Powerpoint Templates
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Powerpoint Templates Page 1 Powerpoint Templates Electromagnetic Radiation
description
Electromagnetic Radiation. Powerpoint Templates. Hertz (1857-1894). 1887 (22 years after Maxwell equations) discovered radio waves. Maxwell Equations. Constitutive Equations. Vacuum D = ε o E B = μ o H Conductors J = σ E. Linear media D = ε E B = μ H. Non linear media. - PowerPoint PPT Presentation
Transcript of Powerpoint Templates
Powerpoint Templates Page 1Powerpoint Templates
Electromagnetic
Radiation
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1887 (22 years after Maxwell equations) discovered radio waves
Hertz(1857-1894)
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Maxwell Equations∇.𝐷= 𝜌
∇.𝐵= 0
∇𝑥𝐻= 𝐽+ 𝜕𝐷𝜕𝑡
∇𝑥𝐸= − 𝜕𝐵𝜕𝑡
𝐹= 𝑞(𝐸+ 𝑣Ԧ 𝑥 𝐵)
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Constitutive Equations
VacuumD = εo EB = μo H
ConductorsJ = σ E
Linear mediaD = ε EB = μ H
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Non linear media
P = χ(1)E + χ(2)E2 + χ(3)E3 + …
D = εo(E + χ(1)E + χ(2)E2 + χ(3)E3 + …)
D = εo(E + P)
∇𝑥𝐻= 𝐽+ 𝜀𝑜 𝜕𝐸𝜕𝑡 + 𝜀𝑜 𝜕𝑃𝜕𝑡
∇𝑥𝐻= 𝐽+ 𝜕𝐷𝜕𝑡
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Example SHG
E = 𝐸𝑜𝑒𝑖𝜔𝑡
𝑃= 𝐸2 = 𝐸𝑜2𝑒𝑖2𝜔𝑡
∇𝑥𝐻= 𝐽+ 𝜀𝑜 𝜕𝐸𝜕𝑡 + 𝜀𝑜 𝜕𝑃𝜕𝑡
∇𝑥𝐸= −𝜇 𝜕𝐻𝜕𝑡
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Gauss Theorem
∇.𝐷= 𝜌
න ∇.𝐷V 𝑑𝑉= 𝐷.𝑛ሬԦ 𝑑𝑆𝜕𝑉 = 𝑞
Isotropic media => Spherical field
4πr2𝐷= 𝑞 => 𝐸= 14𝜋𝜀𝑜𝑞𝑟2
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න ∇.𝐵V 𝑑𝑉= 𝐵.𝑛ሬԦ 𝑑𝑆𝜕𝑉 = 0
∇.𝐵= 0
Nº of field lines entering a volume must be equal
to the nº of lines living the volume (no magnetic
charge accumulation allowed). => Field lines are
closed loops.
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Stokes Theorem
න ∇𝑥𝐻S .𝑑𝑆= ර 𝐻.𝑑𝑙𝜕𝑆
න ( 𝐽+ 𝜕𝐷𝜕𝑡S ).𝑑𝑆= ර 𝐻.𝑑𝑙𝜕𝑆
I = 2𝜋𝑟𝐻 => 𝐵= 12𝜋𝜇𝐼𝑟
DC current
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Free Space∇.𝐸= 0
∇.𝐵= 0
∇𝑥𝐵= 𝜇𝑜𝜀𝑜 𝜕𝐸𝜕𝑡 = 1𝑐2 𝜕𝐸𝜕𝑡
∇𝑥𝐸= − 𝜕𝐵𝜕𝑡
∇𝑥∇𝑥𝐸= − ∇𝑥𝜕𝐵𝜕𝑡 = − 𝜕𝜕𝑡(∇𝑥𝐵)
∇𝑥∇𝑥𝐸= ∇(∇.E) − ∇2𝐸
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∇2𝐸= 1𝑐2 𝜕2𝐸𝜕𝑡2
∇2𝐵= 1𝑐2 𝜕2𝐵𝜕𝑡2
Possible solution: Plane waves
E= Eocos (kሬԦ.r ሬሬԦ– ωt + ∅)
B= Bocos (kሬԦ.r ሬሬԦ– ωt + ∅)
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∇.𝐸= 0 ∇.𝐵= 0
E= Eocos (kሬԦ.r ሬሬԦ– ωt + ∅)
∇.𝐸= ൫∇.𝐸𝑜 + 𝐸o.𝑘ሬԦ൯cos൫kሬԦ.r ሬሬԦ– ωt+ ∅൯
B= Bocos (kሬԦ.r ሬሬԦ– ωt + ∅)
𝑘ሬԦ.𝐸o = 0 𝑘ሬԦ.𝐵o = 0 𝑘ሬԦ𝑥𝐸o = 𝜔𝐵𝑜
𝑘ሬԦ𝑥𝐵o = − 𝜔𝑐2 𝐸𝑜
ȁ<𝐸ȁ<ȁ<𝐵ȁ<= 𝑐
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Potentials𝐵= ∇𝑥𝐴
𝐸= −∇𝑉− 𝜕𝐴𝜕𝑡
∇.𝐴+ 1𝑐2 𝜕𝑉𝜕𝑡 = 0
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Law of reflection: θi = θr
Snell’s law: n1 sin θi = n2 sin θt
R
2
2( )( )
i t
i t
tgRtg
2
2( )( )
i t
i t
sinRsin
2 2(2 ) (2 )
( ) ( )i t
i t i t
sin sinTsin cos
2(2 ) (2 )
( )i t
i r
sin sinTsin
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Brewster's angle: 56.4º
Total reflection: 42º
