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Chapter Review

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In this chapter our understanding of the connection between electricity and magnetism culminates in the treatment of electromagnetic waves. As will be mentioned below, visible light is one example of an electromagnetic wave; these waves are sometimes referred to as light even when they cannot be seen with the naked eye (electromagnetic radiation is also a commonly used term).

25-1   The Production of Electromagnetic Waves

Electromagnetic waves are generated by accelerating charges. One of the most common ways of doing this is by connecting an antenna to an AC circuit. The charges accelerating back-an-forth in the antenna produce electromagnetic waves that travel away from the antenna at the speed of light. The electromagnetic wave consists of electric and magnetic fields that are perpendicular to one another and are in phase with one another. Electromagnetic waves are transverse waves; the direction of propagation is perpendicular to both E and B. Given E and B the direction in which the wave propagates can be found from a right-hand rule:

Point the fingers of your right hand in the direction of E so that they would curl toward B. Your thumb then gives the direction of propagation.

Practice Quiz

Consider an electromagnetic wave for which the electric field points toward the top of this page, , and the magnetic field points into this page, . What is the direction of propagation of the electromagnetic wave?

Physlet Illustration: Production of an Electromagnetic Wave
In this simulation, a single, positive electric charge oscillates up and down along the z direction. The electric field lines, radiating outward from the charge, are shown. How does this produce an electromagnetic wave? How does the frequency of the oscillation affect the wave produced?
Hints: Concentrate on the region of space to the right of the oscillating charge. Watch the electric field lines, and observe the "wave" created, which moves along the positive y direction (from left to right in the region to the right of the charge). As the charge moves upward, what is the direction of the current produced? What, then, is the direction of the magnetic field to the right of the charge? (Remember the right-hand rule.) As the charge moves back downward, what is the direction of the current produced? What, then, is the direction of the magnetic field to the right of the charge? Along what direction (x, y or z), then, does the magnetic field oscillate? In what direction do you see the wave moving? What, then, is the relationship between the directions of the oscillating E and B fields and the direction of propagation of the wave? Vary the frequency of the oscillation using the slider bar. What happens to the wavelength of the outgoing wave?

25-2   The Propagation of Electromagnetic Waves

Unlike the other waves we have studied, electromagnetic waves do not require a medium, they can propagate in vacuum. This is because an electromagnetic wave is self-sustaining by electromagnetic induction. The changing magnetic field produces the changing electric field, and this changing electric field produces the changing magnetic field. The speed of an electromagnetic wave in vacuum is a fundamental constant of nature called the speed of light

c = 3.00 x 10⁸ m/s.

Using Maxwell's theory of electricity and magnetism, which predicted electromagnetic waves, we know that the speed of light in vacuum is related to the permittivity and permeability of free space,

.

Physlet Illustration: Propagation of an Electromagnetic Wave
In this simulation, an electromagnetic wave propagates along the z direction. The electric field is displayed in blue and the magnetic field is displayed in red. The lengths of the E and B vectors are arbitrary, only direction and relative phase are important. You may click-drag in the animation to change the viewing angle. How are these varying fields related to each other and to the propagation of electromagnetic energy?
Hints: In what direction (x, y, or z) does the electric field oscillate? In what direction does the magnetic field oscillate? When E is a maximum (or minimum) at some point along the z axis, what is the value of B at that same point? What, then, do we call the phase relationship between E and B? In what direction do you see energy (and momentum) moving?

The propagation of electromagnetic waves exhibits the Doppler effect similar to sound waves. In the case of sound waves the speed of the wave depends on the motion of the source; however, the speed of an electromagnetic wave is independent of the motion of the source. If the relative speed between the source and observer, u, is small compared to the speed of light, the the frequency received f' is given by

,

where f is the frequency emitted by the source. The + sign is used if the source is approaching the observer, and the - sign is used if the source is receding from the observer.

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Example 21.1   The Doppler Effect:   Some frequencies of the light from another galaxy are found to be 0.65% lower than the corresponding frequencies from stationary sources on Earth. Is this galaxy moving toward or away from us? Determine the speed at which it is moving toward or away from us.

Picture the Problem   The picture shows the galaxy moving relative to Earth, either toward or away.

Strategy   Given that the frequency is smaller, we can deduce whether or not the galaxy is moving toward or away from us.

Solution

1. Since the frequency decreases, this implies:
2. The observed frequency can be written as:	f' = f - 0.0065f = (0.9935)f
3. The expression for the Doppler effect becomes:
4. Solving for u gives:	u = (0.0065)c = (0.0065)(3.00 x 108 m/s) = 2.0 x 106 m/s

Insight   This may look like an unreasonably high speed, but many distant galaxies are moving away from us very rapidly.

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Practice Quiz

If an observer determines the frequency of the light given off by a source to be 10% higher than expected, what velocity, relative to the observer, must the source of the light have? 3.0 x 108 m/s 3.0 x 107 m/s 3.0 x 106 m/s 3.0 x 105 m/s 3.0 x 109 m/s

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