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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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