# Electromagnetic Induction, AC Circuits, and Electrical Technologies

## Electrical Safety: Systems and Devices

### Learning Objectives

By the end of this section, you will be able to:
1. Explain how various modern safety features in electric circuits work, with an emphasis on how induction is employed.
Electricity has two hazards. A thermal hazard occurs when there is electrical overheating. A shock hazard occurs when electric current passes through a person. Both hazards have already been discussed. Here we will concentrate on systems and devices that prevent electrical hazards.
 shows the schematic for a simple AC circuit with no safety features. This is not how power is distributed in practice. Modern household and industrial wiring requires the three-wire system, shown schematically in , which has several safety features. First is the familiar circuit breaker (or fuse) to prevent thermal overload. Second, there is a protective case around the appliance, such as a toaster or refrigerator. The case’s safety feature is that it prevents a person from touching exposed wires and coming into electrical contact with the circuit, helping prevent shocks.
There are three connections to earth or ground (hereafter referred to as “earth/ground”) shown in . Recall that an earth/ground connection is a low-resistance path directly to the earth. The two earth/ground connections on the neutral wire force it to be at zero volts relative to the earth, giving the wire its name. This wire is therefore safe to touch even if its insulation, usually white, is missing. The neutral wire is the return path for the current to follow to complete the circuit. Furthermore, the two earth/ground connections supply an alternative path through the earth, a good conductor, to complete the circuit. The earth/ground connection closest to the power source could be at the generating plant, while the other is at the user’s location. The third earth/ground is to the case of the appliance, through the green earth/ground wire, forcing the case, too, to be at zero volts. The live or hot wire (hereafter referred to as “live/hot”) supplies voltage and current to operate the appliance.  shows a more pictorial version of how the three-wire system is connected through a three-prong plug to an appliance.
A note on insulation color-coding: Insulating plastic is color-coded to identify live/hot, neutral and ground wires but these codes vary around the world. Live/hot wires may be brown, red, black, blue or grey. Neutral wire may be blue, black or white. Since the same color may be used for live/hot or neutral in different parts of the world, it is essential to determine the color code in your region. The only exception is the earth/ground wire which is often green but may be yellow or just bare wire. Striped coatings are sometimes used for the benefit of those who are colorblind.
The three-wire system replaced the older two-wire system, which lacks an earth/ground wire. Under ordinary circumstances, insulation on the live/hot and neutral wires prevents the case from being directly in the circuit, so that the earth/ground wire may seem like double protection. Grounding the case solves more than one problem, however. The simplest problem is worn insulation on the live/hot wire that allows it to contact the case, as shown in . Lacking an earth/ground connection (some people cut the third prong off the plug because they only have outdated two hole receptacles), a severe shock is possible. This is particularly dangerous in the kitchen, where a good connection to earth/ground is available through water on the floor or a water faucet. With the earth/ground connection intact, the circuit breaker will trip, forcing repair of the appliance. Why are some appliances still sold with two-prong plugs? These have nonconducting cases, such as power tools with impact resistant plastic cases, and are called doubly insulated. Modern two-prong plugs can be inserted into the asymmetric standard outlet in only one way, to ensure proper connection of live/hot and neutral wires.
Electromagnetic induction causes a more subtle problem that is solved by grounding the case. The AC current in appliances can induce an emf on the case. If grounded, the case voltage is kept near zero, but if the case is not grounded, a shock can occur as pictured in . Current driven by the induced case emf is called a leakage current, although current does not necessarily pass from the resistor to the case.
A ground fault interrupter (GFI) is a safety device found in updated kitchen and bathroom wiring that works based on electromagnetic induction. GFIs compare the currents in the live/hot and neutral wires. When live/hot and neutral currents are not equal, it is almost always because current in the neutral is less than in the live/hot wire. Then some of the current, again called a leakage current, is returning to the voltage source by a path other than through the neutral wire. It is assumed that this path presents a hazard, such as shown in . GFIs are usually set to interrupt the circuit if the leakage current is greater than 5 mA, the accepted maximum harmless shock. Even if the leakage current goes safely to earth/ground through an intact earth/ground wire, the GFI will trip, forcing repair of the leakage.
 shows how a GFI works. If the currents in the live/hot and neutral wires are equal, then they induce equal and opposite emfs in the coil. If not, then the circuit breaker will trip.
Another induction-based safety device is the isolation transformer, shown in . Most isolation transformers have equal input and output voltages. Their function is to put a large resistance between the original voltage source and the device being operated. This prevents a complete circuit between them, even in the circumstance shown. There is a complete circuit through the appliance. But there is not a complete circuit for current to flow through the person in the figure, who is touching only one of the transformer’s output wires, and neither output wire is grounded. The appliance is isolated from the original voltage source by the high resistance of the material between the transformer coils, hence the name isolation transformer. For current to flow through the person, it must pass through the high-resistance material between the coils, through the wire, the person, and back through the earth—a path with such a large resistance that the current is negligible.
The basics of electrical safety presented here help prevent many electrical hazards. Electrical safety can be pursued to greater depths. There are, for example, problems related to different earth/ground connections for appliances in close proximity. Many other examples are found in hospitals. Microshock-sensitive patients, for instance, require special protection. For these people, currents as low as 0.1 mA may cause ventricular fibrillation. The interested reader can use the material presented here as a basis for further study.

### Test Prep for AP Courses

### Section Summary

1. Electrical safety systems and devices are employed to prevent thermal and shock hazards.
2. Circuit breakers and fuses interrupt excessive currents to prevent thermal hazards.
3. The three-wire system guards against thermal and shock hazards, utilizing live/hot, neutral, and earth/ground wires, and grounding the neutral wire and case of the appliance.
4. A ground fault interrupter (GFI) prevents shock by detecting the loss of current to unintentional paths.
5. An isolation transformer insulates the device being powered from the original source, also to prevent shock.
6. Many of these devices use induction to perform their basic function.

### Conceptual Questions

### Problems & Exercises


# Electromagnetic Induction, AC Circuits, and Electrical Technologies

## Inductance

### Learning Objectives

By the end of this section, you will be able to:
1. Calculate the inductance of an inductor.
2. Calculate the energy stored in an inductor.
3. Calculate the emf generated in an inductor.

### Inductors

Induction is the process in which an emf is induced by changing magnetic flux. Many examples have been discussed so far, some more effective than others. Transformers, for example, are designed to be particularly effective at inducing a desired voltage and current with very little loss of energy to other forms. Is there a useful physical quantity related to how “effective” a given device is? The answer is yes, and that physical quantity is called inductance.
Mutual inductance is the effect of Faraday’s law of induction for one device upon another, such as the primary coil in transmitting energy to the secondary in a transformer. See , where simple coils induce emfs in one another.
In the many cases where the geometry of the devices is fixed, flux is changed by varying current. We therefore concentrate on the rate of change of current, , as the cause of induction. A change in the current  in one device, coil 1 in the figure, induces an  in the other. We express this in equation form as
where  is defined to be the mutual inductance between the two devices. The minus sign is an expression of Lenz’s law. The larger the mutual inductance , the more effective the coupling. For example, the coils in  have a small  compared with the transformer coils in . Units for  are , which is named a henry (H), after Joseph Henry. That is, .
Nature is symmetric here. If we change the current  in coil 2, we induce an  in coil 1, which is given by
where  is the same as for the reverse process. Transformers run backward with the same effectiveness, or mutual inductance .
A large mutual inductance  may or may not be desirable. We want a transformer to have a large mutual inductance. But an appliance, such as an electric clothes dryer, can induce a dangerous emf on its case if the mutual inductance between its coils and the case is large. One way to reduce mutual inductance  is to counterwind coils to cancel the magnetic field produced. (See .)
Self-inductance, the effect of Faraday’s law of induction of a device on itself, also exists. When, for example, current through a coil is increased, the magnetic field and flux also increase, inducing a counter emf, as required by Lenz’s law. Conversely, if the current is decreased, an emf is induced that opposes the decrease. Most devices have a fixed geometry, and so the change in flux is due entirely to the change in current  through the device. The induced emf is related to the physical geometry of the device and the rate of change of current. It is given by
where  is the self-inductance of the device. A device that exhibits significant self-inductance is called an inductor, and given the symbol in .The minus sign is an expression of Lenz’s law, indicating that emf opposes the change in current. Units of self-inductance are henries (H) just as for mutual inductance. The larger the self-inductance  of a device, the greater its opposition to any change in current through it. For example, a large coil with many turns and an iron core has a large  and will not allow current to change quickly. To avoid this effect, a small  must be achieved, such as by counterwinding coils as in .
A 1 H inductor is a large inductor. To illustrate this, consider a device with  that has a 10 A current flowing through it. What happens if we try to shut off the current rapidly, perhaps in only 1.0 ms? An emf, given by , will oppose the change. Thus an emf will be induced given by . The positive sign means this large voltage is in the same direction as the current, opposing its decrease. Such large emfs can cause arcs, damaging switching equipment, and so it may be necessary to change current more slowly.
There are uses for such a large induced voltage. Camera flashes use a battery, two inductors that function as a transformer, and a switching system or oscillator to induce large voltages. (Remember that we need a changing magnetic field, brought about by a changing current, to induce a voltage in another coil.) The oscillator system will do this many times as the battery voltage is boosted to over one thousand volts. (You may hear the high pitched whine from the transformer as the capacitor is being charged.) A capacitor stores the high voltage for later use in powering the flash. (See .)
It is possible to calculate  for an inductor given its geometry (size and shape) and knowing the magnetic field that it produces. This is difficult in most cases, because of the complexity of the field created. So in this text the inductance  is usually a given quantity. One exception is the solenoid, because it has a very uniform field inside, a nearly zero field outside, and a simple shape. It is instructive to derive an equation for its inductance. We start by noting that the induced emf is given by Faraday’s law of induction as  and, by the definition of self-inductance, as . Equating these yields
Solving for  gives
This equation for the self-inductance  of a device is always valid. It means that self-inductance  depends on how effective the current is in creating flux; the more effective, the greater /  is.
Let us use this last equation to find an expression for the inductance of a solenoid. Since the area

   of a solenoid is fixed, the change in flux is

  .

To find

  , we note that the magnetic field of a solenoid is given by . (Here , where

   is the number of coils and

   is the solenoid’s length.) Only the current changes, so that . Substituting

   into  gives
This simplifies to
This is the self-inductance of a solenoid of cross-sectional area

   and length

  . Note that the inductance depends only on the physical characteristics of the solenoid, consistent with its definition.
One common application of inductance is used in traffic lights that can tell when vehicles are waiting at the intersection. An electrical circuit with an inductor is placed in the road under the place a waiting car will stop over. The body of the car increases the inductance and the circuit changes sending a signal to the traffic lights to change colors. Similarly, metal detectors used for airport security employ the same technique. A coil or inductor in the metal detector frame acts as both a transmitter and a receiver. The pulsed signal in the transmitter coil induces a signal in the receiver. The self-inductance of the circuit is affected by any metal object in the path. Such detectors can be adjusted for sensitivity and also can indicate the approximate location of metal found on a person. See .

### Energy Stored in an Inductor

We know from Lenz’s law that inductances oppose changes in current. There is an alternative way to look at this opposition that is based on energy. Energy is stored in a magnetic field. It takes time to build up energy, and it also takes time to deplete energy; hence, there is an opposition to rapid change. In an inductor, the magnetic field is directly proportional to current and to the inductance of the device. It can be shown that the energy stored in an inductor  is given by
This expression is similar to that for the energy stored in a capacitor.

### Section Summary

1. Inductance is the property of a device that tells how effectively it induces an emf  in another device.
2. Mutual inductance is the effect of two devices in inducing emfs in each other.
3. A change in current  in one induces an emf  in the second:

        where

   is defined to be the mutual inductance between the two devices, and the minus sign is due to Lenz’s law.
4. Symmetrically, a change in current  through the second device induces an emf  in the first:

        where

   is the same mutual inductance as in the reverse process.
5. Current changes in a device induce an emf in the device itself.
6. Self-inductance is the effect of the device inducing emf in itself.
7. The device is called an inductor, and the emf induced in it by a change in current through it is

where  is the self-inductance of the inductor, and  is the rate of change of current through it. The minus sign indicates that emf opposes the change in current, as required by Lenz’s law.
8. The unit of self- and mutual inductance is the henry (H), where .
9. The self-inductance  of an inductor is proportional to how much flux changes with current. For an -turn inductor,

10. The self-inductance of a solenoid is

where  is its number of turns in the solenoid,  is its cross-sectional area,  is its length, and  is the permeability of free space.
11. The energy stored in an inductor  is

### Conceptual Questions

### Problems & Exercises


# Electromagnetic Induction, AC Circuits, and Electrical Technologies

## RL Circuits

### Learning Objectives

By the end of this section, you will be able to:
1. Calculate the current in an RL circuit after a specified number of characteristic time steps.
2. Calculate the characteristic time of an RL circuit.
3. Sketch the current in an RL circuit over time.
We know that the current through an inductor  cannot be turned on or off instantaneously. The change in current changes flux, inducing an emf opposing the change (Lenz’s law). How long does the opposition last? Current will flow and can be turned off, but how long does it take?  shows a switching circuit that can be used to examine current through an inductor as a function of time.
When the switch is first moved to position 1 (at ), the current is zero and it eventually rises to , where

   is the total resistance of the circuit. The opposition of the inductor  is greatest at the beginning, because the amount of change is greatest. The opposition it poses is in the form of an induced emf, which decreases to zero as the current approaches its final value. The opposing emf is proportional to the amount of change left. This is the hallmark of an exponential behavior, and it can be shown with calculus that
is the current in an RL circuit when switched on (Note the similarity to the exponential behavior of the voltage on a charging capacitor). The initial current is zero and approaches  with a characteristic time constant

for an RL circuit, given by
where  has units of seconds, since

  .
 In the first period of time , the current rises from zero to , since . The current will go 0.632 of the remainder in the next time . A well-known property of the exponential is that the final value is never exactly reached, but 0.632 of the remainder to that value is achieved in every characteristic time . In just a few multiples of the time , the final value is very nearly achieved, as the graph in (b) illustrates.
The characteristic time  depends on only two factors, the inductance  and the resistance . The greater the inductance , the greater  is, which makes sense since a large inductance is very effective in opposing change. The smaller the resistance , the greater  is. Again this makes sense, since a small resistance means a large final current and a greater change to get there. In both cases—large  and small  —more energy is stored in the inductor and more time is required to get it in and out.
When the switch in (a) is moved to position 2 and cuts the battery out of the circuit, the current drops because of energy dissipation by the resistor. But this is also not instantaneous, since the inductor opposes the decrease in current by inducing an emf in the same direction as the battery that drove the current. Furthermore, there is a certain amount of energy, , stored in the inductor, and it is dissipated at a finite rate. As the current approaches zero, the rate of decrease slows, since the energy dissipation rate is . Once again the behavior is exponential, and

 is found to be
(See (c).) In the first period of time  after the switch is closed, the current falls to 0.368 of its initial value, since . In each successive time , the current falls to 0.368 of the preceding value, and in a few multiples of , the current becomes very close to zero, as seen in the graph in (c).
In summary, when the voltage applied to an inductor is changed, the current also changes, but the change in current lags the change in voltage in an RL circuit. In Reactance, Inductive and Capacitive, we explore how an RL circuit behaves when a sinusoidal AC voltage is applied.

### Section Summary

1. When a series connection of a resistor and an inductor—an RL circuit—is connected to a voltage source, the time variation of the current is

 where
  is the final current.
2. The characteristic time constant  is  , where

    is the inductance and

   is the resistance.
3. In the first time constant , the current rises from zero to , and 0.632 of the remainder in every subsequent time interval .
4. When the inductor is shorted through a resistor, current decreases as

Here  is the initial current.
5. Current falls to  in the first time interval , and 0.368 of the remainder toward zero in each subsequent time .

### Problem Exercises


# Electromagnetic Induction, AC Circuits, and Electrical Technologies

## Reactance, Inductive and Capacitive

### Learning Objectives

By the end of this section, you will be able to:
1. Sketch voltage and current versus time in simple inductive, capacitive, and resistive circuits.
2. Calculate inductive and capacitive reactance.
3. Calculate current and/or voltage in simple inductive, capacitive, and resistive circuits.
Many circuits also contain capacitors and inductors, in addition to resistors and an AC voltage source. We have seen how capacitors and inductors respond to DC voltage when it is switched on and off. We will now explore how inductors and capacitors react to sinusoidal AC voltage.

### Inductors and Inductive Reactance

Suppose an inductor is connected directly to an AC voltage source, as shown in . It is reasonable to assume negligible resistance, since in practice we can make the resistance of an inductor so small that it has a negligible effect on the circuit. Also shown is a graph of voltage and current as functions of time.
The graph in (b) starts with voltage at a maximum. Note that the current starts at zero and rises to its peak after the voltage that drives it, just as was the case when DC voltage was switched on in the preceding section. When the voltage becomes negative at point a, the current begins to decrease; it becomes zero at point b, where voltage is its most negative. The current then becomes negative, again following the voltage. The voltage becomes positive at point c and begins to make the current less negative. At point d, the current goes through zero just as the voltage reaches its positive peak to start another cycle. This behavior is summarized as follows:
Current lags behind voltage, since inductors oppose change in current. Changing current induces a back emf . This is considered to be an effective resistance of the inductor to AC. The rms current  through an inductor  is given by a version of Ohm’s law:
where

   is the rms voltage across the inductor and  is defined to be
with  the frequency of the AC voltage source in hertz (An analysis of the circuit using Kirchhoff’s loop rule and calculus actually produces this expression).  is called the inductive reactance, because the inductor reacts to impede the current.  has units of ohms (, so that frequency times inductance has units of ), consistent with its role as an effective resistance. It makes sense that  is proportional to , since the greater the induction the greater its resistance to change. It is also reasonable that  is proportional to frequency , since greater frequency means greater change in current. That is,  is large for large frequencies (large , small ). The greater the change, the greater the opposition of an inductor.
Note that although the resistance in the circuit considered is negligible, the AC current is not extremely large because inductive reactance impedes its flow. With AC, there is no time for the current to become extremely large.

### Capacitors and Capacitive Reactance

Consider the capacitor connected directly to an AC voltage source as shown in . The resistance of a circuit like this can be made so small that it has a negligible effect compared with the capacitor, and so we can assume negligible resistance. Voltage across the capacitor and current are graphed as functions of time in the figure.
The graph in  starts with voltage across the capacitor at a maximum. The current is zero at this point, because the capacitor is fully charged and halts the flow. Then voltage drops and the current becomes negative as the capacitor discharges. At point a, the capacitor has fully discharged ( on it) and the voltage across it is zero. The current remains negative between points a and b, causing the voltage on the capacitor to reverse. This is complete at point b, where the current is zero and the voltage has its most negative value. The current becomes positive after point b, neutralizing the charge on the capacitor and bringing the voltage to zero at point c, which allows the current to reach its maximum. Between points c and d, the current drops to zero as the voltage rises to its peak, and the process starts to repeat. Throughout the cycle, the voltage follows what the current is doing by one-fourth of a cycle:
The capacitor is affecting the current, having the ability to stop it altogether when fully charged. Since an AC voltage is applied, there is an rms current, but it is limited by the capacitor. This is considered to be an effective resistance of the capacitor to AC, and so the rms current  in the circuit containing only a capacitor  is given by another version of Ohm’s law to be
where  is the rms voltage and  is defined (As with , this expression for  results from an analysis of the circuit using Kirchhoff’s rules and calculus) to be
where  is called the capacitive reactance, because the capacitor reacts to impede the current.  has units of ohms (verification left as an exercise for the reader).  is inversely proportional to the capacitance ; the larger the capacitor, the greater the charge it can store and the greater the current that can flow. It is also inversely proportional to the frequency ; the greater the frequency, the less time there is to fully charge the capacitor, and so it impedes current less.
Although a capacitor is basically an open circuit, there is an rms current in a circuit with an AC voltage applied to a capacitor. This is because the voltage is continually reversing, charging and discharging the capacitor. If the frequency goes to zero (DC),  tends to infinity, and the current is zero once the capacitor is charged. At very high frequencies, the capacitor’s reactance tends to zero—it has a negligible reactance and does not impede the current (it acts like a simple wire). Capacitors have the opposite effect on AC circuits that inductors have.

### Resistors in an AC Circuit

Just as a reminder, consider , which shows an AC voltage applied to a resistor and a graph of voltage and current versus time. The voltage and current are exactly in phase in a resistor. There is no frequency dependence to the behavior of plain resistance in a circuit:

### Section Summary

1. For inductors in AC circuits, we find that when a sinusoidal voltage is applied to an inductor, the voltage leads the current by one-fourth of a cycle, or by a

   phase angle.
2. The opposition of an inductor to a change in current is expressed as a type of AC resistance.
3. Ohm’s law for an inductor is

where  is the rms voltage across the inductor.
4.  is defined to be the inductive reactance, given by

with  the frequency of the AC voltage source in hertz.
5. Inductive reactance  has units of ohms and is greatest at high frequencies.
6. For capacitors, we find that when a sinusoidal voltage is applied to a capacitor, the voltage follows the current by one-fourth of a cycle, or by a

   phase angle.
7. Since a capacitor can stop current when fully charged, it limits current and offers another form of AC resistance; Ohm’s law for a capacitor is

where  is the rms voltage across the capacitor.
8.  is defined to be the capacitive reactance, given by

9.  has units of ohms and is greatest at low frequencies.

### Conceptual Questions

### Problems & Exercises


# Electromagnetic Induction, AC Circuits, and Electrical Technologies

## RLC Series AC Circuits

### Learning Objectives

By the end of this section, you will be able to:
1. Calculate the impedance, phase angle, resonant frequency, power, power factor, voltage, and/or current in a RLC series circuit.
2. Draw the circuit diagram for an RLC series circuit.
3. Explain the significance of the resonant frequency.

### Impedance

When alone in an AC circuit, inductors, capacitors, and resistors all impede current. How do they behave when all three occur together? Interestingly, their individual resistances in ohms do not simply add. Because inductors and capacitors behave in opposite ways, they partially to totally cancel each other’s effect.  shows an RLC series circuit with an AC voltage source, the behavior of which is the subject of this section. The crux of the analysis of an RLC circuit is the frequency dependence of  and , and the effect they have on the phase of voltage versus current (established in the preceding section). These give rise to the frequency dependence of the circuit, with important “resonance” features that are the basis of many applications, such as radio tuners.
The combined effect of resistance , inductive reactance , and capacitive reactance  is defined to be impedance, an AC analogue to resistance in a DC circuit. Current, voltage, and impedance in an RLC circuit are related by an AC version of Ohm’s law:
Here  is the peak current,  the peak source voltage, and

   is the impedance of the circuit. The units of impedance are ohms, and its effect on the circuit is as you might expect: the greater the impedance, the smaller the current. To get an expression for  in terms of

  , , and , we will now examine how the voltages across the various components are related to the source voltage. Those voltages are labeled , , and  in .
Conservation of charge requires current to be the same in each part of the circuit at all times, so that we can say the currents in , , and  are equal and in phase. But we know from the preceding section that the voltage across the inductor  leads the current by one-fourth of a cycle, the voltage across the capacitor  follows the current by one-fourth of a cycle, and the voltage across the resistor  is exactly in phase with the current.  shows these relationships in one graph, as well as showing the total voltage around the circuit , where all four voltages are the instantaneous values. According to Kirchhoff’s loop rule, the total voltage around the circuit

   is also the voltage of the source.
You can see from  that while  is in phase with the current,  leads by

  , and  follows by

  . Thus  and  are

   out of phase (crest to trough) and tend to cancel, although not completely unless they have the same magnitude. Since the peak voltages are not aligned (not in phase), the peak voltage  of the source does not equal the sum of the peak voltages across , , and . The actual relationship is
where , , and  are the peak voltages across , , and , respectively. Now, using Ohm’s law and definitions from Reactance, Inductive and Capacitive, we substitute  into the above, as well as , , and , yielding
 cancels to yield an expression for

  :
which is the impedance of an RLC series AC circuit. For circuits without a resistor, take

; for those without an inductor, take ; and for those without a capacitor, take .

### Resonance in RLC Series AC Circuits

How does an RLC circuit behave as a function of the frequency of the driving voltage source? Combining Ohm’s law, , and the expression for impedance

   from  gives
The reactances vary with frequency, with  large at high frequencies and  large at low frequencies, as we have seen in three previous examples. At some intermediate frequency , the reactances will be equal and cancel, giving  —this is a minimum value for impedance, and a maximum value for  results. We can get an expression for  by taking
Substituting the definitions of  and ,
Solving this expression for  yields
where  is the resonant frequency of an RLC series circuit. This is also the natural frequency at which the circuit would oscillate if not driven by the voltage source. At , the effects of the inductor and capacitor cancel, so that , and  is a maximum.
Resonance in AC circuits is analogous to mechanical resonance, where resonance is defined to be a forced oscillation—in this case, forced by the voltage source—at the natural frequency of the system. The receiver in a radio is an RLC circuit that oscillates best at its . A variable capacitor is often used to adjust  to receive a desired frequency and to reject others.  is a graph of current as a function of frequency, illustrating a resonant peak in  at . The two curves are for two different circuits, which differ only in the amount of resistance in them. The peak is lower and broader for the higher-resistance circuit. Thus the higher-resistance circuit does not resonate as strongly and would not be as selective in a radio receiver, for example.

### Power in RLC Series AC Circuits

If current varies with frequency in an RLC circuit, then the power delivered to it also varies with frequency. But the average power is not simply current times voltage, as it is in purely resistive circuits. As was seen in , voltage and current are out of phase in an RLC circuit. There is a phase angle  between the source voltage  and the current , which can be found from
For example, at the resonant frequency or in a purely resistive circuit , so that . This implies that  and that voltage and current are in phase, as expected for resistors. At other frequencies, average power is less than at resonance. This is both because voltage and current are out of phase and because  is lower. The fact that source voltage and current are out of phase affects the power delivered to the circuit. It can be shown that the average power is
Thus  is called the power factor, which can range from 0 to 1. Power factors near 1 are desirable when designing an efficient motor, for example. At the resonant frequency, .
Power delivered to an RLC series AC circuit is dissipated by the resistance alone. The inductor and capacitor have energy input and output but do not dissipate it out of the circuit. Rather they transfer energy back and forth to one another, with the resistor dissipating exactly what the voltage source puts into the circuit. This assumes no significant electromagnetic radiation from the inductor and capacitor, such as radio waves. Such radiation can happen and may even be desired, as we will see in the next chapter on electromagnetic radiation, but it can also be suppressed as is the case in this chapter. The circuit is analogous to the wheel of a car driven over a corrugated road as shown in . The regularly spaced bumps in the road are analogous to the voltage source, driving the wheel up and down. The shock absorber is analogous to the resistance damping and limiting the amplitude of the oscillation. Energy within the system goes back and forth between kinetic (analogous to maximum current, and energy stored in an inductor) and potential energy stored in the car spring (analogous to no current, and energy stored in the electric field of a capacitor). The amplitude of the wheels’ motion is a maximum if the bumps in the road are hit at the resonant frequency.
A pure LC circuit with negligible resistance oscillates at , the same resonant frequency as an RLC circuit. It can serve as a frequency standard or clock circuit—for example, in a digital wristwatch. With a very small resistance, only a very small energy input is necessary to maintain the oscillations. The circuit is analogous to a car with no shock absorbers. Once it starts oscillating, it continues at its natural frequency for some time.  shows the analogy between an LC circuit and a mass on a spring.

### Section Summary

1. The AC analogy to resistance is impedance

  , the combined effect of resistors, inductors, and capacitors, defined by the AC version of Ohm’s law:
    where  is the peak current and  is the peak source voltage.
2. Impedance has units of ohms and is given by .
3. The resonant frequency , at which , is

4. In an AC circuit, there is a phase angle  between source voltage  and the current , which can be found from

5.  for a purely resistive circuit or an RLC circuit at resonance.
6. The average power delivered to an RLC circuit is affected by the phase angle and is given by
     is called the power factor, which ranges from 0 to 1.

### Conceptual Questions

### Problems & Exercises


# Electromagnetic Waves

## Connection for AP® Courses

Electromagnetic waves are all around us. The beauty of a coral reef, the warmth of sunshine, sunburn, an X-ray image revealing a broken bone, even microwave popcorn—all involve electromagnetic waves. The list of the various types of electromagnetic waves, ranging from radio transmission waves to nuclear gamma-rays (γ-rays), is interesting in itself. Even more intriguing is that all of these widely varied phenomena are different manifestations of the same thing—electromagnetic waves. (See .)
What are electromagnetic waves? How are they created, and how do they travel? How can we understand and conceptualize their widely varying properties? What is their relationship to electric and magnetic effects? These and other questions will be explored in this chapter.
Electromagnetic waves support Big Idea 6 that waves can transport energy and momentum. In general, electromagnetic waves behave like any other wave, as they are traveling disturbances (Enduring Understanding 6.A). They consist of oscillating electric and magnetic fields, which can be conceived of as transverse waves (Essential Knowledge 6.A.1). They are periodic and can be described by their amplitude, frequency, wavelength, speed, and energy (Enduring Understanding 6.B).
Simple waves can be modeled mathematically using sine or cosine functions involving the wavelength, amplitude, and frequency of the wave. (Essential Knowledge 6.B.3). However, electromagnetic waves also have some unique properties compared to other waves. They can travel through both matter and a vacuum (Essential Knowledge 6.F.2), unlike mechanical waves, including sound, that require a medium (Essential Knowledge 6.A.2).
Maxwell’s equations define the relationship between electric permittivity, the magnetic permeability of free space (vacuum), and the speed of light, which is the speed of propagation of all electromagnetic waves in a vacuum. This chapter uses the properties electric permittivity (Essential Knowledge 1.E.4) and magnetic permeability (Essential Knowledge 1.E.5) to support Big Idea 1 that objects and systems have certain properties and may have internal structure.
The particular properties mentioned are the macroscopic results of the atomic and molecular structure of materials (Enduring Understanding 1.E). Electromagnetic radiation can be modeled as a wave or as fundamental particles (Enduring Understanding 6.F). This chapter also introduces different types of electromagnetic radiation that are characterized by their wavelengths (Essential Knowledge 6.F.1) and have been given specific names (see ).
Big Idea 1 Objects and systems have properties such as mass and charge. Systems may have internal structure.
Enduring Understanding 1.E Materials have many macroscopic properties that result from the arrangement and interactions of the atoms and molecules that make up the material.
Essential Knowledge 1.E.4 Matter has a property called electric permittivity.
Essential Knowledge 1.E.5 Matter has a property called magnetic permeability.
Big Idea 6. Waves can transfer energy and momentum from one location to another without the permanent transfer of mass and serve as a mathematical model for the description of other phenomena.
Enduring Understanding 6.A A wave is a traveling disturbance that transfers energy and momentum.
Essential Knowledge 6.A.1 Waves can propagate via different oscillation modes such as transverse and longitudinal.
Essential Knowledge 6.A.2 For propagation, mechanical waves require a medium, while electromagnetic waves do not require a physical medium. Examples include light traveling through a vacuum and sound not traveling through a vacuum.
Enduring Understanding 6.B A periodic wave is one that repeats as a function of both time and position and can be described by its amplitude, frequency, wavelength, speed, and energy.
Essential Knowledge 6.B.3 A simple wave can be described by an equation involving one sine or cosine function involving the wavelength, amplitude, and frequency of the wave.
Enduring Understanding 6.F Electromagnetic radiation can be modeled as waves or as fundamental particles.
Essential Knowledge 6.F.1 Types of electromagnetic radiation are characterized by their wavelengths, and certain ranges of wavelength have been given specific names. These include (in order of increasing wavelength spanning a range from picometers to kilometers) gamma rays, x-rays, ultraviolet, visible light, infrared, microwaves, and radio waves.
Essential Knowledge 6.F.2 Electromagnetic waves can transmit energy through a medium and through a vacuum.

### Discovering a New Phenomenon

It is worth noting at the outset that the general phenomenon of electromagnetic waves was predicted by theory before it was realized that light is a form of electromagnetic wave. The prediction was made by James Clerk Maxwell in the mid-19th century when he formulated a single theory combining all the electric and magnetic effects known by scientists at that time. “Electromagnetic waves” was the name he gave to the phenomena his theory predicted.
Such a theoretical prediction followed by experimental verification is an indication of the power of science in general, and physics in particular. The underlying connections and unity of physics allow certain great minds to solve puzzles without having all the pieces. The prediction of electromagnetic waves is one of the most spectacular examples of this power. Certain others, such as the prediction of antimatter, will be discussed in later modules.


# Electromagnetic Waves

## Maxwell’s Equations: Electromagnetic Waves Predicted and Observed

### Learning Objectives

By the end of this section, you will be able to:
1. Restate Maxwell’s equations.
The Scotsman James Clerk Maxwell (1831–1879) is regarded as the greatest theoretical physicist of the 19th century. (See .) Although he died young, Maxwell not only formulated a complete electromagnetic theory, represented by Maxwell’s equations, he also developed the kinetic theory of gases and made significant contributions to the understanding of color vision and the nature of Saturn’s rings.
Maxwell brought together all the work that had been done by brilliant physicists such as Oersted, Coulomb, Gauss, and Faraday, and added his own insights to develop the overarching theory of electromagnetism. Maxwell’s equations are paraphrased here in words because their mathematical statement is beyond the level of this text. However, the equations illustrate how apparently simple mathematical statements can elegantly unite and express a multitude of concepts—why mathematics is the language of science.
Maxwell’s equations encompass the major laws of electricity and magnetism. What is not so apparent is the symmetry that Maxwell introduced in his mathematical framework. Especially important is his addition of the hypothesis that changing electric fields create magnetic fields. This is exactly analogous (and symmetric) to Faraday’s law of induction and had been suspected for some time, but fits beautifully into Maxwell’s equations.
Symmetry is apparent in nature in a wide range of situations. In contemporary research, symmetry plays a major part in the search for sub-atomic particles using massive multinational particle accelerators such as the new Large Hadron Collider at CERN.
Since changing electric fields create relatively weak magnetic fields, they could not be easily detected at the time of Maxwell’s hypothesis. Maxwell realized, however, that oscillating charges, like those in AC circuits, produce changing electric fields. He predicted that these changing fields would propagate from the source like waves generated on a lake by a jumping fish.
The waves predicted by Maxwell would consist of oscillating electric and magnetic fields—defined to be an electromagnetic wave (EM wave). Electromagnetic waves would be capable of exerting forces on charges great distances from their source, and they might thus be detectable. Maxwell calculated that electromagnetic waves would propagate at a speed given by the equation
When the values for  and  are entered into the equation for
 , we find that
which is the speed of light. In fact, Maxwell concluded that light is an electromagnetic wave having such wavelengths that it can be detected by the eye.
Other wavelengths should exist—it remained to be seen if they did. If so, Maxwell’s theory and remarkable predictions would be verified, the greatest triumph of physics since Newton. Experimental verification came within a few years, but not before Maxwell’s death.

### Hertz’s Observations

The German physicist Heinrich Hertz (1857–1894) was the first to generate and detect certain types of electromagnetic waves in the laboratory. Starting in 1887, he performed a series of experiments that not only confirmed the existence of electromagnetic waves, but also verified that they travel at the speed of light.
Hertz used an AC  (resistor-inductor-capacitor) circuit that resonates at a known frequency  and connected it to a loop of wire as shown in . High voltages induced across the gap in the loop produced sparks that were visible evidence of the current in the circuit and that helped generate electromagnetic waves.
Across the laboratory, Hertz had another loop attached to another  circuit, which could be tuned (as the dial on a radio) to the same resonant frequency as the first and could, thus, be made to receive electromagnetic waves. This loop also had a gap across which sparks were generated, giving solid evidence that electromagnetic waves had been received.
Hertz also studied the reflection, refraction, and interference patterns of the electromagnetic waves he generated, verifying their wave character. He was able to determine wavelength from the interference patterns, and knowing their frequency, he could calculate the propagation speed using the equation  (velocity—or speed—equals frequency times wavelength). Hertz was thus able to prove that electromagnetic waves travel at the speed of light. The SI unit for frequency, the hertz (), is named in his honor.

### Section Summary

1. Electromagnetic waves consist of oscillating electric and magnetic fields and propagate at the speed of light

  . They were predicted by Maxwell, who also showed that
        where

2. Maxwell’s prediction of electromagnetic waves resulted from his formulation of a complete and symmetric theory of electricity and magnetism, known as Maxwell’s equations.
3. These four equations are paraphrased in this text, rather than presented numerically, and encompass the major laws of electricity and magnetism. First is Gauss’s law for electricity, second is Gauss’s law for magnetism, third is Faraday’s law of induction, including Lenz’s law, and fourth is Ampere’s law in a symmetric formulation that adds another source of magnetism—changing electric fields.

### Problems & Exercises


# Electromagnetic Waves

## Production of Electromagnetic Waves

### Learning Objectives

By the end of this section, you will be able to:
1. Describe the electric and magnetic waves as they move out from a source, such as an AC generator.
2. Explain the mathematical relationship between the magnetic field strength and the electrical field strength.
3. Calculate the maximum strength of the magnetic field in an electromagnetic wave, given the maximum electric field strength.
We can get a good understanding of electromagnetic waves (EM) by considering how they are produced. Whenever a current varies, associated electric and magnetic fields vary, moving out from the source like waves. Perhaps the easiest situation to visualize is a varying current in a long straight wire, produced by an AC generator at its center, as illustrated in .

The electric field () shown surrounding the wire is produced by the charge distribution on the wire. Both the  and the charge distribution vary as the current changes. The changing field propagates outward at the speed of light.
There is an associated magnetic field () which propagates outward as well (see ). The electric and magnetic fields are closely related and propagate as an electromagnetic wave. This is what happens in broadcast antennae such as those in radio and TV stations.
Closer examination of the one complete cycle shown in  reveals the periodic nature of the generator-driven charges oscillating up and down in the antenna and the electric field produced. At time , there is the maximum separation of charge, with negative charges at the top and positive charges at the bottom, producing the maximum magnitude of the electric field (or -field) in the upward direction. One-fourth of a cycle later, there is no charge separation and the field next to the antenna is zero, while the maximum -field has moved away at speed .
As the process continues, the charge separation reverses and the field reaches its maximum downward value, returns to zero, and rises to its maximum upward value at the end of one complete cycle. The outgoing wave has an amplitude proportional to the maximum separation of charge. Its wavelength is proportional to the period of the oscillation and, hence, is smaller for short periods or high frequencies. (As usual, wavelength and frequency are inversely proportional.)

### Electric and Magnetic Waves: Moving Together

Following Ampere’s law, current in the antenna produces a magnetic field, as shown in . The relationship between  and  is shown at one instant in  (a). As the current varies, the magnetic field varies in magnitude and direction.

The magnetic field lines also propagate away from the antenna at the speed of light, forming the other part of the electromagnetic wave, as seen in  (b). The magnetic part of the wave has the same period and wavelength as the electric part, since they are both produced by the same movement and separation of charges in the antenna.
The electric and magnetic waves are shown together at one instant in time in . The electric and magnetic fields produced by a long straight wire antenna are exactly in phase. Note that they are perpendicular to one another and to the direction of propagation, making this a transverse wave.
Electromagnetic waves generally propagate out from a source in all directions, sometimes forming a complex radiation pattern. A linear antenna like this one will not radiate parallel to its length, for example. The wave is shown in one direction from the antenna in  to illustrate its basic characteristics.
Instead of the AC generator, the antenna can also be driven by an AC circuit. In fact, charges radiate whenever they are accelerated. But while a current in a circuit needs a complete path, an antenna has a varying charge distribution forming a standing wave, driven by the AC. The dimensions of the antenna are critical for determining the frequency of the radiated electromagnetic waves. This is a resonant phenomenon and when we tune radios or TV, we vary electrical properties to achieve appropriate resonant conditions in the antenna.

### Receiving Electromagnetic Waves

Electromagnetic waves carry energy away from their source, similar to a sound wave carrying energy away from a standing wave on a guitar string. An antenna for receiving EM signals works in reverse. And like antennas that produce EM waves, receiver antennas are specially designed to resonate at particular frequencies.
An incoming electromagnetic wave accelerates electrons in the antenna, setting up a standing wave. If the radio or TV is switched on, electrical components pick up and amplify the signal formed by the accelerating electrons. The signal is then converted to audio and/or video format. Sometimes big receiver dishes are used to focus the signal onto an antenna.
In fact, charges radiate whenever they are accelerated. When designing circuits, we often assume that energy does not quickly escape AC circuits, and mostly this is true. A broadcast antenna is specially designed to enhance the rate of electromagnetic radiation, and shielding is necessary to keep the radiation close to zero. Some familiar phenomena are based on the production of electromagnetic waves by varying currents. Your microwave oven, for example, sends electromagnetic waves, called microwaves, from a concealed antenna that has an oscillating current imposed on it.

### Relating -Field and -Field Strengths

There is a relationship between the - and -field strengths in an electromagnetic wave. This can be understood by again considering the antenna just described. The stronger the -field created by a separation of charge, the greater the current and, hence, the greater the -field created.
Since current is directly proportional to voltage (Ohm’s law) and voltage is directly proportional to -field strength, the two should be directly proportional. It can be shown that the magnitudes of the fields do have a constant ratio, equal to the speed of light. That is,
is the ratio of -field strength to -field strength in any electromagnetic wave. This is true at all times and at all locations in space. A simple and elegant result.
The result of this example is consistent with the statement made in the module Maxwell’s Equations: Electromagnetic Waves Predicted and Observed that changing electric fields create relatively weak magnetic fields. They can be detected in electromagnetic waves, however, by taking advantage of the phenomenon of resonance, as Hertz did. A system with the same natural frequency as the electromagnetic wave can be made to oscillate. All radio and TV receivers use this principle to pick up and then amplify weak electromagnetic waves, while rejecting all others not at their resonant frequency.

### Test Prep for AP Courses

### Section Summary

1. Electromagnetic waves are created by oscillating charges (which radiate whenever accelerated) and have the same frequency as the oscillation.
2. Since the electric and magnetic fields in most electromagnetic waves are perpendicular to the direction in which the wave moves, it is ordinarily a transverse wave.
3. The strengths of the electric and magnetic parts of the wave are related by
    which implies that the magnetic field

### Conceptual Questions

### Problems & Exercises


# Electromagnetic Waves

## The Electromagnetic Spectrum

### Learning Objectives

By the end of this section, you will be able to:
1. List three “rules of thumb” that apply to the different frequencies along the electromagnetic spectrum.
2. Explain why the higher the frequency, the shorter the wavelength of an electromagnetic wave.
3. Draw a simplified electromagnetic spectrum, indicating the relative positions, frequencies, and spacing of the different types of radiation bands.
4. List and explain the different methods by which electromagnetic waves are produced across the spectrum.
In this module we examine how electromagnetic waves are classified into categories such as radio, infrared, ultraviolet, and so on, so that we can understand some of their similarities as well as some of their differences. We will also find that there are many connections with previously discussed topics, such as wavelength and resonance. A brief overview of the production and utilization of electromagnetic waves is found in .
As noted before, an electromagnetic wave has a frequency and a wavelength associated with it and travels at the speed of light, or . The relationship among these wave characteristics can be described by , where  is the propagation speed of the wave,  is the frequency, and  is the wavelength. Here , so that for all electromagnetic waves,
Thus, for all electromagnetic waves, the greater the frequency, the smaller the wavelength.
 shows how the various types of electromagnetic waves are categorized according to their wavelengths and frequencies—that is, it shows the electromagnetic spectrum. Many of the characteristics of the various types of electromagnetic waves are related to their frequencies and wavelengths, as we shall see.

### Transmission, Reflection, and Absorption

What happens when an electromagnetic wave impinges on a material? If the material is transparent to the particular frequency, then the wave can largely be transmitted. If the material is opaque to the frequency, then the wave can be totally reflected. The wave can also be absorbed by the material, indicating that there is some interaction between the wave and the material, such as the thermal agitation of molecules.
Of course it is possible to have partial transmission, reflection, and absorption. We normally associate these properties with visible light, but they do apply to all electromagnetic waves. What is not obvious is that something that is transparent to light may be opaque at other frequencies. For example, ordinary glass is transparent to visible light but largely opaque to ultraviolet radiation. Human skin is opaque to visible light—we cannot see through people—but transparent to X-rays.

### Radio and TV Waves

The broad category of radio waves is defined to contain any electromagnetic wave produced by currents in wires and circuits. Its name derives from their most common use as a carrier of audio information (i.e., radio). The name is applied to electromagnetic waves of similar frequencies regardless of source. Radio waves from outer space, for example, do not come from alien radio stations. They are created by many astronomical phenomena, and their study has revealed much about nature on the largest scales.
There are many uses for radio waves, and so the category is divided into many subcategories, including microwaves and those electromagnetic waves used for AM and FM radio, cellular telephones, and TV.
The lowest commonly encountered radio frequencies are produced by high-voltage AC power transmission lines at frequencies of 50 or 60 Hz. (See .) These extremely long wavelength electromagnetic waves (about 6000 km!) are one means of energy loss in long-distance power transmission.
There was a concern regarding potential health hazards associated with exposure to these electromagnetic fields (-fields). Some people suspect that living near such transmission lines may cause a variety of illnesses, including cancer. But these power lines produce non-ionizing radiation, which government environmental organizations, medical researchers, and cancer organizations indicate are not risk factors for illness. Recent reports that have looked at many European and American epidemiological studies have found no increase in risk for cancer due to exposure to -fields.
Extremely low frequency (ELF) radio waves of about 1 kHz are used to communicate with submerged submarines. The ability of radio waves to penetrate salt water is related to their wavelength (much like ultrasound penetrating tissue)—the longer the wavelength, the farther they penetrate. Since salt water is a good conductor, radio waves are strongly absorbed by it, and very long wavelengths are needed to reach a submarine under the surface. (See .)
AM radio waves are used to carry commercial radio signals in the frequency range from 540 to 1600 kHz. The abbreviation AM stands for amplitude modulation, which is the method for placing information on these waves. (See .) A carrier wave having the basic frequency of the radio station, say 1530 kHz, is varied or modulated in amplitude by an audio signal. The resulting wave has a constant frequency, but a varying amplitude.
A radio receiver tuned to have the same resonant frequency as the carrier wave can pick up the signal, while rejecting the many other frequencies impinging on its antenna. The receiver’s circuitry is designed to respond to variations in amplitude of the carrier wave to replicate the original audio signal. That audio signal is amplified to drive a speaker or perhaps to be recorded.

### FM Radio Waves

FM radio waves are also used for commercial radio transmission, but in the frequency range of 88 to 108 MHz. FM stands for frequency modulation, another method of carrying information. (See .) Here a carrier wave having the basic frequency of the radio station, perhaps 105.1 MHz, is modulated in frequency by the audio signal, producing a wave of constant amplitude but varying frequency.
Since audible frequencies range up to 20 kHz (or 0.020 MHz) at most, the frequency of the FM radio wave can vary from the carrier by as much as 0.020 MHz. Thus the carrier frequencies of two different radio stations cannot be closer than 0.020 MHz. An FM receiver is tuned to resonate at the carrier frequency and has circuitry that responds to variations in frequency, reproducing the audio information.
FM radio is inherently less subject to noise from stray radio sources than AM radio. The reason is that amplitudes of waves add. So an AM receiver would interpret noise added onto the amplitude of its carrier wave as part of the information. An FM receiver can be made to reject amplitudes other than that of the basic carrier wave and only look for variations in frequency. It is thus easier to reject noise from FM, since noise produces a variation in amplitude.
Television is also broadcast on electromagnetic waves. Since the waves must carry a great deal of visual as well as audio information, each channel requires a larger range of frequencies than simple radio transmission. TV channels utilize frequencies in the range of 54 to 88 MHz and 174 to 222 MHz. (The entire FM radio band lies between channels 88 MHz and 174 MHz.) These TV channels are called VHF (for very high frequency). Other channels called UHF (for ultra high frequency) utilize an even higher frequency range of 470 to 1000 MHz.
The TV video signal is AM, while the TV audio is FM. Note that these frequencies are those of free transmission with the user utilizing an old-fashioned roof antenna. Satellite dishes and cable transmission of TV occurs at significantly higher frequencies and is rapidly evolving with the use of the high-definition or HD format.
The wavelengths found in the preceding example are representative of AM, FM, and cell phones, and account for some of the differences in how they are broadcast and how well they travel. The most efficient length for a linear antenna, such as discussed in Production of Electromagnetic Waves, is , half the wavelength of the electromagnetic wave. Thus a very large antenna is needed to efficiently broadcast typical AM radio with its carrier wavelengths on the order of hundreds of meters.
One benefit to these long AM wavelengths is that they can go over and around rather large obstacles (like buildings and hills), just as ocean waves can go around large rocks. FM and TV are best received when there is a line of sight between the broadcast antenna and receiver, and they are often sent from very tall structures. FM, TV, and mobile phone antennas themselves are much smaller than those used for AM, but they are elevated to achieve an unobstructed line of sight. (See .)

### Radio Wave Interference

Astronomers and astrophysicists collect signals from outer space using electromagnetic waves. A common problem for astrophysicists is the “pollution” from electromagnetic radiation pervading our surroundings from communication systems in general. Even everyday gadgets like our car keyless entry devices and remote starters and being able to turn TVs on and off using remotes involve radio-wave frequencies. In order to prevent interference between all these electromagnetic signals, strict regulations are drawn up for different organizations to utilize different radio frequency bands.
One reason why we are sometimes asked to switch off our mobile phones (operating in the range of 1.9 GHz) or put them into a noncommunicative mode on airplanes and in hospitals is that important communications or medical equipment often uses similar radio frequencies and their operation can be affected by frequencies used in the communication devices.
For example, radio waves used in magnetic resonance imaging (MRI) have frequencies on the order of 100 MHz, although this varies significantly depending on the strength of the magnetic field used and the nuclear type being scanned. MRI is an important medical imaging and research tool, producing highly detailed two- and three-dimensional images. Radio waves are broadcast, absorbed, and reemitted in a resonance process that is sensitive to the density of nuclei (usually protons or hydrogen nuclei).
The wavelength of 100-MHz radio waves is 3 m, yet using the sensitivity of the resonant frequency to the magnetic field strength, details smaller than a millimeter can be imaged. This is a good example of an exception to a rule of thumb (in this case, the rubric that details much smaller than the probe’s wavelength cannot be detected). The intensity of the radio waves used in MRI presents little or no hazard to human health.

### Microwaves

Microwaves are the highest-frequency electromagnetic waves that can be produced by currents in macroscopic circuits and devices. Microwave frequencies range from about  to the highest practical  resonance at nearly . Since they have high frequencies, their wavelengths are short compared with those of other radio waves—hence the name “microwave.”
Microwaves can also be produced by atoms and molecules. They are, for example, a component of electromagnetic radiation generated by thermal agitation. The thermal motion of atoms and molecules in any object at a temperature above absolute zero causes them to emit and absorb radiation.
Since it is possible to carry more information per unit time on high frequencies, microwaves are quite suitable for communications. Most satellite-transmitted information is carried on microwaves, as are land-based long-distance transmissions. A clear line of sight between transmitter and receiver is needed because of the short wavelengths involved.
Radar is a common application of microwaves that was first developed in World War II. By detecting and timing microwave echoes, radar systems can determine the distance to objects as diverse as clouds and aircraft. A Doppler shift in the radar echo can be used to determine the speed of a car or the intensity of a rainstorm. Sophisticated radar systems are used to map the Earth and other planets, with a resolution limited by wavelength. (See .) The shorter the wavelength of any probe, the smaller the detail it is possible to observe.

### Heating with Microwaves

How does the ubiquitous microwave oven produce microwaves electronically, and why does food absorb them preferentially? Microwaves at a frequency of 2.45 GHz are produced by accelerating electrons. The microwaves are then used to induce an alternating electric field in the oven.
Water and some other constituents of food have a slightly negative charge at one end and a slightly positive charge at one end (called polar molecules). The range of microwave frequencies is specially selected so that the polar molecules, in trying to keep orienting themselves with the electric field, absorb these energies and increase their temperatures—called dielectric heating.
The energy thereby absorbed results in thermal agitation heating food and not the plate, which does not contain water. Hot spots in the food are related to constructive and destructive interference patterns. Rotating antennas and food turntables help spread out the hot spots.
Another use of microwaves for heating is within the human body. Microwaves will penetrate more than shorter wavelengths into tissue and so can accomplish “deep heating” (called microwave diathermy). This is used for treating muscular pains, spasms, tendonitis, and rheumatoid arthritis.
Microwaves generated by atoms and molecules far away in time and space can be received and detected by electronic circuits. Deep space acts like a blackbody with a 2.7 K temperature, radiating most of its energy in the microwave frequency range. In 1964, Penzias and Wilson detected this radiation and eventually recognized that it was the radiation of the Big Bang’s cooled remnants.

### Infrared Radiation

The microwave and infrared regions of the electromagnetic spectrum overlap (see ). Infrared radiation is generally produced by thermal motion and the vibration and rotation of atoms and molecules. Electronic transitions in atoms and molecules can also produce infrared radiation.
The range of infrared frequencies extends up to the lower limit of visible light, just below red. In fact, infrared means “below red.” Frequencies at its upper limit are too high to be produced by accelerating electrons in circuits, but small systems, such as atoms and molecules, can vibrate fast enough to produce these waves.
Water molecules rotate and vibrate particularly well at infrared frequencies, emitting and absorbing them so efficiently that the emissivity for skin is  in the infrared. Night-vision scopes can detect the infrared emitted by various warm objects, including humans, and convert it to visible light.
We can examine radiant heat transfer from a house by using a camera capable of detecting infrared radiation. Reconnaissance satellites can detect buildings, vehicles, and even individual humans by their infrared emissions, whose power radiation is proportional to the fourth power of the absolute temperature. More mundanely, we use infrared lamps, some of which are called quartz heaters, to preferentially warm us because we absorb infrared better than our surroundings.
The Sun radiates like a nearly perfect blackbody (that is, it has ), with a 6000 K surface temperature. About half of the solar energy arriving at the Earth is in the infrared region, with most of the rest in the visible part of the spectrum, and a relatively small amount in the ultraviolet. On average, 50 percent of the incident solar energy is absorbed by the Earth.
The relatively constant temperature of the Earth is a result of the energy balance between the incoming solar radiation and the energy radiated from the Earth. Most of the infrared radiation emitted from the Earth is absorbed by  and  in the atmosphere and then radiated back to Earth or into outer space. This radiation back to Earth is known as the greenhouse effect, and it maintains the surface temperature of the Earth about  higher than it would be if there is no absorption. Some scientists think that the increased concentration of  and other greenhouse gases in the atmosphere, resulting from increases in fossil fuel burning, has increased global average temperatures.

### Visible Light

Visible light is the narrow segment of the electromagnetic spectrum to which the normal human eye responds. Visible light is produced by vibrations and rotations of atoms and molecules, as well as by electronic transitions within atoms and molecules. The receivers or detectors of light largely utilize electronic transitions. We say the atoms and molecules are excited when they absorb and relax when they emit through electronic transitions.
 shows this part of the spectrum, together with the colors associated with particular pure wavelengths. We usually refer to visible light as having wavelengths of between 400 nm and 750 nm. (The retina of the eye actually responds to the lowest ultraviolet frequencies, but these do not normally reach the retina because they are absorbed by the cornea and lens of the eye.)
Red light has the lowest frequencies and longest wavelengths, while violet has the highest frequencies and shortest wavelengths. Blackbody radiation from the Sun peaks in the visible part of the spectrum but is more intense in the red than in the violet, making the Sun yellowish in appearance.
Living things—plants and animals—have evolved to utilize and respond to parts of the electromagnetic spectrum they are embedded in. Visible light is the most predominant and we enjoy the beauty of nature through visible light. Plants are more selective. Photosynthesis makes use of parts of the visible spectrum to make sugars.
Optics is the study of the behavior of visible light and other forms of electromagnetic waves. Optics falls into two distinct categories. When electromagnetic radiation, such as visible light, interacts with objects that are large compared with its wavelength, its motion can be represented by straight lines like rays. Ray optics is the study of such situations and includes lenses and mirrors.
When electromagnetic radiation interacts with objects about the same size as the wavelength or smaller, its wave nature becomes apparent. For example, observable detail is limited by the wavelength, and so visible light can never detect individual atoms, because they are so much smaller than its wavelength. Physical or wave optics is the study of such situations and includes all wave characteristics.

### Ultraviolet Radiation

Ultraviolet means “above violet.” The electromagnetic frequencies of ultraviolet radiation (UV) extend upward from violet, the highest-frequency visible light. Ultraviolet is also produced by atomic and molecular motions and electronic transitions. The wavelengths of ultraviolet extend from 400 nm down to about 10 nm at its highest frequencies, which overlap with the lowest X-ray frequencies. It was recognized as early as 1801 by Johann Ritter that the solar spectrum had an invisible component beyond the violet range.
Solar UV radiation is broadly subdivided into three regions: UV-A (320–400 nm), UV-B (290–320 nm), and UV-C (220–290 nm), ranked from long to shorter wavelengths (from smaller to larger energies). Most UV-B and all UV-C is absorbed by ozone () molecules in the upper atmosphere. Consequently, 99% of the solar UV radiation reaching the Earth’s surface is UV-A.
One of the first illustrations of UV light’s impact on Earth occurred during the Apollo 16 mission in 1972. The mission included the first astronomical images taken from the moon, using a compact and resilient Far Ultraviolet Camera/Spectrograph designed for moon use by scientist and inventor George Robert Carruthers. Designed to capture UV images without the obscuring effects of the Earth’s atmosphere, its most famous image was of the planet itself. Carruthers, who also trained the astronauts on the device’s use, mentioned afterward that “the most immediately obvious and spectacular results were really for the Earth observations, because this was the first time that the Earth had been photographed from a distance in ultraviolet (UV) light, so that you could see the full extent of the hydrogen atmosphere, the polar auroris and what we call the tropical airglow belt.”

### Human Exposure to UV Radiation

It is largely exposure to UV-B that causes skin cancer. It is estimated that as many as 20% of adults will develop skin cancer over the course of their lifetime. Again, treatment is often successful if caught early. Despite very little UV-B reaching the Earth’s surface, there are substantial increases in skin-cancer rates in countries such as Australia, indicating how important it is that UV-B and UV-C continue to be absorbed by the upper atmosphere.
All UV radiation can damage collagen fibers, resulting in an acceleration of the aging process of skin and the formation of wrinkles. Because there is so little UV-B and UV-C reaching the Earth’s surface, sunburn is caused by large exposures, and skin cancer from repeated exposure. Some studies indicate a link between overexposure to the Sun when young and melanoma later in life.
The tanning response is a defense mechanism in which the body produces pigments to absorb future exposures in inert skin layers above living cells. Basically UV-B radiation excites DNA molecules, distorting the DNA helix, leading to mutations and the possible formation of cancerous cells.
Repeated exposure to UV-B may also lead to the formation of cataracts in the eyes—a cause of blindness among people living in the equatorial belt where medical treatment is limited. Cataracts, clouding in the eye’s lens and a loss of vision, are age related; 60% of those between the ages of 65 and 74 will develop cataracts. However, treatment is easy and successful, as one replaces the lens of the eye with a plastic lens. Prevention is important. Eye protection from UV is more effective with plastic sunglasses than those made of glass.
A major acute effect of extreme UV exposure is the suppression of the immune system, both locally and throughout the body.
Low-intensity ultraviolet is used to sterilize haircutting implements, implying that the energy associated with ultraviolet is deposited in a manner different from lower-frequency electromagnetic waves. (Actually this is true for all electromagnetic waves with frequencies greater than visible light.)
Flash photography is generally not allowed of precious artworks and colored prints because the UV radiation from the flash can cause photo-degradation in the artworks. Often artworks will have an extra-thick layer of glass in front of them, which is especially designed to absorb UV radiation.

### UV Light and the Ozone Layer

If all of the Sun’s ultraviolet radiation reached the Earth’s surface, there would be extremely grave effects on the biosphere from the severe cell damage it causes. However, the layer of ozone () in our upper atmosphere (10 to 50 km above the Earth) protects life by absorbing most of the dangerous UV radiation.
Unfortunately, today we are observing a depletion in ozone concentrations in the upper atmosphere. This depletion has led to the formation of an “ozone hole” in the upper atmosphere. The hole is more centered over the southern hemisphere, and changes with the seasons, being largest in the spring. This depletion is attributed to the breakdown of ozone molecules by refrigerant gases called chlorofluorocarbons (CFCs).
The UV radiation helps dissociate the CFC’s, releasing highly reactive chlorine (Cl) atoms, which catalyze the destruction of the ozone layer. For example, the reaction of  with a photon of light  can be written as:
The Cl atom then catalyzes the breakdown of ozone as follows:
A single chlorine atom could destroy ozone molecules for up to two years before being transported down to the surface. The CFCs are relatively stable and will contribute to ozone depletion for years to come. CFCs are found in refrigerants, air conditioning systems, foams, and aerosols.
International concern over this problem led to the establishment of the “Montreal Protocol” agreement (1987) to phase out CFC production in most countries. However, developing-country participation is needed if worldwide production and elimination of CFCs is to be achieved. Probably the largest contributor to CFC emissions today is China. And while there are indicators that the Protocol has been a success, there is still substantial risk and variability in the ozone layer. The 2019 Antarctic ozone hole was small and short-lived, continuing the general trend toward recovery. But the 2020 Antarctic ozone hole was the largest and longest-lasting on record, partially due to atmospheric conditions. Furthermore, emissions are not the only concern. Susan Solomon and her colleagues at MIT have uncovered the substantial impact of CFC “banks,” in certain regions, where outdated and deteriorating equipment (such as air conditioners) or materials can release enough CFCs to be detectable in the atmosphere and deplete the ozone layer. (See .)

### Benefits of UV Light

Besides the adverse effects of ultraviolet radiation, there are also benefits of exposure in nature and uses in technology. Vitamin D production in the skin (epidermis) results from exposure to UVB radiation, generally from sunlight. A number of studies indicate lack of vitamin D can result in the development of a range of cancers (prostate, breast, colon), so a certain amount of UV exposure is helpful. Lack of vitamin D is also linked to osteoporosis. Exposures (with no sunscreen) of 10 minutes a day to arms, face, and legs might be sufficient to provide the accepted dietary level. However, in the winter time north of about  latitude, most UVB gets blocked by the atmosphere.
UV radiation is used in the treatment of infantile jaundice and in some skin conditions. It is also used in sterilizing workspaces and tools, and killing germs in a wide range of applications. It is also used as an analytical tool to identify substances.
When exposed to ultraviolet, some substances, such as minerals, glow in characteristic visible wavelengths, a process called fluorescence. So-called black lights emit ultraviolet to cause posters and clothing to fluoresce in the visible. Ultraviolet is also used in special microscopes to detect details smaller than those observable with longer-wavelength visible-light microscopes.

### X-Rays

In the 1850s, scientists (such as Faraday) began experimenting with high-voltage electrical discharges in tubes filled with rarefied gases. It was later found that these discharges created an invisible, penetrating form of very high frequency electromagnetic radiation. This radiation was called an X-ray, because its identity and nature were unknown.
As described in Things Great and Small, there are two methods by which X-rays are created—both are submicroscopic processes and can be caused by high-voltage discharges. While the low-frequency end of the X-ray range overlaps with the ultraviolet, X-rays extend to much higher frequencies (and energies).
X-rays have adverse effects on living cells similar to those of ultraviolet radiation, and they have the additional liability of being more penetrating, affecting more than the surface layers of cells. Cancer and genetic defects can be induced by exposure to X-rays. Because of their effect on rapidly dividing cells, X-rays can also be used to treat and even cure cancer.
The widest use of X-rays is for imaging objects that are opaque to visible light, such as the human body or aircraft parts. In humans, the risk of cell damage is weighed carefully against the benefit of the diagnostic information obtained. However, questions have risen in recent years as to accidental overexposure of some people during CT scans—a mistake at least in part due to poor monitoring of radiation dose.
The ability of X-rays to penetrate matter depends on density, and so an X-ray image can reveal very detailed density information.  shows an example of the simplest type of X-ray image, an X-ray shadow on film. The amount of information in a simple X-ray image is impressive, but more sophisticated techniques, such as CT scans, can reveal three-dimensional information with details smaller than a millimeter.
The use of X-ray technology in medicine is called radiology—an established and relatively cheap tool in comparison to more sophisticated technologies. Consequently, X-rays are widely available and used extensively in medical diagnostics. During World War I, mobile X-ray units, advocated by Marie Curie, were used to diagnose soldiers.
Because they can have wavelengths less than 0.01 nm, X-rays can be scattered (a process called X-ray diffraction) to detect the shape of molecules and the structure of crystals. X-ray diffraction was crucial to Crick, Watson, and Wilkins in the determination of the shape of the double-helix DNA molecule.
X-rays are also used as a precise tool for trace-metal analysis in X-ray induced fluorescence, in which the energy of the X-ray emissions are related to the specific types of elements and amounts of materials present.

### Gamma Rays

Soon after nuclear radioactivity was first detected in 1896, it was found that at least three distinct types of radiation were being emitted. The most penetrating nuclear radiation was called a gamma ray ( (again a name given because its identity and character were unknown), and it was later found to be an extremely high frequency electromagnetic wave.
In fact,  rays are any electromagnetic radiation emitted by a nucleus. This can be from natural nuclear decay or induced nuclear processes in nuclear reactors and weapons. The lower end of the  frequency range overlaps the upper end of the X-ray range, but  rays can have the highest frequency of any electromagnetic radiation.
Gamma rays have characteristics identical to X-rays of the same frequency—they differ only in source. At higher frequencies,  rays are more penetrating and more damaging to living tissue. They have many of the same uses as X-rays, including cancer therapy. Gamma radiation from radioactive materials is used in nuclear medicine.
 shows a medical image based on  rays. Food spoilage can be greatly inhibited by exposing it to large doses of  radiation, thereby obliterating responsible microorganisms. Damage to food cells through irradiation occurs as well, and the long-term hazards of consuming radiation-preserved food are unknown and controversial for some groups. Both X-ray and  technologies are also used in scanning luggage at airports.

### Detecting Electromagnetic Waves from Space

The entire electromagnetic spectrum is used by researchers for investigating stars, space, and time. Arthur B. C. Walker was a pioneer in X-ray and ultraviolet observations, and designed specialized telescopes and instruments to observe the Sun’s atmosphere and corona. His developments significantly advanced our understanding of stars, and some of his developments are currently in use in space telescopes as well as in microchip manufacturing. As noted earlier, Penzias and Wilson detected microwaves to identify the background radiation originating from the Big Bang. Radio telescopes such as the Arecibo Radio Telescope in Puerto Rico and Parkes Observatory in Australia were designed to detect radio waves.
Infrared telescopes need to have their detectors cooled by liquid nitrogen to be able to gather useful signals. Since infrared radiation is predominantly from thermal agitation, if the detectors were not cooled, the vibrations of the molecules in the antenna would be stronger than the signal being collected.
The most famous of these infrared sensitive telescopes is the James Clerk Maxwell Telescope in Hawaii. The earliest telescopes, developed in the seventeenth century, were optical telescopes, collecting visible light. Telescopes in the ultraviolet, X-ray, and -ray regions are placed outside the atmosphere on satellites orbiting the Earth.
The Hubble Space Telescope (launched in 1990) gathers ultraviolet radiation as well as visible light. In the X-ray region, there is the Chandra X-ray Observatory (launched in 1999), and in the -ray region, there is the new Fermi Gamma-ray Space Telescope (launched in 2008—taking the place of the Compton Gamma Ray Observatory, 1991–2000). The James Webb Space Telescope, launched in late 2021, observes in a lower-frequency portion of the spectrum compared to Hubble. The JWST observes in long-wavelength visible light (red) through infrared, enabling it to detect objects that are further away, older, and fainter than previous telescopes could detect.

### Test Prep for AP Courses

### Section Summary

1. The relationship among the speed of propagation, wavelength, and frequency for any wave is given by , so that for electromagnetic waves,
        where  is the frequency,  is the wavelength, and  is the speed of light.
2. The electromagnetic spectrum is separated into many categories and subcategories, based on the frequency and wavelength, source, and uses of the electromagnetic waves.
3. Any electromagnetic wave produced by currents in wires is classified as a radio wave, the lowest frequency electromagnetic waves. Radio waves are divided into many types, depending on their applications, ranging up to microwaves at their highest frequencies.
4. Infrared radiation lies below visible light in frequency and is produced by thermal motion and the vibration and rotation of atoms and molecules. Infrared’s lower frequencies overlap with the highest-frequency microwaves.
5. Visible light is largely produced by electronic transitions in atoms and molecules, and is defined as being detectable by the human eye. Its colors vary with frequency, from red at the lowest to violet at the highest.
6. Ultraviolet radiation starts with frequencies just above violet in the visible range and is produced primarily by electronic transitions in atoms and molecules.
7. X-rays are created in high-voltage discharges and by electron bombardment of metal targets. Their lowest frequencies overlap the ultraviolet range but extend to much higher values, overlapping at the high end with gamma rays.
8. Gamma rays are nuclear in origin and are defined to include the highest-frequency electromagnetic radiation of any type.

### Conceptual Questions

### Problems & Exercises


# Electromagnetic Waves

## Energy in Electromagnetic Waves

### Learning Objectives

By the end of this section, you will be able to:
1. Explain how the energy and amplitude of an electromagnetic wave are related.
2. Given its power output and the heating area, calculate the intensity of a microwave oven’s electromagnetic field, as well as its peak electric and magnetic field strengths
Anyone who has used a microwave oven knows there is energy in electromagnetic waves. Sometimes this energy is obvious, such as in the warmth of the summer sun. Other times it is subtle, such as the unfelt energy of gamma rays, which can destroy living cells.
Electromagnetic waves can bring energy into a system by virtue of their electric and magnetic fields. These fields can exert forces and move charges in the system and, thus, do work on them. If the frequency of the electromagnetic wave is the same as the natural frequencies of the system (such as microwaves at the resonant frequency of water molecules), the transfer of energy is much more efficient.
But there is energy in an electromagnetic wave, whether it is absorbed or not. Once created, the fields carry energy away from a source. If absorbed, the field strengths are diminished and anything left travels on. Clearly, the larger the strength of the electric and magnetic fields, the more work they can do and the greater the energy the electromagnetic wave carries.
A wave’s energy is proportional to its amplitude squared ( or ). This is true for waves on guitar strings, for water waves, and for sound waves, where amplitude is proportional to pressure. In electromagnetic waves, the amplitude is the maximum field strength of the electric and magnetic fields. (See .)
Thus the energy carried and the intensity  of an electromagnetic wave is proportional to  and . In fact, for a continuous sinusoidal electromagnetic wave, the average intensity  is given by
where  is the speed of light,  is the permittivity of free space, and  is the maximum electric field strength; intensity, as always, is power per unit area (here in ).
The average intensity of an electromagnetic wave  can also be expressed in terms of the magnetic field strength by using the relationship , and the fact that , where  is the permeability of free space. Algebraic manipulation produces the relationship
where  is the maximum magnetic field strength.
One more expression for  in terms of both electric and magnetic field strengths is useful. Substituting the fact that , the previous expression becomes
Whichever of the three preceding equations is most convenient can be used, since they are really just different versions of the same principle: Energy in a wave is related to amplitude squared. Furthermore, since these equations are based on the assumption that the electromagnetic waves are sinusoidal, peak intensity is twice the average; that is, .

### Test Prep for AP Courses

### Section Summary

1. The energy carried by any wave is proportional to its amplitude squared. For electromagnetic waves, this means intensity can be expressed as
    where

2. This can also be expressed in terms of the maximum magnetic field strength  as
    and in terms of both electric and magnetic fields as

3. The three expressions for  are all equivalent.

### Problems & Exercises


# Geometric Optics

## Connection for AP® Courses

Many visual aspects of light result from the transfer of energy in the form of electromagnetic waves (Big Idea 6). Light from this page or screen is formed into an image by the lens of your eye, much like the lens of the camera that make a photograph. Mirrors, like lenses, can also form images that in turn are captured by your eye (Essential Knowledge 6.E.2, Essential Knowledge 6.E.4). In this chapter, you will explore the behavior of light as an electromagnetic wave and learn:
1. what makes a diamond sparkle (Essential Knowledge 6.E.3),
2. how images are formed by lenses for the purposes of magnification or photography (Essential Knowledge 6.E.5),
3. why objects in some mirrors are closer than they appear (Essential Knowledge 6.E.2), and
4. why clear mountain streams are always a little bit deeper than they appear to be.
You will examine different ways of thinking about and modeling light and when each method is most appropriate (Enduring Understanding 6.F, Essential Knowledge 6.F.4). You will also learn how to use simple geometry to predict how light will move when crossing from one medium to another, or when passing through a lens, or when reflecting off a curved surface (Enduring Understanding 6.E, Essential Knowledge 6.E.1). With this knowledge, you will be able to predict what kind of image will form when light interacts with matter.
Big Idea 6 Waves can transfer energy and momentum from one location to another without the permanent transfer of mass and serve as a mathematical model for the description of other phenomena.
Enduring Understanding 6.E The direction of propagation of a wave such as light may be changed when the wave encounters an interface between two media.
Essential Knowledge 6.E.1 When light travels from one medium to another, some of the light is transmitted, some is reﬂected, and some is absorbed. (Qualitative understanding only.)
Essential Knowledge 6.E.2 When light hits a smooth reﬂecting surface at an angle, it reﬂects at the same angle on the other side of the line perpendicular to the surface (specular reﬂection); and this law of reﬂection accounts for the size and location of images seen in plane mirrors.
Essential Knowledge 6.E.3 When light travels across a boundary from one transparent material to another, the speed of propagation changes. At a non–normal incident angle, the path of the light ray bends closer to the perpendicular in the optically slower substance. This is called refraction.
Essential Knowledge 6.E.4 The reﬂection of light from surfaces can be used to form images.
Essential Knowledge 6.E.5 The refraction of light as it travels from one transparent medium to another can be used to form images.
Enduring Understanding 6.F Electromagnetic radiation can be modeled as waves or as fundamental particles.
Essential Knowledge 6.F.4 The nature of light requires that different models of light are most appropriate at different scales.


# Geometric Optics

## The Ray Aspect of Light

### Learning Objectives

By the end of this section, you will be able to:
1. List the ways by which light travels from a source to another location.
There are three ways in which light can travel from a source to another location. (See .) It can come directly from the source through empty space, such as from the Sun to Earth. Or light can travel through various media, such as air and glass, to the person. Light can also arrive after being reflected, such as by a mirror. In all of these cases, light is modeled as traveling in straight lines called rays. Light may change direction when it encounters objects (such as a mirror) or in passing from one material to another (such as in passing from air to glass), but it then continues in a straight line or as a ray. The word ray comes from mathematics and here means a straight line that originates at some point. It is acceptable to visualize light rays as laser rays (or even science fiction depictions of ray guns).
Experiments, as well as our own experiences, show that when light interacts with objects several times as large as its wavelength, it travels in straight lines and acts like a ray. Its wave characteristics are not pronounced in such situations. Since the wavelength of light is less than a micron (a thousandth of a millimeter), it acts like a ray in the many common situations in which it encounters objects larger than a micron. For example, when light encounters anything we can observe with unaided eyes, such as a mirror, it acts like a ray, with only subtle wave characteristics. We will concentrate on the ray characteristics in this chapter.
Since light moves in straight lines, changing directions when it interacts with materials, it is described by geometry and simple trigonometry. This part of optics, where the ray aspect of light dominates, is therefore called geometric optics. There are two laws that govern how light changes direction when it interacts with matter. These are the law of reflection, for situations in which light bounces off matter, and the law of refraction, for situations in which light passes through matter.

### Test Prep for AP Courses

### Section Summary

1. A straight line that originates at some point is called a ray.
2. The part of optics dealing with the ray aspect of light is called geometric optics.
3. Light can travel in three ways from a source to another location: (1) directly from the source through empty space; (2) through various media; (3) after being reflected from a mirror.

### Problems & Exercises