24. Electromagnetic Wavesphome.postech.ac.kr/user/genphys/download/phy102-24.pdf · Production of...

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24. Electromagnetic Waves 24. Electromagnetic Waves 24-1. Displacement Current and the Ampere’s Law • Ampere’s Law I = 0 I s d B 0 µ = r r • Induction dt d B Φ = ε dt d s d E B Φ = r r Faraday’s Law • Similar form for B-field d E E I dt d dt d s d B 0 0 0 µ = Φ ε µ = Φ r r dt d I E d Φ ε 0 : Displacement current

Transcript of 24. Electromagnetic Wavesphome.postech.ac.kr/user/genphys/download/phy102-24.pdf · Production of...

Page 1: 24. Electromagnetic Wavesphome.postech.ac.kr/user/genphys/download/phy102-24.pdf · Production of Electromagnetic Wave by an Antenna. A charged particle undergoes an acceleration,

24. Electromagnetic Waves24. Electromagnetic Waves24-1. Displacement Current and the Ampere’s Law

• Ampere’s Law

I = 0

IsdB 0µ=⋅∫rr

• Induction

dtd BΦ

−=ε

dtdsdE BΦ

−=⋅∫rr

Faraday’s Law

• Similar form for B-field

dEE I

dtd

dtdsdB 000 µ=

Φεµ=

Φ∝⋅∫

rr

dtdI E

ε≡ 0 : Displacement current

Page 2: 24. Electromagnetic Wavesphome.postech.ac.kr/user/genphys/download/phy102-24.pdf · Production of Electromagnetic Wave by an Antenna. A charged particle undergoes an acceleration,

+ −

Er

ii

C

ε

R⊗ ⊗

Br

?

Q

t

c/te τ−−1

( ) RC,er

ieCQ C/t/t cc =τ=→−= τ−τ− εε 1

riBIsdBπµ

=→µ=⋅∫ 20

0rr

( )c/teCQV τ−−== ε 1 ( )c/te

ddVE τ−−==

ε 1

( )c/tE e

dA τ−−=Φ ε 1

d/t

/t

/t

c

E

ieR

ed

ARC

ed

Adt

drBsdB

c

c

c

00

00

00

00

1

1

2

µ=µ=

εµ=

τεµ=

Φεµ=π⋅=⋅

τ−

τ−

τ−

ε

ε

ε∫

rr

dAC 0ε=

c/td e

Ri τ−ε=

Page 3: 24. Electromagnetic Wavesphome.postech.ac.kr/user/genphys/download/phy102-24.pdf · Production of Electromagnetic Wave by an Antenna. A charged particle undergoes an acceleration,

Er

r

Br

R

dirBsdB 02 µ=π⋅=⋅∫rr

CtEd e

Rdtdi τε

ε= 0

dirB

πµ2

0=

riR

B d20

2πµ

=

for r > R

for r < R

dtdiiisdB E

+=+=⋅∫ 00000 εµµµµrr id : displacement current

0εQAEE =⋅=Φ Gauss’s Law

ε

=ε=⋅

==A

QdA

dQ

CdQE,

CQV

00

dE i

dtdQ

dtd

==Φ

• Magnetic fields are produced both by conduction currents and by changing electric fields.

Page 4: 24. Electromagnetic Wavesphome.postech.ac.kr/user/genphys/download/phy102-24.pdf · Production of Electromagnetic Wave by an Antenna. A charged particle undergoes an acceleration,

24-2. Maxwell’s Equation

0εQAdE =⋅∫

rrGauss’s Law

0=⋅∫ AdBrr

No magnetic monopole

dtdsdE BΦ

−=⋅∫rr

dtdisdB EΦ

µε+µ=⋅∫ 000rr

Faraday’s Law (Induction)

Ampere-Maxwell’s Law

∫∫∫ ερ

=⋅∇=⋅ dvdvEAdE0

rrrrCf)

0ερ

=⋅∇ Err

0=⋅∇=⋅ ∫∫ dvBAdBrrrr

∫∫∫ ⋅−=⋅×∇=⋅ AdBdtdAdEsdE

rrrrrrr

∫∫

∫∫

⋅εµ+⋅µ=

Φεµ+µ=⋅×∇=⋅

AdEdtdAdj

dtdiAdBsdB E

rrrr

rrrrr

000

000

tEjB∂∂

εµ+µ=×∇r

rrr000

tBE∂∂

−=×∇r

rr⇒

0=⋅∇ Brr

djtE rr

=∂∂

ε0 ( )djjBrrrr

+µ=×∇ 0⇒

Lorentz force: force law for a particle of charge q

BvqEqFrrrr

×+=

Page 5: 24. Electromagnetic Wavesphome.postech.ac.kr/user/genphys/download/phy102-24.pdf · Production of Electromagnetic Wave by an Antenna. A charged particle undergoes an acceleration,

24-3. Electromagnetic WaveMaxwell’s equations

dtdsdE

dtd BB Φ

−=⋅⇒Φ

−= ∫ε rr

tBE∂∂

−=×∇r

rr

dtdildB E

incΦ

+=⋅∫ 000 εµµrr

dtdldB EΦ

=⋅∫ 00εµrr

tEB∂∂

=×∇r

rr00εµ

iinc = 0

( )

∂∂

−∂∂

=×∇∂∂

=×∇×∇tB

tE

tB

rrrrrr

0000 εµεµ

( ) BBrrrr

2−∇=×∇×∇

kz

jy

ix

ˆˆˆ∂∂

+∂∂

+∂∂

=∇r

( ) ( ) BBBBrrrrrrrr

22 −∇=∇−⋅∇∇=×∇×∇

( ) ( ) ( )CBABCACBArrrrrrrrr

⋅−⋅=××

2

2

002

tBB

∂∂

=∇r

rεµ

2

2

002

tEE

∂∂

=∇r

rεµ

02

2

002

2

=∂∂

−∂∂

tB

xB εµ

02

2

002

2

=∂∂

−∂∂

tE

xE εµ

Wave equations

Page 6: 24. Electromagnetic Wavesphome.postech.ac.kr/user/genphys/download/phy102-24.pdf · Production of Electromagnetic Wave by an Antenna. A charged particle undergoes an acceleration,

02

2

002

2

=∂∂

−∂∂

tB

xB εµ 02

2

002

2

=∂∂

−∂∂

tE

xE εµ

)cos(0 tkxBB ω−=

)cos(0 tkxEE ω−=

0200

2 =ωεµ−k

cvk

≡==00

1εµ

ω

BEtEB

rrr

rr⊥⇒

∂∂

εµ−=×∇ 00

Speed of Light

EkB ωεµ 00=

ckk

kkBE

==

==

ωωωεµω

2

00

1

smmc /103sec/1099792.2 88 ×≈×=

Electromagnetic Wave

Page 7: 24. Electromagnetic Wavesphome.postech.ac.kr/user/genphys/download/phy102-24.pdf · Production of Electromagnetic Wave by an Antenna. A charged particle undergoes an acceleration,

24-4. Electromagnetic Oscillators and Alternating Current

LC-Circuit -- Oscillator

CQU E 2

20=i Energy

+ + +− − − CQ0

L

S

0=+−CQ

dtdiL

02

2

=+⇒CQ

dtQdL

dtdQi −=

02

2

=+LCQ

dtQd

General solution

LC12 =ω)cos(0 φω += tQQ

tcosQQQQ ω=⇒= 00

tsinQdtdQi ωω=−= 0

CQ

tsinC

QtcosC

Q

tsinQLtcosC

Q

LiC

QUUU BE

2

22

21

2

21

2

20

2202

20

220

2220

22

=

ω+ω=

ωω+ω=

+=+=

Energy conserved

At t = 0,

Total Energy

Page 8: 24. Electromagnetic Wavesphome.postech.ac.kr/user/genphys/download/phy102-24.pdf · Production of Electromagnetic Wave by an Antenna. A charged particle undergoes an acceleration,

QUE UB

CQ2

20

i

Page 9: 24. Electromagnetic Wavesphome.postech.ac.kr/user/genphys/download/phy102-24.pdf · Production of Electromagnetic Wave by an Antenna. A charged particle undergoes an acceleration,

• Alternating Currenttsinω= εε 0

iR=εtsin

Ri ω=ε0

tsinR

RiP ω==ε 2

202

21sinsin

2

0

22 == ∫π

θθθ dave

rmsrmsrmsrms

ave

iRiR

RiR

P

εε

ε

===

==

22

20

20

21

21

20εε =rms 2

0iirms =

Power

Power

( Root mean square )

ε

i

Page 10: 24. Electromagnetic Wavesphome.postech.ac.kr/user/genphys/download/phy102-24.pdf · Production of Electromagnetic Wave by an Antenna. A charged particle undergoes an acceleration,

• Hertz’s Apparatus

+

+

Transmitter Receiver

LCf

π21

= Resonance frequency

24-5. Production of Electromagnetic Wave by an Antenna

A charged particle undergoes an acceleration, and it must radiate energy via electromagnetic radiation.

Magnetic field lines are always perpendicular to Electric field lines.

Page 11: 24. Electromagnetic Wavesphome.postech.ac.kr/user/genphys/download/phy102-24.pdf · Production of Electromagnetic Wave by an Antenna. A charged particle undergoes an acceleration,

24-6. Energy carried by Electromagnetic WavesPoynting Vector : Intensity of an electromagnetic wave

BESrrr

×=0

2

0

2

0

0

1

1

BcEc

EBS

µ=

µ=

µ=

(Watt/m2)

= c

EB

2m

rmsEE = 2

0

2

0

12

1rms

m Ec

Ec

Sµµ

==

202

1 EuE ε=Energy density associated with an Electric field :

2

021 BuB µ

=Energy density associated with a Magnetic field :

For an electromagnetic wave

BE uBEu =µ

=ε=0

22

0 221

Total instantaneous energy density :

0

22

0 µ=ε=+=

BEuuu BE aveave cuSI ==

00

2 1εµ

=c( )0

22

0212

0 2µ=ε=ε= m

maveaveBEEu

Page 12: 24. Electromagnetic Wavesphome.postech.ac.kr/user/genphys/download/phy102-24.pdf · Production of Electromagnetic Wave by an Antenna. A charged particle undergoes an acceleration,

Example 24.2 Fields due to a Point Source

Average power of the source : Pave

Intensity at a distance r

24 rPI ave

π=

2

00

2

22 maxmax

ave Bcc

ESIµ

==

At a distance r

cE

crP

cIB

rcPcIE

maxavemax

avemax

µ=

µ=

πµ

=µ=

200

20

0

22

22

For Pave = 800 W and r = 3.5 m

Emax = 62.6 V/mBmax = 2.09 × 10-7 T

Page 13: 24. Electromagnetic Wavesphome.postech.ac.kr/user/genphys/download/phy102-24.pdf · Production of Electromagnetic Wave by an Antenna. A charged particle undergoes an acceleration,

24-7. Momentum and Radiation Pressure

reflectedincident Total absorption caseMomentum change:

cUp =∆ U: incident energy

Total reflection caseMomentum change:

cUp 2

=∆

tpF∆∆

= tIAU ∆=

Total absorption case

cIAF =⇒ctIAp /∆=∆

cI

AFP .A.T ==Pressure

Total reflection case

cIAF 2

=⇒c/tIAp ∆=∆ 2

cI

AFP .R.T

2==Pressure

Page 14: 24. Electromagnetic Wavesphome.postech.ac.kr/user/genphys/download/phy102-24.pdf · Production of Electromagnetic Wave by an Antenna. A charged particle undergoes an acceleration,

24-8. The Spectrum of Electromagnetic Wave

f : frquencyc = f λ

mmfc 300

101103

6

8

=××

==λ1 MHz :

Page 15: 24. Electromagnetic Wavesphome.postech.ac.kr/user/genphys/download/phy102-24.pdf · Production of Electromagnetic Wave by an Antenna. A charged particle undergoes an acceleration,

24-9. Polarization

0I 021 II = θ= 2

021 cosII

θθ= 2202

1 sincosII

θcosEEy = θ∝ 2cosI