Classical Optics Prof D Rich

Study of light Wave Particle duality of photons

E = h Einstein Photoelectric Effect h = 663 x 10-34 Js

Wave aspect of light stems from the unification of

Through Maxwellrsquos equations in vacuum

BE

amp

o

E

0 B

t

BE

t

EjB ooo

Gaussrsquos Law

Magnetic Flux Law (ie no magnetic monopoles)

dvsdEo

1

0sdB

Sdt

BrdE

Sdt

ESdjrdB ooo

Generalized Circuital (Amperersquos) Law

Faradayrsquos Law of Induction

Development of the idea of E-M wave propagation

Let

o

ooo

ooo

BE

tEj

t

BB

t

EEEj

EBBE

22

)()(

22

Letrsquos use the differential vector identity )()()( RQQRRQ

Let BREQ

then

o

oo

o

BE

tEjBE

22

1 22

Take an integral d 3r and use the Divergence theorem

sdGrdGV

3

G

V

dVBE

tdVEjsdBE

V o

o

V

o

22

22

uE uB

1 2 3

2 Rate at which the Kinetic Energy of the particles change

3 Rate at which Energy stored in the Fields increase units in Joulessec or Watts

dVvrFrnvnqj

)()(

DefineoBES

Poynting vector points in the direction in

which the fields E and B transport Energy)Units Wm2(

Power = Force Velocity

1 Rate at which total energy in V increases Note that = rate at which energy flows out of V across the boundary

sdBE o

We want to search for plane E-M wave solutions in a vacuum So from Maxwellrsquos equations using = 0 and j = 0 we have

t

EB

t

BEBE oo

00

Let

ktzBjtzBitzBB

ktzEjtzEitzEE

zyx

zyx

ˆ)(ˆ)(ˆ)(

ˆ)(ˆ)(ˆ)(

Fields in a sinusoidal electromagnetic wave The E-Field is parallel to the x-axis and to the x-z plane and the B-Field is parallel to the y -axis and to the y-z plane The propagation direction is along z

0000

z

BB

z

EE zz

This assumption leads to conditions on the E and B components

00ˆˆ

ˆˆˆ

t

B

t

Bk

y

E

x

Ek

t

B

EEEzyx

kji

t

BE

zzxy

zyx

0

t

E

t

EB z

oo

0 0

Similarly

So we can take Ez = Const = 0 and Bz = Const = 0 without a loss of generality

Also from above

t

E

z

B

t

B

z

Eii

t

E

z

B

t

B

z

Ei

xoo

yyx

yoo

xxy

)(

)(

For (i) take z and t

2

2

2

22

2

22

2

21

z

B

t

B

zt

E

t

E

zt

B

z

Ex

oo

xyyoo

xy

We arrive immediately to expressions of the 1D Wave Equation

We can now identify the constant representing the speed of light

sm

ckgmkgmcs

coo

1003

10410858

11 8

273

2212

)as predicted by Maxwell in the year 1861(

The 3D expressions are as follows

2

22

2

22

t

BB

t

EE oooo

In general the 3D wave equation has the form

)()()(

)(1)(

21

2

2

22

vtkkrgcvtkkrfctr

t

tr

vtr

For case (i) above )cos()( tkzEtzE yoy ie a simple harmonic plane wave solution

Insertion into the 1D wave eq yields

kcckk

tkzEtkzEk

oo

yoooyo

)cos()cos( 22

Note that k is the wavevector and is the direction of propagation k without the arrow is the magnitude and is called the wavenumber

zkkkk ˆˆ

Again use the result

)cos(

)sin(

tkzkE

B

tkzkEz

E

t

B

yox

yoyx

Thus if

thenEkBEkitkz

kEB

jtkzEE

yo

yo

1

1ˆ)cos(

ˆ)cos(

)in general(

Consider a 1D propagation of an arbitrary wave of the form f = f (z-vt)

z1

t1

z2

t2

z3

t3

f(z)

z

If z-vt = const then f(const) = const1

z 3gt z 2gt z1

t 3gt t 2gt t1

z1

t1

z2

t2

z3

t3

Ey

z

If z-vt = const then Ey(const) = const1

z 3gt z 2gt z1

t 3gt t 2gt t1

The same analysis can be performed of course for harmonic waves

2

2ˆcosˆ)cos( ck

vjtk

zkEjtkzEE phyoyo

Same analysis can be applied in 3D for a plane wave Surfaces on which the amplitude has a constant phase form a set of planes which are perpendicular to the propagation direction For harmonic plane waves

))(()()( tkkrkitrki AeAer

2

1 2

k

ee iki

Planes are such that the phase defines a set of planes

ˆ

constrk

constrk

Use the vector identity

kB

k

cBk

kE

kEkkEEkkEkkBk

BACCABCBA

2

2

)()(1

)(1

)()()(

0

using

BEcEc

BkE

BEkB xy 11

As shown before itrsquos possible to express in 3D using a complex field representation

BB

EE

Re)(

Re)()(

)(

BeBtr

EeEtrtrki

o

trkio

oo

oo

ootrki

o

EkB

tt

BEand

BkBkBalso

EkEkeEki

E

1

ReRe

00

0

0Re0)(

BE

E

E

With the complex representations it is possible to derive explicit relations between E B and k

Letrsquos examine the flow of energy again using the Poynting vector S

EkBEkitkzk

EB

jtkzEEBES

yo

yoo

1

1ˆ)cos(

ˆ)cos(1

zEczE

cSbecause

ckczk

Etkz

ijk

EtkzS

yooo

yo

tt

ooyo

o

yoo

ˆ2

1ˆ

2

1

2

1()cos

1ˆ)(cos

1

ˆˆ)(cos1

22

2

222

22

A

A

WWatt

s

J

Sec

EnergyAS

Therefore we can define irradiance as

2

2 oo

tE

cSI

In older texts (and in discussion) the term ldquointensityrdquo is also used

Average Energy

Areatime

The energy per unit volume or energy density stored in the fields can be written as before

222

2222 1

2222BE

c

EEBEuuu

oo

o

o

o

oBE

Note again a factor of frac12 must be added for the time averages

22

2

1

2 oo

ootBEt

BE

uuu

The units

volume

momentum

cm

J

c

Eo 1

3

2

Thus similar to the Poynting vector the E-M momentum P per unit volume that exerts a radiation pressure is given by

kEc

P

ktkzEc

BEP

yoo

t

yoo

o

ˆ2

1

ˆ)(cos

2

22

Consider a slab of E-M radiation whose thickness is ct and cross-sectional area A

ct

Area A

Vol=Act k

If the light is absorbed by an object the momentum transfer is given by the impulse forcetime

tBE

yooyoo

tr

t

uu

EcEcA

FP

tAcPptF

22

2

1

2

1

Thus the energyvol contained in the E-M propagation also represents the pressure exerted on an object

For example if Eyo= 1 Vm then ltPrgt = 44 x 10-12 Nm2 10-17 atm

Note also that cPIorccE

E

c

P

Sr

ooyoo

o

yo

r

1

21

21

2

2

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15

Development of the idea of E-M wave propagation

Let

o

ooo

ooo

BE

tEj

t

BB

t

EEEj

EBBE

22

)()(

22

Letrsquos use the differential vector identity )()()( RQQRRQ

Let BREQ

then

o

oo

o

BE

tEjBE

22

1 22

Take an integral d 3r and use the Divergence theorem

sdGrdGV

3

G

V

dVBE

tdVEjsdBE

V o

o

V

o

22

22

uE uB

1 2 3

2 Rate at which the Kinetic Energy of the particles change

3 Rate at which Energy stored in the Fields increase units in Joulessec or Watts

dVvrFrnvnqj

)()(

DefineoBES

Poynting vector points in the direction in

which the fields E and B transport Energy)Units Wm2(

Power = Force Velocity

1 Rate at which total energy in V increases Note that = rate at which energy flows out of V across the boundary

sdBE o

We want to search for plane E-M wave solutions in a vacuum So from Maxwellrsquos equations using = 0 and j = 0 we have

t

EB

t

BEBE oo

00

Let

ktzBjtzBitzBB

ktzEjtzEitzEE

zyx

zyx

ˆ)(ˆ)(ˆ)(

ˆ)(ˆ)(ˆ)(

Fields in a sinusoidal electromagnetic wave The E-Field is parallel to the x-axis and to the x-z plane and the B-Field is parallel to the y -axis and to the y-z plane The propagation direction is along z

0000

z

BB

z

EE zz

This assumption leads to conditions on the E and B components

00ˆˆ

ˆˆˆ

t

B

t

Bk

y

E

x

Ek

t

B

EEEzyx

kji

t

BE

zzxy

zyx

0

t

E

t

EB z

oo

0 0

Similarly

So we can take Ez = Const = 0 and Bz = Const = 0 without a loss of generality

Also from above

t

E

z

B

t

B

z

Eii

t

E

z

B

t

B

z

Ei

xoo

yyx

yoo

xxy

)(

)(

For (i) take z and t

2

2

2

22

2

22

2

21

z

B

t

B

zt

E

t

E

zt

B

z

Ex

oo

xyyoo

xy

We arrive immediately to expressions of the 1D Wave Equation

We can now identify the constant representing the speed of light

sm

ckgmkgmcs

coo

1003

10410858

11 8

273

2212

)as predicted by Maxwell in the year 1861(

The 3D expressions are as follows

2

22

2

22

t

BB

t

EE oooo

In general the 3D wave equation has the form

)()()(

)(1)(

21

2

2

22

vtkkrgcvtkkrfctr

t

tr

vtr

For case (i) above )cos()( tkzEtzE yoy ie a simple harmonic plane wave solution

Insertion into the 1D wave eq yields

kcckk

tkzEtkzEk

oo

yoooyo

)cos()cos( 22

Note that k is the wavevector and is the direction of propagation k without the arrow is the magnitude and is called the wavenumber

zkkkk ˆˆ

Again use the result

)cos(

)sin(

tkzkE

B

tkzkEz

E

t

B

yox

yoyx

Thus if

thenEkBEkitkz

kEB

jtkzEE

yo

yo

1

1ˆ)cos(

ˆ)cos(

)in general(

Consider a 1D propagation of an arbitrary wave of the form f = f (z-vt)

z1

t1

z2

t2

z3

t3

f(z)

z

If z-vt = const then f(const) = const1

z 3gt z 2gt z1

t 3gt t 2gt t1

z1

t1

z2

t2

z3

t3

Ey

z

If z-vt = const then Ey(const) = const1

z 3gt z 2gt z1

t 3gt t 2gt t1

The same analysis can be performed of course for harmonic waves

2

2ˆcosˆ)cos( ck

vjtk

zkEjtkzEE phyoyo

Same analysis can be applied in 3D for a plane wave Surfaces on which the amplitude has a constant phase form a set of planes which are perpendicular to the propagation direction For harmonic plane waves

))(()()( tkkrkitrki AeAer

2

1 2

k

ee iki

Planes are such that the phase defines a set of planes

ˆ

constrk

constrk

Use the vector identity

kB

k

cBk

kE

kEkkEEkkEkkBk

BACCABCBA

2

2

)()(1

)(1

)()()(

0

using

BEcEc

BkE

BEkB xy 11

As shown before itrsquos possible to express in 3D using a complex field representation

BB

EE

Re)(

Re)()(

)(

BeBtr

EeEtrtrki

o

trkio

oo

oo

ootrki

o

EkB

tt

BEand

BkBkBalso

EkEkeEki

E

1

ReRe

00

0

0Re0)(

BE

E

E

With the complex representations it is possible to derive explicit relations between E B and k

Letrsquos examine the flow of energy again using the Poynting vector S

EkBEkitkzk

EB

jtkzEEBES

yo

yoo

1

1ˆ)cos(

ˆ)cos(1

zEczE

cSbecause

ckczk

Etkz

ijk

EtkzS

yooo

yo

tt

ooyo

o

yoo

ˆ2

1ˆ

2

1

2

1()cos

1ˆ)(cos

1

ˆˆ)(cos1

22

2

222

22

A

A

WWatt

s

J

Sec

EnergyAS

Therefore we can define irradiance as

2

2 oo

tE

cSI

In older texts (and in discussion) the term ldquointensityrdquo is also used

Average Energy

Areatime

The energy per unit volume or energy density stored in the fields can be written as before

222

2222 1

2222BE

c

EEBEuuu

oo

o

o

o

oBE

Note again a factor of frac12 must be added for the time averages

22

2

1

2 oo

ootBEt

BE

uuu

The units

volume

momentum

cm

J

c

Eo 1

3

2

Thus similar to the Poynting vector the E-M momentum P per unit volume that exerts a radiation pressure is given by

kEc

P

ktkzEc

BEP

yoo

t

yoo

o

ˆ2

1

ˆ)(cos

2

22

Consider a slab of E-M radiation whose thickness is ct and cross-sectional area A

ct

Area A

Vol=Act k

If the light is absorbed by an object the momentum transfer is given by the impulse forcetime

tBE

yooyoo

tr

t

uu

EcEcA

FP

tAcPptF

22

2

1

2

1

Thus the energyvol contained in the E-M propagation also represents the pressure exerted on an object

For example if Eyo= 1 Vm then ltPrgt = 44 x 10-12 Nm2 10-17 atm

Note also that cPIorccE

E

c

P

Sr

ooyoo

o

yo

r

1

21

21

2

2

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15

Take an integral d 3r and use the Divergence theorem

sdGrdGV

3

G

V

dVBE

tdVEjsdBE

V o

o

V

o

22

22

uE uB

1 2 3

2 Rate at which the Kinetic Energy of the particles change

3 Rate at which Energy stored in the Fields increase units in Joulessec or Watts

dVvrFrnvnqj

)()(

DefineoBES

Poynting vector points in the direction in

which the fields E and B transport Energy)Units Wm2(

Power = Force Velocity

1 Rate at which total energy in V increases Note that = rate at which energy flows out of V across the boundary

sdBE o

We want to search for plane E-M wave solutions in a vacuum So from Maxwellrsquos equations using = 0 and j = 0 we have

t

EB

t

BEBE oo

00

Let

ktzBjtzBitzBB

ktzEjtzEitzEE

zyx

zyx

ˆ)(ˆ)(ˆ)(

ˆ)(ˆ)(ˆ)(

Fields in a sinusoidal electromagnetic wave The E-Field is parallel to the x-axis and to the x-z plane and the B-Field is parallel to the y -axis and to the y-z plane The propagation direction is along z

0000

z

BB

z

EE zz

This assumption leads to conditions on the E and B components

00ˆˆ

ˆˆˆ

t

B

t

Bk

y

E

x

Ek

t

B

EEEzyx

kji

t

BE

zzxy

zyx

0

t

E

t

EB z

oo

0 0

Similarly

So we can take Ez = Const = 0 and Bz = Const = 0 without a loss of generality

Also from above

t

E

z

B

t

B

z

Eii

t

E

z

B

t

B

z

Ei

xoo

yyx

yoo

xxy

)(

)(

For (i) take z and t

2

2

2

22

2

22

2

21

z

B

t

B

zt

E

t

E

zt

B

z

Ex

oo

xyyoo

xy

We arrive immediately to expressions of the 1D Wave Equation

We can now identify the constant representing the speed of light

sm

ckgmkgmcs

coo

1003

10410858

11 8

273

2212

)as predicted by Maxwell in the year 1861(

The 3D expressions are as follows

2

22

2

22

t

BB

t

EE oooo

In general the 3D wave equation has the form

)()()(

)(1)(

21

2

2

22

vtkkrgcvtkkrfctr

t

tr

vtr

For case (i) above )cos()( tkzEtzE yoy ie a simple harmonic plane wave solution

Insertion into the 1D wave eq yields

kcckk

tkzEtkzEk

oo

yoooyo

)cos()cos( 22

Note that k is the wavevector and is the direction of propagation k without the arrow is the magnitude and is called the wavenumber

zkkkk ˆˆ

Again use the result

)cos(

)sin(

tkzkE

B

tkzkEz

E

t

B

yox

yoyx

Thus if

thenEkBEkitkz

kEB

jtkzEE

yo

yo

1

1ˆ)cos(

ˆ)cos(

)in general(

Consider a 1D propagation of an arbitrary wave of the form f = f (z-vt)

z1

t1

z2

t2

z3

t3

f(z)

z

If z-vt = const then f(const) = const1

z 3gt z 2gt z1

t 3gt t 2gt t1

z1

t1

z2

t2

z3

t3

Ey

z

If z-vt = const then Ey(const) = const1

z 3gt z 2gt z1

t 3gt t 2gt t1

The same analysis can be performed of course for harmonic waves

2

2ˆcosˆ)cos( ck

vjtk

zkEjtkzEE phyoyo

Same analysis can be applied in 3D for a plane wave Surfaces on which the amplitude has a constant phase form a set of planes which are perpendicular to the propagation direction For harmonic plane waves

))(()()( tkkrkitrki AeAer

2

1 2

k

ee iki

Planes are such that the phase defines a set of planes

ˆ

constrk

constrk

Use the vector identity

kB

k

cBk

kE

kEkkEEkkEkkBk

BACCABCBA

2

2

)()(1

)(1

)()()(

0

using

BEcEc

BkE

BEkB xy 11

As shown before itrsquos possible to express in 3D using a complex field representation

BB

EE

Re)(

Re)()(

)(

BeBtr

EeEtrtrki

o

trkio

oo

oo

ootrki

o

EkB

tt

BEand

BkBkBalso

EkEkeEki

E

1

ReRe

00

0

0Re0)(

BE

E

E

With the complex representations it is possible to derive explicit relations between E B and k

Letrsquos examine the flow of energy again using the Poynting vector S

EkBEkitkzk

EB

jtkzEEBES

yo

yoo

1

1ˆ)cos(

ˆ)cos(1

zEczE

cSbecause

ckczk

Etkz

ijk

EtkzS

yooo

yo

tt

ooyo

o

yoo

ˆ2

1ˆ

2

1

2

1()cos

1ˆ)(cos

1

ˆˆ)(cos1

22

2

222

22

A

A

WWatt

s

J

Sec

EnergyAS

Therefore we can define irradiance as

2

2 oo

tE

cSI

In older texts (and in discussion) the term ldquointensityrdquo is also used

Average Energy

Areatime

The energy per unit volume or energy density stored in the fields can be written as before

222

2222 1

2222BE

c

EEBEuuu

oo

o

o

o

oBE

Note again a factor of frac12 must be added for the time averages

22

2

1

2 oo

ootBEt

BE

uuu

The units

volume

momentum

cm

J

c

Eo 1

3

2

Thus similar to the Poynting vector the E-M momentum P per unit volume that exerts a radiation pressure is given by

kEc

P

ktkzEc

BEP

yoo

t

yoo

o

ˆ2

1

ˆ)(cos

2

22

Consider a slab of E-M radiation whose thickness is ct and cross-sectional area A

ct

Area A

Vol=Act k

If the light is absorbed by an object the momentum transfer is given by the impulse forcetime

tBE

yooyoo

tr

t

uu

EcEcA

FP

tAcPptF

22

2

1

2

1

Thus the energyvol contained in the E-M propagation also represents the pressure exerted on an object

For example if Eyo= 1 Vm then ltPrgt = 44 x 10-12 Nm2 10-17 atm

Note also that cPIorccE

E

c

P

Sr

ooyoo

o

yo

r

1

21

21

2

2

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15

We want to search for plane E-M wave solutions in a vacuum So from Maxwellrsquos equations using = 0 and j = 0 we have

t

EB

t

BEBE oo

00

Let

ktzBjtzBitzBB

ktzEjtzEitzEE

zyx

zyx

ˆ)(ˆ)(ˆ)(

ˆ)(ˆ)(ˆ)(

Fields in a sinusoidal electromagnetic wave The E-Field is parallel to the x-axis and to the x-z plane and the B-Field is parallel to the y -axis and to the y-z plane The propagation direction is along z

0000

z

BB

z

EE zz

This assumption leads to conditions on the E and B components

00ˆˆ

ˆˆˆ

t

B

t

Bk

y

E

x

Ek

t

B

EEEzyx

kji

t

BE

zzxy

zyx

0

t

E

t

EB z

oo

0 0

Similarly

So we can take Ez = Const = 0 and Bz = Const = 0 without a loss of generality

Also from above

t

E

z

B

t

B

z

Eii

t

E

z

B

t

B

z

Ei

xoo

yyx

yoo

xxy

)(

)(

For (i) take z and t

2

2

2

22

2

22

2

21

z

B

t

B

zt

E

t

E

zt

B

z

Ex

oo

xyyoo

xy

We arrive immediately to expressions of the 1D Wave Equation

We can now identify the constant representing the speed of light

sm

ckgmkgmcs

coo

1003

10410858

11 8

273

2212

)as predicted by Maxwell in the year 1861(

The 3D expressions are as follows

2

22

2

22

t

BB

t

EE oooo

In general the 3D wave equation has the form

)()()(

)(1)(

21

2

2

22

vtkkrgcvtkkrfctr

t

tr

vtr

For case (i) above )cos()( tkzEtzE yoy ie a simple harmonic plane wave solution

Insertion into the 1D wave eq yields

kcckk

tkzEtkzEk

oo

yoooyo

)cos()cos( 22

Note that k is the wavevector and is the direction of propagation k without the arrow is the magnitude and is called the wavenumber

zkkkk ˆˆ

Again use the result

)cos(

)sin(

tkzkE

B

tkzkEz

E

t

B

yox

yoyx

Thus if

thenEkBEkitkz

kEB

jtkzEE

yo

yo

1

1ˆ)cos(

ˆ)cos(

)in general(

Consider a 1D propagation of an arbitrary wave of the form f = f (z-vt)

z1

t1

z2

t2

z3

t3

f(z)

z

If z-vt = const then f(const) = const1

z 3gt z 2gt z1

t 3gt t 2gt t1

z1

t1

z2

t2

z3

t3

Ey

z

If z-vt = const then Ey(const) = const1

z 3gt z 2gt z1

t 3gt t 2gt t1

The same analysis can be performed of course for harmonic waves

2

2ˆcosˆ)cos( ck

vjtk

zkEjtkzEE phyoyo

Same analysis can be applied in 3D for a plane wave Surfaces on which the amplitude has a constant phase form a set of planes which are perpendicular to the propagation direction For harmonic plane waves

))(()()( tkkrkitrki AeAer

2

1 2

k

ee iki

Planes are such that the phase defines a set of planes

ˆ

constrk

constrk

Use the vector identity

kB

k

cBk

kE

kEkkEEkkEkkBk

BACCABCBA

2

2

)()(1

)(1

)()()(

0

using

BEcEc

BkE

BEkB xy 11

As shown before itrsquos possible to express in 3D using a complex field representation

BB

EE

Re)(

Re)()(

)(

BeBtr

EeEtrtrki

o

trkio

oo

oo

ootrki

o

EkB

tt

BEand

BkBkBalso

EkEkeEki

E

1

ReRe

00

0

0Re0)(

BE

E

E

With the complex representations it is possible to derive explicit relations between E B and k

Letrsquos examine the flow of energy again using the Poynting vector S

EkBEkitkzk

EB

jtkzEEBES

yo

yoo

1

1ˆ)cos(

ˆ)cos(1

zEczE

cSbecause

ckczk

Etkz

ijk

EtkzS

yooo

yo

tt

ooyo

o

yoo

ˆ2

1ˆ

2

1

2

1()cos

1ˆ)(cos

1

ˆˆ)(cos1

22

2

222

22

A

A

WWatt

s

J

Sec

EnergyAS

Therefore we can define irradiance as

2

2 oo

tE

cSI

In older texts (and in discussion) the term ldquointensityrdquo is also used

Average Energy

Areatime

The energy per unit volume or energy density stored in the fields can be written as before

222

2222 1

2222BE

c

EEBEuuu

oo

o

o

o

oBE

Note again a factor of frac12 must be added for the time averages

22

2

1

2 oo

ootBEt

BE

uuu

The units

volume

momentum

cm

J

c

Eo 1

3

2

Thus similar to the Poynting vector the E-M momentum P per unit volume that exerts a radiation pressure is given by

kEc

P

ktkzEc

BEP

yoo

t

yoo

o

ˆ2

1

ˆ)(cos

2

22

Consider a slab of E-M radiation whose thickness is ct and cross-sectional area A

ct

Area A

Vol=Act k

If the light is absorbed by an object the momentum transfer is given by the impulse forcetime

tBE

yooyoo

tr

t

uu

EcEcA

FP

tAcPptF

22

2

1

2

1

Thus the energyvol contained in the E-M propagation also represents the pressure exerted on an object

For example if Eyo= 1 Vm then ltPrgt = 44 x 10-12 Nm2 10-17 atm

Note also that cPIorccE

E

c

P

Sr

ooyoo

o

yo

r

1

21

21

2

2

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15

00ˆˆ

ˆˆˆ

t

B

t

Bk

y

E

x

Ek

t

B

EEEzyx

kji

t

BE

zzxy

zyx

0

t

E

t

EB z

oo

0 0

Similarly

So we can take Ez = Const = 0 and Bz = Const = 0 without a loss of generality

Also from above

t

E

z

B

t

B

z

Eii

t

E

z

B

t

B

z

Ei

xoo

yyx

yoo

xxy

)(

)(

For (i) take z and t

2

2

2

22

2

22

2

21

z

B

t

B

zt

E

t

E

zt

B

z

Ex

oo

xyyoo

xy

We arrive immediately to expressions of the 1D Wave Equation

We can now identify the constant representing the speed of light

sm

ckgmkgmcs

coo

1003

10410858

11 8

273

2212

)as predicted by Maxwell in the year 1861(

The 3D expressions are as follows

2

22

2

22

t

BB

t

EE oooo

In general the 3D wave equation has the form

)()()(

)(1)(

21

2

2

22

vtkkrgcvtkkrfctr

t

tr

vtr

For case (i) above )cos()( tkzEtzE yoy ie a simple harmonic plane wave solution

Insertion into the 1D wave eq yields

kcckk

tkzEtkzEk

oo

yoooyo

)cos()cos( 22

Note that k is the wavevector and is the direction of propagation k without the arrow is the magnitude and is called the wavenumber

zkkkk ˆˆ

Again use the result

)cos(

)sin(

tkzkE

B

tkzkEz

E

t

B

yox

yoyx

Thus if

thenEkBEkitkz

kEB

jtkzEE

yo

yo

1

1ˆ)cos(

ˆ)cos(

)in general(

Consider a 1D propagation of an arbitrary wave of the form f = f (z-vt)

z1

t1

z2

t2

z3

t3

f(z)

z

If z-vt = const then f(const) = const1

z 3gt z 2gt z1

t 3gt t 2gt t1

z1

t1

z2

t2

z3

t3

Ey

z

If z-vt = const then Ey(const) = const1

z 3gt z 2gt z1

t 3gt t 2gt t1

The same analysis can be performed of course for harmonic waves

2

2ˆcosˆ)cos( ck

vjtk

zkEjtkzEE phyoyo

Same analysis can be applied in 3D for a plane wave Surfaces on which the amplitude has a constant phase form a set of planes which are perpendicular to the propagation direction For harmonic plane waves

))(()()( tkkrkitrki AeAer

2

1 2

k

ee iki

Planes are such that the phase defines a set of planes

ˆ

constrk

constrk

Use the vector identity

kB

k

cBk

kE

kEkkEEkkEkkBk

BACCABCBA

2

2

)()(1

)(1

)()()(

0

using

BEcEc

BkE

BEkB xy 11

As shown before itrsquos possible to express in 3D using a complex field representation

BB

EE

Re)(

Re)()(

)(

BeBtr

EeEtrtrki

o

trkio

oo

oo

ootrki

o

EkB

tt

BEand

BkBkBalso

EkEkeEki

E

1

ReRe

00

0

0Re0)(

BE

E

E

With the complex representations it is possible to derive explicit relations between E B and k

Letrsquos examine the flow of energy again using the Poynting vector S

EkBEkitkzk

EB

jtkzEEBES

yo

yoo

1

1ˆ)cos(

ˆ)cos(1

zEczE

cSbecause

ckczk

Etkz

ijk

EtkzS

yooo

yo

tt

ooyo

o

yoo

ˆ2

1ˆ

2

1

2

1()cos

1ˆ)(cos

1

ˆˆ)(cos1

22

2

222

22

A

A

WWatt

s

J

Sec

EnergyAS

Therefore we can define irradiance as

2

2 oo

tE

cSI

In older texts (and in discussion) the term ldquointensityrdquo is also used

Average Energy

Areatime

The energy per unit volume or energy density stored in the fields can be written as before

222

2222 1

2222BE

c

EEBEuuu

oo

o

o

o

oBE

Note again a factor of frac12 must be added for the time averages

22

2

1

2 oo

ootBEt

BE

uuu

The units

volume

momentum

cm

J

c

Eo 1

3

2

Thus similar to the Poynting vector the E-M momentum P per unit volume that exerts a radiation pressure is given by

kEc

P

ktkzEc

BEP

yoo

t

yoo

o

ˆ2

1

ˆ)(cos

2

22

Consider a slab of E-M radiation whose thickness is ct and cross-sectional area A

ct

Area A

Vol=Act k

If the light is absorbed by an object the momentum transfer is given by the impulse forcetime

tBE

yooyoo

tr

t

uu

EcEcA

FP

tAcPptF

22

2

1

2

1

Thus the energyvol contained in the E-M propagation also represents the pressure exerted on an object

For example if Eyo= 1 Vm then ltPrgt = 44 x 10-12 Nm2 10-17 atm

Note also that cPIorccE

E

c

P

Sr

ooyoo

o

yo

r

1

21

21

2

2

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15

For (i) take z and t

2

2

2

22

2

22

2

21

z

B

t

B

zt

E

t

E

zt

B

z

Ex

oo

xyyoo

xy

We arrive immediately to expressions of the 1D Wave Equation

We can now identify the constant representing the speed of light

sm

ckgmkgmcs

coo

1003

10410858

11 8

273

2212

)as predicted by Maxwell in the year 1861(

The 3D expressions are as follows

2

22

2

22

t

BB

t

EE oooo

In general the 3D wave equation has the form

)()()(

)(1)(

21

2

2

22

vtkkrgcvtkkrfctr

t

tr

vtr

For case (i) above )cos()( tkzEtzE yoy ie a simple harmonic plane wave solution

Insertion into the 1D wave eq yields

kcckk

tkzEtkzEk

oo

yoooyo

)cos()cos( 22

Note that k is the wavevector and is the direction of propagation k without the arrow is the magnitude and is called the wavenumber

zkkkk ˆˆ

Again use the result

)cos(

)sin(

tkzkE

B

tkzkEz

E

t

B

yox

yoyx

Thus if

thenEkBEkitkz

kEB

jtkzEE

yo

yo

1

1ˆ)cos(

ˆ)cos(

)in general(

Consider a 1D propagation of an arbitrary wave of the form f = f (z-vt)

z1

t1

z2

t2

z3

t3

f(z)

z

If z-vt = const then f(const) = const1

z 3gt z 2gt z1

t 3gt t 2gt t1

z1

t1

z2

t2

z3

t3

Ey

z

If z-vt = const then Ey(const) = const1

z 3gt z 2gt z1

t 3gt t 2gt t1

The same analysis can be performed of course for harmonic waves

2

2ˆcosˆ)cos( ck

vjtk

zkEjtkzEE phyoyo

Same analysis can be applied in 3D for a plane wave Surfaces on which the amplitude has a constant phase form a set of planes which are perpendicular to the propagation direction For harmonic plane waves

))(()()( tkkrkitrki AeAer

2

1 2

k

ee iki

Planes are such that the phase defines a set of planes

ˆ

constrk

constrk

Use the vector identity

kB

k

cBk

kE

kEkkEEkkEkkBk

BACCABCBA

2

2

)()(1

)(1

)()()(

0

using

BEcEc

BkE

BEkB xy 11

As shown before itrsquos possible to express in 3D using a complex field representation

BB

EE

Re)(

Re)()(

)(

BeBtr

EeEtrtrki

o

trkio

oo

oo

ootrki

o

EkB

tt

BEand

BkBkBalso

EkEkeEki

E

1

ReRe

00

0

0Re0)(

BE

E

E

With the complex representations it is possible to derive explicit relations between E B and k

Letrsquos examine the flow of energy again using the Poynting vector S

EkBEkitkzk

EB

jtkzEEBES

yo

yoo

1

1ˆ)cos(

ˆ)cos(1

zEczE

cSbecause

ckczk

Etkz

ijk

EtkzS

yooo

yo

tt

ooyo

o

yoo

ˆ2

1ˆ

2

1

2

1()cos

1ˆ)(cos

1

ˆˆ)(cos1

22

2

222

22

A

A

WWatt

s

J

Sec

EnergyAS

Therefore we can define irradiance as

2

2 oo

tE

cSI

In older texts (and in discussion) the term ldquointensityrdquo is also used

Average Energy

Areatime

The energy per unit volume or energy density stored in the fields can be written as before

222

2222 1

2222BE

c

EEBEuuu

oo

o

o

o

oBE

Note again a factor of frac12 must be added for the time averages

22

2

1

2 oo

ootBEt

BE

uuu

The units

volume

momentum

cm

J

c

Eo 1

3

2

Thus similar to the Poynting vector the E-M momentum P per unit volume that exerts a radiation pressure is given by

kEc

P

ktkzEc

BEP

yoo

t

yoo

o

ˆ2

1

ˆ)(cos

2

22

Consider a slab of E-M radiation whose thickness is ct and cross-sectional area A

ct

Area A

Vol=Act k

If the light is absorbed by an object the momentum transfer is given by the impulse forcetime

tBE

yooyoo

tr

t

uu

EcEcA

FP

tAcPptF

22

2

1

2

1

Thus the energyvol contained in the E-M propagation also represents the pressure exerted on an object

For example if Eyo= 1 Vm then ltPrgt = 44 x 10-12 Nm2 10-17 atm

Note also that cPIorccE

E

c

P

Sr

ooyoo

o

yo

r

1

21

21

2

2

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15

In general the 3D wave equation has the form

)()()(

)(1)(

21

2

2

22

vtkkrgcvtkkrfctr

t

tr

vtr

For case (i) above )cos()( tkzEtzE yoy ie a simple harmonic plane wave solution

Insertion into the 1D wave eq yields

kcckk

tkzEtkzEk

oo

yoooyo

)cos()cos( 22

Note that k is the wavevector and is the direction of propagation k without the arrow is the magnitude and is called the wavenumber

zkkkk ˆˆ

Again use the result

)cos(

)sin(

tkzkE

B

tkzkEz

E

t

B

yox

yoyx

Thus if

thenEkBEkitkz

kEB

jtkzEE

yo

yo

1

1ˆ)cos(

ˆ)cos(

)in general(

Consider a 1D propagation of an arbitrary wave of the form f = f (z-vt)

z1

t1

z2

t2

z3

t3

f(z)

z

If z-vt = const then f(const) = const1

z 3gt z 2gt z1

t 3gt t 2gt t1

z1

t1

z2

t2

z3

t3

Ey

z

If z-vt = const then Ey(const) = const1

z 3gt z 2gt z1

t 3gt t 2gt t1

The same analysis can be performed of course for harmonic waves

2

2ˆcosˆ)cos( ck

vjtk

zkEjtkzEE phyoyo

Same analysis can be applied in 3D for a plane wave Surfaces on which the amplitude has a constant phase form a set of planes which are perpendicular to the propagation direction For harmonic plane waves

))(()()( tkkrkitrki AeAer

2

1 2

k

ee iki

Planes are such that the phase defines a set of planes

ˆ

constrk

constrk

Use the vector identity

kB

k

cBk

kE

kEkkEEkkEkkBk

BACCABCBA

2

2

)()(1

)(1

)()()(

0

using

BEcEc

BkE

BEkB xy 11

As shown before itrsquos possible to express in 3D using a complex field representation

BB

EE

Re)(

Re)()(

)(

BeBtr

EeEtrtrki

o

trkio

oo

oo

ootrki

o

EkB

tt

BEand

BkBkBalso

EkEkeEki

E

1

ReRe

00

0

0Re0)(

BE

E

E

With the complex representations it is possible to derive explicit relations between E B and k

Letrsquos examine the flow of energy again using the Poynting vector S

EkBEkitkzk

EB

jtkzEEBES

yo

yoo

1

1ˆ)cos(

ˆ)cos(1

zEczE

cSbecause

ckczk

Etkz

ijk

EtkzS

yooo

yo

tt

ooyo

o

yoo

ˆ2

1ˆ

2

1

2

1()cos

1ˆ)(cos

1

ˆˆ)(cos1

22

2

222

22

A

A

WWatt

s

J

Sec

EnergyAS

Therefore we can define irradiance as

2

2 oo

tE

cSI

In older texts (and in discussion) the term ldquointensityrdquo is also used

Average Energy

Areatime

The energy per unit volume or energy density stored in the fields can be written as before

222

2222 1

2222BE

c

EEBEuuu

oo

o

o

o

oBE

Note again a factor of frac12 must be added for the time averages

22

2

1

2 oo

ootBEt

BE

uuu

The units

volume

momentum

cm

J

c

Eo 1

3

2

Thus similar to the Poynting vector the E-M momentum P per unit volume that exerts a radiation pressure is given by

kEc

P

ktkzEc

BEP

yoo

t

yoo

o

ˆ2

1

ˆ)(cos

2

22

Consider a slab of E-M radiation whose thickness is ct and cross-sectional area A

ct

Area A

Vol=Act k

If the light is absorbed by an object the momentum transfer is given by the impulse forcetime

tBE

yooyoo

tr

t

uu

EcEcA

FP

tAcPptF

22

2

1

2

1

Thus the energyvol contained in the E-M propagation also represents the pressure exerted on an object

For example if Eyo= 1 Vm then ltPrgt = 44 x 10-12 Nm2 10-17 atm

Note also that cPIorccE

E

c

P

Sr

ooyoo

o

yo

r

1

21

21

2

2

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15

Again use the result

)cos(

)sin(

tkzkE

B

tkzkEz

E

t

B

yox

yoyx

Thus if

thenEkBEkitkz

kEB

jtkzEE

yo

yo

1

1ˆ)cos(

ˆ)cos(

)in general(

Consider a 1D propagation of an arbitrary wave of the form f = f (z-vt)

z1

t1

z2

t2

z3

t3

f(z)

z

If z-vt = const then f(const) = const1

z 3gt z 2gt z1

t 3gt t 2gt t1

z1

t1

z2

t2

z3

t3

Ey

z

If z-vt = const then Ey(const) = const1

z 3gt z 2gt z1

t 3gt t 2gt t1

The same analysis can be performed of course for harmonic waves

2

2ˆcosˆ)cos( ck

vjtk

zkEjtkzEE phyoyo

Same analysis can be applied in 3D for a plane wave Surfaces on which the amplitude has a constant phase form a set of planes which are perpendicular to the propagation direction For harmonic plane waves

))(()()( tkkrkitrki AeAer

2

1 2

k

ee iki

Planes are such that the phase defines a set of planes

ˆ

constrk

constrk

Use the vector identity

kB

k

cBk

kE

kEkkEEkkEkkBk

BACCABCBA

2

2

)()(1

)(1

)()()(

0

using

BEcEc

BkE

BEkB xy 11

As shown before itrsquos possible to express in 3D using a complex field representation

BB

EE

Re)(

Re)()(

)(

BeBtr

EeEtrtrki

o

trkio

oo

oo

ootrki

o

EkB

tt

BEand

BkBkBalso

EkEkeEki

E

1

ReRe

00

0

0Re0)(

BE

E

E

With the complex representations it is possible to derive explicit relations between E B and k

Letrsquos examine the flow of energy again using the Poynting vector S

EkBEkitkzk

EB

jtkzEEBES

yo

yoo

1

1ˆ)cos(

ˆ)cos(1

zEczE

cSbecause

ckczk

Etkz

ijk

EtkzS

yooo

yo

tt

ooyo

o

yoo

ˆ2

1ˆ

2

1

2

1()cos

1ˆ)(cos

1

ˆˆ)(cos1

22

2

222

22

A

A

WWatt

s

J

Sec

EnergyAS

Therefore we can define irradiance as

2

2 oo

tE

cSI

In older texts (and in discussion) the term ldquointensityrdquo is also used

Average Energy

Areatime

The energy per unit volume or energy density stored in the fields can be written as before

222

2222 1

2222BE

c

EEBEuuu

oo

o

o

o

oBE

Note again a factor of frac12 must be added for the time averages

22

2

1

2 oo

ootBEt

BE

uuu

The units

volume

momentum

cm

J

c

Eo 1

3

2

Thus similar to the Poynting vector the E-M momentum P per unit volume that exerts a radiation pressure is given by

kEc

P

ktkzEc

BEP

yoo

t

yoo

o

ˆ2

1

ˆ)(cos

2

22

Consider a slab of E-M radiation whose thickness is ct and cross-sectional area A

ct

Area A

Vol=Act k

If the light is absorbed by an object the momentum transfer is given by the impulse forcetime

tBE

yooyoo

tr

t

uu

EcEcA

FP

tAcPptF

22

2

1

2

1

Thus the energyvol contained in the E-M propagation also represents the pressure exerted on an object

For example if Eyo= 1 Vm then ltPrgt = 44 x 10-12 Nm2 10-17 atm

Note also that cPIorccE

E

c

P

Sr

ooyoo

o

yo

r

1

21

21

2

2

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15

z1

t1

z2

t2

z3

t3

Ey

z

If z-vt = const then Ey(const) = const1

z 3gt z 2gt z1

t 3gt t 2gt t1

The same analysis can be performed of course for harmonic waves

2

2ˆcosˆ)cos( ck

vjtk

zkEjtkzEE phyoyo

Same analysis can be applied in 3D for a plane wave Surfaces on which the amplitude has a constant phase form a set of planes which are perpendicular to the propagation direction For harmonic plane waves

))(()()( tkkrkitrki AeAer

2

1 2

k

ee iki

Planes are such that the phase defines a set of planes

ˆ

constrk

constrk

Use the vector identity

kB

k

cBk

kE

kEkkEEkkEkkBk

BACCABCBA

2

2

)()(1

)(1

)()()(

0

using

BEcEc

BkE

BEkB xy 11

As shown before itrsquos possible to express in 3D using a complex field representation

BB

EE

Re)(

Re)()(

)(

BeBtr

EeEtrtrki

o

trkio

oo

oo

ootrki

o

EkB

tt

BEand

BkBkBalso

EkEkeEki

E

1

ReRe

00

0

0Re0)(

BE

E

E

With the complex representations it is possible to derive explicit relations between E B and k

Letrsquos examine the flow of energy again using the Poynting vector S

EkBEkitkzk

EB

jtkzEEBES

yo

yoo

1

1ˆ)cos(

ˆ)cos(1

zEczE

cSbecause

ckczk

Etkz

ijk

EtkzS

yooo

yo

tt

ooyo

o

yoo

ˆ2

1ˆ

2

1

2

1()cos

1ˆ)(cos

1

ˆˆ)(cos1

22

2

222

22

A

A

WWatt

s

J

Sec

EnergyAS

Therefore we can define irradiance as

2

2 oo

tE

cSI

In older texts (and in discussion) the term ldquointensityrdquo is also used

Average Energy

Areatime

The energy per unit volume or energy density stored in the fields can be written as before

222

2222 1

2222BE

c

EEBEuuu

oo

o

o

o

oBE

Note again a factor of frac12 must be added for the time averages

22

2

1

2 oo

ootBEt

BE

uuu

The units

volume

momentum

cm

J

c

Eo 1

3

2

Thus similar to the Poynting vector the E-M momentum P per unit volume that exerts a radiation pressure is given by

kEc

P

ktkzEc

BEP

yoo

t

yoo

o

ˆ2

1

ˆ)(cos

2

22

Consider a slab of E-M radiation whose thickness is ct and cross-sectional area A

ct

Area A

Vol=Act k

If the light is absorbed by an object the momentum transfer is given by the impulse forcetime

tBE

yooyoo

tr

t

uu

EcEcA

FP

tAcPptF

22

2

1

2

1

Thus the energyvol contained in the E-M propagation also represents the pressure exerted on an object

For example if Eyo= 1 Vm then ltPrgt = 44 x 10-12 Nm2 10-17 atm

Note also that cPIorccE

E

c

P

Sr

ooyoo

o

yo

r

1

21

21

2

2

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15

2

1 2

k

ee iki

Planes are such that the phase defines a set of planes

ˆ

constrk

constrk

Use the vector identity

kB

k

cBk

kE

kEkkEEkkEkkBk

BACCABCBA

2

2

)()(1

)(1

)()()(

0

using

BEcEc

BkE

BEkB xy 11

As shown before itrsquos possible to express in 3D using a complex field representation

BB

EE

Re)(

Re)()(

)(

BeBtr

EeEtrtrki

o

trkio

oo

oo

ootrki

o

EkB

tt

BEand

BkBkBalso

EkEkeEki

E

1

ReRe

00

0

0Re0)(

BE

E

E

With the complex representations it is possible to derive explicit relations between E B and k

Letrsquos examine the flow of energy again using the Poynting vector S

EkBEkitkzk

EB

jtkzEEBES

yo

yoo

1

1ˆ)cos(

ˆ)cos(1

zEczE

cSbecause

ckczk

Etkz

ijk

EtkzS

yooo

yo

tt

ooyo

o

yoo

ˆ2

1ˆ

2

1

2

1()cos

1ˆ)(cos

1

ˆˆ)(cos1

22

2

222

22

A

A

WWatt

s

J

Sec

EnergyAS

Therefore we can define irradiance as

2

2 oo

tE

cSI

In older texts (and in discussion) the term ldquointensityrdquo is also used

Average Energy

Areatime

The energy per unit volume or energy density stored in the fields can be written as before

222

2222 1

2222BE

c

EEBEuuu

oo

o

o

o

oBE

Note again a factor of frac12 must be added for the time averages

22

2

1

2 oo

ootBEt

BE

uuu

The units

volume

momentum

cm

J

c

Eo 1

3

2

Thus similar to the Poynting vector the E-M momentum P per unit volume that exerts a radiation pressure is given by

kEc

P

ktkzEc

BEP

yoo

t

yoo

o

ˆ2

1

ˆ)(cos

2

22

Consider a slab of E-M radiation whose thickness is ct and cross-sectional area A

ct

Area A

Vol=Act k

If the light is absorbed by an object the momentum transfer is given by the impulse forcetime

tBE

yooyoo

tr

t

uu

EcEcA

FP

tAcPptF

22

2

1

2

1

Thus the energyvol contained in the E-M propagation also represents the pressure exerted on an object

For example if Eyo= 1 Vm then ltPrgt = 44 x 10-12 Nm2 10-17 atm

Note also that cPIorccE

E

c

P

Sr

ooyoo

o

yo

r

1

21

21

2

2

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15

Use the vector identity

kB

k

cBk

kE

kEkkEEkkEkkBk

BACCABCBA

2

2

)()(1

)(1

)()()(

0

using

BEcEc

BkE

BEkB xy 11

As shown before itrsquos possible to express in 3D using a complex field representation

BB

EE

Re)(

Re)()(

)(

BeBtr

EeEtrtrki

o

trkio

oo

oo

ootrki

o

EkB

tt

BEand

BkBkBalso

EkEkeEki

E

1

ReRe

00

0

0Re0)(

BE

E

E

With the complex representations it is possible to derive explicit relations between E B and k

Letrsquos examine the flow of energy again using the Poynting vector S

EkBEkitkzk

EB

jtkzEEBES

yo

yoo

1

1ˆ)cos(

ˆ)cos(1

zEczE

cSbecause

ckczk

Etkz

ijk

EtkzS

yooo

yo

tt

ooyo

o

yoo

ˆ2

1ˆ

2

1

2

1()cos

1ˆ)(cos

1

ˆˆ)(cos1

22

2

222

22

A

A

WWatt

s

J

Sec

EnergyAS

Therefore we can define irradiance as

2

2 oo

tE

cSI

In older texts (and in discussion) the term ldquointensityrdquo is also used

Average Energy

Areatime

The energy per unit volume or energy density stored in the fields can be written as before

222

2222 1

2222BE

c

EEBEuuu

oo

o

o

o

oBE

Note again a factor of frac12 must be added for the time averages

22

2

1

2 oo

ootBEt

BE

uuu

The units

volume

momentum

cm

J

c

Eo 1

3

2

Thus similar to the Poynting vector the E-M momentum P per unit volume that exerts a radiation pressure is given by

kEc

P

ktkzEc

BEP

yoo

t

yoo

o

ˆ2

1

ˆ)(cos

2

22

Consider a slab of E-M radiation whose thickness is ct and cross-sectional area A

ct

Area A

Vol=Act k

If the light is absorbed by an object the momentum transfer is given by the impulse forcetime

tBE

yooyoo

tr

t

uu

EcEcA

FP

tAcPptF

22

2

1

2

1

Thus the energyvol contained in the E-M propagation also represents the pressure exerted on an object

For example if Eyo= 1 Vm then ltPrgt = 44 x 10-12 Nm2 10-17 atm

Note also that cPIorccE

E

c

P

Sr

ooyoo

o

yo

r

1

21

21

2

2

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15

oo

oo

ootrki

o

EkB

tt

BEand

BkBkBalso

EkEkeEki

E

1

ReRe

00

0

0Re0)(

BE

E

E

With the complex representations it is possible to derive explicit relations between E B and k

Letrsquos examine the flow of energy again using the Poynting vector S

EkBEkitkzk

EB

jtkzEEBES

yo

yoo

1

1ˆ)cos(

ˆ)cos(1

zEczE

cSbecause

ckczk

Etkz

ijk

EtkzS

yooo

yo

tt

ooyo

o

yoo

ˆ2

1ˆ

2

1

2

1()cos

1ˆ)(cos

1

ˆˆ)(cos1

22

2

222

22

A

A

WWatt

s

J

Sec

EnergyAS

Therefore we can define irradiance as

2

2 oo

tE

cSI

In older texts (and in discussion) the term ldquointensityrdquo is also used

Average Energy

Areatime

The energy per unit volume or energy density stored in the fields can be written as before

222

2222 1

2222BE

c

EEBEuuu

oo

o

o

o

oBE

Note again a factor of frac12 must be added for the time averages

22

2

1

2 oo

ootBEt

BE

uuu

The units

volume

momentum

cm

J

c

Eo 1

3

2

Thus similar to the Poynting vector the E-M momentum P per unit volume that exerts a radiation pressure is given by

kEc

P

ktkzEc

BEP

yoo

t

yoo

o

ˆ2

1

ˆ)(cos

2

22

Consider a slab of E-M radiation whose thickness is ct and cross-sectional area A

ct

Area A

Vol=Act k

If the light is absorbed by an object the momentum transfer is given by the impulse forcetime

tBE

yooyoo

tr

t

uu

EcEcA

FP

tAcPptF

22

2

1

2

1

Thus the energyvol contained in the E-M propagation also represents the pressure exerted on an object

For example if Eyo= 1 Vm then ltPrgt = 44 x 10-12 Nm2 10-17 atm

Note also that cPIorccE

E

c

P

Sr

ooyoo

o

yo

r

1

21

21

2

2

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15

zEczE

cSbecause

ckczk

Etkz

ijk

EtkzS

yooo

yo

tt

ooyo

o

yoo

ˆ2

1ˆ

2

1

2

1()cos

1ˆ)(cos

1

ˆˆ)(cos1

22

2

222

22

A

A

WWatt

s

J

Sec

EnergyAS

Therefore we can define irradiance as

2

2 oo

tE

cSI

In older texts (and in discussion) the term ldquointensityrdquo is also used

Average Energy

Areatime

The energy per unit volume or energy density stored in the fields can be written as before

222

2222 1

2222BE

c

EEBEuuu

oo

o

o

o

oBE

Note again a factor of frac12 must be added for the time averages

22

2

1

2 oo

ootBEt

BE

uuu

The units

volume

momentum

cm

J

c

Eo 1

3

2

Thus similar to the Poynting vector the E-M momentum P per unit volume that exerts a radiation pressure is given by

kEc

P

ktkzEc

BEP

yoo

t

yoo

o

ˆ2

1

ˆ)(cos

2

22

Consider a slab of E-M radiation whose thickness is ct and cross-sectional area A

ct

Area A

Vol=Act k

If the light is absorbed by an object the momentum transfer is given by the impulse forcetime

tBE

yooyoo

tr

t

uu

EcEcA

FP

tAcPptF

22

2

1

2

1

Thus the energyvol contained in the E-M propagation also represents the pressure exerted on an object

For example if Eyo= 1 Vm then ltPrgt = 44 x 10-12 Nm2 10-17 atm

Note also that cPIorccE

E

c

P

Sr

ooyoo

o

yo

r

1

21

21

2

2

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15

The energy per unit volume or energy density stored in the fields can be written as before

222

2222 1

2222BE

c

EEBEuuu

oo

o

o

o

oBE

Note again a factor of frac12 must be added for the time averages

22

2

1

2 oo

ootBEt

BE

uuu

The units

volume

momentum

cm

J

c

Eo 1

3

2

Thus similar to the Poynting vector the E-M momentum P per unit volume that exerts a radiation pressure is given by

kEc

P

ktkzEc

BEP

yoo

t

yoo

o

ˆ2

1

ˆ)(cos

2

22

Consider a slab of E-M radiation whose thickness is ct and cross-sectional area A

ct

Area A

Vol=Act k

If the light is absorbed by an object the momentum transfer is given by the impulse forcetime

tBE

yooyoo

tr

t

uu

EcEcA

FP

tAcPptF

22

2

1

2

1

Thus the energyvol contained in the E-M propagation also represents the pressure exerted on an object

For example if Eyo= 1 Vm then ltPrgt = 44 x 10-12 Nm2 10-17 atm

Note also that cPIorccE

E

c

P

Sr

ooyoo

o

yo

r

1

21

21

2

2

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15

Consider a slab of E-M radiation whose thickness is ct and cross-sectional area A

ct

Area A

Vol=Act k

If the light is absorbed by an object the momentum transfer is given by the impulse forcetime

tBE

yooyoo

tr

t

uu

EcEcA

FP

tAcPptF

22

2

1

2

1

Thus the energyvol contained in the E-M propagation also represents the pressure exerted on an object

For example if Eyo= 1 Vm then ltPrgt = 44 x 10-12 Nm2 10-17 atm

Note also that cPIorccE

E

c

P

Sr

ooyoo

o

yo

r

1

21

21

2

2

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15