can move away, such as via an electric current or electrical discharge.

Rubbing certain materials against one another can transfer negative charges (electrons). For example, when you rub your shoe on the carpet, your body collects extra electrons. The electrons cling to your body until they can be released. When you reach out and touch an animal, you get a shock as you release the surplus electrons to your unsuspecting pet. An accumulation and discharge of static electricity can create a spark that can be dangerous, especially when a flammable fluid like gasoline is around. Lightning is an example of static electricity. Read this text, which explores static electricity and charge.

The reading in our textbook explained what a battery is at a basic level. The important thing to keep in mind is that batteries can be charged and discharged, but they do not create new charges. Batteries contain a mechanism (usually chemical) that can maintain a charge imbalance between the positive and negative terminals even if you allow charges to flow between the two terminals through an external circuit like a light bulb. When a battery "loses its charge", that means that the chemical process that was designed to maintain the charge imbalance has stopped working.

Charge is a property of fundamental particles found in all the matter around us, and in that sense, charge is not so different from mass. The big difference between the mass and charge of a particle is that there are no negative masses. On the other hand, whereas the protons in the nucleus of an atom carry positive charge, the electrons that inhabit the outer regions of an atom are negatively charged.

To create a charge on a macroscopic object such as a balloon, we just transfer electrons from one object to another while leaving the positively-charged atomic nuclei where they are. When this is achieved by friction, it is called triboelectricity.

Because charge is ultimately carried by fundamental particles, it also obeys a fundamental conservation law: charge cannot be created or destroyed. Watch this video, which gives several examples of conservation of charge.

Read this text, which discusses conductors and insulators.

Neutral objects are not "devoid" of charge – they just contain balanced amounts of positive and negative charges. Watch this video for more illustrations.

The analogies between gravitational and electrostatic force go even further, in that the force between two objects is proportional to the charge of each object individually. So when you double one of the charges, the force doubles. If you double both charges, the force between them quadruples.

The most obvious difference between gravity and electricity is that gravity is always attractive because mass does not come with two different signs. According to Coulomb's Law, the electric force is always attractive between charges of opposite sign and always repulsive between charges of the same sign!

We can discover a less obvious difference between gravity and electricity if we try to quantify the forces in units of Newtons. It then turns out that the force of gravity is unimaginably weaker than the electrostatic force at the atomic level. Read this text, which gives details.

In the quantitative discussion, we encountered the standard unit of charge, the Coulomb, named after the person who discovered the electric force law. Watch this video, which drives home the point that a Coulomb is actually a huge amount of charge if you count the number of fundamental microscopic particles that would have to be gathered to make that amount of charge.

Coulomb's force law contains the assumption that the two charged objects involved in the interaction are either spherical (charged uniformly) or are so far apart that it is okay to consider both of them as point-like (that is, objects without a shape altogether). Then the direction of the electric force is always along the straight line connecting the objects. In the case of spheres, we consider that to be from center to center.

It can be confusing to figure out in what direction the force should point. To figure out the correct force direction, remember this simple rule: the forces at opposite ends of the straight line connecting the charges must always point in opposite directions. That is the same as for the tension forces at both ends of a spring. Either both ends are pulling inward, or both ends are pushing outward. Since two charges with opposite signs attract, the forces in Coulomb's Law point inward at both ends of the line connecting the charges. And since two charges with equal sign repel, the Coulomb force, in that case, points outward at both ends of the connecting line.

You may ask yourself how Coulomb's Law can be true for the electrons and protons inside a material if it is electrically neutral overall. After all, even though electric forces fall off with increasing distance, the force is never precisely zero, no matter how far away you are from an individual point-like charge.

In fact, when we say an object is electrically neutral, it is really a statement about the electrical forces that the object is (or is not) able to create over relatively large distances. To measure such a force, you would approach the neutral object with an electrically charged test object and observe if it feels a push or a pull. When you are far enough away from a neutral object, the repulsive and attractive forces created by the positively- and negatively-charged fundamental particles inside of it cancel out when their combined effect on your test particle is measured. The combined effect of several simultaneous forces is what we call the resultant, and the resultant of two opposing forces of equal strength is zero.

Almost all applications of electrostatics (including the explanation of what "neutral" means) involve large numbers of charges. In contrast, Coulomb's Law only applies to the limited special scenario of two charged objects that essentially behave like points. This is why we need additional tools to make practical use of Coulomb's Law.

Now, according to Coulomb's Law, the electrostatic force on a test object due to some other object is proportional not to the mass but to the charge of the test object. Following the same logic as for gravity, what do you get if you divide the Coulomb force by the charge of the test object? The result is again a quantity that is independent of the test object itself. But this time, it is not the acceleration. It is called the electric field (or E-field), and it is wholly determined by the charge of the other object involved in the electrical interaction.

As we mentioned earlier, electric fields are a tool that helps us deal with situations where Coulomb's Law has to be applied to collections of several charged particles at once. Read this section, which discusses this concept and introduces a powerful way of visualizing the effects of charges on other charges: electric field lines. There is a new level of abstraction here because although the lines representing the electrostatic field begin and end on charged objects (more or less tangible entities), the lines themselves fill the empty space between the charges. So, we are drawing something that is not tangible because there appears to be nothing there.

To put it briefly: electric fields are would-be electric forces. As a physical quantity, the electric field at any given point in space tells you what the electric force on a test object would be if you were to place it there.

Based on how the electric field is constructed (remember the analogy to Newtonian gravity), you get the electric force on an object of charge Q in an electric field of strength E by simply multiplying the two. The purpose of the field lines is to tell you in which direction that electric force would then point.

The use of electric field lines goes beyond mere visualization. Drawing field lines for a collection of electric charges can help you identify patterns and symmetries that make calculations easier in practical applications.

As an example for symmetry in electric field line patterns, recall that the field lines of a point charge form a star shape that looks the same in all directions. The same field line pattern also forms around a uniformly charged sphere, which is why Coulomb's Law makes no distinction between such a sphere and a point charge. It doesn't matter whether the sphere has all the charge sitting on its surface or if the charge is spread out throughout its interior.

When we look at the symmetries of field line patterns, we discover that you can get the same electric fields from seemingly quite different distributions of charges.

Read this text, which discusses some consequences of this approach.

Watch this lecture to review this material and connect it to applications of electrostatics.

Read this section, which explains (among other applications) how inkjet printers work.

Read this text, which discussed these concepts.

Watch this lecture, which gives a visual summary of how electrostatic potential energy can be converted to kinetic energy.

The video we just watched on electrostatic potential also makes use of an important property that electrical conductors show as relates to the electric potential: in equilibrium, the electric potential is the same everywhere along the surface of a conductor. This is closely connected to the observation that electric field lines are perpendicular to the conducting surface. Both of these facts come together in the discussion of the parallel-plate configuration in the next section.

Watch this lecture, which leads us from the parallel-plate configuration back to the point-like charges that we started with when introducing Coulomb's Law. The reason we discussed the potential for the seemingly more complicated situation of densely crowded charges on parallel plates before going back to the individual point charge is math. We found that the potential between two plates changes in direct proportion to the distance from one of the plates. This is connected to the fact that the electric field lines between the plates are parallel straight lines.

For a point charge, recall that the field lines form a radial star shape. As a consequence, the potential in this situation is more difficult to compute. Fortunately, the result is relatively simple, differing only slightly from Coulomb's formula for the electric force of a point charge. The main difference is that the potential decreases with the inverse distance to the point charge, not with the inverse squared distance.

Read this text, which illustrates an example of this concept and gives a written formula.

The second part of the video we just watched addresses the concept of equipotential lines, which you can find in this next section of the text.

As you read, pay attention to the interactive simulation at the bottom of the page. Try recreating the charge configurations of Figures 19.9 and 19.10 by yourself. Then click on the panel on the right to enable the "Voltage" indicator and drag the indicator around the screen. To find an equipotential line, try to move the voltage indicator in just the right way so that its reading stays roughly the same. As you do, note what shape you are tracing out. It ought to be similar to the closed green lines in the figures, indicating the equipotential lines.

Watch this video, which introduces the central characteristics of capacitors.

Read this text, which gives examples of capacitors. Air capacitors power radio tuning circuits; mylar capacitors power timer circuits, such as clocks, alarms, and counters; glass capacitors power high voltage applications; ceramic capacitors are used for high-frequency antennas X-rays, and MRI machines; super-capacitors power electric and hybrid cars.

Capacitance is the amount of charge stored per unit volt. This physical quantity characterizes how capacitors perform in electric circuits, which we will discuss in more detail later.

When viewed as a whole, is a capacitor that stores a certain amount of charge electrically neutral or electrically charged?

The answer is that the two plates of a capacitor store the same amount of charge but with opposite signs. That means that the capacitor is actually neutral, even when it is said to store a large amount of charge! If you could probe the space between the capacitor plates, you would feel electric forces because each plate individually is charged.

But viewed from the outside (at a distance much larger than the plate separation), the forces from the oppositely charged plates would cancel each other, so there is practically no electric field. This is why capacitance, and not field strength, is the quantity that we care most about when using capacitors in practice.

It is not usually feasible to build capacitors just by placing two parallel metal plates opposite each other with an air gap in between. Instead, you fill the space between the conductors with a different insulating material, called a dielectric. This is not just for mechanical stability – it actually enhances the functionality of the capacitor.

Watch this lecture to see how this works.

Although you will typically see capacitors illustrated as two parallel plates arranged like a sandwich, many realistic capacitor designs look nothing like that. All capacitors have two conducting sheets and two-wire terminals that connect to those sheets. But the internal geometry of the sheets can be quite intricate – for example, arranged like stacked cups or rolled-up tape. After this introduction to capacitors, watch this lecture, which closely follows the presentation in the text.

This section gives formulas that combine several capacitors when connected in different ways. The idea is that, from the application point of view, all that matters about a capacitor is the value of its capacitance (as mentioned earlier, the shape of the conductors inside a capacitor can vary widely). That makes it possible to characterize entire collections of capacitors by a single capacitance as if they were just a single unit with two terminals that connect it to the rest of the world.

The two basic ways of combining capacitors are in series and in parallel. Watch this video to see an example calculation for capacitors in series.

Watch this video to see examples for capacitors in parallel.

The ability to store charge in a capacitor is only one aspect of its usefulness. The other aspect is its ability to store energy. The two aspects are related because it takes work to charge a capacitor. Charging really means that we create a charge imbalance between the plates by moving excess electrons (negative charges) onto one plate against their mutual repulsion, and simultaneously removing electrons from the other plate against the attractive force of the positively charged atomic nuclei that remain there.

Working against these forces is analogous to lifting a weight against gravity – except that gravity stays constant, whereas electric forces change depending on the amount of charge that has already been moved.

Read this text, which explores the relationship between stored charge and stored energy in a capacitor.

Watch this video for a brief summary of the main results.

Watch this video for two numerical examples (you may skip the second example at the end).

As you read this text, pay attention to the formula for the current, which will also define its standard unit, the ampere. One ampere corresponds to one Coulomb per second, crossing a given surface.

Read this text, which introduces Ohm's Law and simple circuits.

While the lecture you watched earlier has already covered this material, watch this brief video to review how we define the unit ohm.

Read this section which defines the concepts that quantify the factors that control the current. The electrical resistance of an object measures its opposition to the flow of electric current. Electrical conductance is the reciprocal quantity – it describes how easy it is for an electric current to pass. Electrical resistance shares some conceptual parallels with the notion of mechanical friction. The SI unit of electrical resistance is the ohm (Ω), while electrical conductance is measured in siemens (S) (formerly called "mho"s and represented by ℧).

Get a feeling for the interplay of resistivity and geometry in determining the resistance of a wire by manipulating the interactive simulation at the bottom of the page linked above.

Review the lecture on currents to see how resistance and resistivity are related. Then watch this video, which explains the difference between resistance, which is the ratio of voltage to current, resistor, a device that creates resistance, and resistivity, which is a material-dependent property of the device.

As the following section in the text explains, watt is the standard unit of power – and power is the rate at which energy is transferred.

The text also combines the definition of electrical power with Ohm's Law to get some alternative, but equivalent ways of relating power to the other quantities we have already encountered. Watch this video, which discusses these relationships further.

Watch this video, which leads us from the discussion of electrical power to the definition and properties of AC circuits.

Read this text, which explains that, because the voltage in an AC circuit changes all the time, there is some ambiguity as to what "the" voltage of an AC outlet in your home actually means. Two different ways to characterize the AC voltage are mentioned in the video: the peak voltage and the rms voltage – an averaged value that accounts for the fact that the voltage in an AC circuit spends most of the time at levels that are smaller than the peak value.

As we learned in the text, electricity is delivered to U.S. households at a frequency of 60 hertz. In Europe, that frequency is 50 hertz. Watch this video if you are not sure how frequency is related to the concept of alternating current.

Different countries also employ different voltages in the electricity being delivered to homes. This is described in the video below.

Most of the electrical appliances you use in your home draw their energy from AC outlets. In all likelihood, you have also encountered the problem that too many devices were plugged in at the same time and tripped the circuit breaker. This may seem like a nuisance, but it is designed to save lives. To appreciate this, it is important to have a basic understanding of electrical hazards, as discussed below.

Watch this video for a brief overview of why household wiring includes a third pole, called a ground.

Read this text to further explore this aspect of electricity in the field of biology.

Before we look at different ways to combine resistors, watch this video, which summarizes the graphical symbols used to draw circuit diagrams. We represent capacitors, batteries, and resistors with abstract symbols that are meant to remind you of their basic design.

We have already covered combinations of capacitors in series and parallel. We can make the same two basic types of arrangement with resistors. Just as with capacitors, we can simplify the analysis of a resistor circuit step-by-step, by imagining replacing a combination of resistors by a single ("effective") resistor. We can do this because the internal details of a resistor are irrelevant to its function in a circuit. All we really care about is the value of the resistance that needs to be plugged into Ohm's Law, as applied to the current and voltage found at the two terminals leading into and out of the resistor.

You could say the idea behind analyzing circuits is to get as far as possible by treating parts of the circuit as a black box. And we can indeed treat complicated arrangements of resistors as a black box, so long as there are just two terminals connecting the black box to the outside (because resistors must have two terminals).

It does not matter whether several individual resistors exist between the two terminals in question, or just a single wire, as long as we know the resistance of our black box. Read this text, which implements this strategy, with the aid of two basic rules: one for series and one for parallel resistors.

Watch this video for a brief review of series resistors connected to a battery.

Next, watch this video to see how to deal with resistors connected to a battery in parallel.

To really appreciate the power of the "black-box" thinking that underlies circuit analysis, we have to go one step further by combining three resistors in series and in parallel at the same time. Jennifer Cash discusses this in her next video.

Our textbook also presented an example with three resistors (example 21.3), which is part of this video.

Read this text, which provides an in-depth discussion of batteries and how they act in electric circuits.

Note that you may be confused by the term "electromotive force" (emf) for the voltage the battery creates – and you are right to be confused. It is not a force, but a term we have used historically for so long that it is no longer practical to modernize all the textbooks by inventing a different name. Just think of emf as "internally generated voltage". In a battery, it is caused by the chemical process that separates positive and negative charges to keep individual electrons on the two battery terminals at a constant potential-energy difference.

The reading above addresses issues that arise when you are trying to characterize the performance of a battery, or increase the voltage or current the battery can deliver. This involves stacking batteries to form a series configuration, or putting them in parallel. Put batteries in series to increase the battery voltage. Increasing the current involves overcoming the internal resistance which is ultimately a by-product of the battery's chemistry. Use a parallel configuration to get more current without degrading the battery voltage.

To illustrate the internal resistance of a battery, we can look for the heating that should occur when a large current flows through any resistor. In this video, a large car battery is made to produce a large current by creating a short circuit (that is a circuit between its terminals containing nothing but a good conductor). Indeed, the battery not only produces heat, it sparks!

Watch this video, which explores how to apply Kirchhoff's rules to generate equations to find the unknowns in circuits. These unknowns may be currents, voltages (called emfs when relating to batteries), or resistances.

Read the accompanying textbook so you can carefully go through an example (21.5).

The main purpose of Kirchhoff's Laws is to help predict the values for quantities, such as current and voltage, in a circuit. We would not be able to design electronic circuits, if we could not make these predictions! Anyone working in electronics needs to have a firm grasp of these laws. Watch these videos to get additional practice.

Note that when people talk about junctions in circuits, they typically refer to the points where three or more wires come together. But in principle, even the connection between only two wires is a simple junction. It is just that we do not need any additional laws to figure out what the current and voltage on both sides of a two-wire junction must be: they are simply equal. You only face the difficulty of not being able to directly see what those quantities must be – without having to look to see how the junction is connected to the rest of the circuit – when you have junctions with more than two wires.

In Kirchhoff's second (loop) rule, we deal with voltages. Watch this video for a reminder that voltages are differences in electric potential.

When applying Kirchhoff's second rule, it is important to know when to count a voltage as positive, and when to count it as negative. This depends on the coordinate direction that you have chosen in each branch of the circuit, which also fixes what you call the positive current direction. Watch this video, which shows the specific case of the resistor one more time.

Watch this video for a discussion on how to treat batteries according to Kirchhoff's second rule. The direction in which the electric potential changes is opposite to how it behaves in a resistor. This is why people use a different name for the voltage across a battery: electromotive force. It is not a force, but it is "reminiscent" of a force in that the battery does work on the charges that flow through it, elevating them to a higher potential.

As we mentioned earlier, we need Kirchhoff's rules for circuits that are more complex than simple combinations of series and parallel resistors. It can be instructive to connect these new rules to what you have already learned about series circuits. Watch this video to see how to apply the loop rule to a circuit that you could understand without using this rule.

We will discuss the magnetism on which both of these devices rely in more detail in later parts of this course. Read this text, which introduces voltmeters and ammeters from a practical perspective first, without going into the physics of magnetism. For now, you should just be aware that the force which deflects the needle in a galvanometer is in fact magnetic, and this force obeys a different law than the one discovered by Coulomb for electrostatic forces.

Watch this video to learn about the differences in how to hook up voltmeters versus ammeters.

The problem with the kind of voltage measurements we have described so far is that they are really based on magnetism. It turns out that we can only generate magnetic forces if there is at least some current flow. Consequently, for a voltmeter to function, it must allow some current to flow through it instead of flowing through the intended path in the circuit you are measuring. To minimize this unintentional rerouting of current, one tries to make the internal resistance of a voltmeter as large as possible.

We can use a galvanometer (which is always based on magnetism and hence current flow) to push this idea to the extreme with a special configuration that involves a variable resistor, called a potentiometer. The basic insight behind this setup is that there is one particular scenario in which a voltmeter gives a perfectly accurate reading without drawing any current whatsoever: that is when the reading is precisely zero volts. A vanishing voltage means vanishing current by Ohm's Law. Read this text to learn how we can use a technique called a null measurement to measure an unknown emf (voltage).

The role of the adjustable resistor is crucial because it needs to be "tuned" just right to make the galvanometer show a null reading (zero voltage). Watch this video, which recaps this information. The discussion also includes the Wheatstone bridge, a setup that is used to measure resistances instead of voltages, using the same idea of making a null measurement.

A detailed discussion of how to make precise measurements is not only important for practical applications. It is also a good way to get acquainted with the tremendous predictive power that our physical laws provide. This understanding helps us design devices to reveal what is going on inside a circuit even though it is all happening at a microscopic level, invisible to the naked eye. Measurement devices are the bridge between this micro-world and the macroscopic world of pointer arrows and displays that we can perceive and interpret.

The range of validity of Kirchhoff's rules is broad. Simply put, Kirchhoff's rules are valid whenever the conservation laws on which they are based are valid. The following three energy forms are accounted for in Kirchhoff's rules: electrical potential energy, the energy generating an electromotive force, and the energy lost to heat in a resistor. An energy form that is not included in Kirchoff's rules is electromagnetic radiation. 

With this in mind, we can apply Kirchhoff's Laws to many other types of circuits, containing not just batteries and resistors. In this we add capacitors to the mix. As you recall from our discussion of capacitance, a capacitor is able to store electric charge in an amount proportional to the voltage between its plates.

Read this text to explore a topic we have not yet discussed: the details of the charging and discharging process. Since this process involves current flow, we have to look at capacitors in a complete circuit.

Watch this video to learn about a new mathematical concept that appears in this context: the exponential function. It is a function that appears frequently in physics when growth or decay processes are described. Here, the growth and decay refers to the amount of charge stored on the capacitor. The time it takes to charge and discharge a capacitor is characterized by a time that is directly proportional to the capacitance.

Let's get started by reading these two brief sections in our textbook.

Read this text for a deeper microscopic picture of the substances and devices that show magnetism.

Why couldn't we have made the definitions for electric and magnetic field lines more similar? The answer is that, with magnets you cannot separate positive and negative charges to make a test object that is purely positive. The closest things to "charges" we have in magnetism are the poles of a bar magnet, which are usually called "north" and "south" instead of "positive" and "negative". But, as this video points out, you cannot separate the north and south poles of a magnet from each other.

The fact that bar magnets can deflect compass needles is important because it gave Oersted the crucial hint that electric currents also produce magnetism. He saw that compass needles near a straight wire get deflected as soon as he turned a current on. But when he turned the current off, the compass returned to what it usually does: pointing along the Earth's north-south direction. Read this text to see the magnetic field line patterns we can construct from this behavior of the compass needle.

Watch this video, which begins with a review of the material introduced above. The video shows that a magnetic field not only deflects compass needles or other small bar magnets – it also deflects the motion of charged particles.

Read this text to learn how the Lorentz force on a moving charge is related to the magnetic field, by a formula that also involves the speed of the motion. In principle, having a formula like this would make it possible to quantify the magnetic field by using known values of charge and speed in an experiment that measures the force on the charge (see equation 22.2 in the text).

The Lorentz force does not tell us how magnetic fields are created; it tells us what magnetic fields can do. Watch this video to clarify this and compare it with the electric field.

Coming back to the question how the Lorentz force equation allows us to quantify the magnetic field, watch this video, which specifically talks about the standard unit of the magnetic field, the tesla, and how it is related to the unit of force, the newton.

The formula for the Lorentz force is rather complicated – it is probably the trickiest formula you will encounter in this course due to the strange directionality of the magnetic force. Watch this video, which demonstrates that the force on a moving particle is never in the direction of the magnetic field!

Scientists often use a special symbol for the product of velocity and magnetic field when they must include the angle between them, as in the Lorentz formula. It is called the cross product. Watch this video to see this symbol written down next to the more familiar expression you encountered in the text.

Our main takeaway is that the magnetic force is actually zero if the charged particle is moving parallel to the magnetic field lines – no matter how fast it is moving or how strong the magnetic field is. That is the directional information encoded in the cross product, which shows up in the factor sin 𝜃 in the Lorentz force.

As we saw in reading above, the Lorentz force may be complicated, but it also creates some beautiful effects, such as the Aurora near the Earth's poles. The magnetic force can cause a charged particle to move in a circular or spiral path. Cosmic rays are energetic, charged particles in outer space. As they approach Earth, the Earth's magnetic field can force them into spiral paths. Magnetic forces cause protons in giant accelerators to stay in a circular path.

The curved paths of charged particles in magnetic fields form the basis of a number of phenomena that we use analytically, such as in a mass spectrometer. Read this text, which discusses these applications in more detail.

We have seen effects of a magnetic field on free-moving charges. The magnetic field also affects charges moving in a conductor. The Hall-effect principle is named for physicist Edwin Hall. In 1879 he discovered that when a conductor or semiconductor with current flowing in one direction was introduced perpendicular to a magnetic field, he could measure the voltage at right angles to the current path.

The Hall effect manifests itself as a voltage instead of a mechanical force. This makes it suitable as a compact way to measure magnetic fields by directly converting them to a voltage. We already know how to accurately measure voltages, and we can then apply the same technique to measure a magnetic field.

This video also moves on to our next topic, which is also related to the general idea of how magnetic fields deflect moving charges.

In a galvanometer (the core of an ammeter), a magnetic field is converted into a force that moves a needle. This is done by exploiting the effect discussed in this text – a current-carrying wire feels a force when it is surrounded by a magnetic field.

The law by which a current-carrying conductor is deflected is just an application of the Lorentz force which we discussed earlier, and it follows essentially the same rules. The difference is only that when talking about wires, we usually prefer to replace the speed that appears in the original Lorentz formula by the current, which is directly proportional to the speed. Watch this video for an explanation of this relationship.

Watch this video, which reviews the steps involved in replacing the electron velocity with the current.

Now that we have figured out how the magnetic force on a straight wire depends on the magnetic field and the current, we can now bend the wire into a loop, or coil. This gives rise to two new applications:

Motors are the most common application of magnetic force on current-carrying wires. Motors have loops of wire in a magnetic field. When current is passed through the loops, the magnetic field exerts torque on the loops, which rotates a shaft. Electrical energy is converted to mechanical work in the process.

Meters, such as the galvanometer, are another common application of magnetic torque on a current-carrying loop. This finally answers the question how ammeters actually work (we treated them as black boxes in the discussion of circuits earlier). As with motors, the basic idea is to convert the magnetic force into a twisting action, also called torque. This is how the indicator arrow on a meter is made to rotate to the appropriate position, indicating how many amperes are flowing through the meter. 

 Read this text, which discusses these applications.

Watch this video which shows animations of how a simple motor works.

To quantify the strength and direction of the magnetic field created by flowing currents, it is best to start with the simplest case of a straight wire. As you will see when you read this text, this is the starting point from which we then progress to the magnetic fields of a simple wire loop and finally a solenoid – the kind of wire coil used in real-life electromagnets.

Read about this experiment and its results in this section.

We can distinguish two cases, which these next two videos treat separately. First, let's send the currents through both wires in the same direction.

Next, we reverse the direction of the current in one of the wires.

Let's begin by reading this introductory text.

An important quantity defined here is the magnetic flux. We can visualize magnetic flux as a measure of how many magnetic field lines intersect a given surface (for example the cross-sectional area enclosed by a wire loop). Think of a sail catching the wind, where the wind is analogous to the magnetic field. You can catch a lot of wind by turning the sail perpendicular to the wind, or you can catch almost no wind by turning the sail parallel to the wind. The amount of wind you catch with the surface area of the sail is analogous to the magnetic flux going through that area.

Watch this brief explanation of magnetic flux. In the last part, it mentions two new mathematical constructs, the dot product and integral, which is optional information. Feel free to stop the video at timestamp 3:23.

Next, watch this lecture demonstration by Walter Lewin until timestamp 13.30.

Watch this lecture, which puts the magnetic flux into the context of magnetic induction.

Read this text, which reviews the material from the video you just watched.

Faraday's Law always involves loops that enclose a certain magnetic flux. If the loop is a conductor, then a changing magnetic flux creates a current inside the loop – even though there is no battery at all, and even if the wire has some resistance. This sounds contradictory because Ohm's Law says that you cannot have a current without a voltage; how can you have a voltage if there is no battery?

The changing magnetic flux itself acts like a battery, and the voltage it generates is therefore also called electromotive force (emf), just as for "regular" batteries. In other words, we have got a new type of power source for our electric circuits: the changing magnetic field is able to do work on the electrons inside the conducting loop, and if that loop contains a light bulb, you could make it light up, too. That is a transfer of energy which must come from somewhere.

From the electron's point of view, that energy comes from the magnetic field. But ultimately, work must have been done somewhere else to create the changing magnetic field in the first place. The law of energy conservation is never violated in this process.

Watch this video to explore different factors in Faraday's Law – the law that tells us how large the generated emf will be, depending on the rate at which the magnetic flux changes.

One of these phenomena is the Hall effect which we discussed earlier. Read this text, which discusses how we can view the Hall effect as a manifestation of magnetic induction. More generally, the text studies motion in a magnetic field, which is stationary relative to the earth, producing what we loosely call motional emf.

The paradigm of motional emf is illustrated for a conducting rod rolling on two rails connected to a battery. Watch this video, which explores this scenario further.

Here is a sanity check for the formula used above: it is good to remind ourselves that an emf is just a voltage, so it must also have the unit of electric potential, or voltage. If you recall that potential is just potential energy per unit charge, we can figure out that a volt is the same as one joule per coulomb. Watch this video to see how this unit arises from the formula for the motional emf in terms of the magnetic field.

Watch this video, which discusses how to get started on a problem that involves an airplane flying through the Earth's magnetic field. Note that one of the slides accidentally reads "hand hand rule" when it should say "right-hand rule".

If you care to anthropomorphize the situation, Lenz's Law makes induced currents behave "stubbornly" – always opposing what is being done to them. If you prefer a physics analogy, the behavior has a lot in common with friction forces. Read this text, which explains how this effect is being applied in everyday life.

Many machines use magnetic brakes from the small to the gigantic. The basic idea is that when a conductor moves through a magnetic field, currents are induced that resist the motion. We call these eddy currents. 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 tends 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?

Read this text, which explains how the rotation of a loop (or coil) in a magnetic field causes the magnetic flux through the cross section to vary periodically, and that is what gives rise to an emf (voltage) which periodically changes direction. In other words, the "natural" output of a generator is an AC voltage.

Watch this animation of a generator. Note that you can run the interactive simulation in this video yourself if you have a desktop computer. Go to https://phet.colorado.edu/en/simulation/generator.

The reason why electromagnetic generators produce AC voltage is that the turning wire loop has the magnetic flux going through its cross section in opposite directions every half turn.

Using the concept of induction, we can also answer a question you may have had when we discussed motors and electromagnets: if you connect a solenoid (or coil) directly to a battery, isn't that just a short circuit that will drain the battery really quickly? For an electromagnet operating with DC current, that is indeed a problem, so we have to limit the current it can draw from the battery (e.g. using resistors). But in a motor, the coil does not act the same way. It does not drain the battery even if connected directly to it.

The difference is that the coil in the motor does not see a supply of continuous direct current while it is turning. To keep the rotation going, the current through the coil is reversed every half turn, so there is, in fact, an alternating current in the rotating coil. Now we know that any current produces a magnetic field, and that makes the motor into an electromagnet.

However, unlike the DC electromagnet, we are now dealing with a magnet whose magnetic field is constantly changing. And according to Faraday's Law, this changing field will in turn cause magnetic induction – leading to an emf. But what is the loop in which this emf will appear? It is the same coil that produced the changing magnetic field in the first place. So any electromagnet whose current is changing will produce an emf inside of itself!

Read this text, which explains how Lenz's Law dictates the direction of the induced emf and always opposes the voltage applied to the coil to get the current going in the first place. What does this means for a motor coil? As soon as it starts rotating, it will not act like a short circuit anymore because it will oppose the varying current flowing through it.

Watch this lecture, which accompanies our textbook. We will explore the other topics of this lecture in more detail next.

You may say that this is not really an energy transformation at all. But observe that the two coils of a transformer make up parts of two completely separate electric circuits, and still energy is able to transfer from one circuit to the other. This happens via an intermediate form of energy, stored in the magnetic field lines that permeate both coils simultaneously.

Read this text to review some systems and devices that prevent electrical hazards.

In our discussion of capacitors, we encountered the exponential function when we described how the current and voltage behave during the process of charging and discharging. Read this text where we encounter the exponential function once again – this time in the process of turning the current through an inductor coil on and off.

We see that the current through an inductor always shows a kind of "sluggishness", in the sense that it delays the onset of a current when it is turned on, and perhaps even more surprisingly it also delays the turning-off of a current. This sluggishness is characterized by a time constant which is directly proportional to the inductance. You can loosely understand the inductance as an electronic equivalent to inertia, if you take electric current to be analogous to velocity and electric voltage to be analogous to mechanical force.

Keep this analogy in mind when you read this section, which explores how inductors and capacitors respond when they are placed in an AC circuit. First, an inductor is connected to a voltage that varies in time according to a sine function. This is just the type of voltage that would be produced by a generator. Measuring the current through the inductor coil, you find that it does not reach its maximum value when the voltage is at its maximum. Instead, there is a delay caused by the electronic inertia mentioned above, which is to say – the inductance.

In the second part of the text, the same analysis is shown for a capacitor. In this case, the roles of voltage and current are reversed. The current periodically reaches a maximum, but the voltage across the capacitor reaches its maximum with a delay. The reason is that a large voltage can only build up when the capacitor has accumulated a large stored charge. But the accumulation of charge must be preceded by a charging current. So we first need a large current and then get a large voltage.

In addition to the delay between current and voltage, there is also the question of how large the peaks of current and voltage become, whenever they happen. Read this text, which introduces a new concept to characterize the relative heights of the current and voltage peaks, called the reactance.

It is the generalization of the concept of resistance, in that the reactance is defined as the ratio of voltage to current – but because both are variable we must take their rms values. Recall that rms stands for root-mean-square and represents an average of a time-varying quantity. In fact, you can equivalently get the reactance by dividing the peak voltage by the peak current.

The reactance of a capacitor depends on the capacitance, and the reactance of an inductor depends on the inductance. Both also depend on the frequency of the sinusoidally-varying voltage. Because inductors act like they have inertia, they will produce less peak current for an input voltage that changes direction rapidly, making the ratio of voltage to current (i.e. their reactance) larger for high frequencies.

Capacitors behave the other way around: their reactance becomes smaller at high frequencies. This is because when the charging current changes direction rapidly, the capacitor never gets enough time to charge fully, which never allows the voltage to grow to large values. As a result, the ratio of voltage to current is small.

Even though reactances have the same units as the familiar resistance we know from Ohm's Law, there is one big difference: it is the time delay between voltage and current that we see (in different directions) for inductors and capacitors – but there is no such delay for "normal" resistors.

In the text below we consider the most complex circuit of this course: a combination of resistor, capacitor and inductor in series. The circuit diagram does not look complex, but the difficulties are hidden in the delay between current and voltage. Because that delay is affected in opposite ways by capacitors and inductors, and not at all by a resistor, the combination of all three produces a delay that depends on the parameters L, R and C.

Fortunately, when such a circuit is driven by a sinusoidally-varying voltage, it behaves a lot like a mechanical oscillator driven by a harmonically-varying force. In particular, the current and voltages will all respond by oscillating at the same frequency as the driving voltage. And just as we saw for a driven oscillator, there is a specific driving frequency at which the circuit responds particularly strongly.

To understand what is going on, you can think about the AC voltage supply as the analogue of a mechanical driving force and the peak current through the circuit as the analogue of the maximum speed of the oscillator. Also remember that the inductor acts as if it gives inertia to the current that is trying to flow.

On the other hand, the capacitor acts analogously to a spring. This is because whenever the current through the capacitor stops, it is done charging and has therefore reached its maximum voltage across the plates. This voltage is analogous to the restoring force of a spring. And if you imagine when an oscillating spring reaches its maximum restoring force, that is precisely when it turns around so that its speed is momentarily zero. Zero speed is the analogue of a stopped current.

With these mechanical analogies, it should come as no surprise that a circuit consisting of capacitor and inductor will exhibit the phenomenon of resonance. This is indeed the case: when the frequency of the AC voltage supply hits an optimal value, the circuit will show a current oscillating back and forth with especially large amplitude.

Read this text, which comes to this conclusion using another concept which generalizes the idea of resistance even further (and has the same units): impedance. It is still defined as the ratio of peak voltage to peak current, just allowing for arbitrary delays between when those peaks occur. The resonance phenomenon is identified by an impedance that becomes very small (or zero). This corresponds to very large currents in relation to the voltage.

Maxwell's theoretical prediction of electromagnetic waves was not immediately verified or proven false. It was only ten years after Maxwell's death that Heinrich Hertz finally managed to create these waves in the laboratory. The experiment is described in more detail in the text.

Watch this lecture.

Read this text for a description of how accelerating electrons in an antenna back and forth put energy into an electromagnetic wave. The text also describes how wave propagation is made possible by a kind of "symbiosis" between the electric and magnetic fields, where each field induces the other to extend to ever larger distances from the antenna where they first originated. It is really the coexistence of electric and magnetic induction that makes this possible.

Read this text, which provides an overview over the broad categories that make up the electromagnetic spectrum. The fact that we are dealing with a wave phenomenon also means you will encounter the concepts of amplitude and interference in this new context.

The amplitude of an electromagnetic wave is conventionally defined as the maximum strength of the electric field (the magnetic field has a smaller value that can always be deduced from the electric field). For example, when you listen to an AM station, you are receiving electromagnetic waves whose amplitude is turned up and down in synchrony with the sounds being transmitted (AM stands for amplitude modulation).

Electromagnetic waves satisfy the superposition principle, and therefore it is possible for any number of waves with different amplitude and frequency to coexist in the same region of space. In fact, we are constantly bathed in a jumble of electromagnetic waves at all possible frequencies – especially if you are in a room full of people who are all using their cell phones.

As Heinrich Hertz had already realized in his original experiments, a good way of picking out the electromagnetic waves you would like to detect is by using the phenomenon of resonance. Radios are "tuned" to respond to just one of the many frequencies of electromagnetic waves that surround us, by adjusting the resonance frequency of an LRC circuit (or a comparable device). The circuit then produces large currents only in response to radio waves with frequencies near resonance, and the average value of the current follows the amplitude of those specific waves.

Watch this lecture, which accompanies our textbook.

For simplicity, only electromagnetic waves at large distances from any antennas and with a single frequency are considered. That makes it possible to write down a relationship between the intensity and the squared electric field of the wave. As for all time-varying quantities, one can characterize the wave intensity either by its peak value or by its average. The average is taken over one period of the wave's oscillation. That is usually the most useful quantity to look at, for example when calculating how a focused light wave will raise the temperature of an object it strikes.

The text briefly mentions that there is something mysterious about the nature of light. Although we have just tried very hard to convince ourselves that light is an electromagnetic wave, there are also situations where the same light behaves more like a material object, i.e., a particle. In those situations one refers to light as made up of photons.

Fortunately, the language of energy we use in this section is universal, which means we can apply it no matter what we think the nature of light actually is. Energy can be transmitted by a stream of particles (photons) just as it can be transported by a wave. It also dilutes and concentrates the same way in both descriptions.

The reason this so-called wave particle duality crops up for light, but not for radio waves lies in the details of how light waves are generated and detected. There are no radio antennas small enough to create visible light because the wavelengths are billions of times shorter than for radio waves.

Instead, visible light (and waves of higher frequency) is generated in atoms, by processes that can only be described using the modern theories of matter, based on quantum physics. As the term "quantum" implies, energy is transmitted by these processes in packets that have many of the properties we typically associate with particles – and that is where the concept of the photon comes from.

As we move onto the next unit, it may help you to keep this photon idea in mind. We are going to investigate optics – the science of light – starting with a perspective that has a lot in common with the idea of light as a stream of particles.

Read this text for more information on the the ray aspect of light.

When you see drawings of light rays, do not confuse them with pictures of electric or magnetic field lines. We will not be talking about these fields at all in the context of geometric optics. Light rays always indicate the direction in which the corresponding electromagnetic wave is traveling, and that direction happens to be precisely perpendicular to the orientation of the magnetic and electric field lines at any given point!

Watch this video, which covers the material in this section and introduces the next three sections of our textbook which we will discuss next.

Read this text, which explains why it is important to have a smooth surface to make a good mirror.

Read this text to find some of these results.

The way one usually quotes the material-dependent speed of light, v, is by giving its value in relation to the vacuum speed of light, c. That ratio is called the refractive index, n = c/v.

Watch this video to see some animations of reflection and refraction.

In the material we just reviewed, we saw that the refractive index change between two materials affects how the direction of a light ray changes. The law that describes this change is Snell's Law. It is similar to the law of refraction in that it describes the angles relative to the perpendicular at the interface. But it differs in that this law refers to transmitted light and not to reflection. Also, the outgoing angle is usually different from the incoming angle. When light hits an interface between two media, there is typically a partially reflected ray as well – and that reflection still obeys the law of equal incident and outgoing angles.

Watch this video, which shows different ways of writing Snell's Law, either with or without making use of the wave speeds directly.

Watch this video, which explains the familiar illusion of the "broken straw" in water.

Watch these two videos for some examples of numerical calculations. The second example is more challenging.

Read this text where we learn that under certain conditions, Snell's Law for the direction of the refracted ray can give us an angle of 90º with respect to the perpendicular. This corresponds to a transmitted ray that is directed exactly parallel to the interface between the two media, and therefore is not actually transmitted across the interface at all. This happens when the index of refraction is larger on the incoming side than on the side we are trying to enter, and if the incident angle is made large enough to reach a critical value. This angle of incidence is called the critical angle for total internal reflection, because for any angle equal to, or larger than, this value the light cannot be transmitted.

The law of energy conservation implies that all of the incident light must be reflected back into the medium it came from. This effect of total internal reflection is at the heart of fiber optics, the technology on which most of our long-distance internet communication is based. The idea is to send light rays into a thin strand of glass surrounded by a material of lower index of refraction (typically just a different type of glass). This optical glass fiber is so thin that it becomes flexible and can bend around corners. The light follows the bending of the fiber because it zig-zags along the fiber at such grazing angles that it always meets the condition for total internal reflection.

Watch this video, which discusses the conditions for total internal reflection in water.

Read this section which discusses the effects that arise when the propagation speed depends on the frequency. This is called dispersion.

A rainbow is the most familiar natural phenomenon that demonstrates dispersion. The light from the sun contains waves with all frequencies of the visible spectrum. The combination of all these waves is perceived as white light. If there are water droplets in the air, the light refracts when it enters the droplets, and then refracts again as it leaves the droplets. Since the angle of refraction is different for light of different frequencies, waves of different colors separate. We perceive this as a rainbow.

Watch this short video on differently-colored light rays as they get refracted by different amounts.

Watch this video for an overview of dispersion and the topics covered in the next section.

The way your eye achieves this primary objective is by refraction through the converging lens at the front of the eye. All lenses are based on refraction, but their specific properties depend on the way their surfaces are curved. Without curved surfaces, you cannot make a lens – but not all curved surfaces make a good lens. Read this text to investigate some typical geometries for a lens, and characterize their light-bending properties.

The idea of diverging and converging bundles of rays, as created by lenses, is also related to the concept of intensity which we encountered earlier. Recall that the intensity of an electromagnetic wave decreases as we move away from the source. This is because the energy in the wave gets "diluted" over a larger area.

Compare this to the light rays fanning out from a point-like source in geometric optics. Here it is the density of the rays that gets diluted. So there is a correspondence between the density of rays and the intensity of the light. Converging a bundle of rays onto a focal point is the same as increasing the intensity of the light.

We use the shape and material of the lens to determine its focal point, not by the way we send in the light rays. The distance of the focal point from the lens is not the same as the distance at which the image of a given object forms. That depends on how far away the object is. Equations that relate the distance of the object and the distance at which the image forms can be given in especially simple form if you assume that the lens is very thin.

Watch this video, which illustrates the geometric constructions behind the thin-lens approximation. All the constructions are drawn with a horizontal axis that goes straight through the center of the lens, while the lens itself is drawn vertically upright. We call the axis in this drawing the optical axis. For symmetric lenses of the type we are considering here, a light ray coming in precisely on the optical axis would hit all the interfaces between glass and air in a perpendicular direction. That would mean the ray would not change its direction because it will not experience any refraction.

Lenses work because light rays are refracted once on the way into the glass, and a second time on the way back out. But in the thin-lens approximation, the two surfaces are so close together that you do not have room to draw the two refraction events individually. Instead, one uses a trick.

Watch this video, which details how this trick works. It is based on the idea that rays going through the exact center of a lens do not suffer any refraction at all, whereas light rays coming into the lens parallel to the optical axis must be refracted precisely through the focal point. This is why in the ray diagrams for a lens, we typically draw only two rays (out of the infinitely many rays coming from the object) to completely characterize what the lens does.

In addition to the image distance, another characteristic feature of a lens is the relation between the size of the image and the size of the object. This is called the magnification of the lens, and it also depends on the object distance.

The next two videos are optional information for those of you who would like to understand how to get the formulas in the text by analyzing the geometry of the ray diagrams.

The image that needs to form on our retina to have clear vision is called a real image because it has rays that originally diverged from one point on the object coming together at one single point in the image. But depending on the object distance and the shape of the lens, one can also observe virtual images. A virtual image is formed when light rays emitted from a single point on the object come out the other side of a lens in a diverging fan which would meet at a single point if traced back to the input side of the lens along straight lines.

Watch this view which explains how virtual images occur in concave lenses.

However, you can also get virtual images in convex lenses. See the video below for a step-by-step explanation of virtual images.

In our final video for this section, we see examples of how to apply the lens formulas, paying special attention to the signs of the quantities. They will tell you if you are going to get an upright or inverted image, and whether it is real or virtual.

Read this text, which illustrates how flat mirrors, like the one in your bathroom, produce virtual images.

The formulas in this section are similar to those for thin lenses; they are also based on an approximation. For mirrors, the approximation is that the mirror diameter that captures the light is much smaller than the radius that characterizes the curvature of the mirror. This means we assume the mirror is nearly flat, but not quite.

When a mirror is flat like this, it is impossible to distinguish whether its shape is actually part of a sphere or part of a different curved form, such as a parabolic cross section. To draw ray diagrams, it is actually more convenient to assume we have a mirror of parabolic shape, because these mirrors reflect all rays that come in parallel to the optical axis back into a single focal point.

In an approximate way, we can apply what we learn about parabolic mirrors in the next video to all weakly curved mirrors because they are indistinguishable in shape from a parabolic cross section.

Watch this next video, which provides additional examples of ray diagrams for parabolic mirrors.

In the next video, we explain why you can see a tiny reflection of yourself in a teaspoon when you look at it with the bottom side facing you. That is a concave mirror.

Watch this video, which summarizes the material we have discussed in this section.

Read this section to see how refraction of light occurs in the eye.

Watch this video, which also discusses how we can correct our vision, such as by using eye glasses or by performing surgery on the eye itself.

LASIK is the most common procedure used to surgically correct the way light is refracted onto the retina. It reshapes the cornea that covers the lens on the outside, not the lens itself. The cornea works together with the lens to refract light, and the corrections required for improved vision are usually so small that we only need to adjust less than half a micrometer of thickness (a micrometer is a thousandth of a millimeter).

For comparison, this is about the same as the wavelength of visible light. But this is a problem because the spot size to which a laser beam can be focused is limited by the wavelength of the light. This is called the resolution limit, which we will discuss in the section on wave optics. The solution is to use laser light which has a wavelength much shorter than visible light – ultraviolet light. This provides the required higher resolution.

The resolution limit for visible light is about a micrometer, but that does not mean we are able to perceive objects that are that small with the naked eye. The problem is that the human lens is not perfect in the sense that rays emitted from a single point of light will not recreate a single bright point of light on the retina, but a somewhat blurred spot instead. Moreover, the light-sensitive cells in our retina have their own size, so we cannot distinguish between closely spaced points of focused light that overlap the same cell.

Read this text, which describes two of the most common conditions that may require corrective eyewear – near and farsightedness.

In a microscope, we guide light rays emitted from two very closely spaced points on the object (for example a microorganism) so they converge back together in two spots on the retina that are spaced much farther apart than on the object itself. That allows our eye to distinguish the two points clearly, and as a result we can see details that would have been completely washed out for the unaided eye.

Watch this lecture, which draws the ray diagrams that explain this.

Read this text, which shows an example of the calculation of the magnification which is achieved by a compound microscope with two lenses.

While microscopes allow us to look more deeply into the microcosmos, telescopes let us see into the vast distances of the cosmos on the largest scales. The problem we face when looking at distant objects is mainly related to the concept of intensity. Because light rays coming from distant stars or planets spread out on their way to us, the intensity of the light that can enter our eye from those objects is very low. When the intensity is too low for our retinal cells, they will not tell our brain that there is a point of light there at all.

The main point of a telescope is to simply catch more light by collecting the rays that each star emits with a large-diameter lens or mirror. After collecting the light, the trick is to then bend the rays in just the right way so they re-converge onto a single spot on our retina. When this is achieved, the intensity of the spot on your retina is much larger than before, when only the limited amount of light entering the small opening in your iris contributed to that spot.

And just as for a microscope, the optics in a telescope must also ensure that spots corresponding to different stars will appear at large enough spacings from each other on the retina, so that they can be perceived as individual points of light instead of just a blur. Watch this lecture, which details this process.

Of course, we use telescopes not only in astronomy, but also to observe distant objects on Earth. There are slightly different requirements for different applications – for example you do not want your binoculars (which are essentially paired telescopes) to flip the image upside down. Note that this is perfectly acceptable to astronomers, whose primary objective is not to lose any light intensity, such as from absorption in the lens material. Read this text to see how to achieve these different goals.

Large modern telescopes, including the famous Hubble Space telescope, use mirrors instead of lenses to collect light. Our textbook shows the ray paths in this type of reflecting telescope. Reflectors are used to make astronomical images using electromagnetic waves at invisible wavelengths, such as radio waves, where it is not possible in practice to make refracting lenses.

It is interesting to note that we can also apply the ray-based methods of geometric optics when designing antenna dishes for radio telescopes, which are essentially mirrors, too. This works because the wavelength, as measured in meters (or micrometers), does not limit the applicability of geometric optics. What matters is the wavelength relative to the size of the mirrors, openings and distances encountered by the electromagnetic waves. A ray description is possible whenever the wavelength is short by comparison to these other length scales.

Radio waves are of a length that puts them right at the borderline where ray optics breaks down. Radio dishes are usually designed to operate at wavelengths in the range of centimeters or even millimeters. This is indeed much shorter than the size of the dish mirrors which can be tens of meters wide. But radio waves can also have much longer wavelengths of several hundred meters, and then it is not possible to treat them with the methods of geometric optics.

First, let's read this text to remind ourselves of the fundamental relationship between frequency, wavelength and speed of a light wave.

Watch this video, which introduces us to the most important effects that all waves can produce. The discussion talks specifically about light even though the explanations apply to all waves. The reason why light is our main example here has to do with the ambiguous nature of light which was mentioned when we introduced the concept of the photon: light is different from water waves because it does not require a medium to propagate. This makes it much harder to prove that light is actually a wave, because we cannot directly observe the ripples that make a light wave, as we see the ripples on a pond when a water wave passes through.

What we have to do to prove that light is a wave is: verify that light shows all the same effects that only waves (not particles) can produce. Geometric optics cannot help us here, because the rays we have been talking about could still in principle be interpreted as the trajectories of the little particles we called photons.

The main thing waves can do and particles (in the classical sense) cannot is for two of them to occupy the same region of space. When this happens with waves, we get the effect called interference. Building on this, we can observe the additional effects discussed in this video.

Read this text to explore Huygens' principle of diffraction.

As the reading illustrates, Huygens' principle is not just a philosophical interpretation – it is also a computational tool. In particular, the idea of circular (or spherical) elementary waves makes it relatively easy to explain how a wave can bend around corners and spread out after passing through a constriction. This is called diffraction because it allows wave energy to go around corners in directions that the rays of geometric optics (or the trajectories of classical particles) would not be permitted to go.

Read about the proof that light is a wave in this experiment Thomas Young gave using diffraction by a pair of closely spaced slits.

What makes the double slit experiment better than diffraction by a single slit, if you want to prove the wave nature of light?

It may still be possible to explain why you can see a light beam spreading out behind a single slit, by photon particles ricocheting off the edges of the slit. But when the spreading light waves coming from two slits overlap, they form the characteristic pattern of constructive and destructive interference that only waves can produce.

Moreover, because this interference is created by two waves that are, at any given point, almost parallel, the corresponding interference patterns are spread out spatially. This is important because it magnifies the interference pattern from a microscopic to a macroscopic scale that can be seen by the naked eye.

You can see an analogy of the double-slit interference pattern when you hold two combs on top of each other against the light. The teeth (or tines) of the comb are like the wavelength-spaced ripples of a wave. When the combs are perfectly aligned, you can see through the spaces between both sets of teeth, but when you rotate one comb ever so slightly, you will see the light being blocked in regular patterns that form the analogue of destructive interference between two waves. The more you rotate one comb, the closer the spacing of the dark pattern becomes.

Rotate the combs back to a nearly parallel angle and you see the spacing between the dark lines grow. The latter is what happens in the double-slit experiment if the separation of the slits is decreased, because it brings the diffracted waves from each slit closer and closer into alignment.

Watch this video for a step-by-step construction of the interference pattern.

Watch this video for a discussion of the equation that predicts how the interference pattern depends on the wavelength and the spacing of the two slits.

Watch this video for additional practice with the double-slit setup.

Watch this video, which takes us one step further, from two closely-spaced slits to many closely-spaced slits.

Because diffraction and interference are wave effects, they depend on the wavelength of the light. In particular, the spacings of the multiple-slit diffraction pattern become wider when the wavelength becomes longer.

In Young's double-slit experiment, this actually posed a challenge because you cannot easily see the interference pattern if you shine white light through the slits. Recall that white light is actually a mixture of light with all the colors of the rainbow. If each color shows slightly different interference patterns, the end result will be a washed out intensity distribution where the destructive interference for one color overlaps with constructive interference for another color.

This situation is improved if we use a lot more than two slits to produce the interference pattern. Here is the reason: to get constructive interference between the waves coming from a large number of slits simultaneously, the condition on the angle where this happens becomes more strict, which makes the bright regions of the interference patterns narrower. Then the regions of highest intensity produced by different colors show less overlap.

Read this text, which provides a worked example that calculates the angles where the diffraction pattern shows maximal intensity, for light of different colors.

Diffraction gratings offer a highly-effective method of routing light of different color (wavelength) in different directions. This way, we can split a beam of white light into the rainbow colors, much like what a prism does due to dispersion. But diffraction gratings do not depend on dispersion, the wavelength-dependence of the light speed in a material – they only use the fact that light can diffract and form interference patterns.

Some subtle interference patterns appear when light shines through just a single slit. However, it is more difficult to see them because the contrast between this pattern, and the central part of the light beam that passes straight through, is quite low. However, we have now assembled the tools necessary to calculate this effect as well.

Read more about single slit diffraction in this section of our textbook.

In the context of the microscope, we have briefly mentioned the resolution limit, which makes it impossible to form arbitrarily-sharp image spots on the retina. Read this text as we return to this concept and arrive at the fundamental criterion on which the resolution limit is based – the Rayleigh criterion. The heart of our limit to resolve small objects is the same physics we just covered in the single-slit diffraction pattern.

Watch this lecture, which accompanies this section of our textbook.

Read this text to learn more about how reflections at closely-spaced interfaces of a thin layer create the rainbow colors you see on soap bubbles or oil slicks. As with the diffraction grating, the interference pattern depends on the wavelength, leading to constructive interference in different directions for different colors of light.

Read this text, which discusses several different ways to create light of a specific polarization direction.

Read this text to learn about Einstein's postulates.

It turns out that the speed of light can only be constant for all observers if we also admit that c plays a special role in the mechanics of material objects. This means that c was discovered to be a universal speed limit not just for light but for all objects that show any kind of motion relative to some observer!

Watch this video, which provides an overview over the often counter-intuitive conclusions that must be drawn from this fact.

Read this text to learn about simultaneity and time duration.

The nature of space is changed in Einstein's theory because the length of objects now becomes dependent on their velocity relative to the observer.

Read this text to learn more about this phenomenon.

Because Einstein identified the speed of light as a universal speed limit, it can now no longer be strictly true that the ground speed of a person running forward on a fast-moving airplane will just be the sum of the plane's ground speed and the person's speed relative to the plane. This velocity-addition formula, which goes back all the way to Galileo, would in principle allow ground speeds larger than c to be achieved by adding speeds that are individually smaller than c.

As this text discusses, Einstein's Law for Velocity addition is more complicated, but it becomes approximately the same as Galileo's old law when all the speeds involved are low compared to the speed of light.

Conservation laws help make the world predictable even when we do not know all the details of all the objects we are tracking. This is why it is important to know how the concept of momentum needs to be modified so that the law of momentum conservation survives Einstein's rewriting of mechanics. Read this text for additional explanation.