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Chapter 31

Maxwells Equations andElectromagnetic Waves

James Clerk Maxwell (1831-1879)

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Electromagnetic theory

Maxwell united all phenomena ofelectricity and magnetism in one

magnificent theory

which became the base of physical

(wave) optics

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0 =B

0d =B A

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31-1 Faradays Law

The electromotive force around a closed circuit is

proportional to the rate of change of magnetic fieldflux through the circuit.

ddldt = E

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Stocks theorem relates a line integral around a closed

path to the surface integral over any surface enclosedby that path.

B

dl d

dddt t

= =

= =

E E A

B A

t

=

BE

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31-1 Ampres Law

Ampres law relates the magnetic field around a

current to the current through a surface.

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In order for Ampres

law to hold, it cantmatter 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

31 1 Ch i El t i Fi ld

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Therefore, Ampres law is modified to include thecreation of a magnetic field by a changing electric

field the field between the plates of the capacitor

in this example:

31-1 Changing Electric Fields

Produce Magnetic Fields; Ampres

Law and Displacement Current

Magnetic field is created by a current and a

changing electric field.

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-Produce Magnetic Fields;

Ampres Law and DisplacementCurrent

The second term in Amperes law has thedimensions of a current (after factoring out the

0), and is sometimes called the displacement

current:

where

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Using Stocks theorem again:

0 0dl d d d

t

= = +

B B A j A E A

0 0 0 t

= +

E

B j

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31-3 Maxwells Equations

We now have a complete set of equations thatdescribe electric and magnetic fields, called

Maxwells equations. In the absence of dielectric

or magnetic materials, they are:

0

0 0 0

0

t

t

=

=

=

= +

E

B

B

E

EB j

31 4 Production of Electromagnetic

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Since a changing electric field produces a

magnetic field, and a changing magneticfield 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

31 4 P d ti f El t ti

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Oscillating charges will

produce

electromagnetic waves:

31-4 Production of Electromagnetic

Waves

31 4 P d ti f El t ti

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31-4 Production of Electromagnetic

WavesClose to the antenna,

the fields arecomplicated, and are

called the near field:

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31-5 Electromagnetic waves and

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31-5 Electromagnetic waves, and

their speed, derived from Maxwells

EquationsIn the absence of currents and charges,

Maxwells equations become:

0 0

0

0

t

t

=

=

=

=

E

B

B

E

EB

31 5 Electromagnetic Waves and

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31-5 Electromagnetic Waves, and

Their Speed, Derived fromMaxwells EquationsTo derive the wave equation we are going to use a

vector identity:

Lets first prove this.

( )2( ) = E E E

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2 2 22

2 2 2

2 2 22

2 2 2( ) x y z

x y z

E E E x y z

= + +

= + + + +

E i j k

2( ) E E

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( ) E

y yx xz z

x y z

i j k

E EE EE Ei j k

x y z y z z x x y

E E E

= = +

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( )2( ) E E i ( )2( ) E E j ( )2( ) E E k

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( )2( ) = E E E

0 0

0

0

t

t

=

=

=

=

E

B

B

E

EB

0

( )

2

2

0 0 2

( )

t t t

= =

= = =

E E

B B E

8

0 0

13 10 m/sv

= =

22

0 0 2t

=

EE

22 222

2 2 2 2

yx zEE EE

vt x y z

= + +

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Wave equations22 22

2

2 2 2 2

yx zEE EE

vt x y z

= + +

22 222

2 2 2 2

yx zBB BB

vt x y z

= + +

31 5 Pl W S l ti

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31-5 Plane Waves Solutions

This figure shows an electromagnetic wave of

wavelength and frequency f. The electric andmagnetic fields are given by

where

.

31-5 Electromagnetic Waves, and

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31 5 Electromagnetic Waves, and

Their Speed, Derived from

Maxwells EquationsApplying Faradays law

gives a relationship between Eand B:

.

0 0

0 0

y

y

EE

x x

E

= =

i j k

k

t

=

BE

31-5 Electromagnetic Waves, and

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g ,

Their Speed, Derived from

Maxwells EquationsFrom the plane wave solution:

.

( )0 0sin( ) cos( )E E kx t kE kx t x x = =

( )0 0sin( ) cos( )B

B kx t B kx t

t t

= =

.

0 0kE B=E

cB k

= =

- g as an ec romagne c

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The frequency of an electromagnetic wave is

related to its wavelength and to the speed of

light:

g as an ec romagne cWave and the Electromagnetic

Spectrum

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

measured speed of light.

Example 31-2: Determining E and BE

B

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Example 31 2: Determining E and B

in EM waves.Assume a 60-Hz EM wave is a sinusoidal wave

propagating in the zdirection with pointing in

the x direction, and E0 = 2.0 V/m. Write vectorexpressions for and as functions of

position and time.

E B

E

E

B

The wavelength is c/f = 5.0 x 106m. The wave number is 2/=

1.26 x 10-6 m-1. The angular frequency is 2f = 377 rad/s. Finally, B0 =

E0/c = 6.7 x 10-9 T. B must be in the y direction, as E, v, and B are

mutually perpendicular.

6 1

9 6 1

(2.0V/m)sin (1.26 10 m ) (377rad/s)

(6.7 10 T)sin (1.26 10 m ) (377rad/s)

z t

z t

=

=

E i

B j

31-6 Light as an Electromagnetic

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Electromagnetic waves can have any

wavelength; we have given different names todifferent parts of the wavelength spectrum.

g gWave and the Electromagnetic

Spectrum

Example 31-3: Wavelengths of EM

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p g

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 laserat frequency 4.74 x 1014 Hz.

a. 5.0 x 106 m

b. 3.22 m

c. 6.33 x 10-7 m

Example 31-4: Cell phone antenna

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Example 31 4: Cell phone antenna.

The antenna of a cell phone is often

wavelength long. A particular cell phone has an

8.5-cm-long straight rod for its antenna. Estimatethe operating frequency of this phone.

The frequency is the speed of light divided by the wavelength (which is 4 timesthe antenna length): f = 880 MHz.

Example 31-5: Phone call time lag.

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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 anoticeable delay in either case?

a) Distance from NY to London is 5000 km.The time delay is t =d/c= 0.017 s.

b) Via satellite d=2x36000km and t=d/c=0.24s

31-7 Measuring the Speed of

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The speed of light

was known to be

very large,although careful

studies of the orbits

of Jupiters moonsshowed that it is

finite.

One important

measurement, by

Michelson, used arotating mirror:

31 7 Measuring the Speed of

Light

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Over the years, measurements have become more

and more precise; now the speed of light is definedto be

c = 2.99792458 108 m/s.

This is then used to define the meter.

31 7 Measuring the Speed of

Light