chapter 3. electromagnetic theory, photons, and...

Hecht by YHLEE;100310; 3-1

Chapter 3. Electromagnetic Theory, Photons, and Light Classical Electrodynamics : Energy transfer by electromagnetic waves Quantum Electrodynamics : Energy transfer by massless particles(photons) 3.1 Basic Laws of Electromagnetic Theory Electric charges, time varying magnetic fields ® Electric field Electric currents, time varying electric fields ® Magnetic field The force on a charge q

r r r rF qE qv B= + ´ : Lorentz force

A. Faraday’s Induction Law A time-varying magnetic flux through a loop ® Induced Electromotive Force(emf) or voltage

emf ddt B= - F

r rE dl

C·z .

r rB dS

A·zz

® r r r r r rE dl d

dtB dS dB

dtdS

C A A· = - · Þ - ·z zz zz : Faraday’s law

(Wrong direction of dlr in the book)


B. Gauss’s Law-Electric Electric flux : E A A

D dS D dS^F = = ·òò òòrr

Ò Ò

The total electric flux from a closed surface = The total enclosed charge

A VD dS dVr· =òò òòò

rrÒ ,

where D Ee=r r

, e e e= r o , electric permittivity Permittivity of free space Relative permittivity, Dielectric constant

A point charge at the origin ® D is constant over a sphere, 24E D rpF = . ® E qF =

Coulomb’s law, 24 o

qErpe

=

C. Gauss’s Law-Magnetic No isolated magnetic charge

r rB dS

A· =zz 0

D. Ampere’s Law A current penetrating a closed loop induces magnetic field around the loop

Using current density 2[ / ]J A m

C A

H dl J dS· = ·ò òòr rr r

Ñ : Ampere’s law where B Hm=

r r with m m m= r o , permeability

Permeability of free space Relative permeability


· A capacitor

(1) Current through the surface A1 0¹ ® Magnetic field along curve C ¹ 0 Current through the surface A2 0= ® Magnetic field along curve C=0 (???) (2) Magnetic field around the wire 0¹ . B ¹ 0 between the plates even if there is no current. (They are indistinguishable)

· Between the capacitor plates

E QA

=e

® e¶¶Et

iA

J D= ºr

, Displacement current density

Differentiate both sides The generalized Ampere’s law

C A

DH dl J dSt

¶¶

æ ö· = + ·ç ÷

è øò òò

rr rr rÑ : Maxwell


E. Maxwell’s Equations

In free space : , , 0o o Je e m m r= = = =

C A

BE dl dSt

¶¶

· = - ·ò òòrr rr

Ñ ® BEt

¶¶

Ñ ´ = -rr

(1)

C A

DH dl dSt

¶¶

· = ·ò òòrr rr

Ñ ® DHt

¶¶

Ñ ´ =rr

(2)

0AB dS· =òò

rrÒ ® 0BÑ · =

r (3)

0AD dS· =òò

rrÒ ® 0DÑ · =

r (4)

Differential form

3.2 Electromagnetic Waves Three basic properties

(1) Perpendicularity of the fields A time-varying D

r generates H

r that is perpendicular to D

r [Fig. 3.12]

“ rB “

rE “

rB [Fig. 3.5]

® Transverse nature of rE and H

r

(2) Interdependence of and E H

r r

® Time varying r rE B and regenerate each other endlessly

(3) Symmetry of the equations ® The propagation direction will be symmetrical to both and E H

r r

Differential wave equation From (1)

Ñ ´ Ñ ´ = - Ñ ´r rE

tBe j e j¶

¶

Using (2)

Ñ Ñ · - Ñ = -r r rE E E

to oe j 22

2e m¶

¶ ® Ñ =2

2

2

r rE E

to oe m¶

¶

=0 · In Cartesian coord. by separation of variables

¶

¶

¶

¶

¶

¶e m

¶

¶

2

2

2

2

2

2

2

2Ex

Ey

Ez

Et

x x xo o

x+ + = , : o

Wave equation known long before Maxwell

with 1/ ov e m=

Maxwell calculated

v m so o

= =´ · ´

» ´- -

1 1

8 854 10 4 103 10

12 78

e m p./ :

Fizeau's measurement, c=315,300 /Km s

® Maxwell concluded that light is an electromagnetic wave.


A. Transverse Waves

A plane wave propagating along x-axis

® ¶

¶

¶

¶

r rE x y z t

yE x y z t

z, , , , , ,b g b g

= = 0

® ( ), ( , ) ( , ) ( , )x y zE E x t E x t x E x t y E x t z= Þ + +r r ) ) )

From divergence eq. (4)

Ñ · =rE 0 ® ¶

¶

¶

¶¶¶

Ex

Ey

Ez

x y z+ + = 0 ® ¶¶Ex

x = 0 ,

E constx = (not a traveling wave) = 0, = 0

® 0xE =

( , ) ( , )y zE E x t y E x t z= +r ) ) : Transverse wave

· Assume rE yE x ty= $ ,b g

From (1)

$ $ $x y z

x y zE

Bt

y

¶¶

¶¶

¶¶

¶¶

0 0

= -

r ® $ $ $ $z

Ex

x Bt

yBt

z Bt

y x y z¶

¶¶¶

¶

¶¶¶

= - - - (5)

= 0, = 0, to match $z ® Only E By z and exist : Transverse wave

· A plane harmonic wave

( ) [ ], cosy oE x t E kx t= - w + e The B- field from (5)

[ ]1 cosyz o

EB dt E kx t

x c¶

= - Þ - w + e¶ò

® E cBy z= : r rE B and are in phase

r rE B´ // Beam propagation direction


3.3 Energy and Momentum A. The Poynting Vector The electromagnetic wave transports energy and momentum The energy density of an electric field : u EE o= 1

22e

The energy density of a magnetic field : u BB o= 1

22

m

From the relation E cB= : u uE B= The total energy density of an electromagnetic wave E Bu u u= + · The power density of an electromagnetic wave [W/m2] (energy per unit time per unit area) S E H= ´

r r r : Poynting vector W m/ 2

A harmonic plane wave

r r r rE E k r to= · -cos we j ,

r r r rB B k r to= · -cos we j ®

r r r r rS c E B k r to o o= ´ · -2 2e we j e jcos (1)

Time average of Harmonic Functions

( ) ( )/2

/2

1 t T

T t Tf t f t dt

T+

-º ò ® [ ]2 1 1cos ( ) 1 sinc( )cos(2 )

2 2Tt T tw = + n w » (2)

1/T >> n B. The Irradiance The time average of power density From (1) and (2)

º = e ´ Þ ´ Þ er r r r

2 21 1 1 2 2 2o o o o o o oT

I S c E B E H c E

rE is more efficient on a charge than

rB ®

rE is called “optical field”

The Inverse Square Law A point source at the center

® A spherical wave

From the energy conservation 4 41

21 2

22p pr I r r I rb g b g= ,

® r E r r E r consto o1 1 2 2b g b g= = . Electric field ~ /1 r ,

Irradiance ~ /1 2r : Inverse square law


C. Photons Light is absorbed and emitted in photons

engergy quanta

Photons are stable, chargeless, massless. Photons exist only at the speed c. Photons can only be seen from the result of being created or annihilated. · Experiments on the quantum nature

(1) Atoms emit localized light quanta in random directions. A weak light source surrounded by the identical photodetectors at equal distances. ® Independent detections, one at a time. (2) Atoms recoils when emitting photons. A beam of excited atoms

® Spontaneous emission of photons in random direction ® Atoms are kicked backward ® The beam spreads out.

Radiation Pressure and Momentum Maxwell 1873 : There is a pressure in the direction normal to the wave, which is equal to the energy density.

The electric field of a wave incident on a conductor ® Current on the conductor ® The wave’s magnetic field generates forces on the current. The radiation pressure, .P , by Maxwell

E BSu u uc

= = + Þ.P S : Poynting vector, c : speed of light

powerarea speed

force speedarea speed

forcearea·

=··

=

Example The pressure of solar radiation on earth ® 4 7 10 6 2. /´ - N m (The atmospheric pressure: 105 2N m/ ) ~10 tons

Viking spacecraft would have missed Mars by 15,000 Km without consideration of solar radiation pressure.

Space ship driven by solar radiation pressure can be possible.


3.4 Radiation A stationary charge ® A constant

rE -field, no

rB -field

A uniformly moving charge ® r rE B and fields but not coupled

The source of electromagnetic wave ® Nonuniformly moving charges Accelerated charges in a straight line : Linear accelerator Circulating charges : Cyclotron Oscillating charges : Radio antenna A. Linearly Accelerating Charges (a) A charge at rest ® Uniform distribution of field lines (b) A charge with const. speed ® Nonuniform distribution of field lines

· Accelerating electron from O1 to O4

Uniform acceleration Stationary at O for t < 0 . from O1 to O4 Acceleration from O to O1 . Linear move from O1 to O2 . ® The transverse component starts to appear. (It changes in time in space and therefore magnetic field is associated )

The radial component of the electric field : ~ 12r

The transverse component of the electric field : ~ 1r

As r ® ¥ , the transverse component dominates. Radiation field


· Radiation by a linearly accelerating charge

B. Synchrotron Radiation Charges are accelerated in a circular path by a magnetic field and radiate in a narrow cone.

The frequency of the emission ~ The frequency around the orbit ® Tunable from IR to X-ray

The higher speed ® The shorter backward lobe, the longer forward lobe.

For v c» ® The polarization in the plane of motion.

[Fig. 3.31] Synchrotron radiation from the Crab Nebula Due to circulating charges trapped by magnetic field

C. Electric Dipole Radiation An oscillating dipole : One plus and one minus charges vibrate to and fro along a straight line

The dipole moment p t p tob g = cosw : op qd= , (charge X max. separation)


In the near region : rE has the form of a static electric field

In the closed loop region : No specific wavelength (rE is composed of 5 terms)

In the far region (radiation zone) : A fixed wavelength.

r rE B and are transverse, mutually perpendicular,

and in phase.

E p k kr tr

o

o=

-2

4sin cosq

pe

wb g

The irradiance

I pc r

o

oq

w

p e

qb g =2 4

2 3

2

232sin : The higher the freq. w , the stronger the radiation


D. The Emission of Light from Atoms The source of emission and absorption of light ® The bound charges (particularly, the outer electrons of atoms) Valence electrons Light is emitted during readjustments of the outer charges. An atom is in ground state ® Every electron in the lowest possible energy state. An atom is in excited state ® One or more electrons into a higher energy level At low temperatures ® Atoms in their ground states At higher temperatures ® More atoms in excited state due to atomic collisions (glow discharge, flame, spark and so on) Enough energy to an atom (By collision with another atom, electron, photon) ® Excitation of the atom

The absorbed energy = The energy difference between the initial and final states

The lifetime of the excited atoms : 10 108 9- -~ seconds ® Relaxation to the ground state by emitting light or by releasing thermal energy hnD =E , resonance freq. Interatomic collision

The emission spectra of single atoms or low pressure gases ® Sharp lines The emission spectra of solids or liquids ® Frequency bands


3.5 Light in Matter In a homogeneous, isotropic dielectric (nonconducting material) e e m mo o® ®, ® v = 1/ em The absolute index of refraction

n cv o o

r rº = Þem

e me m ® n r= e , Maxwell’s relation

mr » 1 , except for ferromagnetic materials ( mr = - ´ -1 22 10 5. for diamond)

The index of refraction depends on frequency

® Dispersion

Scattering and Absorption

Interaction of the incident light with atoms (1) Dissipative absorption at resonance freq. Photon’s energy = Excitation energy of the atom ® Absorption of the photon by the atom ® Transfer of the absorbed energy to thermal energy by collisions before emitting photons (2) Nonresonant scattering (Lorentz Model) Incident photon energy < Excitation energy of atom ® Oscillation of electron cloud at the same freq. as the incident light. No excitation, the atom is still in the ground state. (Oscillating dipole) ® Radiation at the same freq. with the same energy as the incident light. (Elastic scattering)

(3) Spontaneous emission Excitation ® Spontaneous emission ® Excitation … » -10 8 s

® An atom can emit 108 photons per second ® The saturation, constantly emitting and reexcited, at I W m» 102 2/ An atom behaves as a source of spherical electromagnetic waves.


A. Dispersion Macroscopic view: A matter responses to the electric or magnetic field via e or m Microscopic view: The atom interacts with electric field via electric dipole Applied electric field ® Distorted charge distribution ® Internal field (External) (Electric dipole moment) The electric polarization

rP : Dipole moment per unit volume

oD E P Ee e= + Þr r r r

· The origin of polarization

(1) Orientational polarization : Alignment of polar molecules in the electric field Permanent dipole moment from unequal sharing of valence electrons (water molecule) Polar molecules ® Rotation by E ® Reduced response (heavy) Higher freq. Example Water er » 80 from 0 Hz to 1010 Hz, er falls off quickly above 1010 Hz (2) Electronic polarization : Distortion of electron cloud in nonpolar molecules. Electrons ® Good response to optical freq. (light) (5 1014´ Hz ) (3) Ionic or atomic polarization : Shift of the positive and negative ions in ionic crystals NaCl,… Ÿ The eq. of motion Restoring force of the electron for small x : F kx= - From Newton’s second law

2

22

i to o

d x dxqE e m x m mdtdt

- w = w + + g : wo k m2 = /

Damping effect (Energy loss during oscillation due to Mass X Acceleration interaction between neighboring atoms)

Restoring force Driving force from the incident wave The solution

( ) ( )2 2

/ i to

o

q mx t E ei

- wé ùê ú=ê úw - w - gwë û

(1) Without the driving force (no incident wave) ® The oscillator vibrates at the resonance freq. wo (2) For ow w= : E tb g and x tb g are in phase

(3) For ow w? : x tb g is 180o out of phase with E tb g


· The electric polarization, The permittivity

( )

2

2 2

/oo

o

q NE mP qx N

i= Þ

w - w - gw,

( )( ) ( )

2

2 2

/o o

o

P t q N mE t i

e = e + Þ e +w - w - gw

The dispersion relation

2

22 2

11o o

Nqnm i

æ ö= + ç ÷ç ÷e w - w - gwè ø

> <1 for w wo

< >1 for w wo · A molecule with many oscillators with different resonant frequencies

® 2

22 21

j

jo oj j

fNqnm ie w w g w

æ ö= + ç ÷ç ÷- -è ø

å , f j : oscillator strength, transition probability

woj : characteristic freq.

For w w<< oj : w can be neglected in the eq. ® constant n

For w w® oj : n gradually increases with frequency (normal dispersion)

For w w» oj : The damping term becomes dominant (strong absorption)

dndw

< 0 (anomalous dispersion)

Shaded regions: absorption bands Shaded region: visible band (Note rise in UV and fall in IR) A material opaque near the resonance frequency can be transparent at other frequencies.


3.6 The Electromagnetic-Photon Spectrum In 1867, Maxwell published his theory. The frequency band known at that time : IR-VIS-UV A. Radiofrequency Waves Experiments by Hertz in 1887

Oscillatory discharge(Oscillating electric dipole)

Transmitter Receiver

Brass knob

Copper point

Spark is seen

Hertz’s experiments : Focusing, reflecting, refracting of radiation. Measurement of polarization. Interference for a standing wave and measurement of the wavelength. Frequency in his experiments : A few Hertz ~ 109 Hz (Radiofrequency) B. Microwaves 9 1110 ~10 Hz · Molecules : Electronic, rotational, vibrational energy levels Polar molecules Water molecules : Strong rotational resonances (Microwave oven 2.45 GHz) C. Infrared 11 1410 ~10Hz Hz Near IR : 780 ~ 3,000 nm Intermediate IR : 3,000 ~ 6,000 nm Far IR : 6,000 ~ 15,000 nm Extreme IR : 15,000 nm ~ 1.0 mm Any material can radiate and absorb IR via thermal agitation of its molecules. Molecules have both vibrational and rotational resonances in the IR Example ~ 1

2 of the electromagnetic energy from the sun is IR

More IR than light from light bulbs Human body : 3,000 nm ® peak near 10,000 nm ® extreme IR


D. Light 3 84 10 7 69 1014 14. ~ .´ ´Hz Hz Sources of light Rearragnement of valence electrons in general Thermal radiation by incandescent materials (hot, glowing metal filaments) Randomly accelerated electrons ® Broad emission spectrum Collisions Gas discharge Atoms of a gas ® Excitation ® Radiation Electric discharge, Series of freq. lines E. Ultraviolet 14 1610 ~10 Hz F. X-rays 16 1910 ~10 Hz Its wavelengths are mostly smaller than an atom. G. Gamma Rays The highest energy, shortest wavelength electromagnetic radiations Difficult to observe wavelike properties

chapter 3. electromagnetic theory, photons, and...

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