Lecture21 Electromagnetic Waves

8/3/2019 Lecture21 Electromagnetic Waves

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http://www.labinitio.com (nz302.jpg)
http://www.labinitio.com/http://www.labinitio.com/


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adapted from http://www.labinitio.com (nz302.jpg)
http://www.labin/http://www.labin/


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Todays lecture is brought to you by the letter P.

http://www.labinitio.com (nz288.jpg)
http://www.nearingzero.net/http://www.nearingzero.net/


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Announcements

There is lots ofnice math in chapter 32! This lecturecalls your attention to those parts of the chapter thatyou need to know for exams. Keep this lecture in mindwhen you study chapter 32.

Exam 3 is two weeks from yesterday. I will need toknow by next Wednesday of any students who havespecial needs different than for exam 2.

Exam 3 will cover material through the end of todayslecture. Material presented during next weeks lectureswill be covered on the final exam.


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Review and Note!

These say integrate over a surface (which has anarea) that encloses (and defines) some volume:

r r BdE ds= - dtr r 0B ds= I

encl

0

qE dA=

rr

0 B dA= rr

These say integrate over a line (which has a length)that encloses (and defines) some surface:

Note: All of these mean average: Saverage Sav Savg


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Todays agenda:

Electromagnetic Waves.

Energy Carried by Electromagnetic Waves.

Momentum and Radiation Pressure of anElectromagnetic Wave.

rarely in the course of human events have so many starting equations been given in so little time


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We began this course by studying fields that didntvary with timethe electric field due to static charges,and the magnetic field due to a constant current.

In case you didnt noticeabout a half dozen lecturesago things started moving!

We found that changingmagnetic field gives rise to anelectric field. Also a changingelectric field gives rise to a

magnetic field.

These time-varying electric and magnetic fields canpropagate through space.


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Electromagnetic Waves

E0 encl 0 0

dB ds= I + dtr r

enclosed

o

qE dA =

rr

These four equations provide a complete description ofelectromagnetism.

0

E =

r r

02

1 dEB= + J

c dt

r

r r r

B dA 0 =rr

B 0 =r r

B

dE ds dt = r r

dBE=-

dt

r

r r

Maxwells Equations


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Production of Electromagnetic Waves

Apply a sinusoidal voltage to an antenna.

Charged particles in the antenna oscillatesinusoidally.

The accelerated charges produce sinusoidallyvarying electric and magnetic fields, which extend

throughout space.The fields do not instantaneously permeate allspace, but propagate at the speed of light.

direction ofpropagation

y

z

x


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http://www.walter-fendt.de/ph11e/emwave.htm

http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=3

This static image doesnt show how the wavepropagates.

Here are a couple of animations, available on-line:

direction of

propagation

y

z

x

Here is a movie.
http://www.walter-fendt.de/ph11e/emwave.htmhttp://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=35http://eb_light.mpeg/http://eb_light.mpeg/http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=35http://www.walter-fendt.de/ph11e/emwave.htm


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Electromagnetic waves are transverse waves, but arenot mechanical waves (they need no medium to vibrate

in).


Therefore, electromagnetic waves can propagate infree space.

At any point, the magnitudes of E and B (of thewave shown) depend only upon x and t, and not ony or z. A collection of such waves is called a planewave.

y

z

x


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Manipulation of Maxwells equations leads to thefollowing plane wave equations for E and B:

These equations have solutions:

ou can verify this by direct substitution.

2 2

y y0 02 2

E E (x,t)=x t

2 2

z z0 02 2B B (x,t)=x t

( )y maxE =E sin kx- t

( )z maxB =B sin kx- t

where , , 2k= =2 f and f = =c.k

Emax and Bmax in these notes are sometimes written by others as E0 and B0.

Emax and Bmax arethe electric andmagnetic field

amplitudes

Equations on this slide are forwaves propagating along x-

direction.


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You can also show that

At every instant, the ratio of the magnitude of theelectric field to the magnitude of the magnetic field inan electromagnetic wave equals the speed of light.

y zE B

=-x t

( ) ( ) max maxE k cos kx- t =B cos kx- t

.

max

max 0 0

E E 1= = =c=

B B k


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y

z

x

Emax (amplitude)

E(x,t)


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Summary of Important Properties of ElectromagneticWaves

The solutions of Maxwells equations are wave-like withboth E and B satisfying a wave equation.


( )z maxB =B sin kx- tElectromagnetic waves travel through empty spacewith the speed of light c = 1/( 0 0).

Emax and Bmax are the electric and magnetic field

amplitudes.


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Summary of Important Properties of ElectromagneticWaves

The components of the electric and magnetic fields ofplane EM waves are perpendicular to each other andperpendicular to the direction of wave propagation. Thelatter property says that EM waves are transversewaves.

The magnitudes of E and B in empty space are relatedby

E/B = c.max

max

E E= = =c

B B k


y

z

x

Spring 2011:

slide 19 next


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Possible Homework Hints (may not be needed every semester)

The speed of light in a nonconducting medium otherthan a vacuum is less than c:

0 0

1

c=

( ) ( ) m 0 0

1v=

where is the relative dielectric constant (remember itfrom capacitors?) and m is called the relativepermeability of the medium.

Because

. m

cv=you can show that

These equations are not on yourequation sheet, but you havepermission to use them for

tomorrows homework (if needed):use v for the wave speed, and

replace 0 by 0 and 0 by .


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Satellite:2

earth2GmMmvF= =R R

2 Rv =

T

Solve the above to get the distance R of the satellitefrom the center of the earth, then subtract 6.38x106 mto get the height of the satellite above the ground.


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Gravitational force of sun: sun2GmMF= R


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Todays agenda:



Momentum and Radiation Pressure of anElectromagnetic Wave.


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The magnitude S represents the rate at which energyflows through a unit surface area perpendicular to thedirection of wave propagation.

Energy Carried by Electromagnetic Waves

Electromagnetic waves carry energy, and as they

propagate through space they can transfer energy toobjects in their path. The rate of flow of energy in anelectromagnetic wave is described by a vector S,calledthe Poynting vector.*

r r r

0

1S= E B

Thus, S represents power per unit area. Thedirection of S is along the direction of wavepropagation. The units of S areJ/(sm2) =W/m2.

*J. H. Poynting, 1884.


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z

x

y

c

SE

B Because B = E/cwe can write

These equations for S apply at any instant of time andrepresent the instantaneous rate at which energy ispassing through a unit area.

r r r

01S= E B

For an EM wave

so

r rE B =EB

.0

EBS=

.

2 2

0 0

E cBS= =

c


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The time average of sin2(kx - t) is , so

2 2

0 0 0

EB E cBS= = =

c

EM waves are sinusoidal.

The average of S over one or more cycles is called thewave intensity I.

2 2max max max max

average0 0 0

E B E cBI =S = S = = =

2 2 c 2

This equation is the same as 32-29 in your text, using c = 1/( 0 0).


( )z maxB =B sin kx- t

Notice the 2sin this

equation.

EM wavepropagating along

x-direction


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Thus,

The magnitude of S is the rate at which energy istransported by a wave across a unit area at anyinstant:

instantaneousinstantaneous

energypowertimeS= =

area area

averageaverage

energy

powertimeI = S = =area area

Note: Saverage and mean the same thing!


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The energy densities (energy per unit volume)

associated with electric and magnetic fields are:

sing B = E/c and c = 1/( 0 0) we can write

Energy Density

2E 01

u = E2

2

B0

1Bu =

2

( )

222

20 0

B 00 0 0

EE1B 1 1 1c

u = = = = E2 2 2 2

22

B E 00

1 1Bu =u = E =

2 2remember: E and B aresinusoidal functions of

time


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For an electromagnetic wave, the instantaneous energydensity associated with the magnetic field equals theinstantaneous energy density associated with theelectric field.

22

B E 00

1 1Bu =u = E =

2 2

22

B E 00

Bu=u +u = E =

Hence, in a given volume the energy is equally sharedby the two fields. The total energy density is equal tothe sum of the energy densities associated with theelectric and magnetic fields:


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When we average this instantaneous energy densityover one or more cycles of an electromagnetic wave,we again get a factor of from the time average ofsin2(kx - t).

so we see thatRecall

The intensity of an electromagnetic wave equals theaverage energy density multiplied by the speed of

light.

22

B E 00

Bu=u +u = E =

22 max

0 max 0

B1 1u = E =

2 2

2 2max max

average0 0

E cB1 1

S = S = =2 c 2 .S =c u

,

2

E 0 max

1u = E

4,

2max

B 0

B1u =

4and


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No Quiz today.

Homework Clarification

Problem 32.23 also calculate the energy density dueto the electric and magnetic fields

, 2E 0 max1

u = E4

.

2max

B0

B1u =

4

this means calculate the average energy densities


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Example: a radio station on the surface of the earthradiates a sinusoidal wave with an average total powerof 50 kW. Assuming the wave is radiated equally in all

directions above the ground, find the amplitude of theelectric and magnetic fields detected by a satellite 100km from the antenna.

R

Station

SatelliteAll the radiated power passesthrough the hemisphericalsurface* so the averagepower per unit area (theintensity) is

( )( )

4

-7 222 5

average

5.00 10 Wpower PI = = = =7.96 10 W marea 2 R 2 1.00 10 m

*In problems like this you need to ask whether the

power is radiated into all space or into just part ofs ace.

Todays lecture isbrought to you by the

letter P.


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R

Station

Satellite

2max

0

E1I = S =

2 c

max 0E = 2 cI

( ) ( ) ( ) -7 8 -7

= 2 4 10 3 10 7.96 10

-2 V=2.45 10 m

( )( )

-2-11max

max 8

V2.45 10E mB = = =8.17 10 Tc 3 10 m s


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Example: for the radio station in the example on theprevious two slides, calculate the average energydensities associated with the electric and magnetic

field. 2E 0 max

1u = E

4

( ) ( ) 2 -12 -2E 1u = 8.85 10 2.45 104

-15E 3J

u =1.33 10

m

2max

B0

B1u =

4

( )( )

2

-11

B -78.17 101u =4 4 10

-15B 3J

u =1.33 10

m


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Todays agenda:



Momentum and Radiation Pressure of an

Electromagnetic Wave.


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Momentum and Radiation Pressure

EM waves carry linear momentum as well as energy.

When this momentum is absorbed at a surfacepressure is exerted on that surface.

If we assume that EM radiation is incident on an objectfor a time t and that the radiation is entirely absorbed

by the object, then the object gains energy U in time t.Maxwell showed that the momentumchange of the object is then:

The direction of the momentum change of the object isin the direction of the incident radiation.

incident

U

p = (total absorption)c

Todays lecture isbrought to you by the

letter P.


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If instead of being totally absorbed the radiation istotally reflected by the object, and the reflection is

along the incident path, then the magnitude of themomentum change of the object is twice that for totalabsorption.

The direction of the momentum change of the object isagain in the direction of the incident radiation.

2 Up = (total reflection along incident path)

c

incident

reflected


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The radiation pressure on the object is defined as the

force per unit area:

From Newtons 2nd Law (F = dp/dt) we

have:For total absorption,

So

Radiation Pressure

FP=

A

F 1 dpP= =

A A dt

U

p=c

dU1 dp 1 d U 1 SdtP= = = =A dt A dt c c A c

incident

(Equations on this slide involve magnitudes of vector quantities.)


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This is the instantaneous radiation pressure in the caseof total absorption:

S

P=c

For the average radiation pressure, replace S by

=Savg =I: averagerad

S IP = =

c c

Electromagnetic waves also carry momentum throughspace with a momentum density of Saverage /c2=I/c2. Thisis not on your equation sheet but you have specialpermission to use it in tomorrows homework, if

necessary.Todays lecture is

brought to you by theletter P.


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rad

IP = (total absorption)

c

rad

2IP = (total reflection)

c

incident

incident

reflected

absorbed

ing the arguments above it can also be shown that:


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Example: a satellite orbiting the earth has solar energycollection panels with a total area of 4.0 m2. If thesuns radiation is incident perpendicular to the panels

and is completely absorbed find the average solarpower absorbed and the average force associated withthe radiation pressure. The intensity (I or Saverage ) ofsunlight prior to passing through the earthsatmosphere is 1.4 kW/m2.

( ) ( ) 3 2 32WPower=IA= 1.4 10 4.0 m =5.6 10 W=5.6 kWmssuming total absorption of the radiation:

( )( )

3 2average -6

rad 8

W1.4 10S I mP = = = =4.7 10 Pamc c 3 10 s

( ) ( ) -6 2 -52rad NF=P A= 4.7 10 4.0 m =1.9 10 N

m

Caution! The letterP (or p) has been

used in this lecturefor power,

pressure, andmomentum!

Thats because todays lecture

is brought to you by the letterP.


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New starting equations from this lecture:

r r r

0

1S= E B

2 2max max

average0 0

E cB1 1S = =

2 c 2

max

max 0 0

E E 1= =c=

B B

22

B E 00

1 1Bu =u = E =

2 2

22 max

0 max0

B1 1u = E =

2 2

U 2 Up = or

c c radI 2I

P = orc c

, ,

2

k= =2 f f = =ck

There are even more on your starting equation sheet; they are derived from the

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Lecture21 Electromagnetic Waves

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