lecture21 gel. em

8/10/2019 Lecture21 Gel. EM

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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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Review and Note!

These say integrate over a surface (which has an area) thatencloses (and defines) some volume:

BdE ds= -dt

0B ds = I

encl

0

qE dA

0 B dA

These say integrate over a line (which has a length) thatencloses (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 an ElectromagneticWave.

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 didnt vary withtimethe electric field due to static charges, and the magneticfield due to a constant current.

In case you didnt noticeabout a half dozen lectures agothings started moving!

We found that changing magneticfield gives rise to an electric field.Also a changing electric field givesrise to a magnetic field.

These time-varying electric and magnetic fields can propagatethrough space.


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Electromagnetic Waves

E0 encl 0 0

dB ds= I + dt

enclosed

o

qE dA

These four equations provide a complete description ofelectromagnetism.

0

E

02

1 dEB= + J

c dt

B dA 0

B 0

B

dE ds

dt

dBE=-

dt

Maxwells Equations


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Production of Electromagnetic Waves

Apply a sinusoidal voltage to an antenna.

Charged particles in the antenna oscillate sinusoidally.

The accelerated charges produce sinusoidally varyingelectric and magnetic fields, which extend throughout

space.

The fields do not instantaneously permeate all space, butpropagate at the speed of light.

direction ofpropagation

y

z

x


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http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=35

This static image doesnt show how the wave propagates.

Here is an animation, available on-line:

direction of

propagation

y

z

x

Here is a movie.
http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=35http://localhost/var/www/apps/conversion/tmp/scratch_7/EB_Light.mpeghttp://localhost/var/www/apps/conversion/tmp/scratch_7/EB_Light.mpeghttp://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=35


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Electromagnetic waves are transverse waves, but are notmechanical waves (they need no medium to vibrate in).


Therefore, electromagnetic waves can propagate in freespace.

At any point, the magnitudes of E and B (of the waveshown) depend only upon x and t, and not on y or z. Acollection of such waves is called a plane wave.

y

z

x


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Manipulation of Maxwells equations leads to the followingplane wave equationsfor E and B:

These equations have solutions:

You 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

Emaxand Bmaxin these notes are sometimes written by others as E0and B0.

Emaxand Bmaxare theelectric and magnetic

field amplitudes

Equations on this slide are for wavespropagating along x-direction.


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You can also show that

At every instant, the ratio of the magnitude of the electric fieldto the magnitude of the magnetic field in an electromagneticwave equals the speed of light.

y z

E 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 Electromagnetic Waves

The solutions of Maxwells equations are wave-like with both Eand B satisfying a wave equation.


z maxB =B sin kx - tElectromagnetic waves travel through empty space with thespeed of light c = 1/(00)

.

Emaxand Bmaxare the electric and magnetic field amplitudes.


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Summary of Important Properties of Electromagnetic Waves

The components of the electric and magnetic fields of plane EMwaves are perpendicular to each other and perpendicular to thedirection of wave propagation. The latter property says that EMwaves are transverse waves.

The magnitudes of E and B in empty space are related by

E/B = c. maxmax

E E= = =c

B B k


y

z

x

Spring 2012:

slide 19 next


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Possible Homework Hints (may not be needed every semester)

The speed of light in a nonconducting medium other than avacuum is less than c:

0 0

1

c =

m 0 0

1v =

where is the relative dielectric constant (remember it fromcapacitors?) and mis called the relative permeability of themedium.

Because

. m

cv =you can show that

These equations are not on yourequation sheet, but you have permissionto use them for tomorrows homework (ifneeded): use v for the wave speed, and

replace 0by 0and 0by m0.


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Satellite:2

earth2

GmMmvF = =R R

2 Rv =

T

Solve the above to get the distance R of the satellite from thecenter of the earth, then subtract 6.38x106m to get the heightof the satellite above the ground.

Gravitational force of sun: sun2

GmMF=

R


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For problem 32.53, the satellite orbits at a height of 2.02x107m above the ground. You dont need to calculate this height;just use the number above


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Todays agenda:



Momentum and Radiation Pressure of an ElectromagneticWave.


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The magnitude S represents the rate at which energy flowsthrough a unit surface area perpendicular to the direction ofwave propagation.

Energy Carried by Electromagnetic Waves

Electromagnetic waves carry energy, and as they propagate

through space they can transfer energy to objects in their path.The rate of flow of energy in an electromagnetic wave isdescribed by a vector S,called the Poynting vector.*

0

1S = E B

Thus, S represents power per unit area. The direction of S isalong the direction of wave propagation. 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/c

we can write

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

0

1S = E B

For an EM wave

so

E B =EB

.0

EB

S =

.

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 the waveintensity I.

2 2max max max max

average

0 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/(00).


z maxB =B sin kx - t

Notice the 2s inthis equation.

EM wave propagatingalong 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 any instant:

instantaneous

instantaneous

energypowertimeS= =

area area

average

average

energy

powertimeI= S = =area area

Note: Saverageand mean the same thing!


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For an electromagnetic wave, the instantaneous energy densityassociated with the magnetic field equals the instantaneousenergy density associated with the electric field.

22

B E 0

0

1 1 Bu =u = E =

2 2

22

B E 0

0

Bu=u +u = E =

Hence, in a given volume the energy is equally shared by thetwo fields. The total energy density is equal to the sum of theenergy densities associated with the electric and magneticfields:


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When we average this instantaneous energy density over oneor more cycles of an electromagnetic wave, we again get afactor of from the time average of sin2(kx - t).

so we see thatRecall

The intensity of an electromagnetic wave equals the averageenergy density multiplied by the speed of light.

22

B E 0

0

Bu=u +u = E =

22 max

0 max0

B1 1u = E =

2 2

2 2max max

average0 0

E cB1 1

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

, 2E 0 max

1u = E

4,

2

max

B0

B1u =

4and

Spring 2012:

slide 29 next


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Homework Clarification

Problem 32.23 also calculate the energy density due to theelectric and magnetic fields

, 2E 0 max1

u = E4

.

2max

B

0

B1u =

4

this means calculate the average energy densities

Not assigned this semester.


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Quiz 8.
http://localhost/var/www/apps/conversion/tmp/scratch_7/03-21-2012.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_7/03-21-2012.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_7/03-21-2012.ppt


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Example: a radio station on the surface of the earth radiates asinusoidal wave with an average total power of 50 kW.Assuming the wave is radiated equally in all directions above

the ground, find the amplitude of the electric and magneticfields detected by a satellite 100 km from the antenna.

R

Station

SatelliteAll the radiated power passesthrough the hemisphericalsurface*so the average powerper unit area (the intensity) is

4

-7 2

22 5average

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 of space.

Todays lecture is broughtto 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

-2V=2.45 10m

-2

-11maxmax 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 the previoustwo slides, calculate the average energy densities associatedwith the electric and magnetic field.

2E 0 max1

u = E4

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

-15E 3J

u =1.33 10

m

2max

B

0

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 surface pressure is exerted onthat surface.

If we assume that EM radiation is incident on an object for atime 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 is in thedirection of the incident radiation.

incident

U

p = (total absorption)c

Todays lecture is brought

to you by the letter P.


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If instead of being totally absorbed the radiation is totallyreflected by the object, and the reflection is along the incident

path, then the magnitude of the momentum change of theobject is twice that for total absorption.

The direction of the momentum change of the object is againin 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 2ndLaw (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 case of totalabsorption:

SP =

c

For the average radiation pressure, replace S by =Savg=I:

average

rad

S IP = =

c c

Electromagnetic waves also carry momentum through spacewith a momentum density of Saverage/c

2=I/c2. This is not on yourequation sheet but you have special permission to use it intomorrows homework, if necessary.

Todays lecture is brought

to you by the letter P.


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rad

IP = (total absorption)

c

rad

2IP = (total reflection)

c

incident

incident

reflected

absorbed

Using 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 the sunsradiation is incident perpendicular to the panels and is

completely absorbed find the average solar power absorbedand the average force associated with the radiation pressure.The intensity (I or Saverage) of sunlight prior to passing throughthe earths atmosphere is 1.4 kW/m2.

3 2 32WPower =IA= 1.4 10 4.0 m =5.6 10 W=5.6 kWmAssuming 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 -52

rad

NF=P A= 4.7 10 4.0 m =1.9 10 Nm

Caution! The letter P(or p) has been used

in this lecture forpower, pressure, and

momentum!

Thats because todays lecture is

brought to you by the letter P.


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lecture21 gel. em

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