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University Physics Volume 2

16.3 Energy Carried by Electromagnetic Waves

University Physics Volume 216.3 Energy Carried by Electromagnetic Waves
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  1. Preface
  2. Unit 1. Thermodynamics
    1. 1 Temperature and Heat
      1. Introduction
      2. 1.1 Temperature and Thermal Equilibrium
      3. 1.2 Thermometers and Temperature Scales
      4. 1.3 Thermal Expansion
      5. 1.4 Heat Transfer, Specific Heat, and Calorimetry
      6. 1.5 Phase Changes
      7. 1.6 Mechanisms of Heat Transfer
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    2. 2 The Kinetic Theory of Gases
      1. Introduction
      2. 2.1 Molecular Model of an Ideal Gas
      3. 2.2 Pressure, Temperature, and RMS Speed
      4. 2.3 Heat Capacity and Equipartition of Energy
      5. 2.4 Distribution of Molecular Speeds
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    3. 3 The First Law of Thermodynamics
      1. Introduction
      2. 3.1 Thermodynamic Systems
      3. 3.2 Work, Heat, and Internal Energy
      4. 3.3 First Law of Thermodynamics
      5. 3.4 Thermodynamic Processes
      6. 3.5 Heat Capacities of an Ideal Gas
      7. 3.6 Adiabatic Processes for an Ideal Gas
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 4 The Second Law of Thermodynamics
      1. Introduction
      2. 4.1 Reversible and Irreversible Processes
      3. 4.2 Heat Engines
      4. 4.3 Refrigerators and Heat Pumps
      5. 4.4 Statements of the Second Law of Thermodynamics
      6. 4.5 The Carnot Cycle
      7. 4.6 Entropy
      8. 4.7 Entropy on a Microscopic Scale
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  3. Unit 2. Electricity and Magnetism
    1. 5 Electric Charges and Fields
      1. Introduction
      2. 5.1 Electric Charge
      3. 5.2 Conductors, Insulators, and Charging by Induction
      4. 5.3 Coulomb's Law
      5. 5.4 Electric Field
      6. 5.5 Calculating Electric Fields of Charge Distributions
      7. 5.6 Electric Field Lines
      8. 5.7 Electric Dipoles
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    2. 6 Gauss's Law
      1. Introduction
      2. 6.1 Electric Flux
      3. 6.2 Explaining Gauss’s Law
      4. 6.3 Applying Gauss’s Law
      5. 6.4 Conductors in Electrostatic Equilibrium
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    3. 7 Electric Potential
      1. Introduction
      2. 7.1 Electric Potential Energy
      3. 7.2 Electric Potential and Potential Difference
      4. 7.3 Calculations of Electric Potential
      5. 7.4 Determining Field from Potential
      6. 7.5 Equipotential Surfaces and Conductors
      7. 7.6 Applications of Electrostatics
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 8 Capacitance
      1. Introduction
      2. 8.1 Capacitors and Capacitance
      3. 8.2 Capacitors in Series and in Parallel
      4. 8.3 Energy Stored in a Capacitor
      5. 8.4 Capacitor with a Dielectric
      6. 8.5 Molecular Model of a Dielectric
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    5. 9 Current and Resistance
      1. Introduction
      2. 9.1 Electrical Current
      3. 9.2 Model of Conduction in Metals
      4. 9.3 Resistivity and Resistance
      5. 9.4 Ohm's Law
      6. 9.5 Electrical Energy and Power
      7. 9.6 Superconductors
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    6. 10 Direct-Current Circuits
      1. Introduction
      2. 10.1 Electromotive Force
      3. 10.2 Resistors in Series and Parallel
      4. 10.3 Kirchhoff's Rules
      5. 10.4 Electrical Measuring Instruments
      6. 10.5 RC Circuits
      7. 10.6 Household Wiring and Electrical Safety
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    7. 11 Magnetic Forces and Fields
      1. Introduction
      2. 11.1 Magnetism and Its Historical Discoveries
      3. 11.2 Magnetic Fields and Lines
      4. 11.3 Motion of a Charged Particle in a Magnetic Field
      5. 11.4 Magnetic Force on a Current-Carrying Conductor
      6. 11.5 Force and Torque on a Current Loop
      7. 11.6 The Hall Effect
      8. 11.7 Applications of Magnetic Forces and Fields
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    8. 12 Sources of Magnetic Fields
      1. Introduction
      2. 12.1 The Biot-Savart Law
      3. 12.2 Magnetic Field Due to a Thin Straight Wire
      4. 12.3 Magnetic Force between Two Parallel Currents
      5. 12.4 Magnetic Field of a Current Loop
      6. 12.5 Ampère’s Law
      7. 12.6 Solenoids and Toroids
      8. 12.7 Magnetism in Matter
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    9. 13 Electromagnetic Induction
      1. Introduction
      2. 13.1 Faraday’s Law
      3. 13.2 Lenz's Law
      4. 13.3 Motional Emf
      5. 13.4 Induced Electric Fields
      6. 13.5 Eddy Currents
      7. 13.6 Electric Generators and Back Emf
      8. 13.7 Applications of Electromagnetic Induction
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    10. 14 Inductance
      1. Introduction
      2. 14.1 Mutual Inductance
      3. 14.2 Self-Inductance and Inductors
      4. 14.3 Energy in a Magnetic Field
      5. 14.4 RL Circuits
      6. 14.5 Oscillations in an LC Circuit
      7. 14.6 RLC Series Circuits
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    11. 15 Alternating-Current Circuits
      1. Introduction
      2. 15.1 AC Sources
      3. 15.2 Simple AC Circuits
      4. 15.3 RLC Series Circuits with AC
      5. 15.4 Power in an AC Circuit
      6. 15.5 Resonance in an AC Circuit
      7. 15.6 Transformers
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    12. 16 Electromagnetic Waves
      1. Introduction
      2. 16.1 Maxwell’s Equations and Electromagnetic Waves
      3. 16.2 Plane Electromagnetic Waves
      4. 16.3 Energy Carried by Electromagnetic Waves
      5. 16.4 Momentum and Radiation Pressure
      6. 16.5 The Electromagnetic Spectrum
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  4. A | Units
  5. B | Conversion Factors
  6. C | Fundamental Constants
  7. D | Astronomical Data
  8. E | Mathematical Formulas
  9. F | Chemistry
  10. G | The Greek Alphabet
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
    14. Chapter 14
    15. Chapter 15
    16. Chapter 16
  12. Index

Learning Objectives

By the end of this section, you will be able to:
  • Express the time-averaged energy density of electromagnetic waves in terms of their electric and magnetic field amplitudes
  • Calculate the Poynting vector and the energy intensity of electromagnetic waves
  • Explain how the energy of an electromagnetic wave depends on its amplitude, whereas the energy of a photon is proportional to its frequency

Anyone who has used a microwave oven knows there is energy in electromagnetic waves. Sometimes this energy is obvious, such as in the warmth of the summer Sun. Other times, it is subtle, such as the unfelt energy of gamma rays, which can destroy living cells.

Electromagnetic waves bring energy into a system by virtue of their electric and magnetic fields. These fields can exert forces and move charges in the system and, thus, do work on them. However, there is energy in an electromagnetic wave itself, whether it is absorbed or not. Once created, the fields carry energy away from a source. If some energy is later absorbed, the field strengths are diminished and anything left travels on.

Clearly, the larger the strength of the electric and magnetic fields, the more work they can do and the greater the energy the electromagnetic wave carries. In electromagnetic waves, the amplitude is the maximum field strength of the electric and magnetic fields (Figure 16.10). The wave energy is determined by the wave amplitude.

Figure on the left shows an electromagnetic wave with electric field E and magnetic field B. It is labeled u. Figure on the right shows an electromagnetic wave with electric field 2E and magnetic field 2B. Here, the amplitudes of the sine waves are doubled. The wave is labeled 4u.
Figure 16.10 Energy carried by a wave depends on its amplitude. With electromagnetic waves, doubling the E fields and B fields quadruples the energy density u and the energy flux uc.

For a plane wave traveling in the direction of the positive x-axis with the phase of the wave chosen so that the wave maximum is at the origin at t=0t=0, the electric and magnetic fields obey the equations

Ey(x,t)=E0cos(kxωt)Bz(x,t)=B0cos(kxωt).Ey(x,t)=E0cos(kxωt)Bz(x,t)=B0cos(kxωt).

The energy in any part of the electromagnetic wave is the sum of the energies of the electric and magnetic fields. This energy per unit volume, or energy density u, is the sum of the energy density from the electric field and the energy density from the magnetic field. Expressions for both field energy densities were discussed earlier (uEuE in Capacitance and uBuB in Inductance). Combining these the contributions, we obtain

u(x,t)=uE+uB=12ε0E2+12μ0B2.u(x,t)=uE+uB=12ε0E2+12μ0B2.

The expression E=cB=1ε0μ0BE=cB=1ε0μ0B then shows that the magnetic energy density uBuB and electric energy density uEuE are equal, despite the fact that changing electric fields generally produce only small magnetic fields. The equality of the electric and magnetic energy densities leads to

u(x,t)=ε0E2=B2μ0.u(x,t)=ε0E2=B2μ0.
(16.27)

The energy density moves with the electric and magnetic fields in a similar manner to the waves themselves.

We can find the rate of transport of energy by considering a small time interval ΔtΔt. As shown in Figure 16.11, the energy contained in a cylinder of length cΔtcΔt and cross-sectional area A passes through the cross-sectional plane in the interval Δt.Δt.

Figure shows a cylinder of length c delta t and cross sectional area A. Arrows indicate the direction of a wave to be along the length of the cylinder. A plane is shown perpendicular to the direction of wave.
Figure 16.11 The energy uAcΔtuAcΔt contained in the electric and magnetic fields of the electromagnetic wave in the volume AcΔtAcΔt passes through the area A in time ΔtΔt.

The energy passing through area A in time ΔtΔt is

u×volume=uAcΔt.u×volume=uAcΔt.

The energy per unit area per unit time passing through a plane perpendicular to the wave, called the energy flux and denoted by S, can be calculated by dividing the energy by the area A and the time interval ΔtΔt.

S=Energy passing areaAin timeΔtAΔt=uc=ε0cE2=1μ0EB.S=Energy passing areaAin timeΔtAΔt=uc=ε0cE2=1μ0EB.

More generally, the flux of energy through any surface also depends on the orientation of the surface. To take the direction into account, we introduce a vector SS, called the Poynting vector, with the following definition:

S=1μ0E×B.S=1μ0E×B.
(16.28)

The cross-product of EE and BB points in the direction perpendicular to both vectors. To confirm that the direction of SS is that of wave propagation, and not its negative, return to Figure 16.7. Note that Lenz’s and Faraday’s laws imply that when the magnetic field shown is increasing in time, the electric field is greater at x than at x+Δxx+Δx. The electric field is decreasing with increasing x at the given time and location. The proportionality between electric and magnetic fields requires the electric field to increase in time along with the magnetic field. This is possible only if the wave is propagating to the right in the diagram, in which case, the relative orientations show that S=1μ0E×BS=1μ0E×B is specifically in the direction of propagation of the electromagnetic wave.

The energy flux at any place also varies in time, as can be seen by substituting u from Equation 16.23 into Equation 16.27.

S(x,t)=cε0E02cos2(kxωt)S(x,t)=cε0E02cos2(kxωt)
(16.29)

Because the frequency of visible light is very high, of the order of 1014Hz,1014Hz, the energy flux for visible light through any area is an extremely rapidly varying quantity. Most measuring devices, including our eyes, detect only an average over many cycles. The time average of the energy flux is the intensity I of the electromagnetic wave and is the power per unit area. It can be expressed by averaging the cosine function in Equation 16.29 over one complete cycle, which is the same as time-averaging over many cycles (here, T is one period):

I=Savg=cε0E021T0Tcos2(2πtT)dt.I=Savg=cε0E021T0Tcos2(2πtT)dt.
(16.30)

We can either evaluate the integral, or else note that because the sine and cosine differ merely in phase, the average over a complete cycle for cos2(ξ)cos2(ξ) is the same as for sin2(ξ)sin2(ξ), to obtain

cos2ξ=12[cos2ξ+sin2ξ]=121=12.cos2ξ=12[cos2ξ+sin2ξ]=121=12.

where the angle brackets stand for the time-averaging operation. The intensity of light moving at speed c in vacuum is then found to be

I=Savg=12cε0E02I=Savg=12cε0E02
(16.31)

in terms of the maximum electric field strength E0,E0, which is also the electric field amplitude. Algebraic manipulation produces the relationship

I=cB022μ0I=cB022μ0
(16.32)

where B0B0 is the magnetic field amplitude, which is the same as the maximum magnetic field strength. One more expression for IavgIavg in terms of both electric and magnetic field strengths is useful. Substituting the fact that cB0=E0,cB0=E0, the previous expression becomes

I=E0B02μ0.I=E0B02μ0.
(16.33)

We can use whichever of the three preceding equations is most convenient, because the three equations are really just different versions of the same result: The energy in a wave is related to amplitude squared. Furthermore, because these equations are based on the assumption that the electromagnetic waves are sinusoidal, the peak intensity is twice the average intensity; that is, I0=2I.I0=2I.

Example 16.3

A Laser Beam The beam from a small laboratory laser typically has an intensity of about 1.0×10−3W/m21.0×10−3W/m2. Assuming that the beam is composed of plane waves, calculate the amplitudes of the electric and magnetic fields in the beam.

Strategy Use the equation expressing intensity in terms of electric field to calculate the electric field from the intensity.

Solution From Equation 16.31, the intensity of the laser beam is

I=12cε0E02.I=12cε0E02.

The amplitude of the electric field is therefore

E0=2cε0I=2(3.00×108m/s)(8.85×10−12F/m)(1.0×10−3W/m2)=0.87V/m.E0=2cε0I=2(3.00×108m/s)(8.85×10−12F/m)(1.0×10−3W/m2)=0.87V/m.

The amplitude of the magnetic field can be obtained from Equation 16.20:

B0=E0c=2.9×10−9T.B0=E0c=2.9×10−9T.

Example 16.4

Light Bulb Fields A light bulb emits 5.00 W of power as visible light. What are the average electric and magnetic fields from the light at a distance of 3.0 m?

Strategy Assume the bulb’s power output P is distributed uniformly over a sphere of radius 3.0 m to calculate the intensity, and from it, the electric field.

Figure shows a light bulb in the centre illuminating a circular area around it. This area has a radius of 3 m.

Solution The power radiated as visible light is then

I=P4πr2=cε0E022,E0=2P4πr2cε0=25.00W4π(3.0m)2(3.00×108m/s)(8.85×10−12C2/N·m2)=5.77N/C,B0=E0/c=1.92×10−8T.I=P4πr2=cε0E022,E0=2P4πr2cε0=25.00W4π(3.0m)2(3.00×108m/s)(8.85×10−12C2/N·m2)=5.77N/C,B0=E0/c=1.92×10−8T.

Significance The intensity I falls off as the distance squared if the radiation is dispersed uniformly in all directions.

Example 16.5

Radio Range A 60-kW radio transmitter on Earth sends its signal to a satellite 100 km away (Figure 16.12). At what distance in the same direction would the signal have the same maximum field strength if the transmitter’s output power were increased to 90 kW?

A point is labeled radio source. A small square labeled A1 is in the path of the lines radiating from the radio source. The lines continue from the corners of A1 and reach A2, a slightly bigger square. A1 is at a distance r1 from the source and A2 is at a distance R2.
Figure 16.12 In three dimensions, a signal spreads over a solid angle as it travels outward from its source.

Strategy The area over which the power in a particular direction is dispersed increases as distance squared, as illustrated in the figure. Change the power output P by a factor of (90 kW/60 kW) and change the area by the same factor to keep I=PA=cε0E022I=PA=cε0E022 the same. Then use the proportion of area A in the diagram to distance squared to find the distance that produces the calculated change in area.

Solution Using the proportionality of the areas to the squares of the distances, and solving, we obtain from the diagram

r22r12=A2A1=90W60W,r2=9060(100km)=122km.r22r12=A2A1=90W60W,r2=9060(100km)=122km.

Significance The range of a radio signal is the maximum distance between the transmitter and receiver that allows for normal operation. In the absence of complications such as reflections from obstacles, the intensity follows an inverse square law, and doubling the range would require multiplying the power by four.

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