24 Electromagnetic Waves

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Transcript Of 24 Electromagnetic Waves

24 ELECTROMAGNETIC WAVES

CHAPTER 24 | ELECTROMAGNETIC WAVES 861

Figure 24.1 Human eyes detect these orange “sea goldie” fish swimming over a coral reef in the blue waters of the Gulf of Eilat (Red Sea) using visible light. (credit: Daviddarom, Wikimedia Commons)
Learning Objectives
24.1. Maxwell’s Equations: Electromagnetic Waves Predicted and Observed • Restate Maxwell’s equations.
24.2. Production of Electromagnetic Waves • Describe the electric and magnetic waves as they move out from a source, such as an AC generator. • Explain the mathematical relationship between the magnetic field strength and the electrical field strength. • Calculate the maximum strength of the magnetic field in an electromagnetic wave, given the maximum electric field strength.
24.3. The Electromagnetic Spectrum • List three “rules of thumb” that apply to the different frequencies along the electromagnetic spectrum. • Explain why the higher the frequency, the shorter the wavelength of an electromagnetic wave. • Draw a simplified electromagnetic spectrum, indicating the relative positions, frequencies, and spacing of the different types of radiation bands. • List and explain the different methods by which electromagnetic waves are produced across the spectrum.
24.4. Energy in Electromagnetic Waves • Explain how the energy and amplitude of an electromagnetic wave are related. • 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
Introduction to Electromagnetic Waves
The beauty of a coral reef, the warm radiance of sunshine, the sting of sunburn, the X-ray revealing a broken bone, even microwave popcorn—all are brought to us by electromagnetic waves. The list of the various types of electromagnetic waves, ranging from radio transmission waves to nuclear
gamma-ray ( γ -ray) emissions, 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 Figure 24.2.) What are electromagnetic waves? How are they created, and how do they travel? How can we understand and organize their widely varying properties? What is their relationship to electric and magnetic effects? These and other questions will be explored.

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Misconception Alert: Sound Waves vs. Radio Waves Many people confuse sound waves with radio waves, one type of electromagnetic (EM) wave. However, sound and radio waves are completely different phenomena. Sound creates pressure variations (waves) in matter, such as air or water, or your eardrum. Conversely, radio waves are electromagnetic waves, like visible light, infrared, ultraviolet, X-rays, and gamma rays. EM waves don’t need a medium in which to propagate; they can travel through a vacuum, such as outer space. A radio works because sound waves played by the D.J. at the radio station are converted into electromagnetic waves, then encoded and transmitted in the radio-frequency range. The radio in your car receives the radio waves, decodes the information, and uses a speaker to change it back into a sound wave, bringing sweet music to your ears.
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.
Figure 24.2 The electromagnetic waves sent and received by this 50-foot radar dish antenna at Kennedy Space Center in Florida are not visible, but help track expendable launch vehicles with high-definition imagery. The first use of this C-band radar dish was for the launch of the Atlas V rocket sending the New Horizons probe toward Pluto. (credit: NASA)
24.1 Maxwell’s Equations: Electromagnetic Waves Predicted and Observed
The Scotsman James Clerk Maxwell (1831–1879) is regarded as the greatest theoretical physicist of the 19th century. (See Figure 24.3.) 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.
Figure 24.3 James Clerk Maxwell, a 19th-century physicist, developed a theory that explained the relationship between electricity and magnetism and correctly predicted that visible light is caused by electromagnetic waves. (credit: G. J. Stodart)
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.
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Maxwell’s Equations 1. Electric field lines originate on positive charges and terminate on negative charges. The electric field is defined as the force per unit
charge on a test charge, and the strength of the force is related to the electric constant ε0 , also known as the permittivity of free space.
From Maxwell’s first equation we obtain a special form of Coulomb’s law known as Gauss’s law for electricity. 2. Magnetic field lines are continuous, having no beginning or end. No magnetic monopoles are known to exist. The strength of the magnetic
force is related to the magnetic constant µ 0 , also known as the permeability of free space. This second of Maxwell’s equations is known
as Gauss’s law for magnetism. 3. A changing magnetic field induces an electromotive force (emf) and, hence, an electric field. The direction of the emf opposes the change.
This third of Maxwell’s equations is Faraday’s law of induction, and includes Lenz’s law. 4. Magnetic fields are generated by moving charges or by changing electric fields. This fourth of Maxwell’s equations encompasses Ampere’s
law and adds another source of magnetism—changing electric fields.
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.
Making Connections: Unification of Forces
Maxwell’s complete and symmetric theory showed that electric and magnetic forces are not separate, but different manifestations of the same thing—the electromagnetic force. This classical unification of forces is one motivation for current attempts to unify the four basic forces in nature—the gravitational, electrical, strong, and weak nuclear forces.

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

c = µ10 ε0.

(24.1)

When the values for µ 0 and ε0 are entered into the equation for c , we find that

c=

1

= 3.00×108 m/s,

(8.85×10−12 C2 )(4π×10−7 T ⋅ m)

N ⋅ m2

A

(24.2)

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

RLC

(resistor-inductor-capacitor) circuit that resonates at a known frequency

f0

=

1 2π LC

and connected it to a loop of wire as

shown in Figure 24.4. 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 RLC 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.

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Figure 24.4 The apparatus used by Hertz in 1887 to generate and detect electromagnetic waves. An RLC circuit connected to the first loop caused sparks across a gap in
the wire loop and generated electromagnetic waves. Sparks across a gap in the second loop located across the laboratory gave evidence that the 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 υ = fλ (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 ( 1 Hz = 1 cycle/sec ), is named in his honor.
24.2 Production of Electromagnetic Waves
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 Figure 24.5.

Figure 24.5 This long straight gray wire with an AC generator at its center becomes a broadcast antenna for electromagnetic waves. Shown here are the charge distributions
at four different times. The electric field ( E ) propagates away from the antenna at the speed of light, forming part of an electromagnetic wave.

The electric field ( E ) shown surrounding the wire is produced by the charge distribution on the wire. Both the E 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 ( B ) which propagates outward as well (see Figure 24.6). 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 Figure 24.5 reveals the periodic nature of the generator-driven charges oscillating up and
down in the antenna and the electric field produced. At time t = 0 , 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 E -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 E -field has moved away at speed c .

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 (λ) is proportional to the period of the oscillation and, hence, is smaller for short periods or high frequencies. (As usual, wavelength

and

frequency

⎛ ⎝

f

⎞ ⎠

are inversely proportional.)

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Electric and Magnetic Waves: Moving Together Following Ampere’s law, current in the antenna produces a magnetic field, as shown in Figure 24.6. The relationship between E and B is shown at
one instant in Figure 24.6 (a). As the current varies, the magnetic field varies in magnitude and direction.
Figure 24.6 (a) The current in the antenna produces the circular magnetic field lines. The current ( I ) produces the separation of charge along the wire, which in turn creates the electric field as shown. (b) The electric and magnetic fields ( E and B ) near the wire are perpendicular; they are shown here for one point in space. (c) The magnetic
field varies with current and propagates away from the antenna at the speed of light.
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 Figure 24.6 (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 Figure 24.7. 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.
Figure 24.7 A part of the electromagnetic wave sent out from the antenna at one instant in time. The electric and magnetic fields ( E and B ) are in phase, and they are
perpendicular to one another and the direction of propagation. For clarity, the waves are shown only along one direction, but they propagate out in other directions too.
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 Figure 24.7 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 E -Field and B -Field Strengths There is a relationship between the E - and B -field strengths in an electromagnetic wave. This can be understood by again considering the antenna just described. The stronger the E -field created by a separation of charge, the greater the current and, hence, the greater the B -field created. Since current is directly proportional to voltage (Ohm’s law) and voltage is directly proportional to E -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,

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E B

=

c

(24.3)

is the ratio of E -field strength to B -field strength in any electromagnetic wave. This is true at all times and at all locations in space. A simple and
elegant result.

Example 24.1 Calculating B -Field Strength in an Electromagnetic Wave

What is the maximum strength of the B -field in an electromagnetic wave that has a maximum E -field strength of 1000 V/m ?

Strategy

To find the B -field strength, we rearrange the above equation to solve for B , yielding

B = Ec .

(24.4)

Solution

We are given E , and c is the speed of light. Entering these into the expression for B yields

B = 3.0100×0010V8/mm/s = 3.33×10-6 T,

(24.5)

Where T stands for Tesla, a measure of magnetic field strength.

Discussion

The B -field strength is less than a tenth of the Earth’s admittedly weak magnetic field. This means that a relatively strong electric field of 1000
V/m is accompanied by a relatively weak magnetic field. Note that as this wave spreads out, say with distance from an antenna, its field strengths become progressively weaker.

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.
Take-Home Experiment: Antennas
For your TV or radio at home, identify the antenna, and sketch its shape. If you don’t have cable, you might have an outdoor or indoor TV antenna. Estimate its size. If the TV signal is between 60 and 216 MHz for basic channels, then what is the wavelength of those EM waves?
Try tuning the radio and note the small range of frequencies at which a reasonable signal for that station is received. (This is easier with digital readout.) If you have a car with a radio and extendable antenna, note the quality of reception as the length of the antenna is changed.

PhET Explorations: Radio Waves and Electromagnetic Fields
Broadcast radio waves from KPhET. Wiggle the transmitter electron manually or have it oscillate automatically. Display the field as a curve or vectors. The strip chart shows the electron positions at the transmitter and at the receiver.

Figure 24.8 Radio Waves and Electromagnetic Fields (http://cnx.org/content/m42440/1.5/radio-waves_en.jar)
24.3 The Electromagnetic 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 Table 24.1.

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Table 24.1 Electromagnetic Waves

Type of EM wave

Production

Accelerating charges

Microwaves Infrared Visible light Ultraviolet X-rays

Accelerating charges & thermal agitation
Thermal agitations & electronic transitions
Thermal agitations & electronic transitions
Thermal agitations & electronic transitions
Inner electronic transitions and fast collisions

Gamma rays

Nuclear decay

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Applications

Life sciences aspect

Communications Remote controls

MRI Deep heating

Thermal imaging Heating Absorbed by atmosphere

All pervasive

Photosynthesis Human vision

Sterilization Cancer control Vitamin D production

Medical Security Nuclear medicineSecurity

Medical diagnosis Cancer therapy
Medical diagnosis Cancer therapy

Issues Requires controls for band use Cell phone use
Greenhouse effect
Ozone depletion Cancer causing Cancer causing Cancer causing Radiation damage

Connections: Waves
There are many types of waves, such as water waves and even earthquakes. Among the many shared attributes of waves are propagation
speed, frequency, and wavelength. These are always related by the expression vW = fλ . This module concentrates on EM waves, but other
modules contain examples of all of these characteristics for sound waves and submicroscopic particles.

As noted before, an electromagnetic wave has a frequency and a wavelength associated with it and travels at the speed of light, or c . The relationship among these wave characteristics can be described by vW = fλ , where vW is the propagation speed of the wave, f is the frequency,
and λ is the wavelength. Here vW = c , so that for all electromagnetic waves,

c = fλ.

(24.6)

Thus, for all electromagnetic waves, the greater the frequency, the smaller the wavelength.

Figure 24.9 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.

Figure 24.9 The electromagnetic spectrum, showing the major categories of electromagnetic waves. The range of frequencies and wavelengths is remarkable. The dividing line between some categories is distinct, whereas other categories overlap.
Electromagnetic Spectrum: Rules of Thumb Three rules that apply to electromagnetic waves in general are as follows:
• High-frequency electromagnetic waves are more energetic and are more able to penetrate than low-frequency waves. • High-frequency electromagnetic waves can carry more information per unit time than low-frequency waves. • The shorter the wavelength of any electromagnetic wave probing a material, the smaller the detail it is possible to resolve. Note that there are exceptions to these rules of thumb.

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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 Figure 24.10.) These extremely long wavelength electromagnetic waves (about 6000 km!) are one means of energy loss in long-distance power transmission.
Figure 24.10 This high-voltage traction power line running to Eutingen Railway Substation in Germany radiates electromagnetic waves with very long wavelengths. (credit: Zonk43, Wikimedia Commons)
There is an ongoing controversy regarding potential health hazards associated with exposure to these electromagnetic fields ( E -fields). Some
people suspect that living near such transmission lines may cause a variety of illnesses, including cancer. But demographic data are either inconclusive or simply do not support the hazard theory. Recent reports that have looked at many European and American epidemiological studies
have found no increase in risk for cancer due to exposure to E -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 Figure 24.11.)
Figure 24.11 Very long wavelength radio waves are needed to reach this submarine, requiring extremely low frequency signals (ELF). Shorter wavelengths do not penetrate to any significant depth.
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 Figure 24.12.) 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.
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Figure 24.12 Amplitude modulation for AM radio. (a) A carrier wave at the station’s basic frequency. (b) An audio signal at much lower audible frequencies. (c) The amplitude of the carrier is modulated by the audio signal without changing its basic frequency.
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 Figure 24.13.) 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.

Figure 24.13 Frequency modulation for FM radio. (a) A carrier wave at the station’s basic frequency. (b) An audio signal at much lower audible frequencies. (c) The frequency of the carrier is modulated by the audio signal without changing its amplitude.
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 oldfashioned 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.

Example 24.2 Calculating Wavelengths of Radio Waves

Calculate the wavelengths of a 1530-kHz AM radio signal, a 105.1-MHz FM radio signal, and a 1.90-GHz cell phone signal. Strategy

The relationship between wavelength and frequency is c = fλ , where c = 3.00×108 m / s is the speed of light (the speed of light is only very
slightly smaller in air than it is in a vacuum). We can rearrange this equation to find the wavelength for all three frequencies.

Solution

Rearranging gives

λ = cf .

(24.7)

(a) For the f = 1530 kHz AM radio signal, then,

870 CHAPTER 24 | ELECTROMAGNETIC WAVES

(b) For the f = 105.1 MHz FM radio signal,

λ = 3.00×108 m/s 1530×103 cycles/s
= 196 m.

(c) And for the f = 1.90 GHz cell phone,

λ = 3.00×108 m/s 105.1×106 cycles/s
= 2.85 m.

λ = 3.00×108 m/s 1.90×109 cycles/s
= 0.158 m.
Discussion
These wavelengths are consistent with the spectrum in Figure 24.9. The wavelengths are also related to other properties of these electromagnetic waves, as we shall see.

(24.8) (24.9) (24.10)

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 λ / 2 , 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 Figure 24.14.)

Figure 24.14 (a) A large tower is used to broadcast TV signals. The actual antennas are small structures on top of the tower—they are placed at great heights to have a clear line of sight over a large broadcast area. (credit: Ozizo, Wikimedia Commons) (b) The NTT Dokomo mobile phone tower at Tokorozawa City, Japan. (credit: tokoroten, Wikimedia Commons)