Additional context

This document introduces electromagnetic waves, fitting into the broader field of electromagnetism, a cornerstone of modern physics and engineering. The study of electromagnetism, initiated by seminal works of scientists like Maxwell, underpins diverse technologies, ranging from radio communications to medical imaging. This work builds upon Maxwell's equations, a set of fundamental laws describing the behavior of electric and magnetic fields, and extends the understanding of wave phenomena from mechanical waves to electromagnetic waves. The principles discussed have broad applications in optics, telecommunications, and the development of new materials with specific electromagnetic properties. Further research in this area may lead to advancements in wireless power transfer, improved sensor technologies, and a deeper understanding of the universe through the study of electromagnetic radiation from celestial objects.

? Metal objects reflect not only visible light but also radio waves. This is because at the surface of a metal, (i) the electric-field component parallel to the surface must be zero; (2) the electric-field component perpendicular to the surface must be zero; (3) the magnetic-field component parallel to the surface must be zero; (4) the magnetic-field component perpendicular to the surface must be zero; (v) more than one of these.

Image summary:This is a photograph of an outdoor art installation. The art is a large, mirrored, bean-shaped sculpture. The sculpture reflects the surrounding cityscape, including several skyscrapers. The presence of people near the sculpture suggests it is a public art installation.

Electromagnetic Waves

2 Definitions

Definition 1: Maxwell's equations: A set of four fundamental equations that describe the behavior of electric and magnetic fields and their interactions.

Definition 2: Electromagnetic wave: A disturbance consisting of time-varying electric and magnetic fields that propagate through space, carrying energy and momentum.

What is light? This question has been asked by humans for centuries, but there was no answer until electricity and magnetism were unified into electromagnetism, as described by Maxwell's equations. These equations show that a time-varying magnetic field acts as a source of electric field and that a time-varying electric field acts as a source of magnetic field. These {E} and {B} fields can sustain each other, forming an electromagnetic wave that propagates through space. Visible light emitted by the glowing filament of a light bulb is one example of an electromagnetic wave; other kinds of electromagnetic waves are produced by wi-fi base stations, x-ray machines, and radioactive nuclei.

Definition

Sinusoidal electromagnetic waves: Electromagnetic waves in which the electric and magnetic fields vary sinusoidally with time and position.

In this chapter we'll use Maxwell's equations as the theoretical basis for understanding electromagnetic waves. We'll find that these waves carry both energy and momentum. In sinusoidal electromagnetic waves, the {E} and {B} fields are sinusoidal functions of time and position, with a definite frequency and wavelength.

Visible light, radio, x rays, and other types of electromagnetic waves differ only in their frequency and wavelength. Our study of optics in the following chapters will be based in part on the electromagnetic nature of light.

Unlike waves on a string or sound waves in a fluid, electromagnetic waves do not require a material medium; the light that you see coming from the stars at night has traveled without difficulty across tens or hundreds of light-years of (nearly) empty space. Nonetheless, electromagnetic waves and mechanical waves have much in common and are described in much the same language. Before reading further in this chapter, you should review the properties of mechanical waves as discussed in Chapters 15 and 16.

32.1 Maxwell's Equations and Electromagnetic Waves

In the last several chapters we studied various aspects of electric and magnetic fields. We learned that when the fields don't vary with time, such as an electric field produced by charges at rest or the magnetic field of a steady current, we can analyze the electric and magnetic fields independently without considering interactions between them. But when the fields vary with time, they are no longer independent. Faraday's law (see

Learning Outcomes

In this chapter, you'll learn...

32.1 How electromagnetic waves are generated.

32.2 How and why the speed of light is related to the fundamental constants of electricity and magnetism.

32.3 How to describe the propagation of a sinusoidal electromagnetic wave.

32.4 What determines the amount of energy and momentum carried by an electro-magnetic wave.

32.5 How to describe standing electromagnetic waves.

You'll need to review...

8.1 Momentum.

Definition

Standing wave: A wave pattern created by the superposition of two waves traveling in opposite directions, resulting in stationary points of maximum and minimum amplitude.

15.3, 15.7 Traveling waves and standing waves on a string.

16.4 Standing sound waves.

23.4 Electric field in a conductor.

3 Definitions

Definition 1: Dielectric: An insulating material characterized by its permittivity, which affects the speed of electromagnetic waves within the material.

Definition 2: Permittivity: A measure of how easily an electric field can form in a medium; it affects the speed of electromagnetic waves in the medium.

Definition 3: Energy density: The amount of energy stored per unit volume in an electric or magnetic field.

24.3, 24.4 Electric energy density; permittivity of a dielectric.

28.1, 28.8 Magnetic field of a moving charge; permeability of a dielectric.

29.2, 29.7 Faraday's law and Maxwell's equations.

Definition

L-C circuits: An electrical circuit consisting of an inductor (L) and a capacitor (C), capable of oscillating at a resonant frequency.

30.3, 30.5 Magnetic energy density; L-C circuits.

Section 29.2) tells us that a time-varying magnetic field acts as a source of electric field, as shown by induced emfs in inductors and transformers. Ampere's law, including the displacement current discovered by James Clerk Maxwell (see Section 29.7), shows that a time-varying electric field acts as a source of magnetic field. This mutual interaction between the two fields is summarized in Maxwell's equations, presented in Section 29.7.

Thus, when either an electric or a magnetic field is changing with time, a field of the other kind is induced in adjacent regions of space. We are led (as Maxwell was) to consider the possibility of an electromagnetic disturbance, consisting of time-varying electric and magnetic fields, that can propagate through space from one region to another, even when there is no matter in the intervening region. Such a disturbance, if it exists, will have the properties of a wave, and an appropriate term is electromagnetic wave.

Such waves do exist; radio and television transmission, light, x rays, and many other kinds of radiation are examples of electromagnetic waves. Our goal in this chapter is to see how such waves are explained by the principles of electromagnetism that we have studied thus far and to examine the properties of these waves.

Electricity, Magnetism, and Light

The theoretical understanding of electromagnetic waves actually evolved along a considerably more devious path than the one just outlined. In the early days of electromagnetic theory (the early 19th century), two different units of electric charge were used: one for electrostatics and the other for magnetic phenomena involving currents. In the system of units used at that time, these two units of charge had different physical dimensions. Their ratio had units of velocity, and measurements showed that the ratio had a numerical value that was precisely equal to the speed of light, 3.00 times 10 ^8 meters per second. At the time, physicists regarded this as an extraordinary coincidence and had no idea how to explain it.

In searching to understand this result, Maxwell (Fig. 32.1) proved in 1865 that an electromagnetic disturbance should propagate in free space with a speed equal to that of light and hence that light waves were likely to be electromagnetic in nature. At the same time, he discovered that the basic principles of electromagnetism can be expressed in terms of the four equations that we now call Maxwell's equations, which we discussed in Section 29.7. These four equations are (1) Gauss's law for electric fields; (2) Gauss's law for magnetic fields, showing the absence of magnetic monopoles; (3) Faraday's law; and (4) Ampere's law, including displacement current:

Figure 32.1 summary: The figure is a portrait of James Clerk Maxwell. The image shows him seated, wearing a suit and bow tie, and holding a device in his hands. The portrait suggests Maxwell was a significant figure in his time, likely a person of importance and accomplishment, as indicated by the formal attire and the presence of what appears to be a scientific instrument or model.

Math summary: These equations describe fundamental laws of electromagnetism. They relate electric and magnetic fields to electric charge, magnetic flux, and their rates of change, providing a basis for understanding electromagnetic phenomena.

These equations apply to electric and magnetic fields in vacuum. If a material is present, the electric constant epsilon sub zero and magnetic constant mu sub zero are replaced by the permittivity epsilon and permeability mu of the material. If the values of epsilon and mu are different at different points in the regions of integration, then epsilon and mu have to be transferred to the left sides of Eqs. (29.18) and (29.21), respectively, and placed inside the integrals. The epsilon in equation (29.21) also has to be included in the integral that gives d, capital phi sub E, divided by dt.

Figure 32.2 (a) When your mobile phone sends a text message or photo, the information is transmitted in the form of electromagnetic waves produced by electrons accelerating within the phone's circuits. (b) Power lines carry a strong alternating current, which means that a substantial amount of charge is accelerating back and forth and generating electromagnetic waves. These waves can produce a buzzing sound from your car radio when you drive near the lines.

Figure 32.2 summary: The figure contains two images. The first image shows a woman taking a selfie with a phone. The second image shows a transmission tower against a sunset. The first image likely represents a person using technology for personal enjoyment and communication, while the second image represents infrastructure related to technology and energy.

According to Maxwell's equations, a point charge at rest produces a static E field but no {B} field, whereas a point charge moving with a constant velocity (see Section 28.1) produces both {E} and {B} fields. Maxwell's equations can also be used to show that in order for a point charge to produce electromagnetic waves, the charge must accelerate. In fact, in every situation where electromagnetic energy is radiated, the source is accelerated charges (Fig. 32.2).

Generating Electromagnetic Radiation

Definition

Electromagnetic radiation: Energy emitted in the form of electromagnetic waves.

One way in which a point charge can be made to emit electromagnetic waves is by making it oscillate in simple harmonic motion, so that it has an acceleration at almost every instant (the exception is when the charge is passing through its equilibrium position). Figure 32.3 shows some of the electric field lines produced by such an oscillating point charge. Field lines are not material objects, but you may nonetheless find it helpful to think of them as behaving somewhat like strings that extend from the point charge off to infinity. Oscillating the charge up and down makes waves that propagate outward from the charge along these “strings.” Note that the charge does not emit waves equally in all directions; the waves are strongest at 90 sup circle to the axis of motion of the charge, while there are no waves along this axis. This is just what the “string” picture would lead you to conclude. There is also a magnetic disturbance that spreads outward from the charge; this is not shown in figure 32.3. Because the electric and magnetic disturbances spread or radiate away from the source, the name electromagnetic radiation is used interchangeably with the phrase “electromagnetic waves.”

Figure 32.3 summary: The figure is a series of diagrams showing the electric field lines of a point charge undergoing simple harmonic motion at different times during one oscillation period. The diagrams illustrate how the electric field lines propagate outward from the oscillating charge. The electric field lines exhibit a "kink" that propagates outward, representing a disturbance in the field caused by the charge's changing position. The pattern of electric field lines changes over time, reflecting the charge's oscillation and the outward propagation of electromagnetic information.

Electromagnetic waves with macroscopic wavelengths were first produced in the laboratory in 1887 by the German physicist Heinrich Hertz (for whom the S.I unit of frequency is named). As a source of waves, he used charges oscillating in L-C circuits (see Section 30.5); he detected the resulting electromagnetic waves with other circuits tuned to the same frequency. Hertz also produced electromagnetic standing waves and measured the distance between adjacent nodes (one half-wavelength) to determine the wavelength. Knowing the resonant frequency of his circuits, he then found the speed of the waves from the wavelength–frequency relationship v = lambda f . He established that their speed was the same as that of light; this verified Maxwell's theoretical prediction directly.

The modern value of the speed of light, c, is 299,792,458 meters per second. For our purposes, c = 3.00 times 10 ^8 meters per second is sufficiently accurate.

In the wake of Hertz's discovery, Guglielmo Marconi and others made radio communication a familiar household experience. In a radio transmitter, electric charges are made to oscillate along the length of the conducting antenna, producing oscillating field disturbances like those shown in figure 32.3. Since many charges oscillate together in the antenna, the disturbances are much stronger than those of a single oscillating charge and can be detected at a much greater distance. In a radio receiver the antenna is also a conductor; the fields of the wave emanating from a distant transmitter exert forces on free charges within the receiver antenna, producing an oscillating current that is detected and amplified by the receiver circuitry.

For the remainder of this chapter our concern will be with electromagnetic waves themselves, not with the rather complex problem of how they are produced.

The Electromagnetic Spectrum

Definition

Electromagnetic spectrum: The entire range of frequencies and wavelengths of electromagnetic radiation.

The electromagnetic spectrum encompasses electromagnetic waves of all frequencies and wavelengths. Figure 32.4 shows approximate wavelength and frequency ranges for the most commonly encountered portion of the spectrum. Despite vast differences in their uses and means of production, these are all electromagnetic waves with the same propagation speed (in vacuum) c equals 299,792,458 meters per second. Electromagnetic waves may differ in frequency f and wavelength lambda, but the relationship c equals lambda times f in vacuum holds for each.

Figure 32.4 summary: This figure shows a spectrum that is similar to a bar chart, illustrating the wavelengths of visible light. The spectrum displays a continuous range of colors, starting with red at a higher wavelength and transitioning through orange, yellow, green, and blue, before ending with violet at a lower wavelength. The figure indicates that different colors correspond to different wavelengths of light, with red having a longer wavelength and violet having a shorter wavelength.

We can detect only a very small segment of this spectrum directly through our sense of sight. We call this range visible light. Its wavelengths range from about 380 to 750 nm (380 to 750 × 10⁻⁹ m), with corresponding frequencies from about 790 to 400 terahertz (7.9 to 4.0 × 10¹⁴ Hz). Different parts of the visible spectrum evoke in humans the sensations of different colors. Table 32.1 gives the approximate wavelengths for colors in the visible spectrum.

Table 32.1 summary: The table describes the different wavelengths of visible light and their corresponding colors. The colors are arranged according to their wavelengths, ranging from violet (shortest wavelength) to red (longest wavelength).

Definition

Monochromatic light: Light of a single wavelength or a very narrow band of wavelengths.

Ordinary white light includes all visible wavelengths. However, by using special sources or filters, we can select a narrow band of wavelengths within a range of a few nm. Such light is approximately monochromatic (single-color) light. Absolutely monochromatic light with only a single wavelength is an unattainable idealization. When we say “monochromatic light with lambda = 550 nm ” with reference to a laboratory experiment, we really mean a small band of wavelengths around 550 nm. Light from a laser is much more nearly monochromatic than is light obtainable in any other way.

Invisible forms of electromagnetic radiation are no less important than visible light. Our system of global communication, for example, depends on radio waves: A.M radio uses waves with frequencies from 5.4 times 10 to the power of 5 Hertz to 1.6 times 10 to the power of 6 Hertz, and F.M radio broadcasts are at frequencies from 8.8 times 10 to the power of 7 Hertz to 1.08 times 10 to the power of 8 Hertz. Microwaves are also used for communication (for example, by mobile phones and wireless networks) and for weather radar (at frequencies near 3 times 10 to the power of 9 Hertz). Many cameras have a device that emits a

Table summary: The table illustrates a spectrum of wavelengths, ranging from very small values on one end to extremely large values on the other. Certain technologies such as radio, microwaves, and TV are placed relative to these wavelengths.

Bio Application Ultraviolet

Vision Many insects and birds can see ultraviolet wavelengths that humans cannot. As an example, the left-hand photo shows how black-eyed Susans (genus Rudbeckia) look to us. The right-hand photo (in false color), taken with an ultraviolet-sensitive camera, shows how these same flowers appear to the bees that pollinate them. Note the prominent central spot that is invisible to humans. Similarly, many birds with ultraviolet vision—including budgies, parrots, and peacocks—have ultraviolet patterns on their bodies that make them even more vivid to each other than they appear to us. beam of infrared radiation; by analyzing the properties of the infrared radiation reflected from the subject, the camera determines the distance to the subject and automatically adjusts the focus. X rays are able to penetrate through flesh, which makes them invaluable in dentistry and medicine. Gamma rays, the shortest-wavelength type of electromagnetic radiation, are used in medicine to destroy cancer cells.

Image summary:This image is a side-by-side comparison. The image shows flowers. One side of the image shows flowers with natural coloring. The other side of the image shows the same flowers with altered coloring. The altered coloring is less natural than the original coloring.

Test Your Understanding of Section 32.1 (a) Is it possible to have a purely electric wave propagate through empty space—that is, a wave made up of an electric field but no magnetic field? (b) What about a purely magnetic wave, with a magnetic field but no electric field?

Answer

(a) no, (b) no A purely electric wave would have a varying electric field. Such a field necessarily generates a magnetic field through Ampere's law, equation (29.21), so a purely electric wave is impossible. In the same way, a purely magnetic wave is impossible: The varying magnetic field in such a wave would automatically give rise to an electric field through Faraday's law,

32.2 Plane Electromagnetic Waves and the Speed of Light

Definition

Wave equation: A second-order linear partial differential equation that describes the propagation of waves, such as electromagnetic waves.

We are now ready to develop the basic ideas of electromagnetic waves and their relationship to the principles of electromagnetism. Our procedure will be to postulate a simple field configuration that has wavelike behavior. We'll assume an electric field {E} that has only a y-component and a magnetic field {B} with only a z-component, and we'll assume that both fields move together in the +x-direction with a speed c that is initially unknown. (As we go along, it will become clear why we choose {E} and {B} to be perpendicular to the direction of propagation as well as to each other.) Then we'll test whether these fields are physically possible by asking whether they are consistent with Maxwell's equations, particularly Ampere's law and Faraday's law. We'll find that the answer is yes, provided that c has a particular value. We'll also show that the wave equation, which we encountered during our study of mechanical waves in Chapter 15, can be derived from Maxwell's equations.

A Simple Plane Electromagnetic Wave

Definition

Wave front: A surface of constant phase in a wave; for a plane wave, it is a plane perpendicular to the direction of propagation.

Using an xyz-coordinate system (Fig. 32.5), we imagine that all space is divided into two regions by a plane perpendicular to the x-axis (parallel to the y-plane). At every point to the left of this plane there are a uniform electric field {E} in the +y-direction and a uniform magnetic field {B} in the +z-direction, as shown. Furthermore, we suppose that the boundary plane, which we call the wave front, moves to the right in the +x-direction with a constant speed c, the value of which we'll leave undetermined for now. Thus the {E} and {B} fields travel to the right into previously field-free regions with a definite speed.

Figure 32.5 summary: The figure is an illustration of an electromagnetic wave front. It depicts a plane representing the wave front moving in a specific direction. The wave front propagates with a certain speed. The figure suggests the presence of electromagnetic fields in the region where the wave front exists, while these fields are absent elsewhere.

Definition

Plane wave: A wave in which the wavefronts are infinite parallel planes, perpendicular to the direction of propagation, and the field vectors are uniform across each plane.

This is a rudimentary electromagnetic wave. Such a wave, in which at any instant the fields are uniform over any plane perpendicular to the direction of propagation, is called a plane wave. In the case shown in figure 32.5, the fields are zero for planes to the right of the wave front and have the same values on all planes to the left of the wave front; later we'll consider more complex plane waves.

We won't concern ourselves with the problem of actually producing such a field configuration. Instead, we simply ask whether it is consistent with the laws of electromagnetism—that is, with all four of Maxwell's equations.

Let us first verify that our wave satisfies Maxwell's first and second equations—that is, Gauss's laws for electric and magnetic fields. To do this, we take as our Gaussian surface a rectangular box with sides parallel to the xy-, xz-, and yz-coordinate planes (Fig. 32.6). The box encloses no electric charge. The total electric flux and magnetic flux through the box are both zero, even if part of the box is in the region where E = B = 0. This would not be the case if {E} or {B} had an x-component, parallel to the direction of propagation; if the wave front were inside the box, there would be flux through the left-hand side of the box (at x = 0) but not the right-hand side (at x greater than 0). Thus to satisfy Maxwell's first and second equations, the electric and magnetic fields must be perpendicular to the direction of propagation; that is, the wave must be transverse.

Figure 32.6 summary: The figure illustrates a Gaussian surface for a transverse plane electromagnetic wave. The image depicts a rectangular prism in three-dimensional space, representing the Gaussian surface, with electric and magnetic fields present. The magnetic field is consistent on both the left and right sides of the Gaussian surface, leading to a net magnetic flux of zero through the entire surface.

The next of Maxwell's equations that we'll consider is Faraday's law:

Math summary: This equation calculates the relationship between a changing magnetic field and the electric field it induces. It states that the line integral of the electric field around a closed loop equals the negative rate of change of the magnetic flux through the area enclosed by the loop.

To test whether our wave satisfies Faraday's law, we apply this law to a rectangle efgh that is parallel to the xy-plane (Fig. 32.7a). As shown in figure 32.7b, a cross section in the xy-plane, this rectangle has height a and width Delta x . At the time shown, the wave front has progressed partway through the rectangle, and {E} is zero along the side ef. In applying Faraday's law we take the vector area d{A} of rectangle efgh to be in the +z-direction. With this choice the right-hand rule requires that we integrate {E} dot d{l} counterclockwise around the rectangle. At every point on side ef, {E} is zero. At every point on sides fg and he, {E} is either zero or perpendicular to d{l} . Only side gh contributes to the integral. On this side, {E} and d{l} are opposite, and we find that the left-hand side of equation (32.1) is nonzero:

Math summary: This expression calculates the closed loop line integral of the electric field. It states that the integral of the electric field around a closed loop is equal to the negative product of the electric field strength and the height.

To satisfy Faraday's law, Equation 32.1, there must be a component of B vector in the z direction, perpendicular to E vector, so that there can be a nonzero magnetic flux Phi sub B through the rectangle efgh and a nonzero derivative d Phi sub B divided by dt. Indeed, in our wave, B vector has only a z component. We have assumed that this component is in the positive z direction; let's see whether this assumption is consistent with Faraday's law. During a time interval dt the wave front, traveling at speed c, moves a distance c times dt to the right in Figure 32.7b, sweeping out an area a times c times dt of the rectangle efgh. During this interval the magnetic flux Phi sub B through the rectangle efgh increases by d Phi sub B equals B times the quantity a times c times dt, so the rate of change of magnetic flux is

Math summary: This equation calculates the rate of change of magnetic flux. It multiplies the magnetic field strength by the area and the speed of light to determine how quickly the magnetic flux is changing.

Now we substitute Eqs. (32.2) and (32.3) into Faraday's law, equation (32.1); we get -Ea = -Bac , so

Math summary: This expression relates the electric field magnitude to the magnetic field magnitude in an electromagnetic wave traveling through a vacuum. The electric field magnitude equals the speed of light in a vacuum multiplied by the magnetic field magnitude.

Our wave is consistent with Faraday's law only if the wave speed c and the magnitudes of vector E and vector B are related as in Equation 32.4. If we had assumed that vector B was in the negative z direction, there would have been an additional minus sign in Equation 32.4; since E, c, and B are all positive magnitudes, no solution would then have been possible. Furthermore, any component of vector B in the y direction (parallel to vector E) would not contribute to the changing magnetic flux Phi sub B through the rectangle e f g h (which is parallel to the x y plane) and so would not be part of the wave.

The electric field is the same on the top and bottom sides of the Gaussian surface, so the total electric flux through the surface is zero.

Figure 32.7 (a) Applying Faraday's law to a plane wave. (b) In a time dt, the magnetic flux through the rectangle in the xy-plane increases by an amount d, capital phi, sub B. This increase equals the flux through the shaded rectangle with area a times c times dt; that is, d, capital phi, sub B, equals B times a times c times dt. Thus d, capital phi, sub B, divided by dt equals B times a times c.

Figure 32.7 summary: The figure is a diagram illustrating a physical setup with a rectangular loop and several surfaces. The diagram depicts both electric and magnetic fields. The magnetic field appears uniform on both sides of a Gaussian surface. The total magnetic flux through the surface is zero, implying that the magnetic field lines entering the surface are equal to those exiting, consistent with Gauss's law for magnetism.

(a) In time dt, the wave front moves a distance c dt in the +x-direction.

(a) In time dt, the wave front moves a distance c dt in the +x-direction.

Finally, let's do a similar calculation with Ampere's law, the last of Maxwell's equations. There is no conduction current i sub C equals 0, so Ampere's law is

Math summary: This equation relates the line integral of the magnetic field around a closed loop to the rate of change of electric flux through the loop. It states that the integral of the magnetic field around a closed path equals the product of constants and the time derivative of the electric flux.

To check whether our wave is consistent with Ampere's law, we move our rectangle so that it lies in the xz-plane (Fig. 32.8), and we again look at the situation at a time when the wave front has traveled partway through the rectangle. We take the vector area d{A} in the +y-direction, and so the right-hand rule requires that we integrate {B} dot d{l} counterclockwise around the rectangle. The {B} field is zero at every point along side ef, and at each point on sides fg and he it is either zero or perpendicular to d{l} . Only side gh, where {B} and d{l} are parallel, contributes to the integral, and

Math summary: This expression calculates the closed loop integral of the magnetic field. It states that the integral of the magnetic field around a closed loop is equal to the magnetic field strength multiplied by the area enclosed by the loop.

Figure 32.8 summary: The figure is an illustration of a side view of a situation involving electric and magnetic fields. The figure shows a rectangular loop, partially within a region containing both electric and magnetic fields. The electric field is oriented along the y-axis, while the magnetic field is directed out of the page. Outside this region, both fields are zero. The figure depicts the induced electric field and its relationship to a changing magnetic flux, illustrating Faraday's Law. The change in magnetic flux through the loop induces an electromotive force, which in turn creates an electric field.

Hence the left-hand side of Equation thirty-two point five is nonzero; the right-hand side must be nonzero as well. Thus, the vector E must have a y-component (perpendicular to the vector B) so that the electric flux Phi sub E through the rectangle and the time derivative d Phi sub E over dt can be nonzero. Just as we inferred from Faraday's law, we conclude that in an electromagnetic wave, the vector E and the vector B must be mutually perpendicular.

In a time interval dt the electric flux Phi sub E through the rectangle increases by d Phi sub E equals E times a times c times dt. Since we chose d vector A to be in the positive y direction, this flux change is positive; the rate of change of electric flux is

Math summary: This expression calculates the rate of change of electric flux. It states that the rate of change of electric flux with respect to time is equal to the electric field multiplied by the area and a constant.

Substituting Equations (32.6) and (32.7) into Ampere's law, Equation (32.5), we find B times a equals epsilon sub 0 times mu sub 0 times E times a times c, so

Math summary: This expression calculates the magnetic field magnitude of an electromagnetic wave in a vacuum. It multiplies the electric field magnitude by the vacuum permittivity, vacuum permeability, and the speed of light.

Our assumed wave obeys Ampere's law only if B, c, and E are related as in Equation thirty-two point eight. The wave must also obey Faraday's law, so Equation thirty-two point four must be satisfied as well. This can happen only if epsilon sub zero times mu sub zero times c equals one divided by c, or

Math summary: This expression calculates the speed of electromagnetic waves in a vacuum. It takes the electric constant and the magnetic constant as inputs, multiplies them, calculates the square root of the result, and then takes the reciprocal to produce the speed.

Inserting the numerical values of these quantities to four significant figures, we find

Math summary: This expression calculates the speed of light. It takes the reciprocal of the square root of the product of two constants, the permittivity of free space and the permeability of free space, resulting in the speed of light.

Our assumed wave is consistent with all of Maxwell's equations, provided that the wave front moves with the speed given above, which is the speed of light! Recall that the exact value of c is defined to be 299,792,458 meters per second; both epsilon 0 and mu 0 have small uncertainties (see Sections 21.3 and 28.4), but their product in equation (32.9) has zero uncertainty.

Key Properties of Electromagnetic Waves

We chose a simple wave for our study in order to avoid mathematical complications, but this special case illustrates several important features of all electromagnetic waves:

1. The wave is transverse; both E and B are perpendicular to the direction of propagation of the wave. The electric and magnetic fields are also perpendicular to each other. The direction of propagation is the direction of the vector product E vector cross B vector (Fig. 32.9).

Figure 32.9 summary: The figure is a vector diagram. It illustrates the relationship between the electric field, magnetic field, and the direction of propagation of an electromagnetic wave. The diagram shows the electric field and magnetic field are perpendicular to each other. The direction of propagation is perpendicular to both the electric and magnetic fields. The direction of propagation is the cross product of the electric field and the magnetic field.

2. There is a definite ratio between the magnitudes of vector E and vector B: E equals c times B.

3. The wave travels in vacuum with a definite and unchanging speed.

4. Unlike mechanical waves, which need the particles of a medium such as air to transmit a wave, electromagnetic waves require no medium.

We can generalize this discussion to a more realistic situation. Suppose we have several wave fronts in the form of parallel planes perpendicular to the x-axis, all of which are moving to the right with speed c. Suppose that the {E} and {B} fields are the same at all points within a single region between two planes, but that the fields differ from region to region. The overall wave is a plane wave, but one in which the fields vary in steps along the x-axis. Such a wave could be constructed by superposing several of the simple step waves we have just discussed (shown in figure 32.5). This is possible because the {E} and {B} fields obey the superposition principle in waves just as in static situations: When two waves are superposed, the total {E} field at each point is the vector sum of the {E} fields of the individual waves, and similarly for the total {B} field.

We can extend the above development to show that a wave with fields that vary in steps is also consistent with Ampere's and Faraday's laws, provided that the wave fronts all move with the speed c given by equation (32.9). In the limit that we make the individual steps infinitesimally small, we have a wave in which the {E} and {B} fields at any instant vary continuously along the x-axis. The entire field pattern moves to the right with speed c. In Section 32.3 we'll consider waves in which {E} and {B} are sinusoidal functions of x and t. Because at each point the magnitudes of {E} and {B} are related by E = cB , the periodic variations of the two fields in any periodic traveling wave must be in phase.

Electromagnetic waves have the property of polarization. In the above discussion the choice of the y-direction for vector E was arbitrary. We could instead have specified the z-axis for vector E; then vector B would have been in the negative y-direction. A wave in which vector E is always parallel to a certain axis is said to be linearly polarized along that axis.

More generally, any wave traveling in the x-direction can be represented as a superposition of waves linearly polarized in the y-and z-directions. We'll study polarization in greater detail in Chapter 33.

Derivation of the Electromagnetic Wave Equation

Here is an alternative derivation of equation (32.9) for the speed of electromagnetic waves. It is more mathematical than our other treatment, but it includes a derivation of the wave equation for electromagnetic waves. This part of the section can be omitted without loss of continuity in the chapter.

During our discussion of mechanical waves in Section 15.3, we showed that a function y(x, t) that represents the displacement of any point in a mechanical wave traveling along the x-axis must satisfy a differential equation, equation (15.12):

Math summary: This equation calculates the relationship between the second spatial derivative and the second time derivative of a wave's displacement. It states that the second spatial derivative of the wave's displacement equals the second time derivative of the wave's displacement scaled by the inverse square of the wave's propagation speed.

This equation is called the wave equation, and v is the speed of propagation of the wave.

To derive the corresponding equation for an electromagnetic wave, we again consider a plane wave. That is, we assume that at each instant, E sub y and B sub z are uniform over any plane. Figure 32.9 A right-hand rule for electromagnetic waves relates the directions of vector E and vector B and the direction of propagation.

Right-hand rule for an electromagnetic wave:

- ① Point the thumb of your right hand in the wave's direction of propagation.

- ② Imagine rotating the E field vector 90 degrees in the sense your fingers curl. That is the direction of the B field.

Figure 32.10 Faraday's law applied to a rectangle with height a and width Delta x parallel to the xy-plane. perpendicular to the x-axis, the direction of propagation. But now we let E sub y and B sub z vary continuously as we go along the x-axis; then each is a function of x and t. We consider the values of E sub y and B sub z on two planes perpendicular to the x-axis, one at x and one at x plus delta x.

Figure 32.10 summary: The figure is a diagram showing the propagation of electromagnetic waves in space. It illustrates how electric and magnetic fields oscillate perpendicular to each other and to the direction of wave propagation. The diagram depicts the electric and magnetic field components at different points in space. The magnetic and electric fields are interrelated and propagate together as an electromagnetic wave.

Following the same procedure as previously, we apply Faraday's law to a rectangle lying parallel to the xy-plane, as in figure 32.10. This figure is similar to figure 32.7. Let the left end gh of the rectangle be at position x, and let the right end ef be at position x plus delta x. At time t, the values of E sub y on these two sides are E sub y of x, t and E sub y of x plus delta x, t, respectively. When we apply Faraday's law to this rectangle, we find that instead of the closed integral of vector E dotted with d vector l equals negative E times a as before, we have

Math summary: This expression calculates the line integral of the electric field around a rectangular loop. It computes the difference of the electric field at two points, scaled by a constant 'a', representing the side length of the rectangle.

To find the magnetic flux Phi sub B through this rectangle, we assume that Delta x is small enough that B sub z is nearly uniform over the rectangle. In that case, Phi sub B equals B sub z of x, t, times A, equals B sub z of x, t, times a, times Delta x, and

Math summary: This expression calculates the rate of change of magnetic flux. It multiplies the time derivative of the magnetic field component by the area, which is the product of a constant and a small change in position.

We use partial-derivative notation because B sub z is a function of both x and t. When we substitute this expression and equation (32.11) into Faraday's law, equation (32.1), we get

Math summary: This expression relates the change in the electric field to the time derivative of the magnetic field. Specifically, it shows that the rate of change of the electric field with respect to position is equal to the negative of the rate of change of the magnetic field with respect to time.

Finally, imagine shrinking the rectangle down to a sliver so that delta x approaches zero. When we take the limit of this equation as delta x approaches zero, we get

Math summary: This equation calculates the relationship between the spatial change of the electric field and the temporal change of the magnetic field. Specifically, the rate of change of the electric field with respect to position is equal to the negative rate of change of the magnetic field with respect to time.

This equation shows that if there is a time-varying component B sub z of magnetic field, there must also be a component E sub y of electric field that varies with x, and conversely. We put this relationship on the shelf for now; we'll return to it soon.

Next we apply Ampere's law to the rectangle shown in figure 32.11. The line integral, the closed integral of vector B dot d vector l becomes

Math summary: This expression calculates the line integral of a magnetic field around a rectangular loop. It subtracts the magnetic field component at one location multiplied by the area from the magnetic field component at another location multiplied by the area to find the total line integral.

Figure 32.11 summary: The figure is a diagram illustrating Ampere's law applied to a rectangular area. The diagram depicts a rectangular loop with height a and width delta x, positioned parallel to the xz-plane. The diagram also illustrates the electric field within the rectangular area. The electric field is oriented in opposing directions on either side of the rectangle.

Again assuming that the rectangle is narrow, we approximate the electric flux Phi sub E through it as Phi sub E equals E sub y of x, t, times A equals E sub y of x, t, times a times delta x. The rate of change of Phi sub E, which we need for Ampere's law, is then Now we substitute this expression and equation (32.13) into Ampere's law, equation (32.5):

Math summary: This expression calculates the difference of a magnetic field component at two spatial points. This difference is then set equal to the product of constants, the time derivative of an electric field component, and a spatial increment.

Again we divide both sides by a times delta x and take the limit as delta x approaches 0. We find

Math summary: This equation calculates the relationship between the spatial change of a magnetic field and the temporal change of an electric field. It states that the negative spatial derivative of the magnetic field is proportional to the product of two constants and the time derivative of the electric field.

Now comes the final step. We take the partial derivatives of both sides of equation (32.12) with respect to x, and we take the partial derivatives of both sides of equation (32.14) with respect to t. The results are

Math summary: The equations relate the second spatial derivative of one electric field component to the mixed spatial and temporal derivative of a magnetic field component. The equations also relate the mixed spatial and temporal derivative of the magnetic field component to the second temporal derivative of the electric field component, scaled by constants.

Combining these two equations to eliminate B sub z, we finally find

Math summary: This equation calculates the second spatial derivative of the electric field with respect to position and equates it to the product of two constants and the second time derivative of the electric field with respect to time. The result demonstrates that the electric field behaves as an electromagnetic wave.

This expression has the same form as the general wave equation, equation (32.10). Because the electric field E sub y must satisfy this equation, it behaves as a wave with a pattern that travels through space with a definite speed. Furthermore, comparison of Eqs. (32.15) and (32.10) shows that the wave speed v is given by

Math summary: This equation calculates the speed of an electromagnetic wave. It determines the wave speed by taking the reciprocal of the square root of the product of the permittivity and permeability constants.

This agrees with equation (32.9) for the speed c of electromagnetic waves.

We can show that B sub z also must satisfy the same wave equation as E sub y , equation (32.15). To prove this, we take the partial derivative of equation (32.12) with respect to t and the partial derivative of equation (32.14) with respect to x and combine the results. We leave this derivation for you to carry out.

Test Your Understanding of Section 32.2 For each of the following electromagnetic waves, state the direction of the magnetic field. (a) The wave is propagating in the positive z-direction, and {E} is in the positive x-direction; (b) the wave is propagating in the positive y-direction, and {E} is in the negative z-direction; (c) the wave is propagating in the negative x-direction, and {E} is in the positive z-direction.

Answer

(a) positive y-direction, (b) negative x-direction, (c) positive y-direction You can verify these answers by using the right-hand rule to show that {E} times {B} in each case is in the direction of propagation, or by using the rule shown in figure 32.9.

32.3 Sinusoidal Electromagnetic Waves

Sinusoidal electromagnetic waves are directly analogous to sinusoidal transverse mechanical waves on a stretched string, which we studied in Section 15.3. In a sinusoidal electromagnetic wave, {E} and {B} at any point in space are sinusoidal functions of time, and at any instant of time the spatial variation of the fields is also sinusoidal.

Some sinusoidal electromagnetic waves are plane waves; they share with the waves described in Section 32.2 the property that at any instant the fields are uniform over any plane perpendicular to the direction of propagation. The entire pattern travels in the direction of propagation with speed c. The directions of {E} and {B} are perpendicular to the direction of propagation (and to each other), so the wave is transverse. Electromagnetic waves produced by an oscillating point charge, shown in figure 32.3, are an example of sinusoidal waves that are not plane waves. But if we restrict our observations to a relatively small region of space at a sufficiently great distance from the source, even these waves are well approximated by plane waves (Fig. 32.12). In the same way, the curved surface of the (nearly) spherical earth appears flat to us because of our small size relative to the earth's radius. In this section we'll restrict our discussion to plane waves.

Figure 32.12 summary: This is an illustration. The image depicts a boat surrounded by several arrows. The arrows point in various directions around the boat. The arrows are not uniform; some are wavy, while others are dotted.

Figure 32.12 Waves passing through a small area at a sufficiently great distance from a source can be treated as plane waves. Waves that pass through a large area propagate in different directions ...

Math summary: This expression describes the source of electromagnetic waves. It states that a matrix related to the waves is greater than or equal to zero.

... but waves that pass through a small area all propagate in nearly the same direction, so we can treat them as plane waves.

the vector E

Application Electromagnetic

Plane waves from Space The moon, sun, planets, and stars are so distant that the light we receive from them is well approximated by plane waves. This simplifies the problem of designing the optics of astronomical telescopes to collect this light. The light from objects in space is not all of a single frequency, however, so the waves are not simple sinusoids as in Equations 32.17. The frequency f, the wavelength lambda, and the speed of propagation c of any periodic wave are related by the usual wavelength–frequency relationship c equals lambda times f. If the frequency f is 10 to the power of 8 Hertz (100 megahertz, typical of commercial F.M radio broadcasts, the wavelength is

Image summary: The image is a photograph of an observatory under a starlit sky. The observatory is in the foreground, and the night sky with numerous stars is in the background. The presence of the observatory suggests that the image relates to astronomy or scientific observation of the sky.

Math summary: This calculation determines the wavelength of a wave. It divides the speed of light, three times ten to the eighth meters per second, by the frequency, one hundred million hertz, resulting in a wavelength of three meters.

Figure 32.4 shows the inverse proportionality between wavelength and frequency.

Fields of a Sinusoidal Wave

Figure 32.13 shows a linearly polarized sinusoidal electromagnetic wave traveling in the plus x direction. The electric and magnetic fields oscillate in phase: E vector is maximum where B vector is maximum and E vector is zero where B vector is zero. Where E vector is in the plus y direction, B vector is in the plus z direction; where E vector is in the minus y direction, B vector is in the minus z direction. At all points the vector product E vector cross B vector is in the direction in which the wave is propagating (the plus x direction). We mentioned this in Section 32.2 in the list of characteristics of electromagnetic waves.

Figure 32.13 summary: The figure is a representation of electric and magnetic fields in a linearly polarized sinusoidal plane electromagnetic wave. The figure depicts the electric and magnetic field components of an electromagnetic wave as they vary along the x-axis at a specific time. The electric and magnetic fields oscillate perpendicularly to each other and to the direction of wave propagation. The direction of the wave is the same as the direction of the cross product of the electric and magnetic fields.

Caution In a plane wave, {E} and {B} are everywhere Figure 32.13 shows {E} and {B} at points on the x-axis only. But, in fact, in a sinusoidal plane wave there are electric and magnetic fields at all points in space. Imagine a plane perpendicular to the x-axis (that is, parallel to the y-plane) at a particular point and time; the fields have the same values at all points in that plane. The values are different on different planes.

Definition

Transverse wave: A wave in which the disturbance is perpendicular to the direction of propagation.

We can describe electromagnetic waves by means of wave functions, just as we did in Section 15.3 for waves on a string. One form of the wave function for a transverse wave traveling in the +x-direction along a stretched string is equation (15.7):

Math summary: This expression calculates the transverse displacement of a wave. It takes the cosine of the wave number times position minus the angular frequency times time, and then scales the result by the amplitude.

where y of x and t is the transverse displacement from equilibrium at time t of a point with coordinate x on the string. Here A is the maximum displacement, or amplitude, of the wave; omega is its angular frequency, equal to 2 times pi times the frequency f; and k equals 2 times pi divided by lambda is the wave number, where lambda is the wavelength.

Let E sub y of x, t and B sub z of x, t represent the instantaneous values of the y component of vector E and the z component of vector B, respectively, in figure 32.13, and let E sub max and B sub max represent the maximum values, or amplitudes, of these fields. The wave functions for the wave are then

Math summary: These equations describe the electric and magnetic field components of an electromagnetic wave as a function of position and time. Each component is modeled as a cosine function with a maximum amplitude, wave number, and angular frequency.

We can also write the wave functions in vector form:

Sinusoidal electromagnetic plane wave, propagating in +x-direction:

Math summary: This expression describes the electric and magnetic fields of a wave. It shows how the electric and magnetic field magnitudes vary sinusoidally with position and time.

Caution Two meanings of the symbol k. Note the two different k's in Equations thirty-two point seventeen: the unit vector k hat in the z direction and the wave number k. Don't get these confused!

The sine curves in figure 32.13 represent the fields as functions of x at time t equals 0—that is, E vector of x, t equals 0 and B vector of x, t equals 0. As the wave travels to the right with speed c, Equations 32.16 and 32.17 show that at any point the oscillations of E vector and B vector are in phase. From Equation 32.4 the amplitudes must be related by

Math summary: This expression likely represents the start of a matrix or array definition. It indicates the beginning structure for organizing data in rows and columns, but is incomplete.

These amplitude and phase relationships are also required for E(x, t) and B(x, t) to satisfy Eqs. (32.12) and (32.14), which came from Faraday's law and Ampere's law, respectively. Can you verify this statement? (See Problem 32.30.)

Figure 32.14 shows the E and B fields of a wave traveling in the negative x-direction. At points where the vector E is in the positive y-direction, the vector B is in the negative z-direction; where the vector E is in the negative y-direction, the vector B is in the positive z-direction. As with waves traveling in the plus x-direction, at any point the oscillations of the vector E and vector B fields of this wave are in phase, and the vector product vector E times vector B points in the propagation direction. The wave functions for this wave are

Math summary: This expression defines the electric and magnetic fields of an electromagnetic wave. It describes how the electric and magnetic field vectors vary with position and time as a cosine function, indicating a wave propagating in the negative x direction.

Figure 32.14 summary: The figure is an electromagnetic wave diagram. It depicts the electric and magnetic field components of an electromagnetic wave propagating along the x-axis. The electric field oscillates along the y-axis, and the magnetic field oscillates along the z-axis. The wave travels in the negative x-direction, and this direction corresponds to the cross product of the electric and magnetic fields. The electric and magnetic fields are perpendicular to each other and to the direction of propagation, illustrating the transverse nature of electromagnetic waves.

(sinusoidal electromagnetic plane wave, propagating in -x-direction)

The sinusoidal waves shown in both Figs. 32.13 and 32.14 are linearly polarized in the y-direction; the {E} field is always parallel to the y-axis. Example 32.1 concerns a wave that is linearly polarized in the z-direction. Figure 32.14 Representation of one wavelength of a linearly polarized sinusoidal plane electromagnetic wave traveling in the negative x-direction at t = 0. The fields are shown only for points along the x-axis. (Compare with figure 32.13.)

Problem-Solving Strategy 32.1 Electromagnetic Waves

Identify the relevant concepts: Many of the same ideas that apply to mechanical waves apply to electromagnetic waves. One difference is that electromagnetic waves are described by two quantities (in this case, electric field {E} and magnetic field {B} ), rather than by a single quantity, such as the displacement of a string.

Set Up the problem using the following steps:

1. Draw a diagram showing the direction of wave propagation and the directions of vector E and vector B.

2. Identify the target variables.

Execute the solution as follows:

1. Review the treatment of sinusoidal mechanical waves in Chapters 15 and 16, and particularly the four problem-solving strategies suggested there. 2. Keep in mind the basic relationships for periodic waves: v equals lambda times f and omega equals v times k. For electromagnetic waves in vacuum, v equals c. Distinguish between ordinary frequency f, usually expressed in hertz, and angular frequency omega equals 2 times pi times f, expressed in radians per second. Remember that the wave number is k equals 2 times pi divided by lambda.

3. Concentrate on basic relationships, such as those between {E} and {B} (magnitude, direction, and relative phase), how the wave speed is determined, and the transverse nature of the waves.

Evaluate your answer: Check that your result is reasonable. For electromagnetic waves in vacuum, the magnitude of the magnetic field in teslas is much smaller (by a factor of 3.00 times 10 sup 8 ) than the magnitude of the electric field in volts per meter. If your answer suggests otherwise, you probably made an error in using the relationship E = cB . (We'll see later in this section that this relationship is different for electromagnetic waves in a material medium.)

Example 32.1 Electric and Magnetic Fields of a Laser Beam

A carbon dioxide laser emits a sinusoidal electromagnetic wave that travels in vacuum in the negative x-direction. The wavelength is 10.6 micrometers (in the infrared; see figure 32.4) and the vector E field is parallel to the z-axis, with E sub max equals 1.5 megavolts per meter. Write vector equations for vector E and vector B as functions of time and position.

Identify and Set Up Equations (32.19) describe a wave traveling in the negative x-direction with {E} along the y-axis—that is, a wave that is linearly polarized along the y-axis. By contrast, the wave in this example is linearly polarized along the z-axis. At points where {E} is in the positive z-direction, {B} must be in the positive y-direction for the vector product {E} times {B} to be in the negative x-direction (the direction of propagation). Figure 32.15 shows a wave that satisfies these requirements.

Figure 32.15 summary: The figure is a three-dimensional representation of an electromagnetic wave. The figure illustrates the electric and magnetic field components of the wave as it propagates through space. The electric field oscillates along one axis, while the magnetic field oscillates along a perpendicular axis. The wave travels in a direction perpendicular to both the electric and magnetic fields, demonstrating that electromagnetic waves are transverse waves. The electric and magnetic field components are always perpendicular to each other and are in phase, meaning they reach their maximum and minimum values at the same time and location.

With Variation Problems

Execute A possible pair of wave functions that describe the wave shown in figure 32.15 are

Math summary: This expression describes the electric and magnetic fields of an electromagnetic wave. It shows that both fields oscillate sinusoidally along the x-axis and in time, with the electric field oriented along the z-axis and the magnetic field along the y-axis.

Figure 32.15 Our sketch for this problem.

The plus sign in the arguments of the cosine functions indicates that the wave is propagating in the negative x-direction, as it should. Faraday's law requires that E sub max = cB sub max , so

Math summary: This expression calculates the maximum magnetic field strength by dividing the maximum electric field strength by the speed of light. The maximum magnetic field strength is found to be five times ten to the negative three Tesla.

(Recall that 1 V = 1 Wb/s and 1 Wb/m² = 1 T.)

We have lambda equals 10.6 times 10 to the power of negative 6 meters, so the wave number and angular frequency are

Math summary: This calculates the wave number by dividing two pi radians by the wavelength in meters, resulting in the wave number in radians per meter. Then, it calculates the angular frequency by multiplying the speed of light by the wave number, resulting in the angular frequency in radians per second.

Substituting these values into the above wave functions, we get

Math summary: These expressions describe the electric and magnetic field components of an electromagnetic wave. Each field is calculated as the cosine of a phase that depends on position and time, scaled by a constant amplitude and oriented along a specific axis.

Evaluate As we expect, the magnitude B sub max in teslas is much smaller than the magnitude E sub max in volts per meter. To check the directions of vector E and vector B, note that vector E times vector B is in the direction of k hat times j hat equals negative i hat. This is as it should be for a wave that propagates in the negative x-direction.

Our expressions for E vector of x, t and B vector of x, t are not the only possible solutions. We could always add a phase angle phi to the arguments of the cosine function, so that k times x plus omega times t would become k times x plus omega times t plus phi. To determine the value of phi we would need to know E vector and B vector either as functions of x at a given time t or as functions of t at a given coordinate x. However, the statement of the problem doesn't include this information.

Keyconcept In a sinusoidal electromagnetic wave in vacuum, the electric field vector E and magnetic field vector B oscillate in phase with each other, are always perpendicular to each other, and are both perpendicular to the direction of propagation. The magnetic-field amplitude B sub max equals the electric-field amplitude E sub max divided by the speed of light in vacuum.

Electromagnetic Waves in Matter

So far, our discussion of electromagnetic waves has been restricted to waves in vacuum. But electromagnetic waves can also travel in matter; think of light traveling through air, water, or glass. In this subsection we extend our analysis to electromagnetic waves in non-conducting materials—that is, dielectrics.

Definition

Relative permeability: The ratio of the permeability of a specific medium to the permeability of free space.

In a dielectric the wave speed is not the same as in vacuum, and we denote it by v instead of c. Faraday's law is unaltered, but in equation (32.4), derived from Faraday's law, the speed c is replaced by v. In Ampere's law the displacement current is given not by epsilon sub 0 times d, capital phi sub E, divided by dt, where capital phi sub E is the flux of vector E through a surface, but by epsilon times d, capital phi sub E, divided by dt equals K times epsilon sub 0 times d, capital phi sub E, divided by dt, where K is the dielectric constant and epsilon is the permittivity of the dielectric. (We introduced these quantities in Section 24.4.) Also, the constant mu sub 0 in Ampere's law must be replaced by mu equals K sub m times mu sub 0, where K sub m is the relative permeability of the dielectric and mu is its permeability (see Section 28.8). Hence Eqs. (32.4) and (32.8) are replaced by

Math summary: The first equation calculates the electric field as the product of the wave speed and the magnetic field. The second equation calculates the magnetic field as the product of the permittivity, permeability, wave speed, and electric field.

Following the same procedure as for waves in vacuum, we find that

Math summary: This expression calculates the speed of electromagnetic waves in a dielectric material. It determines the wave speed based on the permittivity and permeability of the material, relating it to the speed of light in a vacuum and the dielectric and magnetic constants.

Definition

Index of refraction: The ratio of the speed of light in a vacuum to the speed of light in a medium.

For most dielectrics the relative permeability K sub m is nearly equal to unity (except for insulating ferromagnetic materials). When K sub m is approximately 1, v equals c divided by square root of K. Because K is always greater than unity, the speed v of electromagnetic waves in a nonmagnetic dielectric is always less than the speed c in vacuum by a factor of 1 divided by square root of K (Fig. 32.16). The ratio of the speed c in vacuum to the speed v in a material is known in optics as the index of refraction n of the material. When K sub m is approximately 1

Math summary: This expression calculates the index of refraction. It divides the speed of light in a vacuum by the speed of light in a material, which equals the square root of the product of the dielectric constant and the relative permeability, and is approximately equal to the square root of the dielectric constant.

Figure 32.16 summary: The figure is a photograph. The photograph depicts two people swimming underwater. The speed of visible light is slower in water than in a vacuum.

Usually, we can't use the values of K in Table 24.1 in this equation because those values are measured in constant electric fields. When the fields oscillate rapidly, there is usually not time for the reorientation of electric dipoles that occurs with steady fields. Values of K with rapidly varying fields are usually much smaller than the values in the table. For example, K for water is 80.4 for steady fields but only about 1.8 in the frequency range of visible light. Thus the dielectric “constant” K is actually a function of frequency (the dielectric function).

With Variation Problems

Example 32.2 Electromagnetic Waves in Different Materials

(a) Visiting a jewelry store one evening, you hold a diamond up to the light of a sodium-vapor street lamp. The heated sodium vapor emits yellow light with a frequency of 5.09 times 10 sup 14 Hz. Find the wavelength in vacuum and the wave speed and wavelength in diamond, for which K = 5.84 and K m = 1.00 at this frequency. (b) A 90.0 megahertz radio wave (in the F.M radio band) passes from vacuum into an insulating ferrite (a ferromagnetic material used in computer cables to suppress radio interference). Find the wavelength in vacuum and the wave speed and wavelength in the ferrite, for which K = 10.0 and K m = 1000 at this frequency.

Identify and Set Up In each case we find the wavelength in vacuum by using c equals lambda times f. To use the corresponding equation 5 equals lambda times f to find the wavelength in a material medium, we find the speed v of electromagnetic waves in the medium from Equation 32.21, which relates v to the values of dielectric constant K and relative permeability K sub m for the medium.

Execute (a) The wavelength in vacuum of the sodium light is

Math summary: This expression calculates the wavelength of light in a vacuum. It divides the speed of light by the frequency to find the wavelength, resulting in 589 nanometers.

The wave speed and wavelength in diamond are

Math summary: First, the wave speed in diamond is calculated by dividing the speed of light by the square root of the product of two constants. Then, the wavelength in diamond is determined by dividing the calculated wave speed by the frequency.

(b) Following the same steps as in part (a), we find

Math summary: First, the wavelength in a vacuum is calculated by dividing the speed of light by the frequency. Then, the velocity in ferrite is computed by dividing the speed of light by the square root of the product of two constants, and finally, the wavelength in ferrite is found by dividing the ferrite velocity by the frequency.

Evaluate The speed of light in transparent materials is typically between 0.2c and c; our result in part (a) shows that v sub diamond equals 0.414 times c. As our results in part (b) show, the speed of electromagnetic waves in dense materials like ferrite (for which v sub ferrite equals 0.010 times c) can be far slower than in vacuum.

Keyconcept The speed of light in a transparent medium is slower than in vacuum. The greater the dielectric constant K of the medium and the greater the relative permeability K sub m of the medium, the slower the speed. For waves of a given frequency, the slower the wave speed, the shorter the wavelength.

Test Your Understanding of Section 32.3 The first of Eqs. (32.17) gives the electric field for a plane wave as measured at points along the x-axis. For this plane wave, how does the electric field at points off the x-axis differ from the expression in Eqs. (32.17)? (1) The amplitude is different; (2) the phase is different; (3) both the amplitude and phase are different; (4) none of these.

Answer

on the coordinates y and z.

(4) In an ideal electromagnetic plane wave, at any instant the fields are the same anywhere in a plane perpendicular to the direction of propagation. The plane wave described by Eqs. (32.17) is propagating in the x-direction, so the fields depend on the coordinate x and time t but do not depend

32.4 Energy and Momentum in Electromagnetic Waves

Electromagnetic waves carry energy; the energy in sunlight is a familiar example. Microwave ovens, radio transmitters, and lasers for eye surgery all make use of this wave energy. To understand how to utilize this energy, it's helpful to derive detailed relationships for the energy in an electromagnetic wave.

We begin with the expressions derived in Sections 24.3 and 30.3 for the energy densities in electric and magnetic fields; we suggest that you review those derivations now. Equations (24.11) and (30.10) show that in a region of empty space where {E} and {B} fields are present, the total energy density u is

Math summary: This expression calculates the total energy density. It sums one half times the permittivity of free space times the square of the electric field, with one half times the permeability of free space times the square of the magnetic field.

For electromagnetic waves in vacuum, the magnitudes E and B are related by

Math summary: This equation calculates the magnitude of the magnetic field. It divides the electric field by the speed of light, which is equivalent to multiplying the electric field by the square root of the product of the permittivity and permeability of free space.

Combining Eqs. (32.23) and (32.24), we can also express the energy density u in a simple electromagnetic wave in vacuum as

Math summary: This expression calculates the energy density of an electromagnetic wave. It sums one half times the permittivity of free space times the square of the electric field with one half times the inverse of the permeability of free space times the square of the product of the square root of the product of the permittivity and permeability of free space and the electric field, resulting in the permittivity of free space times the square of the electric field.

This shows that in vacuum, the energy density associated with the {E} field in our simple wave is equal to the energy density of the {B} field. In general, the electric-field magnitude E is a function of position and time, as for the sinusoidal wave described by Eqs. (32.16); thus the energy density u of an electromagnetic wave, given by equation (32.25), also depends in general on position and time.

Electromagnetic Energy Flow and the Poynting Vector

Electromagnetic waves such as those we have described are traveling waves that transport energy from one region to another. We can describe this energy transfer in terms of energy transferred per unit time per unit cross-sectional area, or power per unit area, for an area perpendicular to the direction of wave travel.

To see how the energy flow is related to the fields, consider a stationary plane, perpendicular to the x-axis, that coincides with the wave front at a certain time. In a time dt after this, the wave front moves a distance dx = c dt to the right of the plane. Consider an area A on this stationary plane (Fig. 32.17). The energy in the space to the right of this area had to pass through the area to reach the new location. The volume dV of the relevant region is the base area A times the length c dt, and the energy dU in this region is the energy density u times this volume:

Figure 32.17 summary: The figure is a diagram representing a wave front passing through a stationary plane. The diagram illustrates the relationship between the electric field, magnetic field, and the Poynting vector. The Poynting vector indicates the energy flux of the electromagnetic wave. The wave front moves a certain distance after a time interval. The direction of the Poynting vector is related to the directions of the electric and magnetic fields.

Math summary: This expression calculates the change in energy as the product of energy density and a change in volume. The change in energy is computed by multiplying a constant, the square of the electric field, the area, and the change in time.

This energy passes through the area A in time dt. The energy flow per unit time per unit area, which we'll call S, is

Math summary: This equation calculates energy flow per unit time per unit area in a vacuum. It states that this energy flow, S, is equal to a constant times the square of the electric field, E.

Using Eqs. (32.4) and (32.9), you can derive the alternative forms

Math summary: This expression calculates the energy per unit time per unit area in a vacuum. It takes the electric field strength and magnetic field strength as inputs, and scales their product by the vacuum permeability to produce the energy flux.

The units of S are energy per unit time per unit area, or power per unit area. The S.I unit of S is 1 J/s·m² or 1 W/m².

Definition

Poynting vector: A vector representing the directional energy flux (the energy transfer per unit area per unit time) of an electromagnetic field.

We can define a vector quantity that describes both the magnitude and direction of the energy flow rate. Introduced by the British physicist, this quantity is called the Poynting vector:

Math summary: This expression calculates the Poynting vector in a vacuum. It computes the vector by taking the cross product of the electric field and the magnetic field, and then scaling the result by the reciprocal of the magnetic constant.

The vector S vector points in the direction of propagation of the wave (Fig. 32.18). Since E vector and B vector are perpendicular, the magnitude of S vector is S equals E times B divided by mu sub 0; from Eqs. (32.26) and (32.27) this is the energy flow per unit area and per unit time through a cross-sectional area perpendicular to the propagation direction. The total energy flow per unit time (power, P) out of any closed surface is the integral of S vector over the surface:

Figure 32.18 summary: This is a photograph of solar panels installed on a rooftop. The image shows an array of solar panels tilted at an angle. The solar panels are positioned in such a way that they directly face the sun. This orientation allows the panels to maximize the absorption of electromagnetic wave energy from the sun, leading to a more efficient energy conversion.

Math summary: This expression calculates the total power flowing out of a closed surface. It integrates the energy flow vector over the area of the surface to determine the total energy flow per unit time.

Definition

Intensity: The time-averaged power per unit area carried by an electromagnetic wave.

For the sinusoidal waves studied in Section 32.3, as well as for other more complex waves, the electric and magnetic fields at any point vary with time, so the Poynting vector at any point is also a function of time. Because the frequencies of typical electromagnetic waves are very high, the time variation of the Poynting vector is so rapid that it's most appropriate to look at its average value. The magnitude of the average value of {S} at a point is called the intensity of the radiation at that point. The S.I unit of intensity is the same as for S, 1 W/m ^2 .

Let's work out the intensity of the sinusoidal wave described by Equations (32.17). We first substitute the vector E and the vector B into Equation (32.28):

Math summary: This calculates the Poynting vector, which represents the energy flux of an electromagnetic wave. It computes the cross product of the electric field vector and the magnetic field vector, then scales the result by the reciprocal of the permeability of free space.

The vector product of the unit vectors is j hat times k hat equals t hat and cosine squared of (k times x minus omega times t) is never negative, so S vector of (x, t) always points in the positive x-direction (the direction of wave propagation). The x-component of the Poyning vector is

Math summary: This expression calculates the x component of the poynting vector. It takes the maximum electric field and maximum magnetic field, divides by the permeability of free space, and multiplies by the square of cosine of the wave number times position minus angular frequency times time, which simplifies to one half of the maximum electric field times the maximum magnetic field divided by the permeability of free space times one plus the cosine of two times the wave number times position minus the angular frequency times time.

The time average value of cosine of 2 times (k times x minus omega times t) is zero because at any point, it is positive during one half-cycle and negative during the other half. So the average value of the Poynting vector over a full cycle is S vector sub av equals t hat S sub av, where

Math summary: This expression calculates the average magnitude of the Poynting vector. It multiplies the maximum electric field by the maximum magnetic field, then divides by twice the permeability of free space.

That is, the magnitude of the average value of S bar for a sinusoidal wave (the intensity I of the wave) is one half times the maximum value. You can verify that by using the relationships E sub max equals B sub max times c and epsilon sub 0 times mu sub 0 equals 1 over c squared, we can express the intensity in several equivalent forms:

Math summary: This calculates the intensity of a sinusoidal wave. It expresses intensity using the maximum electric field, maximum magnetic field, and other constants, providing several equivalent formulas for the intensity.

For a wave traveling in the negative x direction, represented by Equations 32.19, the Poynting vector is in the negative x direction at every point, but its magnitude is the same as for a wave traveling in the positive x direction. Verifying these statements is left to you.

Throughout this discussion we have considered only electromagnetic waves propagating in vacuum. If the waves are traveling in a dielectric medium, however, the expressions for energy density [Eq. (32.23)], the Poynting vector [Eq. (32.28)], and the intensity of a sinusoidal wave [Eq. (32.29)] must be modified. It turns out that the required modifications are quite simple: Just replace epsilon sub 0 with the permittivity epsilon of the dielectric, replace mu sub 0 with the permeability mu of the dielectric, and replace c with the speed v of electromagnetic waves in the dielectric. Remarkably, the energy densities in the E vector and B vector fields are equal even in a dielectric.

Caution Poynting vector versus intensity At any point 10, the magnitude of the Poynting vector varies with time. Hence, the instantaneous rate at which electromagnetic energy in a sinusoidal plane wave arrives at a surface is not constant. This may seem to contradict everyday experience; the light from the sun, a light bulb, or the laser in a grocery-store scanner appears steady and unvarying in strength. In fact the Poynting vector from these sources does vary in time, but the variation isn't noticeable because the oscillation frequency is so high (around 5 times 10 sup 14 Hz for visible light).

All that you sense is the average rate at which energy reaches your eye, which is why we commonly use intensity (the average value of S) to describe the strength of electromagnetic radiation. B.I.O Application Laser Surgery Lasers are used widely in medicine as ultra-precise, bloodless "scalpels." They can reach and remove tumors with minimal damage to neighboring healthy tissues, as in the brain surgery shown here.

The power output of the laser is typically below 40 W, less than that of a typical light bulb. However, this power is concentrated into a spot from 0.1 to 2.0 millimeters in diameter, so the intensity of the light (equal to the average value of the Poynting vector) can be as high as 5 times 10 sup 9 W/m ^{2} .

Image summary: This is a photograph. The image shows a surgeon performing a procedure on a brain. The surgeon is using a laser to target a specific area of the brain. The procedure appears to be highly precise, suggesting a delicate and complex operation.

Example 32.3 Energy in a Nonsinusoidal Wave

For the nonsinusoidal wave described in Section 32.2, suppose that E equals 100 Volts per meter equals 100 Newtons per Coulomb. Find the value of B, the energy density u, and the rate of energy flow per unit area S.

Identify and Set Up In this wave the vector E and the vector B are uniform behind the wave front, and zero ahead of it. Hence the target variables B, u, and S must also be uniform behind the wave front. Given the magnitude E, we use Equation 32.4 to find B, Equation 32.25 to find u, and Equation 32.27 to find S. We cannot use Equation 32.29, which applies to sinusoidal waves only.

Execute From equation (32.4),

Math summary: This expression calculates the magnetic field by dividing the electric field by the speed of light. The electric field of 100 volts per meter is divided by the speed of light of 3.00 times 10 to the eighth meters per second, resulting in a magnetic field of 3.33 times 10 to the negative seventh Tesla.

From equation (32.25),

Math summary: This expression calculates the energy density. It squares the electric field, multiplies by the permittivity of free space, and outputs the energy density as 8.85 times 10 to the power of negative 8 Joules per cubic meter.

The magnitude of the Poynting vector is

Math summary: This expression calculates the magnitude of the poynting vector. It divides the product of the electric field and magnetic field by the permeability of free space, resulting in a value of 26.5 watts per square meter.

Evaluate We can check our result for S by using equation (32.26):

Math summary: This expression calculates the magnitude of the Poynting vector. It multiplies the permittivity of free space by the speed of light and the square of the electric field strength to yield the power per unit area.

Since the vector E and the vector B have the same values at all points behind the wave front, u and S likewise have the same value everywhere behind the wave front. In front of the wave front, the vector E equals bold 0 and the vector B equals bold 0, and so u equals 0 and S equals 0; where there are no fields, there is no field energy.

Keyconcept The Poynting vector S vector equals, open parenthesis, 1 divided by mu sub 0, close parenthesis, times E vector cross B vector for an electromagnetic wave points in the direction of wave propagation. The average magnitude of S vector equals the intensity (average power per unit area) of the wave.

Example 32.4 Energy in a Sinusoidal Wave

With Variation Problems

A radio station on the earth's surface emits a sinusoidal wave with average total power 50 kilowatt (Fig. 32.19). Assuming that the transmitter radiates equally in all directions above the ground (which is unlikely in real situations), find the electric-field and magnetic-field amplitudes E sub max and B sub max detected by a satellite 100 kilometers from the antenna.

Figure 32.19 summary: The figure is an illustration. It depicts a satellite positioned above a transmitter on the ground, with a curved area between them. The illustration suggests a scenario involving communication or transmission between a ground-based transmitter and a satellite. The satellite is at a certain distance from the transmitter.

Identify and Set Up We are given the transmitter's average total power P. The intensity I is the average power per unit area; to find I at 100 kilometers from the transmitter we divide P by the surface area of the hemisphere in figure 32.19. For a sinusoidal wave, I is also equal to the magnitude of the average value S sub av of the Poynting vector, so we can use equation (32.29) to find E sub max ; equation (32.4) yields B sub max .

Execute The surface area of a hemisphere of radius r equals 100 kilometers equals 1.00 times 10 to the power of 5 meters is

Math summary: This calculates the surface area of a hemisphere. It squares the radius, multiplies it by two pi, and the result is the surface area.

Figure 32.19 A radio station radiates waves into the hemisphere shown. All the radiated power passes through this surface, so the average power per unit area (that is, the intensity) is

Math summary: This expression calculates intensity by dividing power by area. Specifically, it divides the given power by two times pi times the radius squared, resulting in the intensity value.

From equation (32.29), I equals S sub av equals E sub max squared divided by 2 times mu sub 0 times c, so

Math summary: This expression calculates the maximum electric field strength. It takes constants and the average power per unit area as inputs, performs a series of multiplications and a square root operation, and outputs the maximum electric field strength.

Then from equation (32.4),

Math summary: This expression calculates the maximum magnetic field strength. It divides the maximum electric field strength by the speed of light, resulting in a maximum magnetic field strength of 8.17 times 10 to the power of negative 11 Tesla.

Evaluate Note that E sub max is comparable to fields commonly seen in the laboratory, but B sub max is extremely small in comparison to {B} fields we saw in previous chapters. For this reason, most detectors of electromagnetic radiation respond to the effect of the electric field, not the magnetic field. Loop radio antennas are an exception (see the Bridging Problem at the end of this chapter).

Keyconcept The magnitude S sub av of the average Poynting vector of a sinusoidal electromagnetic wave depends on the field amplitudes E sub max and B sub max. The value of S sub av is proportional to the product of E sub max and B sub max.

Electromagnetic Momentum Flow and Radiation Pressure

We've shown that electromagnetic waves transport energy. It can also be shown that electromagnetic waves carry momentum p, with a corresponding momentum density (momentum dp per volume dV) of magnitude

Math summary: This equation calculates momentum density of an electromagnetic wave. It states that momentum density, which is the change in momentum with respect to change in volume, equals the energy density divided by the speed of light squared.

This momentum is a property of the field; it is not associated with the mass of a moving particle in the usual sense.

There is also a corresponding momentum flow rate. The volume dV occupied by an electromagnetic wave (speed c) that passes through an area A in time dt is dV = Ac dt. When we substitute this into equation (32.30) and rearrange, we find that the momentum flow rate per unit area is We obtain the average rate of momentum transfer per unit area by replacing S in equation (32.31) by S sub av = I .

Image summary: This image is a diagram that shows the flow rate of electromagnetic momentum. The diagram presents an equation that relates the flow rate of electromagnetic momentum to the Poynting vector magnitude. The flow rate of electromagnetic momentum is equivalent to the momentum transferred per unit surface area per unit time. The Poynting vector magnitude is equal to the electric-field magnitude times the magnetic-field magnitude divided by the magnetic constant times the speed of light in a vacuum. The Poynting vector magnitude is directly proportional to the flow rate of electromagnetic momentum.

Definition

Radiation pressure: The pressure exerted on a surface due to the transfer of momentum from an electromagnetic wave.

This momentum is responsible for radiation pressure. When an electromagnetic wave is completely absorbed by a surface, the wave's momentum is also transferred to the surface. For simplicity we'll consider a surface perpendicular to the propagation direction.

Using the ideas developed in Section 8.1, we see that the rate dp/dt at which momentum is transferred to the absorbing surface equals the force on the surface. The average force per unit area due to the wave, or radiation pressure p sub rad , is the average value of dp/dt divided by the absorbing area A. (We use the subscript “rad” to distinguish pressure from momentum, for which the symbol p is also used.) From equation (32.31) the radiation pressure is

Math summary: This equation calculates radiation pressure when a wave is totally absorbed. It divides the average value of the pointing vector, also known as intensity, by the speed of light.

If the wave is totally reflected, the momentum change is twice as great, and

Math summary: This equation calculates radiation pressure when a wave is totally reflected. It divides twice the average pointing vector or intensity by the speed of light to determine the radiation pressure.

For example, the value of I, or S sub av, for direct sunlight, before it passes through the earth's atmosphere, is approximately 1.4 kilowatt per meter squared. From Equation 32.32 the corresponding average pressure on a completely absorbing surface is

Math summary: This calculates radiation pressure by dividing light intensity by the speed of light. Specifically, an intensity of 1.4 times 10 to the power of 3 watts per square meter is divided by a speed of 3.0 times 10 to the power of 8 meters per second, resulting in a radiation pressure of 4.7 times 10 to the power of negative 6 Pascals.

From equation (32.33) the average pressure on a totally reflecting surface is twice this: 2 times I divided by c or 9.4 times 10 to the power of negative 6 Pascals. These are very small pressures, of the order of 10 to the power of negative 10 atm, but they can be measured with sensitive instruments.

The radiation pressure of sunlight is much greater inside the sun than at the earth (see Problem 32.35). Inside stars that are much more massive and luminous than the sun, radiation pressure is so great that it substantially augments the gas pressure within the star and so helps to prevent the star from collapsing under its own gravity. In some cases the radiation pressure of stars can have dramatic effects on the material surrounding them (Fig. 32.20). Figure 32.20 At the center of this inter-stellar gas cloud is a group of intensely luminous stars that exert tremendous radiation pressure on their surroundings. Aided by a “wind” of particles emanating from the stars, over the past million years the radiation pressure has carved out a bubble within the cloud 70 light-years across.

Figure 32.20 summary: This is an image of an astrophysical object. The image depicts a nebula or interstellar cloud, characterized by a ring-like structure. The ring appears to be composed of glowing gases and dust. The distribution of the gas and dust is uneven, with some regions appearing brighter and denser than others. The image suggests the complex and dynamic nature of star formation and the interstellar medium.

Example 32.5 Power and Pressure from Sunlight

An earth-orbiting satellite has solar energy–collecting panels with a total area of 4.0 m² (Fig. 32.21). If the sun's radiation is perpendicular to the panels and is completely absorbed, find the average solar power absorbed and the average radiation-pressure force.

Figure 32.21 summary: The figure is an illustration of a satellite. The satellite has two solar panels extended on either side. The satellite also has a sun sensor. The solar panels are positioned to face the sun, as indicated by the arrows, to maximize energy absorption.

Identify and Set Up This problem uses the relationships among intensity, power, radiation pressure, and force. In the previous discussion, we used the intensity I (average power per unit area) of sunlight to find the radiation pressure p sub rad (force per unit area) of sunlight on a completely absorbing surface. (These values are for points above the atmosphere, which is where the satellite orbits.) Multiplying each value by the area of the solar panels gives the average power absorbed and the net radiation force on the panels.

Execute The intensity I (power per unit area) is 1.4 times 10 to the power of 3 W/m. Although the light from the sun is not a simple sinusoidal wave, we can still use the relationship that the average power P is the intensity I times the area A:

Math summary: This expression calculates the average power by multiplying the intensity, which is power per unit area, by the area. The result is the average power in watts, which is then converted to kilowatts.

The radiation pressure of sunlight on an absorbing surface is p sub rad equals 4.7 times 10 to the power of negative 6 Pascals equals 4.7 times 10 to the power of negative 6 Newtons per meter squared. The total force F is the pressure p sub rad times the area A:

Math summary: This expression calculates the total force exerted by radiation pressure on a surface. It multiplies the radiation pressure, which is four point seven times ten to the negative six Newtons per square meter, by the area of the surface, which is four point zero square meters, to find the total force of one point nine times ten to the negative five Newtons.

Evaluate The absorbed power is quite substantial. Part of it can be used to power the equipment aboard the satellite; the rest goes into heating the panels, either directly or due to inefficiencies in the photocells contained in the panels.

The total radiation force is comparable to the weight (on the earth) of a single grain of salt. Over time, however, this small force can noticeably affect the orbit of a satellite like that in figure 32.21, and so radiation pressure must be taken into account.

Keyconcept An electromagnetic wave carries both energy and momentum. As a result an electromagnetic wave exerts pressure on any surface that either absorbs or reflects the wave.

Test Your Understanding of Section 32.4 Figure 32.13 shows one wavelength of a sinusoidal electromagnetic wave at time t = 0. For which of the following four values of x is (a) the energy density a maximum; (b) the energy density a minimum; (c) the magnitude of the instantaneous (not average) Poynting vector a maximum; (d) the magnitude of the instantaneous (not average) Poynting vector a minimum? (1) x = 0; (2) x = λ/4; (3) x = λ/2; (4) x = 3λ/4.

Answer

(a) (1) and (3), (b) (2) and (4), (c) (1) and (3), (d) (2) and (4) Both the energy density u and the Poyning vector magnitude S are maximum where the vector E and vector B fields have their maximum magnitudes. (The directions of the fields don't matter.) From figure 32.13, this occurs at x equals 0 and x equals lambda divided by 2. Both u and S have a minimum value of zero; that occurs where vector E and vector B are both zero. From figure 32.13, this occurs at x equals lambda divided by 4 and x equals 3 times lambda divided by 4.

32.5 Standing Electromagnetic Waves

Electromagnetic waves can be reflected by the surface of a conductor (like a polished sheet of metal) or of a dielectric (such as a sheet of glass). The superposition of an incident wave and a reflected wave forms a standing wave. The situation is analogous to standing waves on a stretched string, discussed in Section 15.7.

Suppose a sheet of a perfect conductor (zero resistivity) is placed in the yz-plane of figure 32.22 and a linearly polarized electromagnetic wave, traveling in the negative x-direction, strikes it. As we discussed in Section 23.4, {E} cannot have a component parallel to the surface of a perfect conductor. Therefore in the present situation, {E} must be zero everywhere in the yz-plane. The electric field of the incident electromagnetic wave is not zero at all times in the yz-plane.

Figure 32.22 summary: The figure is an illustration of the electric and magnetic fields of a linearly polarized electromagnetic standing wave. The figure depicts the spatial relationship between the electric and magnetic field vectors at a particular moment in time. The electric field is at its maximum where the magnetic field is zero, and vice versa. The pattern of the standing wave remains stationary along the x-axis, with the electric and magnetic field vectors oscillating at each point.

But this incident wave induces oscillating currents on the surface of the conductor, and these currents give rise to an additional electric field. The net electric field, which is the vector sum of this field and the incident {E} , is zero everywhere inside and on the surface of the conductor.

The currents induced on the surface of the conductor also produce a reflected wave that travels out from the plane in the +x-direction. Suppose the incident wave is described by the wave functions of Eqs. (32.19) (a sinusoidal wave traveling in the -x-direction) and the reflected wave by the negative of Eqs. (32.16) (a sinusoidal wave traveling in the +x-direction). We take the negative of the wave given by Eqs. (32.16) so that the incident and reflected electric fields cancel at x = 0 (the plane of the conductor, where the total electric field must be zero). The superposition principle states that the total {E} field at any point is the vector sum of the {E} fields of the incident and reflected waves, and similarly for the {B} field. Therefore the wave functions for the superposition of the two waves are

Math summary: These equations calculate the electric and magnetic field components of a wave as a function of position and time. Each equation subtracts two cosine functions with slightly different arguments to determine the field strength.

We can expand and simplify these expressions by using the identities

Math summary: This expression computes the cosine of the sum or difference of two angles. It takes the cosine of each individual angle and multiplies them, then either subtracts or adds the product of the sines of the individual angles, depending on whether the original operation was addition or subtraction.

The results are

Math summary: The equations describe the electric and magnetic field components of a standing electromagnetic wave. The electric field is calculated as a function of position and time, scaling a sine function by a maximum electric field and a factor of negative two, while the magnetic field is similarly calculated using a cosine function scaled by a maximum magnetic field and a factor of negative two.

Equation (32.34) is analogous to equation (15.28) for a stretched string. We see that at x equals 0 the electric field E sub y of x equals 0, t is always zero; this is required by the nature of the ideal conductor, which plays the same role as a fixed point at the end of a string. Furthermore, E sub y of x, t is zero at all times at points in those planes perpendicular to the x-axis for which sine of k times x equals 0 that is, k times x equals 0, pi, 2 times pi, and so on. Since k equals 2 times pi divided by lambda, the positions of these planes are

Math summary: This expression defines the positions of nodal planes for an electric field. It lists the locations where the electric field is always zero, starting at zero and increasing in increments of half the wavelength lambda.

2 Definitions

Definition 1: Nodal planes: Planes in a standing wave where the amplitude of the wave is always zero.

Definition 2: Antinodal planes: Planes in a standing wave where the amplitude of the wave is maximum.

These planes are called the nodal planes of the E field; they are the equivalent of the nodes, or nodal points, of a standing wave on a string. Midway between any two adjacent nodal planes is a plane on which sine of k times x equals plus or minus 1; on each such plane, the magnitude of E of x, t equals the maximum possible value of 2 times E sub max twice per oscillation cycle. These are the antinodal planes of the E vector, corresponding to the antinodes of waves on a string.

The total magnetic field is zero at all times at points in planes on which cos kx = 0 . These are the nodal planes of {B} , and they occur where

Math summary: This expression defines the locations of nodal planes for a magnetic field, where the total magnetic field is always zero. The locations are given as multiples of one-quarter of the wavelength, specifically one-quarter, three-quarters, five-quarters, and so on.

There is an antinodal plane of B vector midway between any two adjacent nodal planes.

Figure 32.22 shows a standing-wave pattern at one instant of time. The magnetic field is not zero at the conducting surface (x = 0) . The surface currents that must be present to make {E} exactly zero at the surface cause magnetic fields at the surface. The nodal planes of each field are separated by one half-wavelength. The nodal planes of {E} are midway between those of {B} , and vice versa; hence the nodes of {E} coincide with the antinodes of {B} , and conversely. Compare this situation to the distinction between pressure nodes and displacement nodes in Section 16.4.

The total electric field is a sine function of t, and the total magnetic field is a cosine function of t. The sinusoidal variations of the two fields are therefore 90 degrees out of phase at each point. At times when sine omega times t equals 0, the electric field is zero everywhere, and the magnetic field is maximum. When cosine omega times t equals 0, the magnetic field is zero everywhere, and the electric field is maximum. This is in contrast to a wave traveling in one direction, as described by Eqs. (32.16) or (32.19) separately, in which the sinusoidal variations of vector E and vector B at any particular point are in phase. You can show that Eqs. (32.34) and (32.35) satisfy the wave equation, equation (32.15). You can also show that they satisfy Eqs. (32.12) and (32.14), the equivalents of Faraday's and Ampere's laws.

Standing Waves in a Cavity

Let's now insert a second conducting plane, parallel to the first and a distance L from it, along the +x-axis. The cavity between the two planes is analogous to a stretched string held at the points x = 0 and x = L. Both conducting planes must be nodal planes for {E} ; a standing wave can exist only when the second plane is placed at one of the positions where E(x, t) = 0 , so L must be an integer multiple of lambda/2 . The wavelengths that satisfy this condition are

Math summary: This expression calculates the allowed wavelengths for standing waves between two conducting planes. It divides twice the distance between the planes by an integer to determine each allowed wavelength.

Figure 32.23 A typical microwave oven sets up a standing electromagnetic wave with lambda = 12.2 centimeters , a wavelength that is strongly absorbed by the water in food. Because the wave has nodes spaced lambda/2 = 6.1 centimeters apart, the food must be rotated while cooking. Otherwise, the portion that lies at a node—where the electric field amplitude is zero—will remain cold.

Figure 32.23 summary: This is an image of a microwave oven. The microwave contains two plates of vegetables, one above the other. The vegetables on the upper plate are more varied, while the lower plate contains only broccoli. This image is likely used to demonstrate the concept of standing waves in a microwave oven, where some areas experience higher intensity, leading to uneven heating of food.

Math summary: This equation calculates the frequencies at which standing waves can form. It determines these frequencies by multiplying an integer, the speed of light, and a scaling factor that depends on the length of the space.

The corresponding frequencies are Thus there is a set of normal modes, each with a characteristic frequency, wave shape, and node pattern (Fig. 32.23). By measuring the node positions, we can measure the wavelength. If the frequency is known, the wave speed can be determined. This technique was first used by Hertz in the 1880s in his pioneering investigations of electromagnetic waves.

Conducting surfaces are not the only reflectors of electromagnetic waves. Reflections also occur at an interface between two insulating materials with different dielectric or magnetic properties. The mechanical analog is a junction of two strings with equal tension but different linear mass density.

In general, a wave incident on such a boundary surface is partly transmitted into the second material and partly reflected back into the first. For example, light is transmitted through a glass window, but its surfaces also reflect light.

Calculate the intensity of the standing wave represented by Eqs. (32.34) and (32.35).

With Variation Problems

Identify and Set Up The intensity I of the wave is the time-averaged value S sub av of the magnitude of the Poyning vector S vector. To find S sub av, we first use equation (32.28) to find the instantaneous value of S vector and then average it over a whole number of cycles of the wave.

Execute Using the wave functions of Eqs. (32.34) and (32.35) in equation (32.28) for the Poyning vector {S} , we find

Math summary: This calculates the poynting vector as a function of position and time by taking the cross product of the electric and magnetic field vectors, scaling by the inverse of the permeability of free space. The result is a vector in a given direction multiplied by a scalar function that depends on both position and time.

Using the identity sine of 2 times A equals 2 times sine of A times cosine of A, we can rewrite S sub x of x, t

Math summary: This expression calculates the x component of the poynting vector. It multiplies the maximum electric field, the maximum magnetic field, the sine of two times a wave number times position, and the sine of two times angular frequency times time, and then divides by the permeability of free space.

The average value of a sine function over any whole number of cycles is zero. Thus the time average of S vector at any point is zero; I equals S sub av equals 0.

Evaluate This result is what we should expect. The standing wave is a superposition of two waves with the same frequency and amplitude, traveling in opposite directions. All the energy transferred by one wave is cancelled by an equal amount transferred in the opposite direction by the other wave. When we use electromagnetic waves to transmit power, it is important to avoid reflections that give rise to standing waves.

Keyconcept While there is energy flow in an electromagnetic standing wave, the intensity (the magnitude of the average Poynting vector) is zero at any point.

Example 32.7 Standing Waves in a Cavity

Electromagnetic standing waves are set up in a cavity with two parallel, highly conducting walls 1.50 centimeters apart. (a) Calculate the longest wavelength lambda and lowest frequency f of these standing waves. (b) For a standing wave of this wavelength, where in the cavity does E vector have maximum magnitude? Where is E vector zero? Where does B vector have maximum magnitude? Where is B vector zero?

Identify and Set Up Only certain normal modes are possible for electromagnetic waves in a cavity, just as only certain normal modes are possible for standing waves on a string. The longest possible wavelength and lowest possible frequency correspond to the n=1 mode in Eqs. (32.38) and (32.39); we use these to find lambda and f. Equations (32.36) and (32.37) then give the locations of the nodal planes of {E} and {B}. The antinodal planes of each field are midway between adjacent nodal planes.

With Variation Problems

Execute (a) From Eqs. (32.38) and (32.39), the n = 1 wavelength and frequency are

Math summary: This calculation determines the fundamental wavelength. It multiplies twice the length, which is one point five centimeters, resulting in a wavelength of three point zero zero centimeters.

(b) With n = 1 there is a single half-wavelength between the walls. The electric field has nodal planes ({E} = 0) at the walls and an anti-nodal plane (where {E} has its maximum magnitude) midway between them. The magnetic field has antinodal planes at the walls and a nodal plane midway between them.

Math summary: This expression calculates the fundamental frequency. It divides the speed of light by twice the length to determine the frequency, which is found to be ten gigahertz.

Evaluate One application of such standing waves is to produce an oscillating {E} field of definite frequency, which is used to probe the behavior of a small sample of material placed in the cavity. To subject the sample to the strongest possible field, it should be placed near the center of the cavity, at the antinode of {E} .

Keyconcept In a sinusoidal electromagnetic standing wave, the electric field vector E and magnetic field vector B are 90 degrees out of phase with each other. The antinodal planes of vector E are the nodal planes of vector B, and the antinodal planes of vector B are the nodal planes of vector E.

Test Your Understanding of Section 32.5 In the Standing Wave Described In

Example 32.7, is there any point in the cavity where the energy density is zero at all times? If so, where? If not, why not? is always nonzero.

Answer

1 no There are places where the vector E equals 0 at all times (at the walls) and the electric energy density 1 divided by 2 times epsilon sub 0 times E squared is always zero. There are also places where the vector B equals 0 at all times (on the plane midway between the walls) and the magnetic energy density B squared divided by 2 times mu sub 0 is always zero. However, there are no locations where both the vector E and the vector B are always zero. Hence the energy density at any point in the standing wave

Chapter 32 Summary

Maxwell's equations and electromagnetic waves: Maxwell's equations predict the existence of electromagnetic waves that propagate in vacuum at the speed of light, c. The electromagnetic spectrum covers frequencies from at least 1 to 10 to the power of 24 Hertz and a correspondingly broad range of wavelengths. Visible light, with wavelengths from 380 to 750 nm, is a very small part of this spectrum. In a plane wave, vector E and vector B are uniform over any plane perpendicular to the propagation direction. Faraday's law and Ampere's law give relationships between the magnitudes of vector E and vector B; requiring that both relationships are satisfied gives an expression for c in terms of epsilon sub 0 and mu sub 0. Electromagnetic waves are transverse; the vector E and vector B fields are perpendicular to the direction of propagation and to each other. The direction of propagation is the direction of vector E cross vector B.

Math summary: The first equation states that a quantity E is equal to the product of c and B. The second equation states that B is equal to the product of three terms: epsilon naught, mu naught, and c, all multiplied by E; the final equation defines c as the reciprocal of the square root of the product of epsilon naught and mu naught.

Image summary: The image is a diagram that illustrates a planar wave front, showing the electric and magnetic field vectors associated with an electromagnetic wave. The diagram depicts the electric and magnetic fields oscillating perpendicularly to each other and to the direction of wave propagation. The electric and magnetic fields are non-zero in the region of the wave front and they are zero elsewhere. The electromagnetic wave propagates in a specific direction, with the electric and magnetic fields maintaining a consistent spatial relationship.

Sinusoidal electromagnetic waves: Equations (32.17) and (32.18) describe a sinusoidal plane electromagnetic wave traveling in vacuum in the plus x direction. If the wave is propagating in the minus x direction, replace k times x minus omega times t by k times x plus omega times t. (See Example 32.1.)

Math summary: These equations describe the electric and magnetic fields of a sinusoidal electromagnetic wave, where the electric field oscillates along the j-hat direction and the magnetic field oscillates along the k-hat direction, with the maximum electric field being equal to the speed of light times the maximum magnetic field.

Image summary: The image is a three dimensional representation of an electromagnetic wave. The image depicts the electric and magnetic field components of the wave, along with the wave's direction of propagation. The electric and magnetic fields oscillate perpendicularly to each other and to the direction of wave propagation. The electric and magnetic fields are in phase, meaning they reach their maximum and minimum values at the same time and location. The electromagnetic wave propagates in a direction perpendicular to both the electric and magnetic fields.

Electromagnetic waves in matter: When an electromagnetic wave travels through a dielectric, the wave speed v is less than the speed of light in vacuum c. (See Example 32.2.)

Math summary: This expression calculates the speed of an electromagnetic wave in a medium. It divides the speed of light in a vacuum by the square root of the product of the dielectric constant and the relative permeability of the medium.

Energy and momentum in electromagnetic waves: The energy flow rate (power per unit area) in an electromagnetic wave in vacuum is given by the Poynting vector {S} . The magnitude of the time-averaged value of the Poynting vector is called the intensity I of the wave. Electromagnetic waves also carry momentum. When an electromagnetic wave strikes a surface, it exerts a radiation pressure p sub rad . If the surface is perpendicular to the wave propagation direction and is totally absorbing, p sub rad = I/c ; if the surface is a perfect reflector, p sub rad = 2.I/c . (See Examples 32.3 to 32.5.)

Math summary: First, the Poynting vector is calculated as the cross product of the electric and magnetic field vectors, scaled by a constant. Then, the intensity is computed using maximum electric and magnetic field values, also scaled by constants.

Image summary: The image is a diagram depicting a wave front moving through space. The diagram illustrates the relationship between the electric field, magnetic field, and Poynting vector. The diagram illustrates the movement of a wave front over a brief period. The Poynting vector is orthogonal to both the electric and magnetic fields. The wave front propagates in the direction of the Poynting vector.

Math summary: This equation equates the rate of change of electromagnetic momentum per unit area to the Poynting vector divided by the speed of light, and also to the product of the electric and magnetic fields divided by the vacuum permeability constant times the speed of light. It describes the flow rate of electromagnetic momentum.

(flow rate of electromagnetic momentum)

Standing electromagnetic waves: If a perfect reflecting surface is placed at x equals 0, the incident and reflected waves form a standing wave. Nodal planes for E vector occur at k times x equals 0, pi, 2 times pi, and so on, and nodal planes for B vector at k times x equals pi divided by 2, 3 times pi divided by 2, 5 times pi divided by 2, and so on. At each point, the sinusoidal variations of E vector and B vector with time are 90 degrees out of phase. (See Examples 32.6 and 32.7.)

Image summary: The image is an illustration. It depicts the behavior of electromagnetic waves near a perfect conductor. The illustration shows that the electric field is perpendicular to the perfect conductor, while the magnetic field is parallel to it. The electric and magnetic fields are orthogonal to each other and to the direction of propagation. The electromagnetic wave propagates along the x-axis. The electric field oscillates in the z-direction, and the magnetic field oscillates in the y-direction. The presence of the perfect conductor causes the electric field to be completely reflected, while the magnetic field is not.

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