Ondes électromagnétiques dans les milieux Notes

•
• −e mene~ve
~
˜
E=˜ E0e−j(ω t−k z)~ux
B
c
B
B
BkBk=kEk
cc
B
me
d~ ˜ve
dt =−j ω me~
˜ve=−e~
˜
E
◦

~
˜ve=−je
ω me
~
˜
E
B
~
j=X
k
nk.qk.~vk=−nee~ve=γ~ E
˜γ=jnee2
ω me
˜γ=j ε0
ω2
p
ω
ωp
e−j(ω t−k z)
e+j(ω t−k z)
~
j~ E
˜γ=j ε0
ω2
p
ω
B
B
B
∆~ E=−−→ grad div ~ E−−→ rot −→ rot ~ E=−−→ rot −∂~
B
∂t !=∂ ∂t −→
rot ~ B
∆˜
~
E=∂
∂t µ0.˜
~
j+1
c2
∂˜
~
E
∂t !
∆˜
~
E=1
c2
∂2˜
~
E
∂t2+µ0.˜γ∂˜
~
E
∂t
B˜
~
E=E0.e−j.(ω.t−˜
kz−ϕ)~u
−˜ k2=−ω2
c2+ωµ0.ε0.ω2
p
ω
˜
k=qω2−ω2
p
c
◦

•ω > ωpk=kr
~
˜
E=˜ E0e−j(ω t−krz)~ux⇒~ E=E0cos (ω t −krz+ϕ0)~ux
•ω < ωpk=j ki
~
˜
E=˜ E0e−j(ω t−j kiz)~ux⇒~ E=E0e−kizcos (ω t +ϕ0)~ux
⇒
•ω > ωp
•ω < ωp
•
• −e mene~ve
◦

˜γ=γr
γr∈R
B B
B
∆~ E=−−→ grad div ~ E−−→ rot −→ rot ~ E=−−→ rot −∂~
B
∂t !=∂ ∂t −→
rot ~ B
∆˜
~
E=∂
∂t µ0.˜
~
j+1
c2
∂˜
~
E
∂t !≈∂ ∂t µ0.˜
~
j
∆˜
~
E=µ0.˜γ∂˜
~
E
∂t
B
˜
~
E=E0.e−j.(ω.t−˜
kz−ϕ)~u
˜
k2=j ω µ0γr
j=ejπ
2
˜
k2=ω µ0γrejπ
2
˜
k=±√ω µ0γrejπ
4=±√ω µ0γr1
√2+j1 √2=±rω µ0γr
2(1 + j) = ±1 + j δ
δ=q2
ω µ0γr
z
~
˜
E=˜ E0e−j(ω t−(1+j
δ)z)~ux⇒~
E=E0e−z
δcos ω t −z
δ+ϕ0~ux
⇒
δ=q2
ω µ0γr
◦

f50 Hz 10 kHz 1 MHz
δ9 mm 0,7 mm 70 µm
γ→∞⇔δλ
q m
• −f. ˙r.~ur
•q. ~ Eext(z, t)
~
Eext(t) = E0.cos (ω.t −k.z)~ux
Bτ=m
f
˜
~
j
˜
~
j= ˜γ(ω).˜
~
E˜γ(ω) = γ0
1−j.τ.ω
Bθ
B~ux
m.¨x=−f. ˙x+q.E0.cos (ω.t −kz)
x=<(˜x) ˜x=x0.ej(kz−ω.t+ϕ)
m. (−j.ω)2.˜x+f. (−j.ω).˜x=q.E0e−j.(ω.t−kz)
˜
~v =−j.ω.˜x~ux=q.E0e−j.(ω.t−kz)
−j.m.ω +f~ux
˜
~
j=n.q.˜v=n.q2
−j.m.ω +f
˜
~
E=
n.q2
f
1−j.m
fω
˜
~
E
◦