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Electromagnetic

Radiation

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1887 (22 years after Maxwell equations) discovered radio waves

Hertz(1857-1894)



Maxwell Equations∇.𝐷= 𝜌

∇.𝐵= 0

∇𝑥𝐻= 𝐽+ 𝜕𝐷𝜕𝑡

∇𝑥𝐸= − 𝜕𝐵𝜕𝑡

𝐹= 𝑞(𝐸+ 𝑣Ԧ 𝑥 𝐵)



Constitutive Equations

VacuumD = εo EB = μo H

ConductorsJ = σ E

Linear mediaD = ε EB = μ H



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



Example SHG

E = 𝐸𝑜𝑒𝑖𝜔𝑡

𝑃= 𝐸2 = 𝐸𝑜2𝑒𝑖2𝜔𝑡

∇𝑥𝐻= 𝐽+ 𝜀𝑜 𝜕𝐸𝜕𝑡 + 𝜀𝑜 𝜕𝑃𝜕𝑡

∇𝑥𝐸= −𝜇 𝜕𝐻𝜕𝑡



Gauss Theorem

∇.𝐷= 𝜌

න ∇.𝐷V 𝑑𝑉= 𝐷.𝑛ሬԦ 𝑑𝑆𝜕𝑉 = 𝑞

Isotropic media => Spherical field

4πr2𝐷= 𝑞 => 𝐸= 14𝜋𝜀𝑜𝑞𝑟2



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.



Stokes Theorem

න ∇𝑥𝐻S .𝑑𝑆= ර 𝐻.𝑑𝑙𝜕𝑆

න ( 𝐽+ 𝜕𝐷𝜕𝑡S ).𝑑𝑆= ර 𝐻.𝑑𝑙𝜕𝑆

I = 2𝜋𝑟𝐻 => 𝐵= 12𝜋𝜇𝐼𝑟

DC current



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



∇2𝐸= 1𝑐2 𝜕2𝐸𝜕𝑡2

∇2𝐵= 1𝑐2 𝜕2𝐵𝜕𝑡2

Possible solution: Plane waves

E= Eocos (kሬԦ.r ሬሬԦ– ωt + ∅)

B= Bocos (kሬԦ.r ሬሬԦ– ωt + ∅)



∇.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



Potentials𝐵= ∇𝑥𝐴

𝐸= −∇𝑉− 𝜕𝐴𝜕𝑡

∇.𝐴+ 1𝑐2 𝜕𝑉𝜕𝑡 = 0



Polarization



Refraction & Reflection



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



Brewster's angle: 56.4º

Total reflection: 42º
