8.07 Lecture 35: December 7, 2012
ELECTROMAGNETIC WAVES
The
The Wave
Wave Equation
Equation in
in 11 Dimension:
Dimension:
The travelling wave:
f (z; t)
= f (z
vt; 0)
g(z
vt) :
(1)
Alan Guth
Massachusetts Institute of Technology
8.07 Lecture 35, December 7, 2012
{1{
But waves can move in both directions:
f (z; t) = g1 (z vt) + g2 (z + vt) :
Dierential equation for f (z; t):
@f
@z
@ 2f
@z 2
@f
@t
@ 2f
Wave equation:
@t2
= g10 (z
vt) + g20 (z
+ vt)
= g100 (z
vt) + g200 (z
+ vt)
=
vg10 (z
= v 2 g10 (z
vt) + vg20 (z
+ vt)
(2)
(3)
vt) + v 2 g20 (z + vt) :
@ 2f
1
@ 2f
@z 2
v2
@t2
=0:
(4)
{2{
Sinusoidal Waves:
f (z; t)
where
v
=
!
k
= A cos [k(z
= A cos [kz
vt) + Æ ]
!t + Æ ] ;
(5)
= phase velocity
= angular frequency = 2
= frequency
Æ = phase (or phase constant)
k = wave number
= 2=k = wavelength
T = 2=! = period
A = amplitude.
!
(6)
{3{
Any wave can be constructed by superimposing sinusoidal waves
(Fourier's Theorem, aka Dirichlet's Theorem).
Complex Notation:
~ = AeiÆ . Then
Let A
~ i(kz !t) ];
f (z; t) = Re [Ae
(7)
where we used
ei = cos + i sin :
(8)
~.
Conventions: drop \ Re ", and drop ~ on A
f (z; t) = Aei(kz !t) :
(9)
General solution to wave equation:
Z
f (z; t)
=
1
1
A(k ) ei(kz !t) dk ;
(10)
where !=k = v , v = wave speed = phase velocity.
Alan Guth
Massachusetts Institute of Technology
8.07 Lecture 35, December 7, 2012
{4{
Group Velocity and Phase Velocity:
Not in GriÆths. In Jackson, pp. 324-325.
! can sometimes depend on k : dispersion.
Consider a wave packet centered on k0 :
d!
! (k ) = ! (k0 ) +
(k0 )(k k0 ) + : : :
dk
d!
d!
= ! (k0 ) k0 + k + : : : :
dk
dk
(11)
Alan Guth
Massachusetts Institute of Technology
8.07 Lecture 35, December 7, 2012
{5{
d!
! (k ) = ! (k0 ) +
(k0 )(k k0 ) + : : :
dk
d!
d!
= ! (k0 ) k0 + k + : : : :
dk
dk
Z 1
i[! (k0 ) k0 dd!
t
ik(z dd!
]
k
k t)
f (z; t) = e
dk A(k)e
1
(11)
:
(12)
The integral describes a wave which moves with
d!
(k0 ) :
vgroup =
dk
Alan Guth
Massachusetts Institute of Technology
8.07 Lecture 35, December 7, 2012
(13)
{6{
Envelope moves with v = vgroup .
Waves inside envelope move with vphase = v = ! (k)=k.
If vphase > vgroup , then waves appear at the left of the envelope
and move forward through the envelope, disappearing at the
right.
Alan Guth
Massachusetts Institute of Technology
8.07 Lecture 35, December 7, 2012
{7{
Electromagnetic Plane Waves
Maxwell Equations in Empty Space:
r E~ = 0
r E~ =
r
r
~
B
=0
~
B
=
~
@B
1
@t
~
@E
c2
@t
;
(14)
;
where 1=c2 0 0 . Manipulating,
r~ (r~ E~ ) = r~ |(r~ {z E~ }) r E~
2
=0
~ =r
~
@B
@t
!
=
1
~
@ 2E
c2
@t2
(15)
;
Alan Guth
Massachusetts Institute of Technology
8.07 Lecture 35, December 7, 2012
{8{
r~ (r~ E~ ) = r~ |(r~ {z E~ }) r E~
2
=0
=r
~
~
@B
@t
!
=
1
@ E
c2
@t2
2~
(15)
;
so
r E~
2
1
~
@ 2E
c2
@t2
=0:
(16)
This is the wave equation in 3 dimensions. An identical equation
~:
holds for B
r B~
2
1
~
@ 2B
c2
@t2
=0:
(17)
Alan Guth
Massachusetts Institute of Technology
8.07 Lecture 35, December 7, 2012
{9{
~ and B
~ satises the wave equation. This
Each component of E
implies that waves travel at speed c! But the wave equation is
~ and B
~ are still related by Maxwell's equations.
not all: E
Try
~ (~r ; t)
E
i(~
~
= E0 e k~r
!t)
n
^;
(18)
~ 0 is a complex amplitude, n^ is a unit vector, and
where E
!=j~
k j = vphase = c. Then
r
~
~
E
=
i(~
~
in
^ kE0 e k~r !t) ;
(19)
(transverse wave) :
(20)
so we require
n
^ ~k
=0
Alan Guth
Massachusetts Institute of Technology
8.07 Lecture 35, December 7, 2012
{10{
The magnetic eld satises
~
@B
@t
=
r~ E~ =
~
ik
Integrating,
~
B
=
~
k
!
E~ =
~
ik
n^ E~ ei ~k~r
i(~
~
n^ E0 e k~r
0
!t)
;
(
!t)
:
(21)
(22)
so, remembering that j~kj = !c,
~
B
=
1^
c
k
E~ :
(23)
Alan Guth
Massachusetts Institute of Technology
8.07 Lecture 35, December 7, 2012
{11{
Energy and Momentum:
Energy density:
1
u=
2
j j
~
0 E
2
+
1
jB~ j
2
0
:
(24)
~ and B
~ contributions are equal.
The E
u = 0 E02 cos2 (kz !t + Æ ) ; (~
k = kz
^)
{z
|
}
averages to 1/2
=
~
S
P
EM
I
1
0
B~ = uc z^
~
E
=
1~
c2
S
=
(intensity) =
u
c
(25)
^z
D
E
j j
~
S
1 2
= 0 E0
2
:
Alan Guth
Massachusetts Institute of Technology
8.07 Lecture 35, December 7, 2012
{12{
Electromagnetic Waves in Matter
For linear, homogeneous materials, Maxwell's equations are unchanged except for the replacement 0 0 ! . Dene
n
r
0 0
= index of refraction.
(26)
Then
v
= phase velocity =
c
n
:
(27)
Alan Guth
Massachusetts Institute of Technology
8.07 Lecture 35, December 7, 2012
{13{
~ and B
~ , everything carries over, with
When expressed in terms of E
these substitutions:
1
u=
2
~
B
~
S
=
=
n
c
^
k
1~
j j
~
E
E
2
+
1
~j
j
B
2
E~
~
B
(28)
=
uc
n
^z :
Alan Guth
Massachusetts Institute of Technology
8.07 Lecture 35, December 7, 2012
{14{
Boundary Conditions, Transmission
and Reflection
X
ET
El
V2
V1
Bl
BT
Z
ER
BR
V1
Interface
Y
Image by MIT OpenCourseWare.
Alan Guth
Massachusetts Institute of Technology
8.07 Lecture 35, December 7, 2012
{15{
Boundary Conditions:
1 E ? = 2 E ?
1
B? = B?
1
2
2
1
1
k
~
E1
=
k
~
E2 ;
1 ~k
k
~
B =
B
1
2
2
:
(29)
Alan Guth
Massachusetts Institute of Technology
8.07 Lecture 35, December 7, 2012
{16{
Incident wave (z < 0):
~ 0;I ei(k1 z !t) x^
=E
1~
~
BI (z; t) =
E0;I ei(k1 z !t) y
^:
~ I (z; t)
E
(30)
v1
Transmitted wave (z > 0):
~ 0;T ei(k2 z !t) x^
=E
1~
~
BT (z; t) =
E0;T ei(k2 z !t) y
^:
~ T (z; t)
E
(31)
v2
!
must be the same on both sides, so
!
k1
= v1 =
c
n1
;
!
k2
= v2 =
c
n2
:
(32)
Alan Guth
Massachusetts Institute of Technology
8.07 Lecture 35, December 7, 2012
{17{
Reected wave (z < 0):
~ 0;R ei( k1 z
=E
1~
~
BR (z; t) =
E0;R ei(
~ R (z; t)
E
v1
!t)
x
^
k1 z !t)
(33)
y
^:
Boundary conditions:
~k
E
1
1
1
k
~
B1
=
1
2
k
~
B2
~k
=E
2
=)
=)
1
1
~ 0;I
E
~ 0;R = E
~ 0;T
+E
1~
E0;I
v1
(34)
;
1~
1 1~
E0;R =
E0;T
v1
2 v2
:
(35)
Alan Guth
Massachusetts Institute of Technology
8.07 Lecture 35, December 7, 2012
{18{
~ 0;R and E
~ 0;T .
Two equations in two unknowns: E
Solution:
~ 0;R =
E
n1
n2 n1 + n2
~ 0;I
E
E0;T
2n1
=
n1 + n2
~ 0;I :
E
(36)
Alan Guth
Massachusetts Institute of Technology
8.07 Lecture 35, December 7, 2012
{19{
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8.07 Electromagnetism II
Fall 2012
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