8.022 Lecture Notes Class 42 - 12/6/2006
1-D:
∂ 2f
1 ∂ 2f
= 2 2
∂x2
v ∂t
f (x, t)
1 ∂ 2f
v 2 ∂t2
f (�x, t)
�2 f =
Wave equation can be extended!
˜ i(kz−ωt) n(z,
f˜(z, t) = Ae
ˆ t)
f a vector function
transverse: n̂ · ẑ = 0 ←− EM waves
longitudal: n̂ × ẑ = 0 ←− sound waves
�
�
� = E × B ⇒ S ⊥ E, B
S
µ0
Assume n̂ =
� n̂(z, t) for now.
n̂ = cos θx̂ + sin θŷ
f˜(z, t) = Ã cos θei(kz−ωt) x̂ + Ã sin θei(kz−ωt) ŷ
θ is the polarization angle of the wave
2
No local charge and current:
� ·E
� =0
�
� ×E
� = − ∂B
�
∂t
� ·B
�
= 0
�
� ×B
� = µ0 �0 ∂E
�
∂t
� ·E
� × (�
� × E)
� = �(
� �
� ) − �2 E
�
=
� × (− ∂B
∂t )
=
∂
− ∂t
(� × B)
=
−µ0 �0 ∂∂tE2
2
So,
�2 E = µ 0 � 0
∂ 2E
∂t2
=⇒
|| �2 B = µ0 �0
∂ 2B
∂t2
1
= � 0 µ0
v2
v=√
1
=c
� 0 µ0
Since these E and B waves travel at the speed of light. it must be
what light is made of.
Since
� ·E
� = 0, (Ẽ0 )z = 0
�
3
Assumptions
• ω - given frequency ⇔
Ẽ(z, t) = Ẽ0 ei(kz−ωt)
monochromatic
B̃(z, t) = B̃0 ei(kz−ωt)
• plane wave in ẑ - direc­
tion
� ·B
� = 0, (B̃0 )z = 0
�
this means waves are transverse waves
� ×E ?
What about �
� ×E
� = − ∂B
�
∂t
−k(Ẽ0 )y x̂ + k(Ẽ0 )x ŷ = ω(B̃0 )x x̂ + ω(B̃0 )y ŷ
=⇒ B̃0 =
k
(ẑ × Ê0 )
ω
Tells us 2 things !
� ⊥E
�
•B
• In phase !
(we can get from B0 to E0 merely by cross-product)
|B0 | =
|B0 | =
k �
ω |E0 |
=
2π
λ
2πv
= v1
|E�0 |
1 �
c |E0 |
4
Extend to 3-D:
Ẽ(�r, t) =
B̃(�r, t) =
�
Ẽ0 ei(k·�r−ωt) n̂
E˜0 i(�k·�r−ωt)
(�v
c e
× �n)
Energy, Momentum
U=
=
U=
=
2
2
1
[SI] 1
2
2
[cgs] E0 +B0
[�
E
+
B
]
=
0
2
µ0
8π
1
1 1
2
2
2 �0 E + 2 µ0 µ0 �0 E
�0 E 2
�0 cos2 (�k · �r − ωt + δ)E02
Energy in EM wave is equally distributed
• Poynting time!
� ×B
�
E
1
�
S=
= mu
ẑ · E0 cos(...) · Ec0 cos(...)
0
µ0
√
= ẑ · E02 cos2 (...) µ10 ( �0 µ0 )2 · c
= ẑ · c�0 E 2 cos2 (�k · �r − ωt + δ)
0
Know ẑ · c = �v and U = �0 E02 cos2 (�k · �r − ωt + δ)
� = u · �v
S
5
• Momentum
p� =
=
< u >=
� >=
<S
< p� >=
I= <
�
S
c2
( from eq. P�em = µ0 �0
1
c
· uẑ
1
2
2 �0 E0
1
2
2 c�0 E0 ẑ
11
2
2 c �0 E0 ẑ
|S| >= 12 c�0 E02
�
V
� )
Sdτ
( Let ẑ = �v )
Only difference
�,
between energy , S
and momentum:
factors of c.