❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❖ P❆❘➪ ■◆❙❚■❚❯❚❖ ❉❊ ❚❊❈◆❖▲❖●■❆
P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ❊◆●❊◆❍❆❘■❆ ❊▲➱❚❘■❈❆
❯▼ ▼➱❚❖❉❖ P❆❘❆ ▼❖❉❊▲❆●❊▼ ◆❯▼➱❘■❈❆ P▲❊◆❆ ❉❆ ❉■❙P❊❘❙➹❖ ❊▲➱❚❘■❈❆ ❉❖ ❙❖▲❖ ❯❙❆◆❉❖ ❆P❘❖❳■▼❆❉❖❘❊❙ ❉❊ P❆❉➱ ❊ ❖ ▼➱❚❖❉❖ ❉❆❙
❉■❋❊❘❊◆➬❆❙ ❋■◆■❚❆❙ ◆❖ ❉❖▼❮◆■❖ ❉❖ ❚❊▼P❖
❏Ú▲■❖ ❆◆❚Ô◆■❖ ❙❆▲❍❊❇ ❉❖ ◆❆❙❈■▼❊◆❚❖
❉▼ ✵✻ ✴ ✷✵✶✻
❯❋P❆ ✴ ■❚❊❈ ✴ PP●❊❊ ❈❛♠♣✉s ❯♥✐✈❡rs✐tár✐♦ ❞♦ ●✉❛♠á
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❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❖ P❆❘➪ ■◆❙❚■❚❯❚❖ ❉❊ ❚❊❈◆❖▲❖●■❆
P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ❊◆●❊◆❍❆❘■❆ ❊▲➱❚❘■❈❆
❏Ú▲■❖ ❆◆❚Ô◆■❖ ❙❆▲❍❊❇ ❉❖ ◆❆❙❈■▼❊◆❚❖
❯▼ ▼➱❚❖❉❖ P❆❘❆ ▼❖❉❊▲❆●❊▼ ◆❯▼➱❘■❈❆ P▲❊◆❆ ❉❆ ❉■❙P❊❘❙➹❖ ❊▲➱❚❘■❈❆ ❉❖ ❙❖▲❖ ❯❙❆◆❉❖ ❆P❘❖❳■▼❆❉❖❘❊❙ ❉❊ P❆❉➱ ❊ ❖ ▼➱❚❖❉❖ ❉❆❙
❉■❋❊❘❊◆➬❆❙ ❋■◆■❚❆❙ ◆❖ ❉❖▼❮◆■❖ ❉❖ ❚❊▼P❖
❉▼ ✵✻ ✴ ✷✵✶✻
❯❋P❆ ✴ ■❚❊❈ ✴ PP●❊❊ ❈❛♠♣✉s ❯♥✐✈❡rs✐tár✐♦ ❞♦ ●✉❛♠á
■❱
Nascimento, Júlio Antonio Salheb do, 1973-
Um método para modelagem numérica plena da dispersão elétrica do solo usando aproximadores de padé e o método das diferenças finitas no domínio do tempo / Júlio Antonio Salheb do Nascimento. - 2016.
Orientador: Rodrigo Melo e Silva de Oliveira.
Dissertação (Mestrado) - Universidade Federal do Pará, Instituto de Tecnologia, Programa de Pós-Graduação em Engenharia Elétrica, Belém, 2016.
1. Diferenças finitas. 2. Linhas elétricas subterrâneas. 3. Maxwell, equações de. I. Título.
CDD 23. ed. 515.62
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❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❖ P❆❘➪ ■◆❙❚■❚❯❚❖ ❉❊ ❚❊❈◆❖▲❖●■❆
P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ❊◆●❊◆❍❆❘■❆ ❊▲➱❚❘■❈❆
❏Ú▲■❖ ❆◆❚Ô◆■❖ ❙❆▲❍❊❇ ❉❖ ◆❆❙❈■▼❊◆❚❖
❯▼ ▼➱❚❖❉❖ P❆❘❆ ▼❖❉❊▲❆●❊▼ ◆❯▼➱❘■❈❆ P▲❊◆❆ ❉❆ ❉■❙P❊❘❙➹❖ ❊▲➱❚❘■❈❆ ❉❖ ❙❖▲❖ ❯❙❆◆❉❖ ❆P❘❖❳■▼❆❉❖❘❊❙ ❉❊ P❆❉➱ ❊ ❖ ▼➱❚❖❉❖ ❉❆❙
❉■❋❊❘❊◆➬❆❙ ❋■◆■❚❆❙ ◆❖ ❉❖▼❮◆■❖ ❉❖ ❚❊▼P❖
❉✐ss❡rt❛çã♦ s✉❜♠❡t✐❞❛ à ❇❛♥❝❛ ❊①❛♠✐♥❛❞♦r❛ ❞♦ Pr♦✲ ❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ❊♥❣❡♥❤❛r✐❛ ❊❧étr✐❝❛ ❞❛ ❯❋P❆ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ●r❛✉ ❞❡ ▼❡str❡ ❡♠ ❊♥❣❡✲ ♥❤❛r✐❛ ❊❧étr✐❝❛ ♥❛ ár❡❛ ❞❡ ❚❡❧❡❝♦♠✉♥✐❝❛çõ❡s✳
❯❋P❆ ✴ ■❚❊❈ ✴ PP●❊❊ ❈❛♠♣✉s ❯♥✐✈❡rs✐tár✐♦ ❞♦ ●✉❛♠á
❱
❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❖ P❆❘➪ ■◆❙❚■❚❯❚❖ ❉❊ ❚❊❈◆❖▲❖●■❆
P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ❊◆●❊◆❍❆❘■❆ ❊▲➱❚❘■❈❆
❯▼ ▼➱❚❖❉❖ P❆❘❆ ▼❖❉❊▲❆●❊▼ ◆❯▼➱❘■❈❆ P▲❊◆❆ ❉❆ ❉■❙P❊❘❙➹❖ ❊▲➱❚❘■❈❆ ❉❖ ❙❖▲❖ ❯❙❆◆❉❖ ❆P❘❖❳■▼❆❉❖❘❊❙ ❉❊ P❆❉➱ ❊ ❖ ▼➱❚❖❉❖ ❉❆❙
❉■❋❊❘❊◆➬❆❙ ❋■◆■❚❆❙ ◆❖ ❉❖▼❮◆■❖ ❉❖ ❚❊▼P❖
❆❯❚❖❘✿ ❏Ú▲■❖ ❆◆❚Ô◆■❖ ❙❆▲❍❊❇ ❉❖ ◆❆❙❈■▼❊◆❚❖
❉■❙❙❊❘❚❆➬➹❖ ❉❊ ▼❊❙❚❘❆❉❖ ❙❯❇▼❊❚■❉❆ ➚ ❆❱❆▲■❆➬➹❖ ❉❆ ❇❆◆❈❆ ❊❳❆▼■✲ ◆❆❉❖❘❆ ❆P❘❖❱❆❉❆ P❊▲❖ ❈❖▲❊●■❆❉❖ ❉❖ P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ❊◆●❊◆❍❆❘■❆ ❊▲➱❚❘■❈❆ ❉❆ ❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❖ P❆❘➪ ❊ ❏❯▲✲ ●❆❉❆ ❆❉❊◗❯❆❉❆ P❆❘❆ ❖❇❚❊◆➬➹❖ ❉❖ ●❘❆❯ ❉❊ ▼❊❙❚❘❊ ❊▼ ❊◆●❊◆❍❆❘■❆ ❊▲➱❚❘■❈❆ ◆❆ ➪❘❊❆ ❉❊ ❚❊▲❊❈❖▼❯◆■❈❆➬Õ❊❙✳
❆P❘❖❱❆❉❆ ❊▼ ✷✾✴✵✷✴✷✵✶✻ ❇❆◆❈❆ ❊❳❆▼■◆❆❉❖❘❆✿
Pr♦❢✳ ❉r✳ ❘♦❞r✐❣♦ ▼❡❧♦ ❡ ❙✐❧✈❛ ❞❡ ❖❧✐✈❡✐r❛ ✭❖❘■❊◆❚❆❉❖❘ ✲ PP●❊❊✴■❚❊❈✴❯❋P❆✮
Pr♦❢✳ ❉r✳ ▼❛r❝✉s ❱✐♥í❝✐✉s ❆❧✈❡s ◆✉♥❡s ✭▼❊▼❇❘❖ ✲ PP●❊❊✴■❚❊❈✴❯❋P❆✮
Pr♦❢✳ ❉r✳ ❋❛❜rí❝✐♦ ❏♦sé ❇r✐t♦ ❇❛rr♦s ✭▼❊▼❇❘❖ ✲ ❋❊❊✴❯❋P❆✮
Pr♦❢✳ ❉r✳ ❏♦s✐✈❛❧❞♦ ❞❡ ❙♦✉③❛ ❆r❛ú❥♦ ✭▼❊▼❇❘❖ ✲ PP●❈❈✴■❈❊◆✴❯❋P❆✮ ❱■❙❚❖✿
❱■
❱■■
❱■■■
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦ ❛ ❉❡✉s ♣❡❧♦ ❞♦♠ ❞❛ ✈✐❞❛ ❡ ♣♦r s❡♠♣r❡ ♠❡ ❞❛r ❛ ❢♦rç❛ ♥❡❝❡ssár✐❛ ♣❛r❛ ❝♦♥s❡❣✉✐r ❡♥❢r❡♥t❛r ♠❡✉s ❞❡s❛✜♦s✳
❆❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ♠✐♥❤❛ ❡s♣♦s❛✱ ♣♦r s❡♠♣r❡ ♠❡ ✐♥❝❡♥t✐✈❛r ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❞❡ ♠✐♥❤❛ ✈✐❞❛✳
■❳
▲✐st❛ ❞❡ ❙í♠❜♦❧♦s
ˆ
x✱ yˆ❡ zˆ ❱❡t♦r❡s ❯♥✐tár✐♦s ♥❛s ❉✐r❡çõ❡s x✱y ❡ z ~
A ❈❛♠♣♦ ❱❡t♦r✐❛❧
Ax, Ay ❡Az ❈♦♠♣♦♥❡♥t❡s ❈❛rt❡s✐❛♥❛s ❞♦ ❱❡t♦rA~
j ❯♥✐❞❛❞❡ ■♠❛❣✐♥ár✐❛
˜
A ❊s❝❛❧❛r ❈♦♠♣❧❡①♦
˜
A ❱❡t♦r ❈♦♠♣❧❡①♦
∇ ×A~ ❘♦t❛❝✐♦♥❛❧ ❞❡A~
∇ ·A~ ❉✐✈❡r❣❡♥t❡ ❞❡A~
∂f
∂α ❉❡r✐✈❛❞❛ P❛r❝✐❛❧ ❞❡ f ❡♠ r❡❧❛çã♦ ❛α
µ P❡r♠❡❛❜✐❧✐❞❛❞❡ ▼❛❣♥ét✐❝❛
µ0 P❡r♠❡❛❜✐❧✐❞❛❞❡ ▼❛❣♥ét✐❝❛ ❞♦ ❱á❝✉♦
ε P❡r♠✐ss✐✈✐❞❛❞❡ ❊❧étr✐❝❛ ε′ ❡
ε′′ P❛rt❡s r❡❛❧ ❡ ✐♠❛❣✐♥ár✐❛ ❞❡ ε
εr P❡r♠✐ss✐✈✐❞❛❞❡ ❊❧étr✐❝❛ ❘❡❧❛t✐✈❛
ε′
r ❡ ε
′′
r P❛rt❡s r❡❛❧ ❡ ✐♠❛❣✐♥ár✐❛ ❞❡εr
ε0 P❡r♠✐ss✐✈✐❞❛❞❡ ❊❧étr✐❝❛ ❞♦ ❱á❝✉♦
σ ❈♦♥❞✉t✐✈✐❞❛❞❡ ❊❧étr✐❝❛
σDC ❈♦♥❞✉t✐✈✐❞❛❞❡ ❊❧étr✐❝❛ ♣❛r❛ ❇❛✐①❛s ❋r❡q✉ê♥❝✐❛s
f ❋r❡q✉ê♥❝✐❛
˜
σ(f) ❈♦♥❞✉t✐✈✐❞❛❞❡ ❊❧étr✐❝❛ ❉❡♣❡♥❞❡♥t❡ ❞❛ ❋r❡q✉ê♥❝✐❛ f ˜
ε(f) P❡r♠✐ss✐✈✐❞❛❞❡ ❊❧étr✐❝❛ ❉❡♣❡♥❞❡♥t❡ ❞❛ ❋r❡q✉ê♥❝✐❛ f ρ ❘❡s✐st✐✈✐❞❛❞❡ ❊❧étr✐❝❛
ρDC ❘❡s✐st✐✈✐❞❛❞❡ ❊❧étr✐❝❛ ♣❛r❛ ❇❛✐①❛s ❋r❡q✉ê♥❝✐❛s
˜
ρ(f) ❘❡s✐st✐✈✐❞❛❞❡ ❊❧étr✐❝❛ ❉❡♣❡♥❞❡♥t❡ ❞❛ ❋r❡q✉ê♥❝✐❛ f
❳
▲✐st❛ ❞❡ ❙✐❣❧❛s
❈P▼▲ ❈♦♥✈♦❧✉t✐♦♥❛❧ P❡r❢❡❝t❧② ▼❛t❝❤❡❞ ▲❛②❡r ❋❉❚❉ ❋✐♥✐t❡✲❉✐✛❡r❡♥❝❡ ❚✐♠❡✲❉♦♠❛✐♥
▼❙❊ ▼❡❛♥ ❙q✉❛r❡ ❊rr♦r
❳■
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶
✶✳✶ ❈♦♥t❡ú❞♦ ❞♦ ❈❛♣ít✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ■♥tr♦❞✉çã♦ ●❡r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✸ ❖❜❥❡t✐✈♦s ❞❡st❡ ❚r❛❜❛❧❤♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✹ ❖r❣❛♥✐③❛çã♦ ❞❡st❡ ❚r❛❜❛❧❤♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹
✷ ❋✉♥❞❛♠❡♥t❛çã♦ ❚❡ór✐❝❛ ✻
✷✳✶ ❈♦♥t❡ú❞♦ ❞♦ ❈❛♣ít✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✷ ❙✐st❡♠❛s ❞❡ ❆t❡rr❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✷✳✶ ❊q✉❛çõ❡s ❞❡ ▼❛①✇❡❧❧ ♥❛ ❢♦r♠❛ ❞✐❢❡r❡♥❝✐❛❧ ♥♦ ❞♦♠í♥✐♦ ❞♦ t❡♠♣♦ ✳ ✻ ✷✳✷✳✷ ❊❧❡tr♦❞✐♥â♠✐❝❛ ❡♠ s✐st❡♠❛s ❞❡ ❛t❡rr❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✸ ❉✐s♣❡rsã♦ ❞♦ ❙♦❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✸✳✶ ▼❛t❡r✐❛✐s ❉✐s♣❡rs✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✸✳✷ ❋♦r♠✉❧❛çã♦ ❞❡ ▲♦♥❣♠✐r❡ ❡ ❙♠✐t❤ ♣❛r❛ ❙♦❧♦s ❉✐s♣❡rs✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✸✳✸ ❋♦r♠✉❧❛çã♦ ❞❡ P♦rt❡❧❛ ♣❛r❛ ❙♦❧♦s ❉✐s♣❡rs✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✸✳✹ ❋♦r♠✉❧❛çã♦ ❞❡ ❱✐s❛❝r♦✲❆❧í♣✐♦ ♣❛r❛ ❙♦❧♦s ❉✐s♣❡rs✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺
✸ ❖ ▼ét♦❞♦ ❞❛s ❉✐❢❡r❡♥ç❛s ❋✐♥✐t❛s ♥♦ ❉♦♠í♥✐♦ ❞♦ ❚❡♠♣♦ ✭❋❉❚❉✮ ♣❛r❛
❙♦❧✉çã♦ ❞❛s ❊q✉❛çõ❡s ❞❡ ▼❛①✇❡❧❧ ✶✾
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✸✳✸ ▼ét♦❞♦ ❞❛s ❉✐❢❡r❡♥ç❛s ❋✐♥✐t❛s ♥♦ ❉♦♠í♥✐♦ ❞♦ ❚❡♠♣♦ ✭❋❉❚❉✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✸✳✸✳✶ ❱✐sã♦ ❣❡r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✸✳✸✳✷ ❈r✐tér✐♦s ❞❡ ♣r❡❝✐sã♦ ❡ ❡st❛❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✹ ▼♦❞❡❧❛❣❡♠ ❞❡ ▼❛t❡r✐❛✐s ❉✐s♣❡rs✐✈♦s ❆r❜✐trár✐♦s ♣❛r❛ ♦ ♠ét♦❞♦ ❋❉❚❉ ❈♦♥s✐❞❡r❛♥❞♦✲s❡ σ+jωε= ˜σ(ω) + jω˜ε(ω) ❆♣❧✐❝❛❞❛ ❛♦ ▼ét♦❞♦ ❋❉❚❉✳ ✷✺
✹✳✶ ❈♦♥t❡ú❞♦ ❞♦ ❈❛♣ít✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✹✳✷ ❘❡❧❛çõ❡s ❈♦♥st✐t✉t✐✈❛s ♣❛r❛ ♠❡✐♦s ❞✐s♣❡rs✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✹✳✸ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ▼♦❞❡❧❛❣❡♠ ❞❡ ▼❛t❡r✐❛✐s ❉✐s♣❡rs✐✈♦s ❆r❜✐trár✐♦s ♣❛r❛
♦ ♠ét♦❞♦ ❋❉❚❉ ❈♦♥s✐❞❡r❛♥❞♦✲s❡ σ +jωε = ˜σ(ω) + jωε(ω)˜ ❆♣❧✐❝❛❞❛ ❛♦
▼ét♦❞♦ ❋❉❚❉✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✺ ❘❡s✉❧t❛❞♦s ◆✉♠ér✐❝♦✲❊①♣❡r✐♠❡♥t❛✐s ✸✵
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✺✳✸✳✾ ❙✐♠✉❧❛çã♦ ♥✉♠ér✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✺✳✸✳✶✵ ❘❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✺✳✹ ❆♥á❧✐s❡ ◗✉❛♥t✐t❛t✐✈❛ ❞♦s ❘❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺
❳■❱
▲✐st❛ ❞❡ ■❧✉str❛çõ❡s
✶ ■❧✉str❛çã♦ ❞♦s ♠❡❝❛♥✐s♠♦s ❞❡ ♣♦❧❛r✐③❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷ ❉❡♣❡♥❞ê♥❝✐❛ ❞❛ P❡r♠✐ss✐✈✐❞❛❞❡ ❈♦♠♣❧❡①❛ ❝♦♠ ❛ ❋r❡q✉ê♥❝✐❛ ■❧✉str❛♥❞♦ ♦s
♠❡❝❛♥✐s♠♦s ❞❡ P♦❧❛r✐③❛çã♦✳ ❆❞❛♣t❛❞♦ ❞❡ ❬✸✹❪ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✸ ❈✉r✈❛s ❞❛ ❢✉♥çã♦ σ(f) ♣❛r❛ σDC = 0,021257 ❙✴♠✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✹ ❈✉r✈❛ ❞❛ ❢✉♥çã♦ εr(f) ♣❛r❛ ❛ ❢♦r♠✉❧❛çã♦ ❞❡ ▲♦♥❣♠✐r❡✲❙♠✐t❤✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✺ ❈✉r✈❛ ❞❛ ❢✉♥çã♦ εr(f) ♣❛r❛ ❛ ❢♦r♠✉❧❛çã♦ ❞❡ P♦rt❡❧❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✻ ❈✉r✈❛ ❞❛ ❢✉♥çã♦ εr(f) ♣❛r❛ ❛ ❢♦r♠✉❧❛çã♦ ❞❡ ❱✐s❛❝r♦✲❆❧í♣✐♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✼ ❈✉r✈❛s ❞❛ ❢✉♥çã♦ ωε(f) ♠❙✴♠✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✽ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ❝é❧✉❧❛ ❞❡ ❨❡❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✾ ❱✐st❛ ❛ér❡❛ ❞♦ ❈❡♥tr♦ ❞❡ ❚❡❝♥♦❧♦❣✐❛ ❞❛ ❊❧❡tr♦❜rás✴❊❧❡tr♦♥♦rt❡✳ ❉❡st❛q✉❡
♥❛ ár❡❛ ♦♥❞❡ ❢♦r❛♠ r❡❛❧✐③❛❞♦s ♦s ❡①♣❡r✐♠❡♥t♦s ❡♠ ❝❛♠♣♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✶✵ ❘❡♣r❡s❡♥t❛çã♦ ❡sq✉❡♠át✐❝❛ ❞❛ ❝♦♥✜❣✉r❛çã♦ ❡①♣❡r✐♠❡♥t❛❧ ❞♦ s✐st❡♠❛ ❞❡
❛t❡rr❛♠❡♥t♦ ❝♦♠♣♦st♦ ♣♦r ✉♠ ú♥✐❝❛ ❤❛st❡ ✈❡rt✐❝❛❧♠❡♥t❡ ❡♥t❡rr❛❞❛✳ ✳ ✳ ✳ ✳ ✸✷ ✶✶ P❛✐♥❡❧ ❢r♦♥t❛❧ ❞♦ ❣❡r❛❞♦r ❞❡ s✉rt♦ ✉t✐❧✐③❛❞♦ ♥♦s ❡①♣❡r✐♠❡♥t♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✶✷ ❙✐♥❛❧ ❞♦ ❣❡r❛❞♦r ♠❡❞✐❞♦ ❡ ♦ s✐♥❛❧ ♠♦❞❡❧❛❞♦ ♠❛t❡♠❛t✐❝❛♠❡♥t❡ ♣❛r❛ ❛s
s✐♠✉❧❛çõ❡s ♥✉♠ér✐❝❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✶✸ ■♠❛❣❡♥s ❞♦ ❡①♣❡r✐♠❡♥t♦ r❡❛❧✐③❛❞♦ ❡♠ ❝❛♠♣♦✳ ✭❛✮ ❉❡st❛q✉❡ ♥♦s ♣♦♥t♦s ❞❡
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✶✹ ❘❡♣r❡s❡♥t❛çã♦ ♥♦ ❛♠❜✐❡♥t❡ ❞♦ s♦❢t✇❛r❡ ❙❆●❙ ❞♦ s❡t✉♣ ❡①♣❡r✐♠❡♥t❛❧ ❞♦ s✐st❡♠❛ ❞❡ ❛t❡rr❛♠❡♥t♦ ❝♦♠♣♦st♦ ♣♦r ✉♠ ú♥✐❝❛ ❤❛st❡ ✈❡rt✐❝❛❧♠❡♥t❡ ❡♥t❡r✲ r❛❞❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✶✺ ❈✉r✈❛ ♦❜t✐❞❛ ♥♦ ❡①♣❡r✐♠❡♥t♦ ❛ ♣❛rt✐r ❞❛ r❛③ã♦ ❡♥tr❡ vR(t) ❡ iR(t) ♣❛r❛
❛✈❛❧✐❛r ♦ ✈❛❧♦r ❞❡RS(ef etiva)✳ ❆ ❧✐♥❤❛ ❝♦♥tí♥✉❛ ✐♥❞✐❝❛ ♦ ✈❛❧♦r ❞❡RS(ef etiva)=
2150 Ω✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✶✻ ❈✉r✈❛ ❞❛ ❚●❘ ♦❜t✐❞❛ ♥♦ ❡①♣❡r✐♠❡♥t♦ ❞❡ ✉♠❛ ❤❛st❡ ❡♥t❡rr❛❞❛ ✈❡rt✐❝❛❧✲ ♠❡♥t❡✳ ❆ ❧✐♥❤❛ ❝♦♥tí♥✉❛ ✐♥❞✐❝❛ ♦ ✈❛❧♦r ❞❡ R = 44,9 Ω✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✶✼ ❙✐♥❛❧ ❞❡ t❡♥sã♦ ✐♥❞✉③✐❞❛ ♠❡❞✐❞♦ ♥♦ ❡①♣❡r✐♠❡♥t♦ ❡♠ ❝❛♠♣♦ ❡ ♦❜t✐❞♦ ♣♦r s✐♠✉❧❛çã♦ ♥✉♠ér✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✶✽ ❈✉r✈❛s ❞❛ ❢✉♥çã♦ σ(f) ♣❛r❛ σDC = 0,021257 ❙✴♠ ♦❜t✐❞❛s ❛ ♣❛rt✐r ❞❛
❡①♣r❡ssã♦ ✭✷✳✷✶✮ ❡ ❛ r❡s♣❡❝t✐✈❛ ❛♣r♦①✐♠❛çã♦ ♣♦r ♣♦❧✐♥ô♠✐♦ ❞❡ P❛❞é✳ ✳ ✳ ✳ ✳ ✹✶ ✶✾ ❈✉r✈❛s ❞❛ ❢✉♥çã♦ εr(f) ♦❜t✐❞❛s ❛ ♣❛rt✐r ❞❛ ❡①♣r❡ssã♦ ✭✷✳✷✷✮ ❡ ❛ r❡s♣❡❝t✐✈❛
❛♣r♦①✐♠❛çã♦ ♣♦r ♣♦❧✐♥ô♠✐♦ ❞❡ P❛❞é✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✵ ❈✉r✈❛s ❞❛ ❢✉♥çã♦ ωε(f)♦❜t✐❞❛s ❛ ♣❛rt✐r ❞❛ ❡①♣r❡ssã♦ ✭✷✳✷✷✮ ❡ ❛ r❡s♣❡❝t✐✈❛
❛♣r♦①✐♠❛çã♦ ♣♦r ♣♦❧✐♥ô♠✐♦ ❞❡ P❛❞é✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✷✶ ❙✐♥❛❧ ❞❡ ❝♦rr❡♥t❡ ✐♥❥❡t❛❞❛ ♠❡❞✐❞♦ ♥♦ ❡①♣❡r✐♠❡♥t♦s ❡♠ ❝❛♠♣♦ ❡ ♦❜t✐❞❛ ♣♦r
s✐♠✉❧❛çã♦ ♥✉♠ér✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✷ ❙✐♥❛❧ ❞❡ t❡♥sã♦ ✐♥❞✉③✐❞❛ ♠❡❞✐❞♦ ♥♦ ❡①♣❡r✐♠❡♥t♦s ❡♠ ❝❛♠♣♦ ❡ ♦❜t✐❞♦ ♣♦r
s✐♠✉❧❛çã♦ ♥✉♠ér✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✷✸ ❘❡♣r❡s❡♥t❛çã♦ ❡sq✉❡♠át✐❝❛ ❞❛ ❝♦♥✜❣✉r❛çã♦ ❡①♣❡r✐♠❡♥t❛❧ ❞♦ s✐st❡♠❛ ❞❡
❛t❡rr❛♠❡♥t♦ ✉t✐❧✐③❛❞♦ ♣❛r❛ ♠❡❞✐çã♦ ❞❡ t❡♥sã♦ ❞❡ t♦q✉❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✷✹ ■♠❛❣❡♠ ❞❡ ❊①♣❡r✐♠❡♥t♦ ❡♠ ❈❛♠♣♦ ✲ ❚❡♥sã♦ ❞❡ ❚♦q✉❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✷✺ ❘❡♣r❡s❡♥t❛çã♦ ♥♦ ❛♠❜✐❡♥t❡ ❞♦ s♦❢t✇❛r❡ ❙❆●❙ ❞♦ ❡①♣❡r✐♠❡♥t♦ ❞❡ ❚❡♥sã♦
❳❱■
✷✻ ❈✉r✈❛ ♦❜t✐❞❛ ♥♦ ❡①♣❡r✐♠❡♥t♦ ❛ ♣❛rt✐r ❞❛ r❛③ã♦ ❡♥tr❡ vR(t) ❡ iR(t) ♣❛r❛
❛✈❛❧✐❛r ♦ ✈❛❧♦r ❞❡RS(ef etiva)✳ ❆ ❧✐♥❤❛ ❝♦♥tí♥✉❛ ✐♥❞✐❝❛ ♦ ✈❛❧♦r ❞❡RS(ef etiva)=
2110 Ω✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼
✷✼ ❙✐♥❛❧ ❞♦ ❣❡r❛❞♦r ♠❡❞✐❞♦ ❡ ♦ s✐♥❛❧ ♠♦❞❡❧❛❞♦ ♠❛t❡♠❛t✐❝❛♠❡♥t❡ ♣❛r❛ ❛s s✐♠✉❧❛çõ❡s ♥✉♠ér✐❝❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✷✽ ❙✐♥❛❧ ❞❡ ❝♦rr❡♥t❡ ✐♥❥❡t❛❞❛ ♠❡❞✐❞♦ ♥♦ ❡①♣❡r✐♠❡♥t♦s ❡♠ ❝❛♠♣♦ ❡ ♦❜t✐❞❛ ♣♦r
s✐♠✉❧❛çã♦ ♥✉♠ér✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✷✾ ❙✐♥❛❧ ❞❡ t❡♥sã♦ ✐♥❞✉③✐❞❛ ♠❡❞✐❞♦ ♥♦ ❡①♣❡r✐♠❡♥t♦s ❡♠ ❝❛♠♣♦ ❡ ♦❜t✐❞♦ ♣♦r
s✐♠✉❧❛çã♦ ♥✉♠ér✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✸✵ ❘❡♣r❡s❡♥t❛çã♦ ❡sq✉❡♠át✐❝❛ ❞❛ ❝♦♥✜❣✉r❛çã♦ ❡①♣❡r✐♠❡♥t❛❧ ❞♦ s✐st❡♠❛ ❞❡
❛t❡rr❛♠❡♥t♦ ✉t✐❧✐③❛❞♦ ♣❛r❛ ♠❡❞✐çã♦ ❞❡ t❡♥sã♦ ❞❡ ♣❛ss♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✸✶ ❘❡♣r❡s❡♥t❛çã♦ ♥♦ ❛♠❜✐❡♥t❡ ❞♦ s♦❢t✇❛r❡ ❙❆●❙ ❞♦ ❡①♣❡r✐♠❡♥t♦ ❞❡ ❚❡♥sã♦
❞❡ P❛ss♦ ♥❛s ♣r♦①✐♠✐❞❛❞❡s ❞❡ ✉♠❛ ❝❡r❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✷ ❈✉r✈❛ ♦❜t✐❞❛ ♥♦ ❡①♣❡r✐♠❡♥t♦ ❛ ♣❛rt✐r ❞❛ r❛③ã♦ ❡♥tr❡ vR(t) ❡ iR(t) ♣❛r❛
❛✈❛❧✐❛r ♦ ✈❛❧♦r ❞❡RS(ef etiva)✳ ❆ ❧✐♥❤❛ ❝♦♥tí♥✉❛ ✐♥❞✐❝❛ ♦ ✈❛❧♦r ❞❡RS(ef etiva)=
2110 Ω✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷
✸✸ ❙✐♥❛❧ ❞♦ ❣❡r❛❞♦r ♠❡❞✐❞♦ ❡ ♦ s✐♥❛❧ ♠♦❞❡❧❛❞♦ ♠❛t❡♠❛t✐❝❛♠❡♥t❡ ♣❛r❛ ❛s s✐♠✉❧❛çõ❡s ♥✉♠ér✐❝❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✸✹ ❙✐♥❛❧ ❞❡ ❝♦rr❡♥t❡ ✐♥❥❡t❛❞❛ ♠❡❞✐❞♦ ♥♦ ❡①♣❡r✐♠❡♥t♦s ❡♠ ❝❛♠♣♦ ❡ ♦❜t✐❞❛ ♣♦r
s✐♠✉❧❛çã♦ ♥✉♠ér✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✸✺ ❙✐♥❛❧ ❞❡ t❡♥sã♦ ✐♥❞✉③✐❞❛ ♠❡❞✐❞♦ ♥♦ ❡①♣❡r✐♠❡♥t♦s ❡♠ ❝❛♠♣♦ ❡ ♦❜t✐❞♦ ♣♦r
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▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✶ ❈♦❡✜❝✐❡♥t❡s an ♣❛r❛ ❛ ❋♦r♠✉❧❛çã♦ ❞❡ ▲♦♥❣♠✐r❡ ❡ ❙♠✐t❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✷ ❱❛❧♦r❡s ❞♦s ♣❛râ♠❡tr♦s Aβ✱toβ ❡ τβ ❞❡ ✭✺✳✶✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹
✸ ❈♦❡✜❝✐❡♥t❡s ❞♦ ❛♣r♦①✐♠❛❞♦r ❞❡ P❛❞é ❋♦r♠✉❧❛çã♦ ▲♦♥❣♠✐r❡✲ ❙♠✐t❤ ✭✹✳✽✮ ♣❛r❛ σDC = 0,021257❙✴♠✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✹ ❈♦❡✜❝✐❡♥t❡s ❞♦ ❛♣r♦①✐♠❛❞♦r ❞❡ P❛❞é ❋♦r♠✉❧❛çã♦ P♦rt❡❧❛ ✭✹✳✽✮ ♣❛r❛σDC =
0,021257❙✴♠✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✺ ❈♦❡✜❝✐❡♥t❡s ❞♦ ❛♣r♦①✐♠❛❞♦r ❞❡ P❛❞é ❋♦r♠✉❧❛çã♦ ❱✐s❛❝r♦✲❆❧í♣✐♦ ✭✹✳✽✮ ♣❛r❛
σDC = 0,021257❙✴♠✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✻ ❈♦♠♣❛r❛çã♦ ❞♦s ✈❛❧♦r❡s ❞❡ Vp ❡ Ip ❡♥tr❡ ♦s ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s ❡ ♦s ❞❛✲
❞♦s ♦❜t✐❞♦s ♥❛s s✐♠✉❧❛çõ❡s ♥✉♠ér✐❝❛s ❝♦♠ s♦❧♦s ❞✐s♣❡rs✐✈♦s ✭❙❉✮ ❡ ♥ã♦ ❞✐s♣❡rs✐✈♦s ✭❙◆❉✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✼ ❆♥á❧✐s❡ ❞♦ M SE ❝❛❧❝✉❧❛❞♦ ❡♥tr❡ ♦s ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s ❡ ♦s ❞❛❞♦s ♦❜✲
t✐❞♦s ❛ ♣❛rt✐r ❞❛s s✐♠✉❧❛çõ❡s ♥✉♠ér✐❝❛s ❝♦♠ s♦❧♦s ❞✐s♣❡rs✐✈♦s ✭❙❉✮ ❡ ♥ã♦ ❞✐s♣❡rs✐✈♦s ✭❙◆❉✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✽ ❚❡♠♣♦ ❞❡ ❡①❡❝✉çã♦ ❞❛s s✐♠✉❧❛çõ❡s ❝♦♠ ♦ ♠♦❞❡❧♦ ❞❡ s♦❧♦ ❞✐s♣❡rs✐✈♦ ✭❙❉✮
❘❡s✉♠♦
◆❡st❛ ❉✐ss❡rt❛çã♦✱ é ❞❡s❡♥✈♦❧✈✐❞♦ ❡ ✐♠♣❧❡♠❡♥t❛❞♦ ❡♠ s♦❢t✇❛r❡ ✉♠ ♠ét♦❞♦ ♣❛r❛ ♠♦❞❡✲ ❧❛❣❡♠ ❞❛ ❞✐s♣❡rsã♦ ❡❧étr✐❝❛ ❞♦ s♦❧♦ ❛tr❛✈és ❞♦ ♠ét♦❞♦ ❞❛s ❞✐❢❡r❡♥ç❛s ✜♥✐t❛s ♥♦ ❞♦♠í♥✐♦ ❞♦ t❡♠♣♦ ✭❋❉❚❉✮✳ ❆ ♠❡t♦❞♦❧♦❣✐❛ é ❜❛s❡❛❞❛ ♥❛ ❛♣❧✐❝❛çã♦ ❞❡ ❛♣r♦①✐♠❛❞♦r❡s ❞❡ P❛❞é ♥♦ ❞♦♠í♥✐♦ ❞❛ ❢r❡q✉ê♥❝✐❛✱ ❞❡ ❢♦r♠❛ ❛ r❡♣r❡s❡♥t❛r ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❡s♣❡❝tr❛❧ t❛♥t♦ ❞❛ ❝♦♥❞✉t✐✲ ✈✐❞❛❞❡ q✉❛♥t♦ ❞❛ ♣❡r♠✐ss✐✈✐❞❛❞❡ ❡❧étr✐❝❛✳ ❆s ❡q✉❛çõ❡s ♥♦ ❞♦♠í♥✐♦ ❞♦ t❡♠♣♦ sã♦ ♦❜t✐❞❛s ❛tr❛✈és ❞❡ s✐♠♣❧❡s ♠❛♥✐♣✉❧❛çõ❡s ❛❧❣é❜r✐❝❛s ❡ tr❛♥s❢♦r♠❛❞❛s ✐♥✈❡rs❛s ❞❡ ❋♦✉r✐❡r✳ ❘❡s✉❧✲ t❛❞♦s ♦❜t✐❞♦s ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡ ❡♠ s✐st❡♠❛s ❞❡ ❛t❡rr❛♠❡♥t♦ s♦❧✐❝✐t❛❞♦s ♣♦r ❞❡s❝❛r❣❛s ❛t♠♦s❢ér✐❝❛s ❛rt✐✜❝✐❛❧♠❡♥t❡ ♣r♦❞✉③✐❞❛s sã♦ ❝♦♠♣❛r❛❞♦s ❛♦s r❡s✉❧t❛❞♦s ♥✉♠ér✐❝♦s ♦❜t✐❞♦s ❛tr❛✈és ❞♦ ♠ét♦❞♦ ♣r♦♣♦st♦✳ ❊①❝❡❧❡♥t❡ ❝♦♥❝♦r❞â♥❝✐❛ ❡♥tr❡ ♦s s✐♥❛✐s tr❛♥s✐tór✐♦s é ♦❜s❡r✲ ✈❛❞❛✳
❆❜str❛❝t
■♥ t❤✐s ❞✐ss❡rt❛t✐♦♥✱ ✐t ✐s ❞❡✈❡❧♦♣❡❞ ❛♥❞ ✐♠♣❧❡♠❡♥t❡❞ ❛ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞ ❢♦r ♠♦❞❡❧✐♥❣ t❤❡ s♦✐❧ ❡❧❡❝tr✐❝❛❧ ❞✐s♣❡rs✐♦♥ ✉s✐♥❣ t❤❡ ❋✐♥✐t❡✲❉✐✛❡r❡♥❝❡ ❚✐♠❡✲❉♦♠❛✐♥ ♠❡t❤♦❞ ✭❋❉❚❉✮✳ ❚❤❡ ♠❡t❤♦❞♦❧♦❣② ✐s ❜❛s❡❞ ♦♥ t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ P❛❞é ❛♣♣r♦①✐♠❛♥ts ✐♥ ❢r❡q✉❡♥❝② ❞♦♠❛✐♥✱ ✐♥ s✉❝❤ ❛ ✇❛② t♦ r❡♣r❡s❡♥t t❤❡ s♣❡❝tr❛❧ ❞❡♣❡♥❞❡♥❝② ♦❢ s♦✐❧✬s ❝♦♥❞✉❝t✐✈✐t② ❛♥❞ ❡❧❡❝tr✐❝❛❧ ♣❡r♠✐tt✐✈✐t②✳ ❚❤❡ t✐♠❡ ❞♦♠❛✐♥ ❡q✉❛t✐♦♥s ❛r❡ ♦❜t❛✐♥❡❞ ✉s✐♥❣ s✐♠♣❧❡ ❛❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s ❛♥❞ ✐♥✈❡rs❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠s✳ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❢♦r ❣r♦✉♥❞✐♥❣ s②st❡♠s s✉❜❥❡❝t❡❞ t♦ ❛rt✐✜❝✐❛❧ ❧✐❣❤t♥✐♥❣ ♣✉❧s❡s ❛r❡ ❝♦♠♣❛r❡❞ ✇✐t❤ ♥✉♠❡r✐❝❛❧ r❡s✉❧ts ♦❜t❛✐♥❡❞ ✉s✐♥❣ t❤❡ ♣r♦♣♦s❡❞ ♠❡t❤♦❞✳ ■t ✇❛s ♦❜s❡r✈❡❞ ❛♥ ❡①❝❡❧❧❡♥t ❛❣r❡❡♠❡♥t ❜❡t✇❡❡♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ tr❛♥s✐❡♥t s✐❣✲ ♥❛❧s✳
✶
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
✶✳✶ ❈♦♥t❡ú❞♦ ❞♦ ❈❛♣ít✉❧♦
◆❡st❡ ❈❛♣ít✉❧♦✱ ❢❛③✲s❡ ✉♠❛ ❜r❡✈❡ ✐♥tr♦❞✉çã♦ s♦❜r❡ ♦s s✐st❡♠❛s ❞❡ ❛t❡rr❛♠❡♥t♦ ❡✱ t❛♠✲ ❜é♠✱ é ❛♣r❡s❡♥t❛❞❛ ❛ ♠♦t✐✈❛çã♦ ❞❛ ❛♣❧✐❝❛çã♦ ❞❛ ♠♦❞❡❧❛❣❡♠ ❞❡ s♦❧♦s ❞✐s♣❡rs✐✈♦s ❡♠ s✐♠✉❧❛çõ❡s ♥✉♠ér✐❝❛s ♣❛r❛ ❛♥❛❧✐s❛r ❛ r❡s♣♦st❛ tr❛♥s✐tór✐❛ ❡ ❞❡ r❡❣✐♠❡ ♣❡r♠❛♥❡♥t❡ ❞❡ss❡s s✐st❡♠❛s✳ ❆✐♥❞❛✱ sã♦ ❛♣r❡s❡♥t❛❞♦s ♦s ♦❜❥❡t✐✈♦s ❡ ❛ ♦r❣❛♥✐③❛çã♦ ❞♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦✳
✶✳✷ ■♥tr♦❞✉çã♦ ●❡r❛❧
❆t❡rr❛♠❡♥t♦ ❡❧étr✐❝♦ ❝♦♥s✐st❡ ❞❡ ❧✐❣❛çã♦ ✐♥t❡♥❝✐♦♥❛❧ ❞❡ ✉♠ ❡q✉✐♣❛♠❡♥t♦ à t❡rr❛✱ ❛tr❛✈és ❞❡ ✉♠ ♠❛t❡r✐❛❧ ❡❧❡tr✐❝❛♠❡♥t❡ ❝♦♥❞✉t✐✈♦✱ ♦ q✉❛❧ ♥♦r♠❛❧♠❡♥t❡ ❡stá ❡♠ ❝♦♥t❛t♦ ❡❧étr✐❝♦ ❝♦♠ ✉♠❛ ♠❛❧❤❛ ✭♦✉ ♦✉tr❛ ❡str✉t✉r❛✮ ❞❡ ❛t❡rr❛♠❡♥t♦✳ ❆s ♠❛❧❤❛s ♦✉ ❡str✉t✉r❛s ❞❡ ❛t❡rr❛♠❡♥t♦ sã♦ ❢♦r♠❛❞❛s ♣♦r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❝♦♥❞✉t♦r❡s ♥✉s ✐♥t❡r❧✐❣❛❞♦s ❡ ❡♥t❡rr❛❞♦s ♥♦ s♦❧♦ ❬✶❪✳
❆s ♣r✐♥❝✐♣❛✐s ❢✉♥çõ❡s ❞♦ s✐st❡♠❛ ❞❡ ❛t❡rr❛♠❡♥t♦ sã♦✿
• P❡r♠✐t✐r ♦ ❡s❝♦❛♠❡♥t♦ ♣❛r❛ ❛ t❡rr❛ ❞❡ ❝❛r❣❛s ♦✉ ❝♦rr❡♥t❡ ❞❡ ❞❡s❝❛r❣❛✱ ♣r♦✈♦❝❛❞❛s
✷
❡①t❡r♥❛✭❉❡s❝❛r❣❛ ❛t♠♦s❢ér✐❝❛✮ ❡ ✐♥t❡r♥❛ ✭♠❛♥♦❜r❛s ♥♦ s✐st❡♠❛✮ ❡ t❛♠❜é♠ ♣❡r♠✐t✐r ♦ ❡s❝♦❛♠❡♥t♦ ❞❡ ❈❛r❣❛s ❊stát✐❝❛s ❛❝✉♠✉❧❛❞❛s ♥❛ ❝❛r❝❛ç❛ ❞❡ ❡q✉✐♣❛♠❡♥t♦s❀
• ❉✐str✐❜✉✐çã♦ ❞❡ ♣♦t❡♥❝✐❛✐s ♥♦ s♦❧♦ ❡♠ ♣❛t❛♠❛r❡s s❡❣✉r♦s✱ ❞❡ ❢♦r♠❛ q✉❡ ❛ ❡❧❡✈❛çã♦
❞❡ ♣♦t❡♥❝✐❛❧ ♥♦ s♦❧♦ s♦❜r❡ ❛ ♠❛❧❤❛ ❞❡ ❛t❡rr❛♠❡♥t♦ s❡❥❛ ❞❡ ❢♦r♠❛ ♠❛✐s ✉♥✐❢♦r♠❡ ♣♦ssí✈❡❧ ❡✈✐t❛♥❞♦ ❞❡st❛ ❢♦r♠❛ t❡♥sã♦ ❞❡ ♣❛ss♦ ❡ t♦q✉❡ ❡♠ ♣❛t❛♠❛r❡s ✐♥s❡❣✉r♦s❀
• ❯s❛r ❛ t❡rr❛ ❝♦♠♦ ❝♦♥❞✉t♦r ❞❡ r❡t♦r♥♦ ❞❡ ❝♦rr❡♥t❡ ❡♠ s✐st❡♠❛s ♠♦♥♦❢ás✐❝♦s ❝♦♠
r❡t♦r♥♦ ♣❡❧❛ t❡rr❛✱ ♠✉✐t♦ ✉t✐❧✐③❛❞♦ ❡♠ r❡❞❡ ❞❡ ❞✐str✐❜✉✐çã♦ r✉r❛❧✱ ❡♠ s✐st❡♠❛s ❞❡ ❚r❛♥s♠✐ssã♦ ❡♠ ❝♦rr❡♥t❡ ❝♦♥t✐♥✉❛ ❡ ❡♠ ❛❧❣✉♥s s✐st❡♠❛s ❞❡ tr❛çã♦ ❡❧étr✐❝❛❀
• ■♥✢✉❡♥❝✐❛r ♥♦ ❞❡s❡♠♣❡♥❤♦ ❞♦ s✐st❡♠❛✱ s❡❥❛ ♥♦ ❡st❛❜❡❧❡❝✐♠❡♥t♦ ❞❡ r❡❢❡rê♥❝✐❛s ❞❡ ♣♦✲
t❡♥❝✐❛❧ ♦✉ ♣❡❧❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ✐♠♣❧❡♠❡♥t❛çã♦ ❞❡ ✉♠❛ ✜❧♦s♦✜❛ ❞❡ ♣r♦t❡çã♦ ❜❛s❡❛❞❛ ♥❛ ❞❡t❡❝çã♦ ❞❡ ✢✉①♦ ❞❡ ❝♦rr❡♥t❡ ♣❡❧♦ s♦❧♦ ❬✷✱✸❪✳
❆ ♠❛✐♦r✐❛ ❞♦s s✐st❡♠❛s ❞❡ ❛t❡rr❛♠❡♥t♦ sã♦ ♣r♦❥❡t❛❞♦s ♣❛r❛ q✉❡ ❛ r❡s✐stê♥❝✐❛ ❞❡ ❛t❡r✲ r❛♠❡♥t♦ ✜q✉❡ ❛❜❛✐①♦ ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❧✐♠✐t❡✱ ❞❡ ❢♦r♠❛ ❛ ❢❛❝✐❧✐t❛r ♦ ❡s❝♦❛♠❡♥t♦ ❞❛ ❝♦rr❡♥t❡ ♣❛r❛ ♦ s♦❧♦ ❡ ❛❧❝❛♥ç❛r ❝♦♠ ♠❛✐♦r ❢❛❝✐❧✐❞❛❞❡ t❡♥sõ❡s ❞❡ ♣❛ss♦ ❡ t♦q✉❡ ❡♠ ♣❛t❛✲ ♠❛r❡s s❡❣✉r♦s✱ ♦ q✉❡ é ✈á❧✐❞♦ ♣❛r❛ ❡✈❡♥t♦s ❞❡ ❜❛✐①❛s ❢r❡q✉ê♥❝✐❛s ❬✷✱ ✸❪✳ ❈♦♠♦ ♦ ❇r❛s✐❧ é ♦ ♣❛ís ❝♦♠ ❛ ♠❛✐♦r ✐♥❝✐❞ê♥❝✐❛ ❞❡ ❞❡s❝❛r❣❛ ❛t♠♦s❢ér✐❝❛ ❞♦ ♠✉♥❞♦✱ ❡ ❝❡r❝❛ ❞❡ ✼✵✪ ❞♦s ❞❡s❧✐❣❛♠❡♥t♦s ❞♦ s✐st❡♠❛ ❞❡ tr❛♥s♠✐ssã♦ ❡ ✹✵✪ ♥❛ ❞✐str✐❜✉✐çã♦ sã♦ ♣r♦✈♦❝❛❞♦s ♣♦r ❞❡s✲ ❝❛r❣❛ ❛t♠♦s❢ér✐❝❛ ❬✹❪✱ é ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ❛✈❛❧✐❛r ♦ ❞❡s❡♠♣❡♥❤♦ ❞♦ s✐st❡♠❛ ❞❡ ❛t❡rr❛♠❡♥t♦ ♣❛r❛ ❢❡♥ô♠❡♥♦s rá♣✐❞♦s ✭❢r❡q✉ê♥❝✐❛s ❡❧❡✈❛❞❛s✮ ❬✺✕✽❪✳
❉❡♥tr❡ ♦✉tr♦s ❢❛t♦r❡s✱ ❛ r❡s♣♦st❛ ❞❡ss❡s s✐st❡♠❛s ♣❛r❛ ❢❡♥ô♠❡♥♦s rá♣✐❞♦s ❞❡♣❡♥❞❡ ❞❛ ❝♦♥✜❣✉r❛çã♦ ❣❡♦♠étr✐❝❛ ❞♦ ❛t❡rr❛♠❡♥t♦✱ ❞❛ ❢♦r♠❛ ❞❡ ♦♥❞❛ ❞♦ ♣✉❧s♦ ✐♥❥❡t❛❞♦ ❡ ❞♦s ♣❛râ♠❡tr♦s ❡❧❡tr♦♠❛❣♥ét✐❝♦s ❞♦ s♦❧♦ ❬✻✱✼❪✳
✸
❞❡♥tr❡ ♦✉tr♦s ❢❛t♦r❡s✱ ❞❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞♦s s❡✉s ♣❛râ♠❡tr♦s ❡❧❡tr♦♠❛❣♥ét✐❝♦s ✭❝♦♥❞✉t✐✲ ✈✐❞❛❞❡ ❡ ♣❡r♠✐ss✐✈✐❞❛❞❡✮ ❝♦♠ ❛ ❢r❡q✉ê♥❝✐❛✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ♣❛r❛ ♦ s♦❧♦✱ q✉❡ ♣♦❞❡ ❛❢❡t❛r s✐❣♥✐✜❝❛t✐✈❛♠❡♥t❡ ❛ r❡s♣♦st❛ tr❛♥s✐tór✐❛ ❞♦ ❛t❡rr❛♠❡♥t♦ ❡❧étr✐❝♦ ❬✶✷❪✳
❱❛❧♦r❡s ❝♦♥st❛♥t❡s ♣❛r❛ ❛ r❡s✐st✐✈✐❞❛❞❡ ❡ ♣❡r♠✐ss✐✈✐❞❛❞❡ r❡❧❛t✐✈❛ ❞♦ s♦❧♦ sã♦ ♥♦r♠❛❧✲ ♠❡♥t❡ ✉t✐❧✐③❛❞♦s✳ ❆ r❡s✐st✐✈✐❞❛❞❡ é ♠❡❞✐❞❛ ♣♦r ♠❡✐♦ ❞❡ ❡①♣❡r✐♠❡♥t♦s ✉t✐❧✐③❛♥❞♦ t❡rr♦✲ ♠❡tr♦ ♦✉ ❞❡ ❢♦r♠❛ ✐♥❞✐r❡t❛ ❛tr❛✈és ❞❛ ❢ór♠✉❧❛ ❞❡ ❙✉♥❞❡ ❬✶✸❪✱ ❡ ❞❡♣❡♥❞❡♥❞♦ ❞♦ t❡♦r ❞❡ ✉♠✐❞❛❞❡ ❞♦ s♦❧♦ ❛❞♦t❛✲s❡ ✉♠ ✈❛❧♦r ❡♥tr❡ ✸ ❡ ✽✶ ♣❛r❛ ❛ ♣❡r♠✐ss✐✈✐❞❛❞❡ r❡❧❛t✐✈❛✱ ❬✻❪✳
P♦ré♠✱ ♦❜s❡r✈❛✲s❡ q✉❡ ❛ ❝♦♥❞✉t✐✈✐❞❛❞❡ ❡ ♣❡r♠✐ss✐✈✐❞❛❞❡ ❡❧étr✐❝❛ ♣♦ss✉❡♠ ✉♠❛ s✐❣✲ ♥✐✜❝❛t✐✈❛ ❞❡♣❡♥❞ê♥❝✐❛ ❝♦♠ ❛ ❢r❡q✉ê♥❝✐❛✳ ❖s tr❛❜❛❧❤♦s ♠❛✐s ❝✐t❛❞♦s s♦❜r❡ ❝♦rr❡çã♦ ❞❛ ❝♦♥❞✉t✐✈✐❞❛❞❡ ❡ ♣❡r♠✐ss✐✈✐❞❛❞❡ ❞♦ s♦❧♦ ❝♦♠ ❛ ❢r❡q✉ê♥❝✐❛ sã♦ ♦s ♣r♦♣♦st♦s ♣♦r✿ ❙❝♦tt ❡♠ ✶✾✻✻ ❡ ✶✾✻✼ ❬✶✹✱ ✶✺❪❀ ▲♦♥❣♠✐r❡ ❡ ❙♠✐t❤ ❡♠ ✶✾✼✺ ❬✶✻❪ ❀ P♦rt❡❧❛ ❡♠ ✶✾✾✾ ❬✶✼❪❀ ❱✐s❛❝r♦ ❡♠ ✷✵✵✼ ❬✻❪ ❡ ❱✐s❛❝r♦ ❡ ❆❧í♣✐♦ ❡♠ ✷✵✶✷ ❬✶✽❪✳
❆❧❣✉♥s tr❛❜❛❧❤♦s ♠♦str❛♠ q✉❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ tr❛♥s✐tór✐♦ ❞❡ s✐st❡♠❛s ❞❡ ❛t❡rr❛✲ ♠❡♥t♦ ❡❧étr✐❝♦ s✉❜♠❡t✐❞♦s ❛ ❞❡s❝❛r❣❛s ❛t♠♦s❢ér✐❝❛s ❛♣r❡s❡♥t❛♠ ❞❡s✈✐♦s s✐❣♥✐✜❝❛t✐✈♦s ❡♥tr❡ ♦s s✐♥❛✐s ♦❜t✐❞♦s ♥✉♠❡r✐❝❛♠❡♥t❡ ❝♦♠ ♦s ♣❛râ♠❡tr♦s ❞♦ s♦❧♦ ❝♦♥st❛♥t❡s ❡ q✉❛♥❞♦ ♦s ♣❛râ✲ ♠❡tr♦s ❞♦ s♦❧♦ sã♦ ❢✉♥çõ❡s ❞❛ ❢r❡q✉ê♥❝✐❛✱ s❡♥❞♦ q✉❡ ❡st❡s ú❧t✐♠♦s sã♦ ♦s ♠❛✐s ♣ró①✐♠♦s ❞♦s s✐♥❛✐s ♠❡❞✐❞♦s ❡♠ ❝❛♠♣♦ ❬✻✱✶✾✱✷✵❪✳
◆♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦✱ é ♣r♦♣♦st❛ ✉♠❛ ♠♦❞❡❧❛❣❡♠ ✉t✐❧✐③❛♥❞♦ ♦ ♠ét♦❞♦ ❋❉❚❉ ✭❋✐♥✐t❡✲ ❉✐✛❡r❡♥❝❡ ❚✐♠❡✲❉♦♠❛✐♥✮ ❬✷✶❪ ♣❛r❛ ♠❛t❡r✐❛✐s ❞✐s♣❡rs✐✈♦s✱ ❛tr❛✈és ❞❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞❛ ❝♦♥✲ ❞✉t✐✈✐❞❛❞❡ ❡ ♣❡r♠✐ss✐✈❛❞❡ ❡❧étr✐❝❛ ❝♦♠ ❛ ❢r❡q✉ê♥❝✐❛✱ ˜σ(f) ❡ ˜ε(f)✳ P❛r❛ r❡♣r❡s❡♥t❛r ♦
❝♦♠♣♦rt❛♠❡♥t♦ ❞❡˜σ(f)❡˜ε(f)✱ ❢♦r❛♠ ✉t✐❧✐③❛❞❛s ❡ ❛❞❛♣t❛❞❛s ❛s ❢♦r♠✉❧❛çõ❡s ❞❡ ▲♦♥❣♠✐r❡✲
❙♠✐t❤ ❬✶✻❪✱ P♦rt❡❧❛ ❬✶✼❪ ❡ ❛ ❞❡ ❱✐s❛❝r♦ ❡ ❆❧í♣✐♦ ❬✶✽❪ ♣❛r❛ ✈❡r✐✜❝❛r ❛ q✉❡ ♠❛✐s s❡ ❛❞éq✉❛ ❛♦ s♦❧♦ ❞♦ ❧♦❝❛❧ ❞♦s ❡①♣❡r✐♠❡♥t♦s✳
❆ ✐♥s❡rçã♦ ❞❡ ˜σ(f) ❡ ˜ε(f) ❞❡ q✉❛❧q✉❡r ❢♦r♠✉❧❛çã♦ ❞❡s❝r✐t❛ ❛❝✐♠❛ ♥❛s ❡q✉❛çõ❡s ❞❡
▼❛①✇❡❧❧ é ❢❡✐t❛ ❛tr❛✈és ❞❛ ❛♣r♦①✐♠❛çã♦ ♣♦r ♣♦❧✐♥ô♠✐♦s ❞❡ P❛❞é ❡♠ t❡r♠♦s ❞❡jω✱ ❢❛❝✐❧✐✲
✹
❛ ❞✐s♣❡rsã♦ ❞❛ ❝♦♥❞✉t✐✈✐❞❛❞❡✱ ✉t✐❧✐③❛♥❞♦ ❋❉❚❉ ❢♦✐ ✈❛❧✐❞❛❞❛ ❡♠ ❬✷✷❪✳
P❛r❛ r❡❛❧✐③❛r ❛s s✐♠✉❧❛çõ❡s ♥✉♠ér✐❝❛s q✉❡ ✈❛❧✐❞❛♠ ♦s ❡①♣❡r✐♠❡♥t♦s r❡❛❧✐③❛❞♦s✱ ❢♦✐ ✐♠✲ ♣❧❡♠❡♥t❛❞❛ ✉♠❛ r♦t✐♥❛ ❝♦♠♣✉t❛❝✐♦♥❛❧ ♣❛r❛ ✐♥❝❧✉✐r ♦s ❡❢❡✐t♦s ❞❛ ♣❡r♠✐ss✐✈✐❞❛❞❡ ❡ ❝♦♥❞✉✲ t✐✈✐❞❛❞❡ ✈❛r✐❛♥❞♦ ❝♦♠ ❛ ❢r❡q✉ê♥❝✐❛ ♥♦ s♦❢t✇❛r❡ ❙❆●❙ ✭❙②♥t❤❡s✐s ❛♥❞ ❆♥❛❧②s✐s ♦❢ ●r♦✉♥✲ ❞✐♥❣ ❙②st❡♠s✮ ❬✷✸❪✱ q✉❡ r❡s♦❧✈❡ ♥✉♠❡r✐❝❛♠❡♥t❡ ❛s ❡q✉❛çõ❡s ❞❡ ▼❛①✇❡❧❧ ❛tr❛✈és ❞♦ ♠ét♦❞♦ ❋❉❚❉ ✭❋✐♥✐t❡✲❉✐✛❡r❡♥❝❡ ❚✐♠❡✲❉♦♠❛✐♥✮ ❬✷✶❪ ♣❛r❛ ✉♠ ❛♠❜✐❡♥t❡ ✸✲❉ ❡♠ ❝♦♥❥✉♥t♦ ❝♦♠ ❛ té❝♥✐❝❛ ❞❡ tr✉♥❝❛❣❡♠ ❞♦ ❞♦♠í♥✐♦ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❈P▼▲ ✭❈♦♥✈♦❧✉t✐♦♥❛❧ P❡r❢❡❝t❧② ▼❛t❝❤❡❞ ▲❛②❡r✮ ❬✷✹❪✳ ❆❧é♠ ❞✐ss♦✱ ♦ ❙❆●❙ ✉t✐❧✐③❛ ❛ ❢♦r♠✉❧❛çã♦ ❞❡ ✜♦✲✜♥♦ ❬✷✺❪ ♣❛r❛ r❡♣r❡s❡♥t❛r ❛s ❤❛st❡s✱ ❝❛❜♦s ❡ ♦s ❡❧❡tr♦❞♦s ✉t✐❧✐③❛❞♦s ♥♦s ❡①♣❡r✐♠❡♥t♦s ❞❡ ♠♦❞♦ ❛ ❡✈✐t❛r ♠❛✐♦r ♥í✈❡❧ ❞❡ ❞✐s❝r❡t✐③❛çã♦ ❞♦ ❞♦♠í♥✐♦ ❝♦♠♣✉t❛❝✐♦♥❛❧✳
❆ ✈❛❧✐❞❛çã♦ ❞❛ ♠♦❞❡❧❛❣❡♠ ♣r♦♣♦st❛ ♥❡st❡ tr❛❜❛❧❤♦ é ❢❡✐t❛ ♣♦r ♠❡✐♦ ❞❡ t❡st❡s ❡①♣❡r✐✲ ♠❡♥t❛✐s ✉t✐❧✐③❛♥❞♦ s✉rt♦s ❞❡ t❡♥sã♦ ✭❝♦♠ ❛ ❢♦r♠❛ ❞❡ ♦♥❞❛ ❞❡ ❞❡s❝❛r❣❛s ❛t♠♦s❢ér✐❝❛s ❝♦♠ ❛♠♣❧✐t✉❞❡ r❡❞✉③✐❞❛✮ ❡♠ s✐st❡♠❛s ❞❡ ❛t❡rr❛♠❡♥t♦✳ ❋♦✐ ✉t✐❧✐③❛❞♦ ♥❡st❡ tr❛❜❛❧❤♦ ✉♠ ♠ét♦❞♦ ❞❡ ♠❡❞✐çã♦ ❝♦♠ ❝♦♥✜❣✉r❛çã♦ ♣r♦♣♦st❛ ♣♦r ❚❛♥❛❜❡ ❡♠ ❬✺❪✳
✶✳✸ ❖❜❥❡t✐✈♦s ❞❡st❡ ❚r❛❜❛❧❤♦
❉❡s❡♥✈♦❧✈❡r ✉♠❛ ♠♦❞❡❧❛❣❡♠ ✉t✐❧✐③❛♥❞♦ ♦ ♠ét♦❞♦ ❋❉❚❉ ❛♣❧✐❝❛❞♦ às ❡q✉❛çõ❡s ❞❡ ▼❛①✇❡❧❧ ❡♠ ✉♠❛ r❡❣✐ã♦ tr✐❞✐♠❡♥s✐♦♥❛❧ ♣❛r❛ ♠❡✐♦s ❝♦♠ ❝♦♥❞✉t✐✈✐❞❛❞❡ ❡ ♣❡r♠✐ss✐✈✐❞❛❞❡ ❡❧étr✐❝❛ ❞❡♣❡♥❞❡♥t❡s ❞❛ ❢r❡q✉ê♥❝✐❛✱ ✈❛❧✐❞❛r ❛ ♠♦❞❡❧❛❣❡♠ ♣♦r ♠❡✐♦ ❞❡ t❡st❡s ❡①♣❡r✐♠❡♥t❛✐s ❡♠ s✐st❡♠❛s ❞❡ ❛t❡rr❛♠❡♥t♦ ❡ ❞❡t❡r♠✐♥❛r ♦ ♠♦❞❡❧♦ ❞❡ ❞✐s♣❡rsã♦✱ ♣r❡✈✐❛♠❡♥t❡ ♣✉❜❧✐❝❛❞♦ ♥❛ ❧✐t❡r❛t✉r❛✱ q✉❡ ♠❡❧❤♦r r❡♣r❡s❡♥t❛ ❡❧❡tr✐❝❛♠❡♥t❡ ♦ s♦❧♦ ❛♠❛③ô♥✐❝♦ ✭r❡❣✐ã♦ ❞❡ ❇❡❧é♠✮✳
✶✳✹ ❖r❣❛♥✐③❛çã♦ ❞❡st❡ ❚r❛❜❛❧❤♦
❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❡stá ♦r❣❛♥✐③❛❞♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿
✺
• ❈❛♣ít✉❧♦ ✷ ✕ ❆♣r❡s❡♥t❛ ♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s r❡❧❛t✐✈♦s ❛♦s s✐st❡♠❛s ❞❡ ❛t❡rr❛♠❡♥t♦ ❡
❞❡s❝r❡✈❡ ♦s ♠♦❞❡❧♦s ✉t✐❧✐③❛❞♦s ♣❛r❛ r❡♣r❡s❡♥t❛r ♦ ❢❡♥ô♠❡♥♦ ❞❛ ❞✐s♣❡rsã♦ ❞♦ s♦❧♦❀
• ❈❛♣ít✉❧♦ ✸ ✕ ❆♣r❡s❡♥t❛ ♦s ❝♦♥❝❡✐t♦s r❡❧❛❝✐♦♥❛❞♦ ❛♦ ▼ét♦❞♦ ❞❛s ❉✐❢❡r❡♥ç❛s ❋✐♥✐t❛s
♥♦ ❉♦♠í♥✐♦ ❞♦ ❚❡♠♣♦ ✭❋❉❚❉✮✳
• ❈❛♣ít✉❧♦ ✹ ✕ ❆♣r❡s❡♥t❛ ❛ ♠♦❞❡❧❛❣❡♠ ❞❡s❡♥✈♦❧✈✐❞❛ ♣❛r❛ r❡♣r❡s❡♥t❛r ♠❡✐♦s ❝♦♠
❝♦♥❞✉t✐✈✐❞❛❞❡ ❡ ♣❡r♠✐ss✐✈✐❞❛❞❡ ❡❧étr✐❝❛ ❞❡♣❡♥❞❡♥t❡s ❞❛ ❢r❡q✉ê♥❝✐❛ ♣❛r❛ ♦ ♠ét♦❞♦ ❋❉❚❉✱ q✉❡ é ❛ ♣r✐♥❝✐♣❛❧ ❝♦♥tr✐❜✉✐çã♦ ❞❡st❡ tr❛❜❛❧❤♦❀
• ❈❛♣ít✉❧♦ ✺ ✕ ❆♣r❡s❡♥t❛ ❛ ❝♦♠♣❛r❛çã♦ ❡♥tr❡ ♦s s✐♥❛✐s ♠❡❞✐❞♦s ❞❡ t❡♥sã♦ ❡ ❝♦rr❡♥t❡
♥♦s t❡st❡s ❡①♣❡r✐♠❡♥t❛✐s✱ ♦s s✐♥❛✐s ♦❜t✐❞♦s ♥✉♠❡r✐❝❛♠❡♥t❡ ♣❡❧♦s ♠♦❞❡❧♦s ❞❡ s♦❧♦ ❞✐s♣❡rs✐✈♦✱ ✐♠♣❧❡♠❡♥t❛❞♦s ♣❛r❛ ✈❛❧✐❞❛r ❛ ♠♦❞❡❧❛❣❡♠ ♣r♦♣♦st❛ ❞❡st❡ tr❛❜❛❧❤♦ ❡ ♦s s✐♥❛✐s ♦❜t✐❞♦s ♥✉♠❡r✐❝❛♠❡♥t❡ s❡♠ ❝♦♥s✐❞❡r❛r ❛ ❞✐s♣❡rsã♦ ♥♦ s♦❧♦✳
✻
❈❛♣ít✉❧♦ ✷
❋✉♥❞❛♠❡♥t❛çã♦ ❚❡ór✐❝❛
✷✳✶ ❈♦♥t❡ú❞♦ ❞♦ ❈❛♣ít✉❧♦
◆❡st❡ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛✲s❡ ✉♠❛ ❛❜♦r❞❛❣❡♠ s♦❜r❡ ❛ ❛♥á❧✐s❡ ❡❧❡tr♦♠❛❣♥ét✐❝❛ ❡ ❞♦s ❛s♣❡❝t♦s q✉❡ ✐♥✢✉❡♥❝✐❛♠ ♥❛ r❡s♣♦st❛ tr❛♥s✐tór✐❛ ❞♦s s✐st❡♠❛s ❞❡ ❛t❡rr❛♠❡♥t♦✳ ❆✐♥❞❛✱ sã♦ ❛♣r❡s❡♥t❛❞❛s ♣r✐♥❝✐♣❛✐s ❢♦r♠✉❧❛çõ❡s ♣✉❜❧✐❝❛❞❛s ♥❛ ❧✐t❡r❛t✉r❛ ♣❛r❛ r❡♣r❡s❡♥t❛r ♦s ❡❢❡✐t♦s ❞✐s♣❡rs✐✈♦s ♥♦ s♦❧♦✱ ♥❛ ❜❛♥❞❛ ❞❡ ❢r❡q✉ê♥❝✐❛s ❞❡ ❞❡s❝❛r❣❛s ❛t♠♦s❢ér✐❝❛s✳
✷✳✷ ❙✐st❡♠❛s ❞❡ ❆t❡rr❛♠❡♥t♦
✷✳✷✳✶
❊q✉❛çõ❡s ❞❡ ▼❛①✇❡❧❧ ♥❛ ❢♦r♠❛ ❞✐❢❡r❡♥❝✐❛❧ ♥♦ ❞♦♠í♥✐♦ ❞♦
t❡♠♣♦
✼
❞❡ ❛t❡rr❛♠❡♥t♦✳
❆ ❧❡✐ ❞❡ ❋❛r❛❞❛② ❡ ❛ ❧❡✐ ❞❡ ❆♠♣èr❡✲▼❛①✇❡❧❧✱ ♥❛ ❢♦r♠❛ ❞✐❢❡r❡♥❝✐❛❧✱ ❝♦♥s✐❞❡r❛♥❞♦ ♠❡✐♦s ✐s♦tró♣✐❝♦s ❡ ♥ã♦ ❞✐s♣❡rs✐✈♦s sã♦ ❞❛❞❛s r❡s♣❡❝t✐✈❛♠❡♥t❡ ♣♦r✿
∇ ×E~ =−∂µ ~H
∂t ✭✷✳✶✮
❡
∇ ×H~ =J~+ ∂ε ~E
∂t , ✭✷✳✷✮
♦♥❞❡E~ é ♦ ✈❡t♦r ✐♥t❡♥s✐❞❛❞❡ ❞❡ ❝❛♠♣♦ ❡❧étr✐❝♦ ✭❱✴♠✮✱ H~ é ♦ ✈❡t♦r ✐♥t❡♥s✐❞❛❞❡ ❞❡ ❝❛♠♣♦
♠❛❣♥ét✐❝♦ ✭❆✴♠✮✱µé ❛ ♣❡r♠❡❛❜✐❧✐❞❛❞❡ ♠❛❣♥ét✐❝❛ ✭❍✴♠✮ ❡ε é ❛ ♣❡r♠✐ss✐✈✐❞❛❞❡ ❡❧étr✐❝❛
✭❋✴♠✮✳ ❖ t❡r♠♦ J~= σ ~E é ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❝♦rr❡♥t❡ ❞❡ ❝♦♥❞✉çã♦ ✭❆✴m2✮✱ ♥❛ q✉❛❧ σ é ❛
❝♦♥❞✉t✐✈✐❞❛❞❡ ❡❧étr✐❝❛ ✭❙✴♠✮✳
❆ ❧❡✐ ❞❡ ❋❛r❛❞❛② ♠♦str❛ q✉❡ ❛ ✈❛r✐❛çã♦ t❡♠♣♦r❛❧ ❞♦ ✈❡t♦rB~ =µ ~H✭✈❡t♦r ❞❡♥s✐❞❛❞❡ ❞❡
✢✉①♦ ♠❛❣♥ét✐❝♦✮ ♣r♦✈♦❝❛ ♦ s✉r❣✐♠❡♥t♦ ❞❡ ❝❛♠♣♦ ❡❧étr✐❝♦ ❝✐r❝✉❧❛♥t❡ ❡♠ t♦r♥♦ ❞♦ ♣ró♣r✐♦ ✈❡t♦r B~ ✈❛r✐❛♥t❡ ♥♦ t❡♠♣♦✳ ❉❡ ❢♦r♠❛ s❡♠❡❧❤❛♥t❡✱ ❛ ❧❡✐ ❞❡ ❆♠♣èr❡ ♠♦str❛ q✉❡ q✉❛♥❞♦
❤á ✈❛r✐❛çã♦ t❡♠♣♦r❛❧ ❞❡ D~ ✭✈❡t♦r ❞❡♥s✐❞❛❞❡ ❞❡ ✢✉①♦ ❡❧étr✐❝♦✮✱ s✉r❣❡ ✉♠❛ ❝✐r❝✉❧❛çã♦ ❞♦
❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❡♠ t♦r♥♦ ❞♦ ✈❡t♦rD~✳ P♦rt❛♥t♦✱ ❛s ❡q✉❛çõ❡s ✭✷✳✷✮ ❡ ✭✷✳✶✮ ♠♦str❛♠ q✉❡✱
♦ ♣r♦❝❡ss♦ ❞❡ ♣r♦♣❛❣❛çã♦ ❞❛ ♦♥❞❛ ❡❧❡tr♦♠❛❣♥ét✐❝❛ ♣♦❞❡ ✐♥✐❝✐❛r ❝♦♠ ❛ ❡①❝✐t❛çã♦ ❞♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ❡♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♣♦♥t♦✱ q✉❡ ♣r♦♠♦✈❡ ❛ ✈❛r✐❛çã♦ t❡♠♣♦r❛❧ ❞❛s ❝♦♠♣♦♥❡♥t❡s ❞♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❛❞❥❛❝❡♥t❡s ❡ ❡st❛s ♣r♦♠♦✈❡♠ ✈❛r✐❛çã♦ t❡♠♣♦r❛❧ ❞♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ❡♠ ♣♦♥t♦s ♣ró①✐♠♦s✳ ◆♦ ❝❛s♦ ❝♦♥tí♥✉♦✱ ❡ss❛s ❞✐stâ♥❝✐❛s sã♦ ✐♥✜♥✐t❡s✐♠❛✐s✳ ❆ r❡♣❡t✐çã♦ ❞❡st❡ ♣r♦❝❡ss♦ ❞❡s❡♥✈♦❧✈❡ ❛ ❡❧❡tr♦❞✐♥â♠✐❝❛ ❛♦ ❧♦♥❣♦ ❞♦ t❡♠♣♦ ♥♦ ❡s♣❛ç♦ tr✐❞✐♠❡♥s✐♦♥❛❧✳
✽
❞❡ ●❛✉ss✿
∇ ·B~ = 0 ✭✷✳✸✮
❡
∇ ·D~ =ρv, ✭✷✳✹✮
♦♥❞❡ρv r❡♣r❡s❡♥t❛ ❛ ❞❡♥s✐❞❛❞❡ ✈♦❧✉♠étr✐❝❛ ❞❡ ❝❛r❣❛s ❡❧étr✐❝❛s ✭❈✴♠3✮✳
❆ ❡q✉❛çã♦ ✭✷✳✸✮✱ ♠♦str❛ q✉❡ ❛s ❧✐♥❤❛s ❞♦ ❝❛♠♣♦ ✈❡t♦r✐❛❧ H~ sã♦ s❡♠♣r❡ ❢❡❝❤❛❞❛s ❡♠
t♦❞♦s ♦s ♣♦♥t♦s ❞♦ ❡s♣❛ç♦ ❡ ❥✉st✐✜❝❛ ❛ ❛✉sê♥❝✐❛ ❞❡ ♠♦♥♦♣ó❧♦s ♠❛❣♥ét✐❝♦s ♥❛ ♥❛t✉r❡③❛✳ ❏á ❛ ❡q✉❛çã♦ ✭✷✳✹✮ ❞✐③ q✉❡ ❛s ❧✐♥❤❛s ❞❡ ❝❛♠♣♦ ✈❡t♦r✐❛❧D~ sã♦ ❢❡❝❤❛❞❛s ❛♣❡♥❛s ❡♠ r❡❣✐õ❡s
s❡♠ ♠♦♥♦♣ó❧♦s ❡❧étr✐❝♦s ❡ q✉❡ ❛ ♣r❡s❡♥ç❛ ❞❡ ❝❛r❣❛s ❡❧étr✐❝❛ ρv ♣r♦✈♦❝❛ ♦ s✉r❣✐♠❡♥t♦ ❞❡
❧✐♥❤❛s ❛❜❡rt❛s ❞♦ ❝❛♠♣♦D~✳
▼❛①✇❡❧❧ ❞❡✜♥✐✉ ❡♠ s❡✉ tr❛❜❛❧❤♦ ❬✷✻❪ ✉♠ ❝❛♠♣♦ ✈❡t♦r✐❛❧ A~ ❞❡♥♦♠✐♥❛❞♦ ❞❡ ✐♠♣✉❧s♦
❛♥❣✉❧❛r✱ q✉❡ ❛t✉❛❧♠❡♥t❡ é ❝♦♥❤❡❝✐❞♦ ♣♦r ✈❡t♦r ♣♦t❡♥❝✐❛❧ ♠❛❣♥ét✐❝♦ ❡ é ❞❡✜♥✐❞♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿
∇ ×A~ =µ ~H. ✭✷✳✺✮
❙✉❜st✐t✉✐♥❞♦ ✭✷✳✺✮ ❡♠ ✭✷✳✶✮✱ t❡♠✲s❡
∇ ×E~ =−∂
∂t
∇ ×A~, ✭✷✳✻✮
q✉❡ ♣♦❞❡ s❡r r❡❡s❝r✐t♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛
∇ × E~ + ∂ ~A ∂t
!
= 0. ✭✷✳✼✮
✾
♥✉❧♦✱ ❞❡✜♥❡✲s❡ ✉♠ ❡s❝❛❧❛rφ✱ ❞❡♥♦♠✐♥❛❞♦ ❞❡ ♣♦t❡♥❝✐❛❧ ❡s❝❛❧❛r ❡❧étr✐❝♦✱ ❞❡ t❛❧ ♠❛♥❡✐r❛ q✉❡
~ E+∂ ~A
∂t =−∇φ, ✭✷✳✽✮
♦✉ ❛✐♥❞❛✱ ♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❡♠ ❢✉♥çã♦ ❞❡ ✉♠ ❝❛♠♣♦ ❡s❝❛❧❛r φ ❡ ❞❡ ✉♠
❝❛♠♣♦ ✈❡t♦r✐❛❧A~
~
E =− ∇φ+ ∂ ~A
∂t !
, ✭✷✳✾✮
q✉❡ ♣♦❞❡ s❡r ✐♥t❡r♣r❡t❛❞❛ ❝♦♠♦ ✉♠❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ ❍❡❧♠❤♦❧t③✳
❖ ❝❛♠♣♦ ❡❧étr✐❝♦E~✱ ❞❛❞♦ ♣♦r ✭✷✳✾✮✱ ♥ã♦ é ❝♦♥s❡r✈❛t✐✈♦✱ ♦ q✉❡ ♣♦❞❡ s❡r ♠♦str❛❞♦ ♣❡❧❛
❡①♣r❡ssã♦ ✭✷✳✺✮✳ ❊♥tr❡t❛♥t♦✱ ♣❛r❛ ♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r q✉❡ ✭∂ ~A
∂t = 0✮✱ ♦❜s❡r✈❛✲s❡ ❛tr❛✈és ❞❡
✭✷✳✼✮ q✉❡ ∇ ×E~ = 0 ✭ ❡❧❡tr♦stát✐❝❛ ✮✱ ♦✉ ❞❡ ❢♦r♠❛ ❡q✉✐✈❛❧❡♥t❡ H
cE~ ·d~l= 0 ❡✱ ♥❡st❡ ❝❛s♦
t❡♠✲s❡ ✉♠ ❝❛♠♣♦ E~ ❝♦♥s❡r✈❛t✐✈♦✳
✷✳✷✳✷ ❊❧❡tr♦❞✐♥â♠✐❝❛ ❡♠ s✐st❡♠❛s ❞❡ ❛t❡rr❛♠❡♥t♦
❯♠❛ ❞❛s ♣r✐♥❝✐♣❛✐s ❢✉♥çõ❡s ❞♦ s✐st❡♠❛ ❞❡ ❛t❡rr❛♠❡♥t♦ é q✉❡ ❛ ❝♦rr❡♥t❡ ✐♥❥❡t❛❞❛ i(t)
♥♦ ♠❡s♠♦ ✢✉❛ ♣❛r❛ ♦ s♦❧♦ ❝♦♠ ❛ ♠❡❧❤♦r ❡✜❝✐ê♥❝✐❛ ♣♦ssí✈❡❧✳ P❛r❛ ❛✈❛❧✐❛r ♦ ❞❡s❡♠♣❡♥❤♦ ❞❡ss❡ s✐st❡♠❛ s♦❜ ❡st❡ ❛s♣❡❝t♦ ♣♦❞❡✲s❡ ❝❛❧❝✉❧❛r ♦✉ ♠❡❞✐r ❛ r❡❧❛çã♦ t❡♥sã♦ ❝♦rr❡♥t❡ ❞❛❞❛
♣♦r v(t)/i(t)✱ t❛♠❜é♠ ❝❤❛♠❛❞❛ ❞❡ ❚●❘ ✭❚r❛♥s✐❡♥t ●r♦✉♥❞✐♥❣ ❘❡s✐st❛♥❝❡✮ ❡♠ r❡❣✐♠❡
tr❛♥s✐tór✐♦✱ ♦✉ ❛✐♥❞❛✱ ♦ ✈❛❧♦r ❞❡ r❡❣✐♠❡ ❡st❛❝✐♦♥ár✐♦ ❞❛ ❚●❘ q✉❡ é ✐❣✉❛❧ à r❡s✐stê♥❝✐❛ ❞❡ ❛t❡rr❛♠❡♥t♦ R✱ ♦♥❞❡ v(t) ❡ i(t) sã♦ r❡s♣❡❝t✐✈❛♠❡♥t❡ ❛ t❡♥sã♦ ✐♥❞✉③✐❞❛ ❡♥tr❡ ♦ ♣♦♥t♦ ❞❡
✐♥❥❡çã♦ ❡ ✉♠ ❡❧❡tr♦❞♦ r❡♠♦t♦ ✭♣♦st♦ ❛ ✉♠❛ ❣r❛♥❞❡ ❞✐stâ♥❝✐❛ ❞♦ ♣♦♥t♦ ❞❡ ✐♥❥❡çã♦✮ ❡ ❛ ❝♦rr❡♥t❡ ✐♥❥❡t❛❞❛ ♥♦ s✐st❡♠❛ ❞❡ ❛t❡rr❛♠❡♥t♦✳
❆ t❡♥sã♦ ❡♥tr❡ ❞♦✐s ♣♦♥t♦sa❡b✱vab✱ é ❞❡✜♥✐❞❛ ♣❡❧❛ ✐♥t❡❣r❛❧ ❞❡ ❧✐♥❤❛ ❞♦ ❝❛♠♣♦ ❡❧étr✐❝♦
~
E ♣♦r ✉♠ ❝❛♠✐♥❤♦ ❛r❜✐trár✐♦✱ ♦✉ s❡❥❛✱
vab =
Z b
a
~
✶✵
◗✉❛♥❞♦ ♦ ♣♦♥t♦ b é ✐♥✜♥✐t❛♠❡♥t❡ ❞✐st❛♥t❡ ❞❡a ♥♦ q✉❛❧ E~ ≈0 t❡♠✲s❡ ❛ t❡♥sã♦ ✐♥❞✉③✐❞❛ v(t) ❝❛❧❝✉❧❛❞❛ ♣♦r✿
v(t) = va(t) =
Z ∞
a
~
E(t)·d~l (❱). ✭✷✳✶✶✮
❆ ❝♦rr❡♥t❡ ✐♥❥❡t❛❞❛i(t)♣♦❞❡ s❡r ❞❡t❡r♠✐♥❛❞❛ ♣❡❧❛ ✐♥t❡❣r❛❧ ❞❡ s✉♣❡r❢í❝✐❡ ❞❛ ❞❡♥s✐❞❛❞❡
❞❡ ❝♦rr❡♥t❡ ❞❡ ❝♦♥❞✉çã♦ J~ q✉❡ ✢✉✐ ❛tr❛✈és ❞❛ s❡çã♦ tr❛♥s✈❡rs❛❧ ❞♦ ❝❛❜♦✱ ✉♠❛ ✈❡③ q✉❡✱
♥♦s ♠❛t❡r✐❛✐s ❝♦♥❞✉t♦r❡s✱ ♦ t❡r♠♦ ∂ ∂t
RR
SD~ ·d~S ❞❛ ❧❡✐ ❞❡ ❆♠♣èr❡ é ✐❣✉❛❧ ❛ ③❡r♦✱ ♣♦✐s✱ ♥♦s
♠❛t❡r✐❛✐s ❝♦♥❞✉t♦r❡s ❛ ❞❡♥s✐❞❛❞❡ ❞❛ ❝♦rr❡♥t❡ ❞❡ ❞❡s❧♦❝❛♠❡♥t♦ é ♥✉❧❛ ✭∂ ~D
∂t = 0✮✳ ❚❡♠✲s❡✱
♣♦rt❛♥t♦✿
i(t) = Z Z
S
~
J ·d~S= I
c
~
H·d~l (❆). ✭✷✳✶✷✮
✷✳✸ ❉✐s♣❡rsã♦ ❞♦ ❙♦❧♦
◗✉❛♥❞♦ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞♦s ♣❛râ♠❡tr♦s ❡❧❡tr♦♠❛❣♥ét✐❝♦s ✭σ✱ ε ♦✉ µ✮ é ❢✉♥çã♦ ❞❛
❢r❡q✉ê♥❝✐❛✱ ♦ s♦❧♦ é ❝♦♥s✐❞❡r❛❞♦ ✉♠ ♠❡✐♦ ❞✐s♣❡rs✐✈♦ ❬✷✼❪✳ ▼✉✐t♦s tr❛❜❛❧❤♦s ❡①✐st❡♥t❡s ♥❛ ❧✐t❡r❛t✉r❛ ♠♦str❛♠ q✉❡ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞❡st❡s ♣❛râ♠❡tr♦s ❝♦♠ ❛ ❢r❡q✉ê♥❝✐❛ é s✐❣♥✐✜❝❛t✐✈❛ ♣❛r❛ r❡s♣♦st❛ ❛♦s s✐♥❛✐s ❞❡ ❛❧t❛s ❢r❡q✉ê♥❝✐❛s ✭❝♦♠♦ ❞❡s❝❛r❣❛s ❛t♠♦s❢ér✐❝❛s✮✱ ❡ ❞❡✈❡ s❡r ❝♦♥s✐❞❡r❛❞❛ ♥❛ ❛♥á❧✐s❡ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❡❧❡tr♦♠❛❣♥ét✐❝♦ ❞♦s s✐st❡♠❛s ❞❡ ❛t❡rr❛♠❡♥t♦ ❬✻✱✶✷✱✶✹✱✶✻✕✶✽✱✷✵✱✷✽✱✷✾❪✳
❯♠ ❣r❛♥❞❡ ❞❡s✈✐♦ ❡♥tr❡ ❛ r❡♣♦st❛ tr❛♥s✐tór✐❛ ❞♦s ❛t❡rr❛♠❡♥t♦s✱ ❞❡ s✐♥❛✐s ♦❜t✐❞♦s ❡①♣❡✲ r✐♠❡♥t❛❧♠❡♥t❡ ❡ ♥✉♠❡r✐❝❛♠❡♥t❡ ♣♦r ♠❡✐♦ ❞❡ s✐♠✉❧❛çõ❡s é ✈❡r✐✜❝❛❞♦✱ ♣♦r ♥ã♦ ❝♦♥s✐❞❡r❛r❡♠ ♦ ❢❡♥ô♠❡♥♦ ❞❡ ❞✐s♣❡rsã♦ ❞♦ s♦❧♦ ❬✶✷✱✶✽✱✷✵✱✸✵✱✸✶❪✳
✶✶
✷✳✸✳✶ ▼❛t❡r✐❛✐s ❉✐s♣❡rs✐✈♦s
❆ ❝♦♥st❛♥t❡ ❞✐❡❧étr✐❝❛ é ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦s ♠❛t❡r✐❛✐s ❡ ❡stá r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ ❛ r❡s✲ ♣♦st❛ ❞♦s ♠❛t❡r✐❛✐s ❛♦s ❝❛♠♣♦s ❡❧étr✐❝♦s✳ ❆ ✐♥t❡r❛çã♦ ❞♦s ❝❛♠♣♦s ❝♦♠ ❛ ♠❛tér✐❛ s❡ ♠❛♥✐❢❡st❛ ♠❡❞✐❛♥t❡ ❛ ♣♦❧❛r✐③❛çã♦ ❞♦s át♦♠♦s ♦✉ ♠♦❧é❝✉❧❛s ❞♦ ♠❛t❡r✐❛❧✳ ❈♦♠♦ r❡s✉❧t❛❞♦ ❞❛ ♣♦❧❛r✐③❛çã♦✱ ♦s ❞✐♣♦❧♦s ♦r✐❡♥t❛❞♦s ❝r✐❛♠ ✉♠ ❝❛♠♣♦ q✉❡ s❡ s✉♣❡r♣õ❡ ❛♦ ❝❛♠♣♦ ❡①t❡r♥♦✱ ❞❡t❡r♠✐♥❛♥❞♦ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞✐❡❧étr✐❝♦ ❞♦ ♠❛t❡r✐❛❧✳ ❆ ♠❡❞✐❞❛ ❞❡ss❡ ❡❢❡✐t♦ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❛ ♣❡r♠✐ss✐✈✐❞❛❞❡ ♦✉ ❝♦♥st❛♥t❡ ❞✐❡❧étr✐❝❛ ❞♦ ♠❡✐♦✱ε˜❬✸✷❪✳
❆ ♣❡r♠✐ss✐✈✐❞❛❞❡ε˜❞♦ ♠❛t❡r✐❛❧ ✈❛r✐❛ ❝♦♠ ❛ ❢r❡q✉ê♥❝✐❛✱ ♣♦✐s q✉❛♥❞♦ ✉♠ ♠❡✐♦ ❞✐❡❧étr✐❝♦
é s✉❜♠❡t✐❞♦ ❛ ✉♠ ❝❛♠♣♦ ❡❧étr✐❝♦ ❛❧t❡r♥❛❞♦ ♦s ❞✐♣♦❧♦s ❞♦ ♠❛t❡r✐❛❧ t❡♥t❛♠ s❡ r❡♦r✐❡♥t❛r ❝♦♠ ♦ ❝❛♠♣♦✱ ♥✉♠ ♣r♦❝❡ss♦ q✉❡ r❡q✉❡r ✉♠ ❞❡t❡r♠✐♥❛❞♦ t❡♠♣♦✱ ❡ ♦ ✐♥✈❡rs♦ ❞❡st❡ t❡♠♣♦ é ❞❡♥♦♠✐♥❛❞♦ ❞❡ ❢r❡q✉ê♥❝✐❛ ❞❡ r❡❧❛①❛çã♦✳ ❊st❛ ❢r❡q✉ê♥❝✐❛ ❞❡♣❡♥❞❡ ❞❛ ❢❛❝✐❧✐❞❛❞❡ ❝♦♠ q✉❡ ♦s ❞✐♣♦❧♦s sã♦ ❝❛♣❛③❡s ❞❡ s❡ r❡❛❧✐♥❤❛r❡♠ ❡ q✉❡ ✈❛r✐❛ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ♠❛t❡r✐❛❧✳ ❯♠ ❞✐♣♦❧♦ ♥ã♦ ❛❧❝❛♥ç❛ ❛ ❞✐r❡çã♦ ❞❡ ♦r✐❡♥t❛çã♦ q✉❛♥❞♦ ❛ ❢r❡q✉ê♥❝✐❛ ❞♦ ❝❛♠♣♦ ❛♣❧✐❝❛❞♦ ✉❧tr❛♣❛ss❛ s✉❛ ❢r❡q✉ê♥❝✐❛ ❞❡ r❡❧❛①❛çã♦ ❡ ❞❡st❛ ❢♦r♠❛ ♥ã♦ ❝♦♥tr✐❜✉✐rá ♣❛r❛ ❛ ❝♦♥st❛♥t❡ ❞✐❡❧étr✐❝❛✳ ❊①✐st❡ ✉♠❛ ♣❡r❞❛ ❞✐❡❧étr✐❝❛ q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛ ❛❜s♦rçã♦ ❞❡ ❡♥❡r❣✐❛ ❡❧étr✐❝❛ ♣♦r ✉♠ ♠❛t❡r✐❛❧ ❞✐❡❧étr✐❝♦ q✉❡ ❡stá s✉❥❡✐t♦ ❛ ✉♠ ❝❛♠♣♦ ❡❧étr✐❝♦ ❛❧t❡r♥❛❞♦✳ ❊ss❛ ♣❡r❞❛ ♣♦❞❡ s❡r ✐♠♣♦rt❛♥t❡ ❡♠ ❢r❡q✉ê♥❝✐❛s ❞♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ♥❛ ✈✐③✐♥❤❛♥ç❛ ❞❛ ❢r❡q✉ê♥❝✐❛ ❞❡ r❡❧❛①❛çã♦ ♣❛r❛ ❝❛❞❛ ✉♠ ❞♦s t✐♣♦s ❞❡ ❞✐♣♦❧♦s ♦♣❡r❛❝✐♦♥❛✐s ❡♠ ✉♠ ♠❛t❡r✐❛❧ ❡s♣❡❝í✜❝♦✳
❆ ♣❡r♠✐ss✐✈✐❞❛❞❡ ❞♦s ♠❡✐♦s ❞✐❡❧étr✐❝♦s ❧✐♥❡❛r❡s ❡ ✐s♦tró♣✐❝♦s ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦✿
˜ ε =ε′
−jε′′
=ε0(1 +χe), ✭✷✳✶✸✮
♦♥❞❡ ❛ ♣❛rt❡ r❡❛❧ ε′
é ❛ ♣❡r♠✐ss✐✈✐❞❛❞❡ ❞♦ ♠❛t❡r✐❛❧ s❡♠ ♣❡r❞❛s✱ ❛ ♣❛rt❡ ✐♠❛❣✐♥ár✐❛ ε′′
r❡♣r❡s❡♥t❛ ❛s ♣❡r❞❛s ♥♦ ♠❛t❡r✐❛❧ ❞❡✈✐❞♦ à r❡❧❛①❛çã♦ tér♠✐❝❛ ❛ss♦❝✐❛❞❛ ❛♦ ♠♦✈✐♠❡♥t♦ ❞♦s ❞✐♣♦❧♦s ♥✉♠ ❝❛♠♣♦ ❡❧étr✐❝♦ ❛❧t❡r♥❛❞♦ ❡ χe é ❛ s✉s❝❡♣t✐❜✐❧✐❞❛❞❡ ❡❧étr✐❝❛✱ q✉❡ é ✉♠❛
✶✷
❆ ❋✐❣✳ ✶ r❡♣r❡s❡♥t❛ ♦s ♣r✐♥❝✐♣❛✐s ♠❡❝❛♥✐s♠♦s ❞❡ ♣♦❧❛r✐③❛çã♦✱ ❡ ❛ ❋✐❣✳ ✷ ✐❧✉str❛ ♦ ❡s♣❡❝tr♦ ❞❛ ♣❡r♠✐ss✐✈✐❞❛❞❡ ❝♦♠♣❧❡①❛ ❡ ♦s ♠❡❝❛♥✐s♠♦s ❞❡ ♣♦❧❛r✐③❛çã♦ ❛ss♦❝✐❛❞♦s ❛ ❝❛❞❛ ❢r❡q✉ê♥❝✐❛✳
◆❛ ♣r✐♠❡✐r❛ ❢❛✐①❛ ❞♦ ❡s♣❡❝tr♦ ❞❡ ❢r❡q✉ê♥❝✐❛ ♦❜s❡r✈❛✲s❡ ❛ ♣♦❧❛r✐③❛çã♦ ✐♥t❡r❢❛❝✐❛❧ ✭❊❢❡✐t♦ ▼❛①✇❡❧❧✲❲❛❣♥❡r✮✱ q✉❡ s✉r❣❡ ❡♠ ❢✉♥çã♦ ❞❛s ❞❡s❝♦♥t✐♥✉✐❞❛❞❡s ❞❡ ❢❛s❡s ❞❡ ♠❛t❡r✐❛✐s ❤❡t❡r♦✲ ❣ê♥❡♦s✱ ♠❛t❡r✐❛✐s ❞✐❢❡r❡♥t❡s✱ ✈❛③✐♦s ❡ ✐♠♣✉r❡③❛s ❬✸✸❪✳ ❉❡✈✐❞♦ às ❞✐❢❡r❡♥t❡s ❝♦♥❞✉t✐✈✐❞❛❞❡s ❡ ♣❡r♠✐ss✐✈✐❞❛❞❡s ❞❛s r❡❣✐õ❡s✱ ♦❝♦rr❡ ❛ ❢♦r♠❛çã♦ ❞❡ ❝❛r❣❛s ❡s♣❛❝✐❛✐s ♥❛s ✐♥t❡r❢❛❝❡s ❞❡st❛s✳ ❊st❡ ❡❢❡✐t♦ é ❢r❡q✉❡♥t❡♠❡♥t❡ ♦❜s❡r✈❛❞♦ ❡♠ s♦❧♦s✱ q✉❡ ♥♦r♠❛❧♠❡♥t❡ sã♦ ❝♦♠♣♦st♦s ♣♦r ✉♠❛ ♠✐st✉r❛ ❞❡ ❝♦♠♣♦♥❡♥t❡s ❝♦♠ ❝♦♥s✐❞❡rá✈❡❧ ❞✐❢❡r❡♥ç❛ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❡❧étr✐❝❛s ❝♦♠♦ ❛r✱ ❧íq✉✐❞♦s ❡ só❧✐❞♦s✱ ❡ s❡♣❛r❛❞♦s ♣♦r ♥✉♠❡r♦s❛s ✐♥t❡r❢❛❝❡s✳
❊♠ s❡❣✉✐❞❛ t❡♠♦s ❛ ❞✐♣♦❧❛r ✭♦✉ ♦r✐❡♥t❛❝✐♦♥❛❧✮✱ q✉❡ ♦❝♦rr❡ só ❡♠ s✉❜stâ♥❝✐❛s q✉❡ ♣♦ss✉❡♠ ♠♦♠❡♥t♦s ❞❡ ❞✐♣♦❧♦ ♣❡r♠❛♥❡♥t❡s ❡ é ♣r♦✈♦❝❛❞❛ ♣❡❧❛ r♦t❛çã♦ ❞♦s ♠♦♠❡♥t♦s ♣❡r♠❛♥❡♥t❡s ♥❛ ❞✐r❡çã♦ ❞♦ ❝❛♠♣♦ ❛♣❧✐❝❛❞♦✳
▲♦❣♦ ❛♣ós ✈❡♠ ❛ ✐ô♥✐❝❛ ✭♦✉ ❛tô♠✐❝❛✮✱ q✉❡ ♦❝♦rr❡ só ❡♠ ♠❛t❡r✐❛✐s ✐ô♥✐❝♦s ❡ é r❡s✉❧t❛♥t❡ ❞♦ ❞❡s❧♦❝❛♠❡♥t♦ ♠út✉♦ ❡♥tr❡ í♦♥s ❝♦♥st✐t✉✐♥t❡s ❞❛ ♠♦❧é❝✉❧❛✳
❋✐♥❛❧♠❡♥t❡ t❡♠✲s❡ ❛ ❡❧❡trô♥✐❝❛✱ ♣r❡s❡♥t❡ ❡♠ t♦❞♦s ♦s át♦♠♦s ❡ ♦❝♦rr❡ ❞❡✈✐❞♦ ❛♦ ❞❡s✲ ❧♦❝❛♠❡♥t♦ ❞❛ ♥✉✈❡♠ ❡❧❡trô♥✐❝❛ ❡♠ r❡❧❛çã♦ ❛♦ ♥ú❝❧❡♦ ❞♦ át♦♠♦ ❬✸✸❪✳
✶✸
❋✐❣✉r❛ ✷✿ ❉❡♣❡♥❞ê♥❝✐❛ ❞❛ P❡r♠✐ss✐✈✐❞❛❞❡ ❈♦♠♣❧❡①❛ ❝♦♠ ❛ ❋r❡q✉ê♥❝✐❛ ■❧✉str❛♥❞♦ ♦s ♠❡❝❛♥✐s♠♦s ❞❡ P♦❧❛r✐③❛çã♦✳ ❆❞❛♣t❛❞♦ ❞❡ ❬✸✹❪
P❛r❛ ♠♦❞❡❧❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❡❧étr✐❝♦ ❞❡s❝r✐t♦ ❛❝✐♠❛ ✉s❛♥❞♦ ❛s ❡q✉❛çõ❡s ❞❡ ▼❛①✇❡❧❧✱ s✉❜st✐t✉✐✲s❡ ✭✷✳✶✸✮ ❡♠ ✭✷✳✷✮ ❡✱ r❡❡s❝r❡✈❡♥❞♦ ♥♦ ❞♦♠í♥✐♦ ❞❛ ❢r❡q✉ê♥❝✐❛✱ ♦❜tê♠✲s❡✿
∇ ×H˜ =σDCE˜ + jω˜εE˜ = (σDC+ωε
′′
)E˜ + jωε′˜
E, ✭✷✳✶✹✮
♦♥❞❡ ❛ ❝♦♥❞✉t✐✈✐❞❛❞❡ t♦t❛❧✱ q✉❡ é ❞❡♣❡♥❞❡♥t❡ ❞❛ ❢r❡q✉ê♥❝✐❛✱ ♣♦❞❡ s❡r ❡①♣r❡ss❛ ♣♦rσ(ω) =˜ σDC +ωε′′✱ q✉❡ é ❝♦♠♣♦st❛ ♣❡❧❛ ❝♦♥❞✉t✐✈✐❞❛❞❡ ❞❡ r❡❣✐♠❡ ❡st❛❝✐♦♥ár✐♦ σDC ❡ ❛ ♣❛r❝❡❧❛
ωε′′ q✉❡ ❢❛③ ❝♦♠ q✉❡ ❤❛❥❛ ❛ ✈❛r✐❛çã♦ ❝♦♠ ❛ ❢r❡q✉ê♥❝✐❛ ❞❡st❛ ❝♦♥❞✉t✐✈✐❞❛❞❡ ˜ σ(ω)✳
P♦rt❛♥t♦ ❞❛ ❡①♣r❡ssã♦ ✭✷✳✶✹✮✱ ♦❜s❡r✈❛✲s❡ q✉❡ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞♦s ♣❛râ♠❡tr♦s ❡❧❡tr♦♠❛❣✲ ♥ét✐❝♦ ❝♦♠ ❛ ❢r❡q✉ê♥❝✐❛ ♦❝♦rr❡ t❛♥t♦ ❡♠ t❡r♠♦s ❞❛ ❝♦♥❞✉t✐✈✐❞❛❞❡ ❡❧étr✐❝❛ σ(ω)˜ q✉❛♥t♦
❡♠ t❡r♠♦s ❞❛ ♣❡r♠✐ss✐✈✐❞❛❞❡ ❡❧étr✐❝❛ε(ω)˜ ❂ε′✳
✶✹
❚❛❜❡❧❛ ✶✿ ❈♦❡✜❝✐❡♥t❡s an ♣❛r❛ ❛ ❋♦r♠✉❧❛çã♦ ❞❡ ▲♦♥❣♠✐r❡ ❡ ❙♠✐t❤
♥ ✶ ✷ ✸ ✹ ✺ ✻ ✼
an 3,4.106 2,74.105 2,58.104 3,38.103 5,26.102 1,33.102 2,72.101
♥ ✽ ✾ ✶✵ ✶✶ ✶✷ ✶✸
an 1,25.101 ✹✱✽ ✷✱✶✼ ✵✱✾✽ ✵✱✸✾✷ ✵✱✶✼✸
✷✳✸✳✷ ❋♦r♠✉❧❛çã♦ ❞❡ ▲♦♥❣♠✐r❡ ❡ ❙♠✐t❤ ♣❛r❛ ❙♦❧♦s ❉✐s♣❡rs✐✈♦s
❊♠ ✶✾✼✺✱ ▲♦♥❣♠✐r❡ ❡ ❙♠✐t❤ ✉t✐❧✐③❛r❛♠ ❞❛❞♦s ❞❡ ♠❡❞✐çã♦ ❞❡ ❛♠♦str❛ ❞❡ ❝♦♥❝r❡t♦ ♣❛r❛ ❢r❡q✉ê♥❝✐❛s ❞❡ ❛té 2×108❍③ ❡ ❡①tr❛♣♦❧❛r❛♠ ❛s ❝✉r✈❛s ❞❡ ♠❡❞✐çã♦ ❛té 1012 ❍③✱ ❡ ❞❡ss❛
❢♦r♠❛ ❡❧❡s ♣r♦♣✉s❡r❛♠ ❛s ❢♦r♠✉❧❛çõ❡s ✭✷✳✶✺✮ ❡ ✭✷✳✶✻✮ ♣❛r❛ ♣❛râ♠❡tr♦s ❞♦ s♦❧♦ ❡♠ ❢✉♥çã♦ ❞❛ ❢r❡q✉ê♥❝✐❛ ❬✶✻❪✱ ❞❛❞♦s ♣♦r✿
˜
σ(ω) = σDC+ε0
N
X
n=1
anβn
(ω βn)
2
1 + (ω βn)
2 ✭✷✳✶✺✮
❡
˜
ε(ω) =ε∞+
N
X
n=1
an
1 + (ω βn)
2, ✭✷✳✶✻✮
♦♥❞❡ ω = 2πf é ❛ ❢r❡q✉ê♥❝✐❛ ❛♥❣✉❧❛r ❡ f é ❛ ❢r❡q✉ê♥❝✐❛ ❛♥❛❧✐s❛❞❛ ❡♠ ❍③✱ ε∞ = 5 é ❛
♣❡r♠✐ss✐✈✐❞❛❞❡ r❡❧❛t✐✈❛ ♣❛r❛ ❛ ❢r❡q✉ê♥❝✐❛ ❞❡1012 ❍③✱β
n= (W10)1,2810n−1✱ s❡♥❞♦ ❲ ♦ ✈❛❧♦r
❞❡ ♣❡r❝❡♥t✉❛❧ ❞❡ ✉♠✐❞❛❞❡ ♥♦ s♦❧♦ ✭❡♠ t❡r♠♦s ❞❡ ✈♦❧✉♠❡✮ ❡ σDC = 8×10−3(W10)1,54 é ❛
❝♦♥❞✉t✐✈✐❞❛❞❡ ❡♠ ❜❛✐①❛s ❢r❡q✉ê♥❝✐❛s✱ ❞❡s❝r✐t❛ ❡♠ t❡r♠♦s ❞♦ ♣❡r❝❡♥t✉❛❧ ❞❡ ✉♠✐❞❛❞❡✳ ❖s ✈❛❧♦r❡s ❞♦s ❝♦❡✜❝✐❡♥t❡s an✱ ❞❡st❛ ❢♦r♠✉❧❛çã♦✱ sã♦ ❛♣r❡s❡♥t❛❞♦s ♥❛ ❚❛❜❡❧❛ ✶
