Download Lecture 20. Electromagnetic Waves

Lecture 20. Electromagnetic Waves
Outline: (Chapter 32 in the book)
Electromagnetic Waves in Free Space
Transfer of EM Energy in Space: Poynting Formalism
Today, we will consider electromagnetic waves in vacuum only!
Cell phones  ~ 0.3 m
Maxwell’s Equations
 Gauss’s Law
𝐸 ∙ 𝑑𝐴 =
𝑠𝑢𝑟𝑓
Different types of solutions, e.g.
𝑄𝑒𝑛𝑐𝑙
𝜀0
-
“Coulomb” electric fields generated
by charges at rest (𝐸 ∝ 1/𝑟 2 );
-
Spherical electromagnetic waves
generated by accelerated charges
(𝐸 ∝ 1/𝑟).
 No magnetic monopoles
𝐵 ∙ 𝑑𝐴 = 0
𝑠𝑢𝑟𝑓
 Faraday’s Law of electromagnetic induction
𝐸 ∙ 𝑑𝑙 = −
𝑙𝑜𝑜𝑝
𝑑
𝑑𝑡
𝐵 ∙ 𝑑𝐴
∝ 1/𝑟 2
𝑠𝑢𝑟𝑓
 Generalized Ampere’s Law
𝐵 ∙ 𝑑𝑙 = 𝜇0
𝑙𝑜𝑜𝑝
𝑠𝑢𝑟𝑓
𝜕𝐸
𝐽 + 𝜖0
∙ 𝑑𝐴
𝜕𝑡
∝ 1/𝑟
𝜆
𝜀0 𝐸 2
𝑢𝐸 =
2
We’ll consider the simplest case of
waves: plane EM waves in vacuum.
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Maxwell’s Equations in Vacuum
𝐸 ∙ 𝑑𝐴 = 0
𝑞 = 0, 𝐽 = 0
𝐵 ∙ 𝑑𝐴 = 0
Do these equations have non-trivial
(𝐸 ≠ 0, 𝐵 ≠ 0) solutions in vacuum?
𝑠𝑢𝑟𝑓
𝑠𝑢𝑟𝑓
𝑑
𝐸 ∙ 𝑑𝑙 = −
𝑑𝑡
𝑙𝑜𝑜𝑝
𝑙𝑜𝑜𝑝
𝐵 ∙ 𝑑𝑙 = 𝜖0 𝜇0
𝐵 ∙ 𝑑𝐴
𝑠𝑢𝑟𝑓
Yes, these solutions describe
electromagnetic waves that propagate at
the speed of light!
𝜕𝐸
∙ 𝑑𝐴
𝜕𝑡
𝑠𝑢𝑟𝑓
Maxwell’s greatest triumph: prediction of the existence of electromagnetic
waves that could travel through empty space at the speed of light and
identification of light as electromagnetic waves.
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Plane EM Waves in Vacuum
Generation of EM waves: by accelerated charges and, thus, by AC currents.
We’ll consider the simplest case of a plane monochromatic wave.
𝐸 ∙ 𝑑𝐴 = 0
ensure transverse
character of EM waves
(E and B are perpendicular
𝐵 ∙ 𝑑𝐴 = 0
𝑠𝑢𝑟𝑓
to dA)
- provide the wave equations for
both E and B waves, e.g. plane E
and B waves that travel along +x:
𝐵 ∙ 𝑑𝐴 = 0
𝑠𝑢𝑟𝑓
𝐸 ∙ 𝑑𝐴 = 0
𝑠𝑢𝑟𝑓
𝑠𝑢𝑟𝑓
𝐸 ∙ 𝑑𝑙 = −
𝑙𝑜𝑜𝑝
𝑙𝑜𝑜𝑝
-
𝑑
𝑑𝑡
𝐵 ∙ 𝑑𝑙 = 𝜖0 𝜇0
p. 1058-1059
𝐵 ∙ 𝑑𝐴
𝑠𝑢𝑟𝑓
𝜕𝐸
∙ 𝑑𝐴
𝜕𝑡
𝑠𝑢𝑟𝑓
𝜕 2 𝐸 𝑥, 𝑡
𝜕 2 𝐸 𝑥, 𝑡
− 𝜖0 𝜇 0
=0
𝜕𝑥 2
𝜕𝑡 2
𝜕 2 𝐵 𝑥, 𝑡
𝜕 2 𝐵 𝑥, 𝑡
− 𝜖0 𝜇 0
=0
𝜕𝑥 2
𝜕𝑡 2
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First important relationship E = c B
𝑑
𝐸 ∙ 𝑑𝑙 = −
𝑑𝑡
𝑙𝑜𝑜𝑝
𝐵 ∙ 𝑑𝐴
𝑠𝑢𝑟𝑓
𝐸 ∙ 𝑑𝑙 = −𝐵𝑎𝑐
𝑙𝑜𝑜𝑝
𝑑
𝐵
𝑑𝑡 𝑠𝑢𝑟𝑓
∙ 𝑑𝐴 = -E a
E = cB !!!
Second important relationship c = 1/sqrt(𝜖0 𝜇0 )
𝐵 ∙ 𝑑𝑙 = 𝜇0
𝑙𝑜𝑜𝑝
Eac=
𝜕𝐸
𝑠𝑢𝑟𝑓 𝜕𝑡
∙ 𝑑𝐴
𝐵 ∙ 𝑑𝑙 = 𝐵 𝑎
𝑙𝑜𝑜𝑝
𝐵 𝑎 = 𝐸𝑎 𝑐 𝜖0 𝜇0
c = 1/sqrt(𝜖0 𝜇0 )
𝑠𝑢𝑟𝑓
𝜕𝐸
𝜖0
∙ 𝑑𝐴
𝜕𝑡
In the same way: The wave equation
𝜕 2 𝐸 𝑥, 𝑡
𝜕 2 𝐸 𝑥, 𝑡
− 𝜖0 𝜇 0
=0
𝜕𝑥 2
𝜕𝑡 2
𝜕 2 𝐵 𝑥, 𝑡
𝜕 2 𝐵 𝑥, 𝑡
− 𝜖0 𝜇 0
=0
𝜕𝑥 2
𝜕𝑡 2
Phase Velocity
𝜕 2 𝐸 𝑥, 𝑡
𝜕 2 𝐸 𝑥, 𝑡
− 𝜖0 𝜇 0
=0
𝜕𝑥 2
𝜕𝑡 2
Units of 𝛼:
𝑠2
𝑚2
- general wave equation in 1D
 1/𝑣 2
𝜕 2 𝑓 𝑥, 𝑡
1 𝜕 2 𝑓 𝑥, 𝑡
− 2
=0
𝜕𝑥 2
𝑣𝑝
𝜕𝑡 2
𝑥 = 𝑣𝑝 𝑡
𝜕 2 𝑓 𝑥, 𝑡
𝜕 2 𝑓 𝑥, 𝑡
−𝛼
=0
𝜕𝑥 2
𝜕𝑡 2
Solutions:
𝑓 𝑥, 𝑡 = 𝑓 𝑥 − 𝑣𝑝 𝑡
- motion of the crests (minima, etc.)
Using dimensionless units:
the wave vector 𝑘 =
the angular velocity 𝜔 = 2𝜋
Phase velocity:
𝑥 − 𝑣𝑝 𝑡
𝜆
𝑥 − 𝑣𝑝 𝑡
𝜆
- phase
wavelength
2𝜋
𝜆
𝑣𝑝
2𝜋
= 2𝜋𝑓
𝑘𝑥 − 𝜔𝑡 = 𝑐𝑜𝑛𝑠𝑡
𝑘𝑥 − 𝜔𝑡  the phase
𝑘𝑑𝑥 − 𝜔𝑑𝑡 = 0
𝑑𝑥 𝜔
𝑣𝑝 ≡
=
𝑑𝑡 𝑘
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Basic Properties of Plane E.-M. Waves in Vacuum
Plane-front wave
traveling in +x direction:
𝐸 𝑥, 𝑡 = 𝐸0 𝑐𝑜𝑠 𝑘𝑥 − 𝜔𝑡 𝑗
𝐵 𝑥, 𝑡 = 𝐵0 𝑐𝑜𝑠 𝑘𝑥 − 𝜔𝑡 𝑘
in phase
𝑘𝑟 − 𝜔𝑡  the phase
𝑘𝑟 = 𝑐𝑜𝑛𝑠𝑡 - the phase front
(the surface of constant phase).
1. In vacuum, the EM waves travel at the speed of light 𝑐 =
𝜕 2 𝐸 𝑥, 𝑡
𝜕 2 𝐸 𝑥, 𝑡
− 𝜖0 𝜇 0
=0
𝜕𝑥 2
𝜕𝑡 2
1
𝜀0 𝜇0
.
𝜕 2 𝑓 𝑥, 𝑡
1 𝜕 2 𝑓 𝑥, 𝑡
− 2
=0
𝜕𝑥 2
𝑣𝑝
𝜕𝑡 2
1
𝑣𝑝 =
=𝑐
𝜀0 𝜇0
2. The e.-m. wave in free space is a transverse wave.
𝑐 ≈ 3 ∙ 108 𝑚/𝑠
𝐸2
𝐸×𝐵 =
𝑘
𝜔
𝐸 and 𝐵 in the traveling e.-m. wave are perpendicular
to the direction of propagation (𝑘).
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