Book cover for Physics

Physics

Alan Giambattista, Betty McCarthy Richardson, Robert C. Richardson

ISBN #9780073404530

2nd Edition

2,795 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section covers the fundamental principles governing electromagnetic waves, beginning with their generation by accelerating charges and the unification of electricity and magnetism under Maxwell’s equations. It explains how antennas, both electric and magnetic dipoles, transmit and receive these waves, and delves into the broad electromagnetic spectrum from radio waves to gamma rays, emphasizing how properties like wavelength, frequency, and dispersion vary. The discussion continues with the manner in which EM waves transport energy and the critical roles of intensity and polarization. Lastly, it provides an introduction to the relativistic Doppler effect for light, distinguishing it from the classical Doppler effect seen in sound.

Learning Objectives

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Key Concepts

CONCEPT

DEFINITION

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Example Problems

Example 1

Apply the Ampère-Maxwell law to one of the circular paths in Fig. 22.3 to find the magnitude of the magnetic field at the locations specified.Find $B$ outside the wire at a distance $r \geq R$ from the central axis. [Hint: The electric field inside the wire is constant, so there is no changing electric flux.]

Example 2

Apply the Ampère-Maxwell law to one of the circular paths in Fig. 22.3 to find the magnitude of the magnetic field at the locations specified.Find $B$ outside the gap in the wire at a distance $r \geq R$ from the central axis. [Hint: What is the rate of change of electric flux through the circle in terms of the current $I ?]$

Example 3

Apply the Ampère-Maxwell law to one of the circular paths in Fig. 22.3 to find the magnitude of the magnetic field at the locations specified.Find $B$ inside the gap in the wire at a distance $r \leq R$ from the central axis. [Hint: Only the rate of change of electric flux $\Delta \Phi_{\mathrm{E}} / \Delta t$ through the interior of the circular path goes into the Ampère-Maxwell law.]

Example 4

Apply the Ampère-Maxwell law to one of the circular paths in Fig. 22.3 to find the magnitude of the magnetic field at the locations specified.Find $B$ inside the wire at a distance $r \leq R$ from the central axis. [Hint: Only the current through the interior of the circular path goes into the Ampère-Maxwell law.]

Example 5

An electric dipole antenna used to transmit radio waves is oriented vertically.At a point due south of the transmitter, what is the direction of the wave's magnetic field?
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Step-by-Step Explanations

QUESTION

Show that the speed of light in vacuum is given by c = 1/\u221a(\u03b5\u2080\u03bc\u2080).\nStep-by-step Answer:\nStep 1: Recognize that Maxwell\u2019s equations lead to a wave equation for the electric and magnetic fields in free space.\nStep 2: Identify that the only constants appearing in the equations are the permittivity of free space (\u03b5\u2080) and the permeability of free space (\u03bc\u2080).\nStep 3: Use dimensional analysis to form a quantity with the dimensions of speed; the combination 1/\u221a(\u03b5\u2080\u03bc\u2080) has units of m/s.\nStep 4: Substitute the measured values of \u03b5\u2080 (8.85 \u00d7 10\u207b\u00b9\u00b2 C\u00b2/(N\u00b7m\u00b2)) and \u03bc\u2080 (4\u03c0 \u00d7 10\u207b\u2077 T\u00b7m/A) into the formula.\nStep 5: Calculate the numerical value to verify that c \u2248 3.00 \u00d7 10\u2078 m/s.\nFinal Answer: The speed of light in vacuum is c = 1/\u221a(\u03b5\u2080\u03bc\u2080) \u2248 3.00 \u00d7 10\u2078 m/s.\n\n- Topic: Calculating Intensity from Energy Density \nQuestion: How do you compute the intensity of an electromagnetic wave given its average energy density?\nStep-by-step Answer:\nStep 1: Note that the energy density (u) of an EM wave is given by u = \u03f5\u2080E\u00b2 (or equivalently in terms of B) and represents energy per unit volume.\nStep 2: Recognize that as the wave propagates at speed c, the energy crossing a unit area per unit time (intensity I) is I = u * c.\nStep 3: Ensure that when using sinusoidal fields, you work with the rms values to obtain the average energy density.\nFinal Answer: The intensity of an electromagnetic wave is given by I = \u27e8u\u27e9 c, where \u27e8u\u27e9 is the average energy density.\n\n"

STEP-BY-STEP ANSWER:

Step 1: Recognize that Maxwell\u2019s equations lead to a wave equation for the electric and magnetic fields in free space.\nStep 2: Identify that the only constants appearing in the equations are the permittivity of free space (\u03b5\u2080) and the permeability of free space (\u03bc\u2080).\nStep 3: Use dimensional analysis to form a quantity with the dimensions of speed; the combination 1/\u221a(\u03b5\u2080\u03bc\u2080) has units of m/s.\nStep 4: Substitute the measured values of \u03b5\u2080 (8.85 \u00d7 10\u207b\u00b9\u00b2 C\u00b2/(N\u00b7m\u00b2)) and \u03bc\u2080 (4\u03c0 \u00d7 10\u207b\u2077 T\u00b7m/A) into the formula.\nStep 5: Calculate the numerical value to verify that c \u2248 3.00 \u00d7 10\u2078 m/s.\nFinal Answer: The speed of light in vacuum is c = 1/\u221a(\u03b5\u2080\u03bc\u2080) \u2248 3.00 \u00d7 10\u2078 m/s.\n\n- Topic: Calculating Intensity from Energy Density \nQuestion: How do you compute the intensity of an electromagnetic wave given its average energy density?\nStep-by-step Answer:\nStep 1: Note that the energy density (u) of an EM wave is given by u = \u03f5\u2080E\u00b2 (or equivalently in terms of B) and represents energy per unit volume.\nStep 2: Recognize that as the wave propagates at speed c, the energy crossing a unit area per unit time (intensity I) is I = u * c.\nStep 3: Ensure that when using sinusoidal fields, you work with the rms values to obtain the average energy density.\nFinal Answer: The intensity of an electromagnetic wave is given by I = \u27e8u\u27e9 c, where \u27e8u\u27e9 is the average energy density.\n\n"
Final Answer: The speed of light in vacuum is c = 1/\u221a(\u03b5\u2080\u03bc\u2080) \u2248 3.00 \u00d7 10\u2078 m/s.\n\n- Topic: Calculating Intensity from Energy Density \nQuestion: How do you compute the intensity of an electromagnetic wave given its average energy density?\nStep-by-step Answer:\nStep 1: Note that the energy density (u) of an EM wave is given by u = \u03f5\u2080E\u00b2 (or equivalently in terms of B) and represents energy per unit volume.\nStep 2: Recognize that as the wave propagates at speed c, the energy crossing a unit area per unit time (intensity I) is I = u * c.\nStep 3: Ensure that when using sinusoidal fields, you work with the rms values to obtain the average energy density.\nFinal Answer: The intensity of an electromagnetic wave is given by I = \u27e8u\u27e9 c, where \u27e8u\u27e9 is the average energy density.\n\n"

"- Topic: Derivation of the Speed of Light in Vacuum \nQuestion: Show that the speed of light in vacuum is given by c = 1/\u221a(\u03b5\u2080\u03bc\u2080).\nStep-by-step Answer:\nStep 1: Recognize that Maxwell\u2019s equations lead to a wave equation for the electric and magnetic fields in free space.\nStep 2: Identify that the only constants appearing in the equations are the permittivity of free space (\u03b5\u2080) and the permeability of free space (\u03bc\u2080).\nStep 3: Use dimensional analysis to form a quantity with the dimensions of speed; the combination 1/\u221a(\u03b5\u2080\u03bc\u2080) has units of m/s.\nStep 4: Substitute the measured values of \u03b5\u2080 (8.85 \u00d7 10\u207b\u00b9\u00b2 C\u00b2/(N\u00b7m\u00b2)) and \u03bc\u2080 (4\u03c0 \u00d7 10\u207b\u2077 T\u00b7m/A) into the formula.\nStep 5: Calculate the numerical value to verify that c \u2248 3.00 \u00d7 10\u2078 m/s.\nFinal Answer: The speed of light in vacuum is c = 1/\u221a(\u03b5\u2080\u03bc\u2080) \u2248 3.00 \u00d7 10\u2078 m/s.\n\n- Topic: Calculating Intensity from Energy Density \nQuestion: How do you compute the intensity of an electromagnetic wave given its average energy density?\nStep-by-step Answer:\nStep 1: Note that the energy density (u) of an EM wave is given by u = \u03f5\u2080E\u00b2 (or equivalently in terms of B) and represents energy per unit volume.\nStep 2: Recognize that as the wave propagates at speed c, the energy crossing a unit area per unit time (intensity I) is I = u * c.\nStep 3: Ensure that when using sinusoidal fields, you work with the rms values to obtain the average energy density.\nFinal Answer: The intensity of an electromagnetic wave is given by I = \u27e8u\u27e9 c, where \u27e8u\u27e9 is the average energy density.\n\n"

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Common Mistakes

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