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Transcript
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
LECTURE NOTES 5
ELECTROMAGNETIC WAVES IN VACUUM
THE WAVE EQUATION(S) FOR E AND B
In regions of free space (i.e. the vacuum), where no electric charges, no electric currents and
no matter of any kind are present, Maxwell’s equations (in differential form) are:
1) E r , t 0
B r ,t
3) E r , t
t
2) B r , t 0
Set of coupled
first-order
partial
differential
equations
E r , t 1 E r , t
2
4) B r , t o o
t
c
t
2
c 1 o o
We can de-couple Maxwell’s equations e.g. by applying the curl operator to equations 3) and 4):
1 E
B
B 2
E
c t
t
0
0
1
E 2 E
B
B 2 B 2
E
t
c t
1 B
1 E
2
2
E 2
B 2
t c t
c t t
1 2 E
1 2 B
2
2
B 2 2
E 2 2
c t
c t
These are three-dimensional de-coupled wave equations for E and B - note that they have
exactly the same structure – both are linear, homogeneous, 2nd order differential equations.
Remember that each of the above equations is explicitly dependent on space and time,
i.e. E E r , t and B B r , t :
or:
2
E
r,t
1
2 E r , t 2
c
t 2
2
B
r,t
1
2 B r , t 2
c
t 2
2
1 E r ,t
E r ,t 2
0
c
t 2
2
1 B r ,t
B r ,t 2
0
c
t 2
2
2
Thus, Maxwell’s equations implies that empty space – the vacuum {which is not empty, at
the microscopic scale} – supports the propagation of {macroscopic} electromagnetic waves,
which propagate at the speed of light {in vacuum}: c 1 o o 3 108 m s .
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
1
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
EM waves have associated with them a frequency f and wavelength , related to each other
via c f . At the microscopic level, EM waves consist of large numbers of {massless} real
photons, each carrying energy E hf hc , linear momentum p h hf c E c and
angular momentum z 1 where h = Planck’s constant = 6.626 1034 Joule-sec and h 2 .
EM waves can have any frequency/any wavelength – the continuum of EM waves over the
frequency region 0 f (c.p.s. or Hertz {aka Hz}), or equivalently, over the wavelength
region 0 (m) is known as the electromagnetic spectrum, which has been divided up
(for convenience) into eight bands as shown in the figure below (kindly provided by Prof. Louis
E. Keiner, of Coastal Carolina University, Conway, SC):
2
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
Monochromatic EM Plane Waves:
Monochromatic EM plane waves propagating in free space/the vacuum are sinusoidal EM
plane waves consisting of a single frequency f , wavelength c f , angular frequency
2 f and wavenumber k 2 . They propagate with speed c f k .
In the visible region of the EM spectrum {~380 nm (violet) ≤ λ ≤ ~ 780 nm (red)}, EM light
waves (consisting of real photons) of a given frequency / wavelength are perceived by the human
eye as having a specific, single color. Hence we call such single-frequency, sinusoidal EM
waves mono-chromatic.
EM waves that propagate e.g. in the ẑ direction but which additionally have no explicit x- or
y-dependence are known as plane waves, because for a given time, t the wave front(s) of the EM
wave lie in a plane which is to the ẑ -axis, as shown in the figure below:
x̂
ŷ
ẑ
The planar wavefront
associated with a plane
EM wave propagating in
the kˆ zˆ direction lies
in the x-y plane.
constant everywhere in
(x,y) on this plane.
Note that there also exist spherical EM waves – e.g. emitted from a point source, such as an
atom, a small antenna or a pinhole aperture – the wavefronts associated with these EM waves are
spherical, and thus do not lie in a plane to the direction of propagation of the EM wave:
Portion of a spherical wavefront
associated with a spherical wave
n.b. If the point source is infinitely far away from observer, then a spherical wave → plane wave.
In this limit, the radius of curvature RC → ∞). i.e. a spherical surface becomes planar as RC → ∞.
A criterion for a {good} approximation of spherical wave as a plane wave is: RC
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
3
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
Monochromatic traveling EM plane waves can be represented by complex E and B fields:
E z , t E o ei kz t
B z , t Bo ei kz t
Propagating in the
Propagating in the
kˆ zˆ direction
kˆ zˆ direction
n.b. complex vectors:
E o E o xˆ E o ei xˆ Eo ei xˆ
e.g.
n.b. complex vectors:
Bo Bo yˆ Bo ei yˆ Bo ei yˆ
e.g.
n.b. The real, physical instantaneous time-domain EM fields are related to their corresponding
complex time-domain fields via:
E r , t Re E r , t
B r , t Re B r , t
Note that Maxwell’s equations for free space impose additional constraints on E o and Bo .
→ Not just any E o and/or Bo is acceptable / allowed !!!
Since: E 0
Re E 0
B 0
Re B 0
These two relations can only be satisfied r , t if E 0 r , t and B 0 r , t .
and:
xˆ
yˆ zˆ
x
y
z
In Cartesian coordinates:
Thus: E 0
and
B 0 become:
i kz t
i kz t
ˆ
ˆ
ˆ
ˆ
ˆ
0
0
x
y
z
E
e
and
x
y
zˆ Bo e
o
y
z
y
z
x
x
Now suppose we do allow: E o Eox xˆ Eoy yˆ Eoz zˆ ei Eo ei
polarization in xˆ yˆ zˆ 3 D
Bo Box xˆ Boy yˆ Boz zˆ ei Bo ei
polarization in xˆ yˆ zˆ 3 D
Then:
4
yˆ zˆ Eox xˆ Eoy yˆ Eoz zˆ ei ei kz t 0
xˆ
y
z
x
yˆ zˆ Box xˆ Boy yˆ Boz zˆ ei ei kz t 0
xˆ
y
z
x
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
yˆ zˆ Eox xˆ Eoy yˆ Eoz zˆ ei kz t ei 0
xˆ
y
z
x
yˆ zˆ Box xˆ Boy yˆ Boz zˆ ei kz t ei 0
xˆ
y
z
x
Or:
Now:
Eox, Eoy, Eoz = Amplitudes (constants) of the electric field components in x, y, z
directions respectively.
Box, Boy, Boz = Amplitudes (constants) of the magnetic field components in x, y, z
directions respectively.
We see that:
And:
ˆ i kz t ei 0 ← has no explicit x-dependence
xˆ Eox xe
x
ˆ i kz t ei 0 ← has no explicit y-dependence
yˆ Eoy ye
y
ˆ i kz t ei 0 ← has no explicit x-dependence
xˆ Box xe
x
ˆ i kz t ei 0 ← has no explicit y-dependence
yˆ Boy ye
y
And:
However:
Thus:
az
e ae az
z
ˆ i kz t ei ikEoz ei kz t ei 0 true iff Eoz 0 !!!
zˆ Eoz ze
z
ˆ i kz t ei ik oz ei kz t ei 0 true iff Boz 0 !!!
zˆ Boz ze
z
Thus, Maxwell’s equations additionally tell us/impose the restriction that an
electromagnetic plane wave cannot have any component of E or B to (or anti- to)
the propagation direction (in this case here, the kˆ zˆ -direction)
Another way of stating this is that an EM plane wave cannot have any longitudinal
components of E and B (i.e. components of E and B lying along the propagation
direction).
Thus, Maxwell’s equations additionally tell us that an EM plane wave is a purely
transverse wave (at least while it is propagating in free space) – i.e. the components of
E and B must be to propagation direction.
The plane of polarization of an EM plane wave is defined (by convention) to be parallel to E .
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
5
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
Furthermore: Maxwell’s equations impose yet another restriction on the allowed form of
E and B for an EM wave:
1 E
B
E
B 2
and/or:
c t
t
B
1 E
Re E Re
Re B Re 2
t
c t
B
E
t
Can only be satisfied r , t iff:
and/or:
1 E
B 2
c t
Thus:
0
0
E z E y
Ex E y
ˆ
E
x
z x
z
y
B
0
0
0
B y
E y E x
B x
B z
yˆ x y zˆ t xˆ t yˆ t ẑ
0
0
0
0
0
B x B y
B y B x
B z B y
1 E x
1 E y
1 E z
xˆ
yˆ x y zˆ c 2 t xˆ c 2 t yˆ c 2 t ẑ
y
z
x
z
0
0
i kz t i
e
With: E E x xˆ E y yˆ E z zˆ Eox xˆ Eoy yˆ Eoz zˆ e
0
0
i kz t i
B B x xˆ B y yˆ B z zˆ Box xˆ Boy yˆ Boz zˆ e
e
i kz t i
E
e
Thus: E x xˆ E y yˆ Eox xˆ Eoy yˆ e
B B x xˆ B y yˆ Box xˆ Boy yˆ ei kz t ei
6
E
B
E
B
E y xˆ x yˆ x xˆ y yˆ
z
z
t
t
B
B
1 E
1 E y
yˆ
B y xˆ x yˆ 2 x xˆ 2
c t
c t
z
z
Can only be satisfied /
can only be true iff the
xˆ and yˆ relations are
separately / independently
satisfied r , t !
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
i.e.
E y
B
xˆ x xˆ
E :
z
t
B y
E
x yˆ
yˆ
z
t
B y
1
xˆ 2
B :
c
z
B
1
x yˆ 2
c
z
From (1):
ikE oy i Box
From (2):
ikE ox i Boy
From (3):
ikBoy
From (4):
ikBox
E :
(1)
(2)
B :
(3)
(4)
E y
z
E x
z
B x
t
B y
t
Lect. Notes 5
Prof. Steven Errede
ikEoy i Box
(1)
ikEox i Boy
(2)
B
E x
1 E
1
xˆ y 2 x ikBoy 2 i Eox (3)
c
t
z c t
E y
B x 1 E y
1
2
yˆ
ikBox 2 i Eoy (4)
c
t
z c t
1
i Eox
c2
1
i Eoy
c2
Now: c f 2 f
2
Fall Semester, 2015
Eoy Box or:
k
Eox Boy or:
k
1
Boy 2 Eox
c k
1
Box 2 Eoy
c k
k
and:
1
Box Eoy
c
1
Boy Eox
c
1
Boy Eox
c
1
Box Eoy
c
k
Box Eoy
k
Boy Eox
k 2
1 k
c
Maxwell’s equations also
have some redundancy
encrypted into them!
1
So we really / actually only have two independent relations: Box Eoy
c
But:
xˆ yˆ zˆ
Very Useful Table: yˆ zˆ xˆ
ẑ xˆ yˆ
ẑ yˆ xˆ
1
and Boy Eox
c
ẑ xˆ yˆ
yˆ xˆ zˆ
ẑ yˆ xˆ
xˆ zˆ yˆ
1 ˆ
We can write the above two relations succinctly/compactly with one relation: Bo c k E o
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
7
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
Physically, the mathematical relation Bo 1c kˆ E o tells us that for a monochromatic EM plane
wave propagating in free space, E and B are:
a.) in phase with each other.
b.) mutually perpendicular to each other .and. each is perpendicular to the
propagation direction: B kˆ ( kˆ zˆ = propagation direction)
The E and B fields associated with this monochromatic plane EM wave are purely transverse
{ n.b. this is as also required by relativity at the microscopic level – for the extreme relativistic
particles – the (massless) real photons traveling at the speed of light c that make up the
macroscopic monochromatic plane EM wave.}
The purely real/physical amplitudes of E and B are {also} related to each other by: Bo 1c Eo
2
2
with Bo Box Boy
2
2
and Eo Eox Eoy
Griffiths Example 9.2:
A monochromatic (single-frequency) plane EM wave that is plane polarized/linearly polarized in
the x̂ direction, propagating in the kˆ zˆ direction in free space, has:
E E xˆ definition of linearly polarized EM wave, polarized in the x̂ direction.
B 1c kˆ E 1c zˆ Exˆ 1c E zˆ xˆ 1c Eyˆ
yˆ
1
1
ˆ
BcE
Bo 1c Eo
With: B c k E ,
and:
i kz t
i kz t i
i kz t
xˆ Eo e
e xˆ Eo e
xˆ
Then: E z , t E o e
i kz t
i kz t i
i kz t
B z , t B e
yˆ B e
e yˆ B e
yˆ
o
o
o
ei cos i sin
The physical instantaneous electric and magnetic fields are given by the following expressions:
imaginary
real
E z, t Re E z, t Re Eo cos kz t xˆ i Eo sin kz t xˆ
E z, t Eo cos kz t xˆ
imaginary
real
B z, t Re B z , t Re Bo cos kz t yˆ i Bo sin kz t yˆ
B z , t Bo cos kz t yˆ 1c Eo cos kz t yˆ
8
The physical
instantaneous
E and B fields
are in-phase
with each other
for a linearly
polarized EM
plane wave
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Note that: E B zˆ E zˆ, B zˆ
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
( ẑ = direction of propagation of EM wave)
Instantaneous Poynting’s vector for a linearly polarized EM plane wave propagating in free space:
S z, t
S z, t
S z, t
1
o
1
o
E z, t B z, t
1
o
Re E z, t Re B z , t
Eo Bo cos 2 kz t xˆ yˆ
zˆ
1
o
Eo Bo cos 2 kz t zˆ
Watts
2
m
EM power flows in the direction of propagation of the EM plane wave (here, kˆ zˆ direction)
Generalization for Propagation of Monochromatic
Plane EM Waves in an Arbitrary Direction
Obviously, there is nothing special / profound with regard to plane EM waves propagating in
a specific direction in free space / the vacuum. They can propagate in any direction. We can
easily generalize the mathematical description for monochromatic plane EM waves traveling in
an arbitrary direction as follows:
Introduce the notion / concept of a wave vector (or propagation vector) k which points in the
direction of propagation, whose magnitude k k . Then the scalar product k r is the
appropriate 3-D generalization of kz:
k kzˆ with k k and: r xxˆ yyˆ zzˆ with r r x 2 y 2 z 2
1-D: If:
Then: k r kzˆ xxˆ yyˆ zzˆ kz
3-D:
If:
Then:
k k x xˆ k y yˆ k z zˆ with k k x2 k y2 k z2 and:
k r k x x k y y k z z
with
r xxˆ yyˆ zzˆ
r x2 y2 z 2
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
9
UIUC Physics 436 EM Fields & Sources II
Now:
Fall Semester, 2015
Prof. Steven Errede
k x k cos x
k y k cos y
where cos x , cos y , cos z = direction cosines w.r.t.
k z k cos z
(with respect to) the xˆ , yˆ , zˆ -axes respectively
cos x kˆ xˆ sin cos
cos kˆ yˆ sin sin
Direction
Cosines:
Lect. Notes 5
in spherical-polar coordinates
y
cos z kˆ zˆ cos
cos 2 x cos 2 y cos 2 z
Note:
sin 2 cos 2 sin 2 sin 2 cos 2
sin 2 cos 2 1
If e.g. k r then: k r kr . We explicitly demonstrate this in spherical polar coordinates:
k x k cos x k sin cos
For k r :
k y k cos y k sin sin
x r cos x r sin cos
y r cos y r sin sin
and:
k z k cos z k cos
Then:
z r cos z r cos
k r k x k y k z kx cos
x
y
z
x
ky cos y kz cos z
kr cos 2 x kr cos 2 y kr cos 2 z
kr sin 2 cos 2 kr sin 2 sin 2 kr cos 2
kr sin 2 cos 2 sin 2 sin 2 cos 2 kr sin 2 cos 2 sin 2 cos 2
kr sin 2 cos 2 kr
Thus, most generally, we can write the E r , t and B r , t -fields as:
i k r t
E r , t E o e
nˆ
where: n̂ polarization vector n̂ kˆ
i k r t
ˆ
ˆ
ˆ
ˆ
B r , t Bo e
kn
i.e. nk 0 because E is transverse
i k r t
i k r t
B r , t 1c kˆ E r , t 1c E o e
kˆ nˆ Bo e
kˆ nˆ
We must have: B r , t E r , t kˆ i.e. E B 0
10
and E kˆ 0
and B kˆ 0
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
The Direction of Propagation of a Monochromatic Plane EM Wave: k̂
The Real/Physical (Instantaneous) EM Fields are:
E r , t Re E r , t Eo cos k r t nˆ
where: n̂ polarization vector E
B r , t Re B r , t Bo cos k r t
Bo 1c Eo
kˆ nˆ
in free space
Instantaneous Energy, Linear & Angular Momentum in EM Plane Waves (Free Space)
Instantaneous Energy Density Associated with an EM Plane Wave (Free Space):
1
1 2
u EM r , t o E 2 r , t
B r , t uelect r , t umag r , t
o
2
1
1 2
1
2
B r ,t oE2 r ,t
where: uelect r , t o E r , t and umag r , t
2
2 o
2
1 2
1
E and 2 o o for EM waves propagating in vacuum/free space
2
c
c
But:
B2
Thus:
o o 2 1
1
uEM r , t o E 2 r , t
E r , t o E 2 r , t o E 2 r , t o E 2 r , t
2
2
o
Or:
uEM r , t o E 2 r , t o Eo2 cos 2 k r t
Joules
3
m
n.b. for EM plane waves propagating in the vacuum:
umag r , t uelect r , t
and/or: umag r , t uelect r , t 1
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
11
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
Instantaneous Poynting’s Vector Associated with an EM Plane Wave (Free Space):
S r,t E r,t H r ,t
1
o
E r ,t B r ,t
Re
E
z
,
t
Re
B
z, t
o
1
Watts
2
m
For a linearly polarized monochromatic plane EM plane wave propagating in the vacuum, e.g.:
E r , t Eo cos kz t xˆ
and: B r , t Bo cos kz t yˆ
Then: S r , t
o
Thus: S r , t
o c
1
1
Eo Bo cos 2 kz t zˆ
but:
Bo 1c Eo for EM plane waves in vacuum.
c
Eo2 cos 2 kz t zˆ ← multiply RHS by 1
c
1 2
1
E cos 2 kz t zˆ but: 2 o o
Hence: S r , t c
2 o
c
o c
o o
Thus: S r , t c
o
But:
2
Eo cos 2 kz t zˆ c o Eo2 cos 2 kz t zˆ
u EM r , t o E 2 r , t o Eo2 cos 2 kz t
S r , t cu EM r , t zˆ Here, the propagation velocity of EM field energy: vE czˆ
Poynting’s Vector = Energy Density * (Energy) Propagation Velocity: S r , t uEM r , t vE
Instantaneous Linear Momentum Density Associated with an EM Plane Wave (Free Space):
1
kg
EM r , t o o S r , t 2 S r , t 2
c
m -sec
For linearly polarized monochromatic plane EM waves propagating in the vacuum:
1
1
EM 2 c o Eo2 cos 2 kz t zˆ o Eo2 cos 2 kz t zˆ
c
c
uEM
But:
u EM r , t o E 2 r , t o Eo2 cos 2 kz t
1
1
kg
EM r , t o o S r , t 2 S r , t uEM r , t zˆ 2
c
c
m -sec
12
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
Instantaneous Angular Momentum Density Associated with an EM Plane Wave (Free Space):
But:
kg
EM r , t r EM r , t
m-sec
1
1
kg
EM r , t o o S r , t 2 S r , t uEM r , t zˆ 2
c
c
m -sec
for an EM plane wave propagating in the ẑ direction:
1
1
kg
EM r , t 2 r S r , t uEM r , t r zˆ
c
c
m-sec
n.b. depends on the choice of origin
The instantaneous EM power flowing into/out of volume v with bounding surface S enclosing
volume v (containing EM fields in the volume v) is:
U EM t
uEM r , t
PEM t
d S r , t da
v
S
t
t
(Watts)
n.b. closed surface S enclosing volume v.
The instantaneous EM power crossing an (imaginary) surface (e.g. a 2-D plane – a window!) is:
PEM t S r , t da
S
The instantaneous total EM energy contained in volume v is: U EM t v uEM r , t d (Joules)
The instantaneous total EM linear momentum contained in the volume v is:
pEM t EM r , t d
v
kg-m
sec
The instantaneous total EM angular momentum contained in the volume v is:
LEM t EM r , t d
v
kg-m 2
sec
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
13
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
3-D Vector Impedance Associated with an EM Plane Wave (Free Space):
Z r ,t E r,t 1 H r ,t
(Ohms) = Ohm’s law for EM fields! (n.b. a vector quantity)
Analog of: Ohm’s law for AC circuits:
Z t V t I t
(n.b. a scalar quantity)
2
*
Complex form of Ohm’s Law: Z t V t I t V t I t I t
What {precisely} is the mathematical meaning of a “generic” reciprocal vector 1 A r , t ???
The magnitude of the reciprocal vector 1 A r , t 1 A r , t is invariant (i.e. cannot
change) for arbitrary rotations & translations of the coordinate system. The direction that the
reciprocal vector 1 A r , t points in space {at time t} is also invariant for arbitrary rotations &
translations of the coordinate system.
Note further that inverse unit vectors {such as 1 xˆ , 1 yˆ , 1 zˆ } are meaningless!
For a purely real “generic” vector A r , t Ax r , t xˆ Ay r , t yˆ Az r , t zˆ {e.g. expressed
in rectangular/Cartesian coordinates}, the mathematical definition of a purely real reciprocal
vector 1 A r , t , satisfying all of the above requirements is:
1
1
Aˆ r , t
Ar ,t Ar ,t
where:
ˆ
ˆ
ˆ
ˆA r , t A r , t Ax r , t x Ay r , t y Az r , t z
Ar ,t
Ar ,t
Hence, we also see that:
Ax r , t xˆ Ay r , t yˆ Az r , t zˆ
Aˆ r , t
Ar ,t
1
2
2
Ar ,t Ar ,t
Ar ,t
Ar ,t
Using
rectangular
/ Cartesian
coordinates
Using
rectangular
/ Cartesian
coordinates
Note: For the more general case of complex reciprocal vectors 1 A r , t these relations become:
1
1
Aˆ * r , t where: Aˆ * r , t
Ar ,t Ar ,t
A* r , t A x* r , t xˆ A *y r , t yˆ A z* r , t zˆ
Ar ,t
Ar ,t
Hence, we also see that:
Aˆ * r , t A* r , t A x* r , t xˆ A *y r , t yˆ A z* r , t zˆ
1
2
2
A r , t A r , t
Ar ,t
Ar ,t
14
Using
rectangular
/ Cartesian
coordinates
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
Using
rectangular
/ Cartesian
coordinates
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
Thus, e.g. for a linearly polarized monochromatic EM plane wave propagating in the vacuum in
the kˆ zˆ direction, with instantaneous/physical purely real time-domain EM fields of:
E r , t Eo cos kz t xˆ and: B r , t Bo cos kz t yˆ
and: H r , t
1
o
B r,t
1
o
Bo cos kz t yˆ
1
o c
1
with: Bo Eo
c
Eo cos kz t yˆ
The vector impedance Z r , t associated with a monochromatic plane EM plane wave
propagating in the kˆ zˆ direction in free space is:
Z r ,t E r ,t 1 H r,t
S r ,t
ˆ
E r ,t H r ,t E r ,t H r ,t
S r,t
2
2
H r ,t
H r,t
H r,t
c o Eo2 cos 2 kz t zˆ
1
o c
2
Eo2 cos 2 kz t
o2 c3 o zˆ o c c 2 o o zˆ
1
1
zˆ
o c
o o zˆ o c zˆ o
o o
o o
Thus, in free space: Z r , t Z o zˆ oo zˆ (Ohms)
where: Z o
o
o
o
zˆ Z o zˆ
o
Note the cancellations!!!
Here, Z has no spatial
and/or temporal
dependence – for a
monochromatic EM
plane wave propagating
in free space!
is known as the {scalar!} characteristic impedance of free space.
The vector impedance Z r , t associated with an EM field is a physical property of the
medium that the EM field is propagating – which in this case {here} – is the vacuum.
Microscopically, the quantum numbers of the {QED} vacuum – free space {which, at the
microscopic level is not empty!} – must all be associated with scalar-type quantities – spin = 0,
even parity (+) for both space inversion operation P and charge conjugation C, i.e. the quantum
numbers of the {QED} vacuum are J PC 0 .
Note further that all of the physical macroscopic (mean-field) parameters of the vacuum must
be invariant {i.e. unchanged} under arbitrary rotations, translations and Lorentz boosts - from
one reference frame to any other. This means that all macroscopic physical parameters of the
vacuum intrinsically must have no spatial and/or temporal dependence – they are constants:
o 8.85 1012 Farads m = electric permittivity of free space
o 4 107 Henrys m = magnetic permeablity of free space
c 1 o o 3 108 m s = speed of EM waves propagating in in free space
Zo
o
o
376.82 = characteristic impedance of EM waves propagating in free space
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
15
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
The vectorial nature of Z r , t is simply associated with the direction of propagation of the
EM wave – here, in this case {i.e. the vacuum} the kˆ zˆ direction, so: Z r , t Z o zˆ oo zˆ .
For EM waves propagating in the vacuum, it can’t physically matter which direction they are
propagating in – any direction k̂ will give Z r , t Z kˆ o kˆ !
o
o
Note also from the above derivations, that we also have a relation between the vector
impedance Z r , t and Poynting’s vector S r , t associated with a propagating EM wave:
Z r ,t E r,t 1 H r ,t
S r ,t
E r , t Hˆ r , t E r , t H r , t
2
H r,t
H r ,t
S r,t
2
H r,t
For the complex time-domain representation of EM fields – at least those associated with
monochromatic (i.e. single-frequency) EM waves, then in general we have:
E r , t ; E r ; e it and: H r , t ; H r ; e it , and thus the complex vector
impedance is:
E r , t ; Hˆ * r , t ; E r , t ; H * r , t ;
Z r , t; E r , t; 1 H r , t;
Ohms
2
H r , t;
H r , t;
E r ; e it H * r ; e it
H r ; e it H * r ; e it
S r ;
*
*
E r ; H r ; E r ; H r ;
2
H r ; H * r ;
H r ;
S r ;
2 Z r ;
H ;
where S r , and Z r ; are the complex frequency-domain Poynting’s vector and vector
impedance, respectively. Note that – at least for monochromatic/single-frequency EM waves –
that: Z r , t ; Z r ; , i.e. the complex vector impedance associated with monochromatic
EM waves has no time dependence! It is a manifestly frequency-domain quantity!
For monochromatic EM plane waves propagating in free space/the vacuum in the kˆ zˆ
direction, the complex vector impedance is a purely real quantity:
Z r , t ; Z r ; Z o zˆ oo zˆ 376.82 zˆ
Physically, the real part of the complex vector impedance e Z r ; is associated with
propagating EM waves/propagating EM wave energy, whereas the imaginary part of the
complex vector impedance m Z r ; is associated with non-propagating EM wave energy
– i.e. EM wave energy that simply sloshes back and forth locally, 2× per cycle of oscillation!
16
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
Time-Averaged Quantities Associated with EM Waves:
Frequently, we are not interested in knowing the instantaneous power P(t), energy / energy density,
Poynting’s vector, linear and angular momentum, etc.- e.g. simply because experimental measurements
of these quantities are very often only averages over many extremely fast cycles of oscillation…
1
(e.g. period of oscillation of a light wave: light 1 flight 15
1015 sec cycle 1 femto-sec )
10 cps
We want/need time-averaged expressions for each of these quantities (e.g. in order to
compare directly with experimental data) e.g. for monochromatic plane EM light waves:
2
If we have e.g. a “generic” instantaneous physical quantity of the form: Q t Qo cos t
The time-average of Q t is defined as: Q t Q
1
t
t 0
Q t dt
Qo
t
t 0
cos 2 t dt
Q(t) = Qocos (t)
2
Qo
1
Q Q t Qo
2
t
The time average of the cos 2 t function:
1
0
But:
cos 2 t dt
2 f
1
0
1 t sin 2t
2
4
and:
cos 2 t dt
t
t 0
1
sin 2
1 sin 2
0
0
2
2
2
2
f 1 2 2 sin sin 2 0
1
1
1
Q t Q Qo
2
2 2
The time-averaged quantities associated with an EM plane wave propagating in free space are:
uEM r , t
S r,t
Poynting’s Vector:
Linear Momentum Density: EM r , t
Angular Momentum Density: EM r , t
EM Energy Density:
uEM r , t Total EM Energy:
U EM t U EM t
S EM r , t EM Power:
PEM t PEM t
EM r , t Linear Momentum: pEM t pEM t
EM r , t Angular Momentum: LEM t LEM t
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
17
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
For a monochromatic EM plane wave propagating in free space / vacuum in the kˆ zˆ direction:
1
Joules
uEM r , t o Eo2
3
2
m
1
Watts
S r , t c o Eo2 zˆ c uEM r , t zˆ
2
2
m
1
1
1
kg
EM r , t o Eo2 zˆ 2 S r , t uEM r , t zˆ 2
c
c
2c
m -sec
1
1
kg
EM r , t r EM r , t 2 r S r , t u EM r , t rˆ zˆ
c
c
m-sec
Time –
averaged
quantities for
EM plane
wave
propagating
in the ẑ
direction
We define the intensity I associated with an EM wave as the time average of the magnitude of
Poynting’s vector:
1
2
Watts
Intensity of an EM wave: I r S r , t S r , t c uEM r , t c o Eo
2
2
m
The intensity of an EM wave is also known as the irradiance of the EM wave – it is the so-called
radiant power incident per unit area on a surface.
When working with time-averaged quantities such as uEM r , t , S r , t , EM r , t ,
EM r , t , etc. it is convenient/useful to define the so-called root-mean-square ( RMS)
values of the E and B electric and magnetic field amplitudes (using the mathematical definition
of RMS from probability and statistics):
For a monochromatic (i.e. single frequency, sinusoidally-varying) EM wave (only):
1
Erms
E
2
1
Brms
B
2
Where:
Eorms
1
Eo 0.707 Eo
2
1
Bo 0.707 Bo
2
Eo = peak (i.e. max) value of the E -field = amplitude of the
Bo = peak (i.e. max) value of the B -field = amplitude of the
Borms
E -field.
B -field.
E z, t
Eorms
18
Eo
1
Eo
2
Eo
Eorms
1
Eo 0.707 Eo
2
z or t
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
Thus we see that:
1 1 1
1 1 1
Erms Erms
E
E E E and Brms Brms
B
B B B
2 2 2
2 2 2
i.e. that:
Then:
For monochromatic
EM plane
waves
(only):
2
Erms
1 2 1 2
1
1
1 2
1 2
2
2
E E peak Eo2rms Eo2 and Brms
B 2 B peak
Borms Bo
2
2
2
2
2
2
1
1 1
1
1
RMS Joules
uEM t o Eo2 o Eo2 o Eo2rms
2
2 2
2
4
m3
1
1
rms
Srms t S t c uEM t zˆ c uEM
t zˆ RMS Watts
2
2
2
m
1
1
1
1 rms
kg
rms
S t
uEM t zˆ 2 Srms t uEM
t zˆ RMS
EM t
2
2
c
c
2c
2c
m -sec
rms
RMS kg
1
1
1 rms
EM t r EM t r rms
uEM
t r zˆ m-sec
EM t 2 r S rms t
c
c
2
1
1
1
rms
I rms S rms t S rms t I
S t c uEM
t c o Eo2rms RMS Watts
2
2
2
2
m
rms
uEM
t
Real world example: Here in the U.S., 120 Vac/60 Hz “wall power” refers to the RMS AC voltage!
The peak voltage (i.e. the voltage amplitude) is: V peak 2Vrms 2 120 169.7 170.0 Volts.
n.b. For EM waves ≠ sinusoidal waves, the root-mean-square (RMS) must be defined properly /
or triangle
wave
mathematically – e.g. the RMS value of square
amplitudes (from Fourier analysis these consist of linear combinations of infinite # of harmonics)
1
1
rms
rms
(See/refer to probability & statistics reference books!!)
2
2
The Relationship(s) Between the Complex Time-Domain Poynting’s Vector
and the Complex Vector Impedance/Admittance of an EM Plane Wave:
Complex Time-Domain Poynting’s Vector:
S r , t ; E r , t ; H r , t ; Watts m 2
Complex Vector Impedance of an EM Plane Wave:
Z r , t ; E r , t ; H 1 r , t ; E r , t ; H * r , t ;
2
H r , t ; Ohms
Complex Vector Admittance = Reciprocal of Complex Vector Impedance:
Y r , t ; Z 1 r , t ; E 1 r , t ; H r , t ; E * r , t ; H r , t ;
2
E r , t ; Siemens
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
19
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
If we start with Poynting’s vector, we show that it is linearly related to vector admittance and/or
reciprocal vector impedance {we suppress (here) the argument r , t ; for notation clarity}:
( E E )
1
S E H H (EE)
E
H ( E E )Y ( E E ) Z 1 Watts m 2
E
Y
2
Note that the units of E Volts m , hence: ( E E ) Z 1 Volts m Ohms Watts m 2 !!!
We can also obtain the alternate relations:
( H H )
1
1
S E H E (H H )
E
H ( H H ) Z ( H H )Y Watts m 2
H
Z
2
Note that the units of H Amps m , hence: ( H H ) Z Amp m Ohms Watts m 2 !!!
Note that the above complex relations are the vector analogs of the complex scalar power and
Ohm’s law relations associated with AC circuits {suppressing the arguments t ; for
notational clarity}:
Complex time-domain AC power:
P V I Volts Amps Watts
Complex Ohm’s Law:
2
Z V I (V I * ) I Volts Amps Ohms
Complex Scalar Admittance = Reciprocal of Complex Scalar Impedance:
2
Y 1 Z I V I V * V Amps Volts Siemens Ohms 1
Starting with complex time-domain AC power, we show that it is linearly related to scalar
admittance and/or reciprocal scalar impedance:
(V V )
I
P V I
I (V V ) (V V )Y (V V ) Z Watts
V
V
Note that the units of V Volts , hence: (V V ) Z Volts 2 Ohms Watts !!!
We can also obtain the alternate relations:
( I I )
V
P V I V
( I I ) ( I I ) Z ( I I ) Y Watts
I
I
Note that the units of I Amps , hence: ( I I ) Z
20
Amp
2
Ohms Watts !!!
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
RMS Newtons
Radiation Pressure: rad
m2
When an EM wave impinges (i.e. is incident) on a perfect absorber (e.g. a totally black object
with absorbance {aka absorption coefficient} A = 1, as “seen” at the frequency of the EM wave),
all of the EM energy (by definition) is absorbed {ultimately winding up as heat…}.
By conservation of energy, linear momentum & angular momentum the object being
irradiated by the incident EM wave acquires energy, linear momentum & angular momentum
from the incident EM wave.
The EM Radiation Pressure acting on a perfect absorber for a normally incident EM wave is
defined as:
net
FEM
t RMS Newtons
Time-Averaged
Force
perfect
Rad
EM absorber A 1
A
Unit Area
m2
However, the time-averaged EM force is defined as:
net
d pEM t
pEM t
FEM t
=
dt
t
the EM Radiation Pressure at normal incidence is:
time rate of change of the timeaveraged linear momentum
Rad
EM
perfect
absorber
A1
pEM t
t
1 RMS Newtons
A
m2
In a time interval t 1 f , the time-averaged magnitude of the EM linear momentum
transfer pEM t at normal incidence to a perfect absorber of EM radiation is:
pEM t EM t V
EM Linear momentum density
Volume of EM wave associated with time interval t
The volume associated with an EM wave propagating in free space over a time interval t is:
V A ct where ct = distance traveled by the EM wave in the time interval t .
Rad
EM
perfect
absorber
A1
pEM t
t
EM t V 1
EM t A c t
1
c EM t
A
t
A
t A
Thus, we see that for a monochromatic EM plane wave propagating in free space normally
incident on a perfect absorber (A = 1):
1
perfect
Rad
o Eo2 uEM I c
EM absorber A 1 c EM t
2
RMS Newtons
m2
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
21
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
For a perfect reflector (e.g. a perfect mirror, with reflection coefficient R = 1{A = 0}), note that:
pEM t
perfect
reflector
2 pEM t
perfect
absorber
initial final
final
initial
Since pEM pEM
pEM and pEM
pEM
for an EM wave reflecting off of a perfect
initial final initial initial
initial
reflector, then pEM pEM pEM pEM pEM 2 pEM
i.e. an EM wave that reflects off of (i.e. “bounces” off of) a perfect reflector delivers twice
(2×) the momentum kick (i.e. impulse) to the perfect reflector than the same EM wave that is
absorbed by a perfect absorber! Thus at normal incidence:
Rad perfect
Rad perfect
EM reflector R 1 2 EM absorber A1 2 I c
RMS Newtons
m2
Note that for a partially reflecting surface, with reflection coefficient R < 1, since R + A = 1,
the radiation pressure associated with an EM wave propagating in free space and reflecting off of
a partially reflecting surface at normal incidence is given by:
c
partial
Rad perfect
Rad perfect
I
Rad
EM reflector R A 1 A EM absorber A1 2 R EM absorber R 1 A 2 R
RMS Newtons
m2
Since A = 1 – R, we can equivalently re-write this relation as:
c 1 R 2R I c 1 R I c
partial
I
Rad
EM reflector R A 1 A 2 R
RMS Newtons
m2
If the EM wave is not at normal incidence on the absorbing/reflecting surface, but instead
makes a finite angle with respect to the unit normal of the surface, these relations need to be
modified, due to the cosine factor S nˆ S cos I cos associated with the flux of EM
energy/momentum EM t nˆ EM t cos o o S t cos c12 S t cos c12 I cos
crossing the surface area A at a finite angle :
c cos
RMS Newtons
m2
c cos
RMS Newtons
m2
perfect
I
Rad
EM absorber A 1
perfect
I
Rad
EM reflector R 1 2
c cos 1 R I c cos
partial
I
Rad
EM reflector R A 1 A 2 R
22
RMS Newtons
m2
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
Maxwell’s equations (and relativity) for the macroscopic E and B fields associated with an
EM plane wave propagating in free space mandate / require that E B propagation direction
(here, kˆ zˆ ) v prop czˆ , as shown in the figure below:
Compare this
microscopic picture
to that of a classical
/ macroscopic EM
plane wave,
polarized in the
x-hat direction:
Macroscopic EM plane waves propagating in free space are purely transverse waves, i.e.
E B , and both of the E and B fields are also to the propagation direction of the EM plane
wave, e.g. v prop czˆ . Thus: E v prop czˆ and: B v prop czˆ .
The behavior of the macroscopic E and B fields associated with e.g. a monochromatic EM
plane wave propagating in free space, at the microscopic scale is simply the sum over (i.e. linear
superposition of) the E and B -field contributions from {large numbers of} individual real
photons making up the EM field.
Each real photon has associated with it, its own E and B field – e.g. a linearly polarized real
photon, polarized in x̂ direction:
x̂
E Eo cos kz t xˆ ( x̂ = polarization direction)
Photon
Real Photon Momentum:
ẑ
p h zˆ
Photon Poynting’s vector: S 1o E B zˆ
ŷ
B Bo cos kz t yˆ
1
1
B kˆ E where the unit wavevector kˆ zˆ {here} and Bo Eo in vacuum.
c
c
0
Real photon energy: E hf p c p c (Total Relativistic Energy2 = E2 p2 c 2 m2 c 4 )
Real photon momentum (deBroglie relation):
p h
and c f
m c 2 0 for real photon
c = speed of light (in vacuum) = 3 × 108 m/sec
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
23
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
Question: How many real visible-light photons per second are emitted e.g. from a EM power =
10 mW laser? (mW = milli-Watt = 103 Watt)
Answer: The rate at which visible-light photons from a 10 mW laser depends on the color
(i.e. the wavelength λ, frequency f, and/or photon energy Eγ) of the laser beam! Eγ = hf = hc/.
When we say a 10 mW power laser, what precisely does this mean/refer to?
It refers to the time-averaged EM power:
Plaser t 10 mW RMS 10 103 Watts RMS 0.010 Watts RMS
Let’s assume that the laser beam points in the ẑ direction.
Also assume that the diameter of the laser beam is D = 1 mm = 0.001 m (typical).
Further assume (for simplicity’s sake): Power flux density = intensity profile I(x,y) is uniform in
x and y over the diameter of the laser beam (not true in real life – laser beams have ~ Gaussian
2
2
intensity profiles in x and y (i.e. I I o e 2 ); note that there also exist e.g. diffraction
{beam-spreading} effects that should/need to be taken into account…)
I x, y S x, y , t
In t = 1 second, the time-averaged energy associated with the 10 (RMS) mW laser beam is:
Elaser t Plaser t t
Elaser t 0.010 RMS Watts 1 sec
Elaser t 0.010
Elaser t
24
RMS Joules 1 sec
sec
0.010 RMS Joules = Time-averaged energy of laser beam
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
The {instantaneous} energy of the laser beam crosses an imaginary planar surface that is to
the laser beam.
If the laser has red light, e.g. λred = 750 nm (n.b. 1 nm = 1 nano-meter = 109 meters)
or if the laser has blue light, e.g. λblue = 400 nm
Since f = c/λ the corresponding photon frequencies associated with red and blue laser light are:
c
3 108 m / s
f red
4.0 1014 cycles/sec (= Hertz, or Hz)
9
red 750 10 m
fblue
c
blue
3 108 m / s
7.5 1014 cycles/sec (= Hertz, or Hz)
9
400 10 m
The energy associated with a single, real photon is: E hf hc , where h = Planck’s
constant: h = 6.626 x 1034 Joule-sec and c = 3 x 108 m/sec (speed of light in vacuum).
Thus, the corresponding photon energies associated with red and blue laser light are:
Ered hf red
hc red
hc blue
and: Eblue hf blue
since f = c/λ
Ered hf red
6.626 1034 Joule / sec 4.0 1014 / sec 2.6504 1019 Joules (red light)
Eblue hfblue
6.626 1034 Joule / sec 7.5 1014 / sec 4.9695 1019 Joules (blue light)
In a time interval of t 1 sec, the time-averaged energy Elaser t N t E where
N t is the {time-averaged} number of photons crossing a area in the time interval t .
Thus, the number of red (blue) photons emitted from a red (blue) laser in a t 1 sec time interval is:
# red photons:
Nred t
# blue photons: Nblue t
Elaser t
red
E
Elaser t
blue
0.010 Joules
3.7730 1016
19
2.6504 10 Joules/photon
0.010 Joules
2.0123 1016
19
4.9695 10 Joules/photon
Thus, the {time-averaged} rate of emission of red (blue) photons from a red (blue) laser is:
red
R
blue
R
t
t
Nred t
t
Nblue t
t
3.7730 1016 red photons/sec
2.0123 1016 blue photons/sec
Note: In a time interval of t 1 sec, photons (of any color / / f / E ) will travel a distance
of d ct 3 108 m/s 1 s 3 108 meters
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
25
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
If the flux of photons is assumed (for simplicity) to be uniform across the D = 1 mm diameter
laser beam, then the time-averaged flux of photons (#/m2/sec) is:
F
F
red
blue
t
t
Rred t
laser
A
Rblue t
laser
A
sec 4.8039 10 red
m / sec
m
3.7730 1016
103
2
22
2
sec 2.562 10 blue
m / sec
m
2.0123 1016
103
2
2
22
2
2
If each photon has E Joules of energy, then power associated with red (blue) laser beam:
Pred t
Alaser
Sred t Ered F
red
t
m 2 /sec
m 2 /sec
2.6504 1019 Joules 4.8039 1022 red
1.2732 104 Watts m 2
Pblue t
laser
A
Sblue t Eblue F
blue
t
4.9695 1019 Joules 2.56211022 blue
1.2732 104 Watts m 2
Thus we see that:
Pred t
Alaser
Pblue t
Alaser
Sred t Sblue t 1.2732 104 Watts/m 2 ←10 mW laser
n.b. This is precisely why you shouldn’t look into a laser beam {with your one remaining eye}!!!
Time-averaged linear momentum density:
1
1
red t o o Sred t 2 Sred t ured t zˆ 1.4147 1013 kg/m 2 -sec
c
c
blue
blue
1 blue
1 blue
t o o S t 2 S t u t zˆ 1.4147 1013 kg/m 2 -se c
c
c
Thus:
red blue 1.4147 1013 kg/m 2 -sec
Momentum density, Poyntings vector, energy density are
independent of frequency / wavelength / photon energy
The time-averaged linear momentum contained in t 1 second’s worth of laser beam:
Time averaged linear momentum: p t = momentum density t x volume V
Volume V Alaser ct
m
3
Distance light travels in t sec.
26
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
Red light momentum:
2
0.001 kg-m
pred t red t ctA 1.4147 1013 3 108 1
2 sec
3.3333 1011 kg-m sec
Blue light momentum:
2
0.001 kg-m
pblue t blue t ctA 1.4147 1013 3 108 1
2 sec
3.3333 1011 kg-m sec
pred t pblue t 3.3333 1011 kg-m sec
Thus:
{“TRICK”}:
For an EM plane wave, the time-averaged energy density uEM t = time-averaged momentum
density EM t c (Since photon energy, E p c ). Thus:
kg
8
5
3
ured t red t c 1.4147 1013 2
3 10 m/s 4.2441 10 Joules/m
m /sec
kg
8
5
3
ublue t blue t c 1.4147 1013 2
3 10 m/s 4.244110 Joules/m
m
/sec
2
kg-m
Joule
kg
Joule
2
2
s
m
m/s 2
The time-averaged energy contained in t = 1 second’s worth of laser beam is:
The time-averaged energy U t = time-averaged energy density u t volume V
V Alaser ct
red
U
t
2
u
red
Joules
0.001
8
3
t A ct 4.244110 3
3 10 1 m
m
2
0.010 Joules 10 mJ
5
2
Joules
0.001
8
3
U blue t ublue t A ct 4.2441105
3 10 1 m
3
m
2
0.010 Joules 10 mJ
The time-averaged power in the laser beam:
Time-averaged Power (Watts) =
d U t
dt
red
Plaser
t
Joules sec
U laser t
t
blue
10mW laser
t
t = 1 sec
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
27
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
Note: Plaser (laser power) is measured by the total time-averaged energy U t deposited in
(a very accurately) known time interval t using an absolutely calibrated photodiode (e.g. by NIST).
A typical time interval t = 10 secs → t (oscillation period) = 1 f !!
red 1 f 2.500 1015 sec 2.500 femto-sec 2.500 fs
red
blue 1 f 1.333 1015 sec 1.333 femto-sec 1.333 fs
blue
1 peak
Plaser t
2
Consider (the time-averaged) energy density associated with this 10 mW laser:
Joules
uEM t 4.2441 105
3
m
1
1 peak
Now: uEM t uelect t umag t o Eo2 uEM
t
2
2
1
1
And because: B t E t for EM plane waves propagating in free space / vacuum ( 2 o o )
c
c
→ The laser power measured is time-averaged power, i.e. Plaser t
We showed that:
uelect t umag t
uelect t
1 1
2
o Eo
2 2
Bo2
1 2
Eo o o Eo2
2
c
1 1 2 1 o o 2 1 1
Bo
Eo o Eo2
2 2o 2 2 o
2 2
E z , t Eo cos kz t xˆ
Now: Eo = amplitude of the macroscopic electric field:
Bo = amplitude of the macroscopic magnetic field: B z , t Bo cos kz t yˆ
umag t
Define the RMS (Root-Mean-Square) amplitudes of the E and B fields:
1
1
Eo Eo2rms Eo2
2
2
1
1
1
Bo Bo2rms Bo2 2 Eo2 in free space / vacuum
2
2c
2
Eorms
Borms
Then:
28
uelect t
1 1
1
2
2
3
o Eo o Eorms (Joules/m )
2 2
2
umag t
1 1 2
1 2
1
Bo
Borms o Eo2rms in free space / vacuum
2 2 o 2 o
2
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
So if:
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
uEM t uelect t umag t 2 uelect t in free space / vacuum
2 o Eo2rms 4.2441105 Joules/m3
Then: Eo2rms
1
uEM t where o 8.85 1012 Farads/m = electric permittivity of free space
o
1
4.2441105 Joules/m 2
U EM t
Thus: E
(Volts/m)
12
o
8.85 10 Farads/m
3
Eolaser
rms 3.0970 10 3097 RMS Volts/m (n.b. same for red vs. blue laser light!)
laser
orms
1
laser
2 Eolaser
Then: Eolaser E peak
rms 4380 Volts/m
1
Eolaser
10.3232 106 RMS Tesla 10.3232 102 RMS Gauss
Then: Bolaser
rms
c rms
Note: 1 Tesla {SI/MKS units} = 104 Gauss {CGS units}
Thus:
1
Bolaser Eolaser 2 Bolaser
14.5970 106 Tesla 14.5970 102 Gauss
rms
c
Now earlier (above) we calculated the (time-averaged) number of photons present in the
{red and blue} laser beams that were emitted in a time interval of t = 1 sec.
# red photons emitted in t = 1 sec:
Nred t 3.7730 1016 red photons
# blue photons emitted in t = 1 sec:
Nblue t 2.0123 1016 blue photons
The volume associated with a D = 1 mm diameter laser beam turned on for t = 1 sec is:
2
2
D
0.001
8
3
V A ct ct
3 10 1 235.6194 m
2
2
The (time-averaged) number density n t
red
n
t
nblue t
Nred t
V
Nblue t
V
N t
V
of {red and blue} photons in the laser beam is:
3.7730 1016
1.6009 1014 red photons/m3
2
2.3562 10
2.0123 1016
8.5405 1013 blue photons/m3
2
2.3562 10
Then the (time-averaged) energy density uEM t of the {red and blue} laser beam is:
Red photon energy:
Ered hfred 2.6504 1019 Joules
Blue photon energy:
Eblue hfblue 4.9695 1019 Joules
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
29
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
red
uEM
t nred t Ered 1.6009 1014 2.6504 1019 4.2442 105 Joules/m3
blue
uEM
t nblue t Eblue 8.5405 1013 4.9695 1019 4.2442 105 Joules/m3
The (time-averaged) energy U EM t uEM t V of the {red and blue} laser beams is:
red
red
U EM
t uEM
t V 4.2442 105 2.3562 102 0.010 Joules 10 mJoules
blue
blue
U EM
t uEM
t V 4.2442 105 2.3562 102 0.010 Joules 10 mJoules
Now here is something quite interesting: Given that Eorms Eo
2 for a monochromatic EM
plane wave propagating in free space/the vacuum, with time-averaged EM energy density:
Joules
uEM t o Eo2rms 12 o Eo2
3
m
Joules
n t = photon number density (#/m3) in laser beam
But: uEM t n t E
3
m
E hf hc = energy/photon (Joules)
o Eo2rms n t E
2
Or: Eorms
n t
o
E
This formula explicitly connects the amplitudes of the
macroscopic E and B fields (since Bo Eo c ) with the
microscopic constituents of the and B fields (i.e. the photons)!!!
n.b. This formula physically says that the number of {real} photons in the EM wave (each of
photon energy E ) is proportional to Eo2 = the square of the macroscopic electric field amplitude!
We can write this as:
n t o Eo2rms E
and note also that: Eorms
n t
o
E
!!!
Thus, we can now see that the {time-averaged} EM energy density:
uEM r , t o Eo2rms n r , t E with:
v
u EM r , t d U EM
plays a role analogous to that of the probability density in quantum mechanics:
2
P r , t r , t | r , t r , t
with:
Since:
n r , t uEM r , t E 2 o Eo2rms r E and:
Then:
P r , t n r , t
P r , t d 1
v
v
n r , t d N ,
2
N r , t | r , t r , t !!!
Thus, we also see that the macroscopic electric field E r , t plays a role analogous to that of the
probability density amplitude r , t in quantum mechanics!!!
30
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
The (real) photon number density in the laser beam is: n t
Then: o Eo2rms A ct N t E
R t
But:
Eo2rms
1
2 o
F t
N t
t
R t
A c
or: Eo2rms
N t
o A ct
N t
V
Prof. Steven Errede
N t
A ct
#
3
m
E
= the time averaged rate of photons in laser beam (#/sec)
E and
N t
A
t
R t
#
A m 2 -s = flux of photons in the laser beam
1
Joules
F t E n r , t E
3
c
m
#
Thus, we see that the {real} photon flux: F t c n r , t 2
m -s
Thus, the intensity {aka irradiance} of the laser beam is:
Eo2rms
1
F t E and
oc
uEM t o Eo2rms
I S t kˆ c uEM t o Eo2rms F t E c n t E
Watts
2
m
The {time-averaged} <longitudinal separation distance> between photons is defined as:
d t
ct
N t
For t = 1 sec: dred t
dblue t
(m)
3 108 m
7.85 109 m ~ 8 109 m 8 nm (1 nm = 109 m)
16
3.773 10 ' s
3 108 m
1.49 108 m ~ 15 109 m 15 nm
16
2.012 10 ' s
Recall that:
red 750 nm and blue 400 nm
Thus:
d t for either red or blue laser light.
The {time-averaged} <transverse separation distance> between photons is defined as:
d t
A
N t
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
31
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 5
Prof. Steven Errede
Thus:
2
d red t
0.001
2
2.35 1020 m
16
3.773 10
2
d blue t
0.001
2
4.40 1020 m
16
2.0123 10
We showed above that the time-averaged/mean number density of photons in the laser beam is:
n r , t uEM r , t E 12 o Eo2 E photons m3 .
The instantaneous number density of photons in the laser beam is:
n r , t uEM r , t E o Eo2 cos 2 kz t E photons m3
We can normalize the instantaneous photon number density to obtain an instantaneous 3-D
photon probability density, P3-D z , t 1 m3 . Recall that the laser beam intensity is
uniform/constant in the cross-sectional area A of the laser beam.
The 3-D photon probability density, P3-D r , t is:
P3-D r , t n r , t N uEM r , t N E o Eo2 cos 2 kz t N E 1 m3
E U EM Joules in time interval of t secs .
Then also note that: U EM uEM r , t Vol 12 o Eo2 A n r , t E A .
But note that:
3-D
Thus: P
N
n r , t u EM r , t
r,t
U EM
N
o Eo2
1
2
o Eo2 A
cos 2 kz t
2
cos 2 kz t 1 m3
A
with:
v
P3-D r , t d
2 A z ct
z ct
2
2
kz
t
dz
da
cos 2 kz t dz
cos
A
z
0
0
z
A
A
2
1 z c t
cos 2 kz t dz 1
0
z
1
2
We thus see that the instantaneous normalized photon number density P3-D r , t n r , t N
plays a role analogous to that of the probability density in quantum mechanics
2
P r , t r , t | r , t r , t , with: P r , t d 1 . Hence, we also see that the
v
32
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
macroscopic electric field amplitude Eo
Fall Semester, 2015
1
o
Lect. Notes 5
Prof. Steven Errede
n r , t E plays a role analogous to the
probability density amplitude in quantum mechanics!
We can calculate the time rate of change of the normalized 3-D photon probability density, P3-D r , t :
P3-D r , t
t
2
4
cos 2 kz t
sin kz t cos kz t
A t
A
We can define a normalized 3-D photon probability current density as: J 3-D r , t c P3-D r , t zˆ .
We calculate the divergence of the normalized 3-D photon probability current density:
2
J3-D r , t c P3-D r , t zˆ c
cos 2 kz t zˆ
A
c
2
4k
2
sin kz t cos kz t
cos kz t c
A z
A
We thus show that the photons in this laser beam obey the continuity equation for photons:
P3-D r , t 3-D
J r , t 0
t
From above:
P3-D r , t
4k
4
sin kz t cos kz t
sin kz t cos kz t and: J3-D r , t c
t
A
A
However, in free space, we have: ck .
Hence:
Thus:
P3-D r , t
t
P3-D r , t
t
4ck
sin kz t cos kz t J3-D r , t
A
J3-D r , t 0
i.e. microscopically, photons neither disappear, nor are they created in propagating as this laser beam!
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
33
