ppg_conducteurs.py

Created by famille-bvc

Created on January 10, 2023

2.45 KB


#      ** Conducteurs **  

# * Eq de Maxwell * H et moy freq

# Pc = 0
# μ = μ0
# ε = ε0
#     ~
# γ = γ (w) = iγi

# SI basses freq :
# γ = γr

#////////////////////////////

# * Eq de propagation *

#    ->
# divE = 0
#    ->
# divB = 0
#            ->
# -> ->     ∂B
# rotE = - ----
#           ∂t
#                       ->
# -> ->      ~ ->      ∂E
# rotB = μ0 (γ E + ε0 ----)
#                      ∂t



#                       ->
#  ->  -> ->     ->    ∂B
# rot(rot(E)) = rot(- ----)
#                      dt
#
#   ->     ->   ->     ∂  -> ->
# grad(div(E))-ΔE = - --(rot(B)
#                 ->  ∂t
#              ->          ->
#  ->      ~  ∂E       ∂^2 B
# ΔE = μ0 (γ ---- + ε0 ------)
#             ∂t        ∂t^2

#//(champ magnétique E -> B)

#/////////////////////////////

# * relation de dispersion *

# //Remplacer x par la dir de U

#    |       d^2 Ex
#    | ΔEx = ------ = -k^2 Ex
#    |        dt^2
# ΔE | ΔEy = 0
#    |
#    | ΔEz = 0
#    |


#    | dEx
#    | --- = -w.Ex
#  ->| dt
# dE |
# ---| 0
# dt |
#    |
#    | 0


#       | d^2 Ex
#       | ------ = -w^2 .Ex
#     ->|  dt^2
# d^2 E |
# ----- | 0
#  dt   |
#       |
#       | 0

# Sur X

#                ~        w^2
# k^2 Ex = -i μ0 γ w Ex - ---Ex
#                         c^2
#
# // Barrer les Ex
#            ~     w^2
# k^2 = i μ0 γ w + ---
#                  c^2

# SI basses freq
# k^2 = i μ0 γr w

#////////////////////////////

# * Comportement de l'onde *

#       w^2 - wp
# k^2 = --------
#         c^2


## SI w < wp :
# 
# k^2 < 0
# donc k = iki =
#           w^2 - wp^2
# = i sqrt(-----------)
#               c
#
#//faire attention au facteur
#  de k
#
# ->
# E = ε0 e^i(kz-wt)
#
# onde croissante


## SI w > wp :
# 
# k^2 > 0
# donc k = kr =
#           w^2 - wp^2
# =   sqrt(-----------)
#               c
#
#//faire attention au facteur
#  de k
#
# ->
# E = ε0 e^i(kz-wt)
#
# propag sans attenuation


## SI w >> wp à haute freq :

#       w^2        w
# k^2 ≈ ---   k ≈ ---
#       c^2        c
#
# milieu transparant où vf=c=z
#
# vf = w / kr
#             avec vfvg = c^2
# vg = dw / dkr
#
# On a un milieu dispersif
#              c
# vf = ----------------
#      sqrt(1-(wp/w)^2)


# BASSES FREQ

# Onde électromagnétique dans
# le conducteur ohmique :

# L'onde va s'atténuer et plus  
# elle se propage plus elle 
# s'atténue

# Conducteur lorsque la 
# conductivité est infinie 
# (conducteur parfait) :

# γr tend vers l'infini donc
# l'épaisseur tend vers 0

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