Chapter 31 Maxwell’s Equations and Electromagnetic Waves

Copyright © 2009 Pearson Education, Inc.

Chapter 31Maxwell’s Equations and Electromagnetic Waves


• Changing Electric Fields Produce Magnetic Fields; Ampère’s Law and Displacement Current

• Gauss’s Law for Magnetism

• Maxwell’s Equations

• Production of Electromagnetic Waves

• Electromagnetic Waves, and Their Speed, Derived from Maxwell’s Equations

• Light as an Electromagnetic Wave and the Electromagnetic Spectrum

Units of Chapter 31


• Measuring the Speed of Light

• Energy in EM Waves; the Poynting Vector

• Radiation Pressure

• Radio and Television; Wireless Communication

Units of Chapter 31


E&M Equations to date

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31-2 Gauss’s Law for Magnetism

Gauss’s law relates the electric field on a closed surface to the net charge enclosed by that surface. The analogous law for magnetic fields is different, as there are no single magnetic point charges (monopoles):


E&M Equations to date - updated

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E&M Equations to date – more updated

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Ampère’s law relates the magnetic field around a current to the current through a surface.

31-1 Changing Electric Fields Produce Magnetic Fields; Ampère’s

Law and Displacement Current


In order for Ampère’s law to hold, it can’t matter which surface we choose. But look at a discharging capacitor; there is a current through surface 1 but none through surface 2:

31-1 Changing Electric Fields Produce Magnetic Fields; Ampère’s Law and

Displacement Current


Therefore, Ampère’s law is modified to include the creation of a magnetic field by a changing electric field – the field between the plates of the capacitor in this example:




Example 31-1: Charging capacitor.

A 30-pF air-gap capacitor has circular plates of area A = 100 cm2. It is charged by a 70-V battery through a 2.0-Ω resistor. At the instant the battery is connected, the electric field between the plates is changing most rapidly. At this instant, calculate (a) the current into the plates, and (b) the rate of change of electric field between the plates. (c) Determine the magnetic field induced between the plates. Assume E is uniform between the plates at any instant and is zero at all points beyond the edges of the plates.



E��




The second term in Ampere’s law has the dimensions of a current (after factoring out the μ0), and is sometimes called the displacement current:

where


31-3 Maxwell’s Equations

We now have a complete set of equations that describe electric and magnetic fields, called Maxwell’s equations. In the absence of dielectric or magnetic materials, they are:


Since a changing electric field produces a magnetic field, and a changing magnetic field produces an electric field, once sinusoidal fields are created they can propagate on their own.

These propagating fields are called electromagnetic waves.

31-4 Production of Electromagnetic Waves

ConcepTest 31.1aConcepTest 31.1a EM Waves IEM Waves I

Plastic

Copper

A loop with an AC current produces

a changing magnetic field. Two

loops have the same area, but one

is made of plastic and the other

copper. In which of the loops is

the induced voltage greater?

1) the plastic loop

2) the copper loop

3) voltage is same in both

Faraday’s law says nothing about the material:

The change in flux is the samechange in flux is the same (and N is the same), so the induced emf induced emf is the sameis the same.

ConcepTest 31.1aConcepTest 31.1a EM Waves IEM Waves I

Plastic

Copper

A loop with an AC current produces

a changing magnetic field. Two

loops have the same area, but one

is made of plastic and the other

copper. In which of the loops is

the induced voltage greater?

1) the plastic loop

2) the copper loop

3) voltage is same in both

BdNdt

%


Oscillating charges will produce electromagnetic waves:




Close to the antenna, the fields are complicated, and are called the near field:


Far from the source, the waves are plane waves:



The electric and magnetic waves are perpendicular to each other, and to the direction of propagation.


ConcepTest 31.2ConcepTest 31.2 OscillationsOscillations

The electric field in an EM The electric field in an EM

wave traveling northeast wave traveling northeast

oscillates up and down. In oscillates up and down. In

what plane does the what plane does the

magnetic field oscillate? magnetic field oscillate?

1) in the north-south planein the north-south plane

2) in the up-down planein the up-down plane

3) in the NE-SW planein the NE-SW plane

4) in the NW-SE plane4) in the NW-SE plane

5) in the east-west plane5) in the east-west plane

The magnetic field oscillates perpendicular to BOTH the electric field and the direction of the wave. Therefore the magnetic field must oscillate in the NW-SE plane.

ConcepTest 31.2ConcepTest 31.2 OscillationsOscillations

The electric field in an EM The electric field in an EM

wave traveling northeast wave traveling northeast

oscillates up and down. In oscillates up and down. In

what plane does the what plane does the

magnetic field oscillate? magnetic field oscillate?

1) in the north-south planein the north-south plane

2) in the up-down planein the up-down plane

3) in the NE-SW planein the NE-SW plane

4) in the NW-SE plane4) in the NW-SE plane

5) in the east-west plane5) in the east-west plane


31-5 Electromagnetic Waves, and Their Speed, Derived from Maxwell’s

EquationsIn the absence of currents and charges, Maxwell’s equations become:



EquationsThis figure shows an electromagnetic wave of wavelength λ and frequency f. The electric and magnetic fields are given by

where

.



EquationsApplying Faraday’s law to the rectangle of height Δy and width dx in the previous figure gives a relationship between E and B:

.



Equations

Similarly, we apply Maxwell’s fourth equation to the rectangle of length Δz and width dx, which gives

.



EquationsUsing these two equations and the equations for B and E as a function of time gives

Here, v is the velocity of the wave. Substituting,

.



Equations

The magnitude of this speed is 3.0 x 108 m/s – precisely equal to the measured speed of light.



Equations

Example 31-2: Determining E and B in EM waves.

Assume a 60-Hz EM wave is a sinusoidal wave propagating in the z direction with E pointing in the x direction, and E0 = 2.0 V/m. Write vector expressions for E and B as functions of position and time.

E��

E��

E��

B��

B��


The frequency of an electromagnetic wave is related to its wavelength and to the speed of light:

31-6 Light as an Electromagnetic Wave and the Electromagnetic Spectrum


Electromagnetic waves can have any wavelength; we have given different names to different parts of the wavelength spectrum.




Example 31-3: Wavelengths of EM waves.

Calculate the wavelength

(a) of a 60-Hz EM wave,

(b) of a 93.3-MHz FM radio wave, and

(c) of a beam of visible red light from a laser at frequency 4.74 x 1014 Hz.



Example 31-4: Cell phone antenna.

The antenna of a cell phone is often ¼ wavelength long. A particular cell phone has an 8.5-cm-long straight rod for its antenna. Estimate the operating frequency of this phone.



Example 31-5: Phone call time lag.

You make a telephone call from New York to a friend in London. Estimate how long it will take the electrical signal generated by your voice to reach London, assuming the signal is (a) carried on a telephone cable under the Atlantic Ocean, and (b) sent via satellite 36,000 km above the ocean. Would this cause a noticeable delay in either case?


The speed of light was known to be very large, although careful studies of the orbits of Jupiter’s moons showed that it is finite.

One important measurement, by Michelson, used a rotating mirror:

31-7 Measuring the Speed of Light


Over the years, measurements have become more and more precise; now the speed of light is defined to be

c = 2.99792458 × 108 m/s.

This is then used to define the meter.

31-7 Measuring the Speed of Light


Energy is stored in both electric and magnetic fields, giving the total energy density of an electromagnetic wave:

Each field contributes half the total energy density:

31-8 Energy in EM Waves; the Poynting Vector


This energy is transported by the wave.



The energy transported through a unit area per unit time is called the intensity:


Its vector form is the Poynting vector:



Typically we are interested in the average value of S:S

.



Example 31-6: E and B from the Sun.

Radiation from the Sun reaches the Earth (above the atmosphere) at a rate of about 1350 J/s·m2 (= 1350 W/m2). Assume that this is a single EM wave, and calculate the maximum values of E and B.


In addition to carrying energy, electromagnetic waves also carry momentum. This means that a force will be exerted by the wave.

The radiation pressure is related to the average intensity. It is a minimum if the wave is fully absorbed:

and a maximum if it is fully reflected:

31-9 Radiation Pressure



Example 31-7: Solar pressure.

Radiation from the Sun that reaches the Earth’s surface (after passing through the atmosphere) transports energy at a rate of about 1000 W/m2. Estimate the pressure and force exerted by the Sun on your outstretched hand.



Example 31-8: A solar sail.

Proposals have been made to use the radiation pressure from the Sun to help propel spacecraft around the solar system. (a) About how much force would be applied on a 1 km x 1 km highly reflective sail, and (b) by how much would this increase the speed of a 5000-kg spacecraft in one year? (c) If the spacecraft started from rest, about how far would it travel in a year?


This figure illustrates the process by which a radio station transmits information. The audio signal is combined with a carrier wave.

31-10 Radio and Television; Wireless Communication


The mixing of signal and carrier can be done two ways. First, by using the signal to modify the amplitude of the carrier (AM):



Second, by using the signal to modify the frequency of the carrier (FM):



At the receiving end, the wave is received, demodulated, amplified, and sent to a loudspeaker.



The receiving antenna is bathed in waves of many frequencies; a tuner is used to select the desired one.




A straight antenna will have a current induced in it by the varying electric fields of a radio wave; a circular antenna will have a current induced by the changing magnetic flux.



Example 31-9: Tuning a station.

Calculate the transmitting wavelength of an FM radio station that transmits at 100 MHz.

ConcepTest 31.3ConcepTest 31.3 TV AntennasTV Antennas

Before the days of cable,

televisions often had two

antennae on them, one straight

and one circular. Which antenna

picked up the magnetic

oscillations?

1) the circular onethe circular one

2) the straight onethe straight one

3) both equally; they were both equally; they were

straight and circular for straight and circular for

different reasonsdifferent reasons

The varying B field in the loop means the flux is changing and therefore an emf is induced.

ConcepTest 31.3ConcepTest 31.3 TV AntennasTV Antennas

Before the days of cable,

televisions often had two

antennae on them, one straight

and one circular. Which antenna

picked up the magnetic

oscillations?

1) the circular onethe circular one

2) the straight onethe straight one

3) both equally; they were both equally; they were

straight and circular for straight and circular for

different reasonsdifferent reasons

ConcepTest 31.4ConcepTest 31.4 Radio AntennasRadio Antennas

If a radio transmitter has a vertical If a radio transmitter has a vertical

antenna, should a receiver’s antenna, should a receiver’s

antenna be antenna be verticalvertical or or horizontalhorizontal

to obtain the best reception? to obtain the best reception?

1) verticalvertical

2) horizontalhorizontal

3) doesn’t matterdoesn’t matter

If a wave is sent out from a vertical vertical

antennaantenna, the electric field oscillates electric field oscillates

up and downup and down. Thus, the receiver’s receiver’s

antenna should also beantenna should also be verticalvertical so that the arriving electric field can set the charges in motion.

ConcepTest 31.4ConcepTest 31.4 Radio AntennasRadio Antennas

E fieldof wave

E fieldof wave

If a radio transmitter has a vertical If a radio transmitter has a vertical

antenna, should a receiver’s antenna, should a receiver’s

antenna be antenna be verticalvertical or or horizontalhorizontal

to obtain the best reception? to obtain the best reception?

1) verticalvertical

2) horizontalhorizontal

3) doesn’t matterdoesn’t matter


• Maxwell’s equations are the basic equations of electromagnetism:

Summary of Chapter 31


• Electromagnetic waves are produced by accelerating charges; the propagation speed is given by

• The fields are perpendicular to each other and to the direction of propagation.



• The wavelength and frequency of EM waves are related:

• The electromagnetic spectrum includes all wavelengths, from radio waves through visible light to gamma rays.

• The Poynting vector describes the energy carried by EM waves:


Chapter 31 Maxwell’s Equations and Electromagnetic Waves

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Transcript of Chapter 31 Maxwell’s Equations and Electromagnetic Waves