଎؅͸ɺਐߦ೾ܕʢ traveling-wave type ʣՃ଎؅

(1)

ཅࢠՃ଎ثͷՃ଎؅

ɹ

͸͡Ίʹ

ిࢠ΍ཅࢠ౳ͷՙిཻࢠͷՃ଎ʹ༻͍ΒΕΔՃ

଎؅͸ɺਐߦ೾ܕʢ traveling-wave type ʣՃ଎؅

ͱఆࡏ೾ܕʢ standing-wave type ʣՃ଎؅ʹ෼͚

ΒΕΔɻਐߦ೾ܕՃ଎؅͸ओʹిࢠͷՃ଎ʹ༻͍

ΒΕɺߴप೾ͷి࣓৔͕ిࢠͷ଎౓ͱಉظͯ͠Ճ

଎؅ͷதΛਐߦ͢Δɻ ^*1 Ұํఆࡏ೾ܕՃ଎؅Ͱ͸ɺ

ۭಎڞৼث಺ʹੜ͡Δఆࡏ೾ͷి৔Λՙిཻࢠͷ Ճ଎ʹ༻͍͍ͯΔɻిࢠͱཅࢠʹ͍ͭͯͦΕΒͷ ΤωϧΪʔͱ଎౓ β = v/c ͷؔ܎Λࣔͨ͠ਤ 1 ͔ ΒΘ͔ΔΑ͏ʹɺΤωϧΪʔΛ༩͑ͯ΋ཅࢠͷ଎

౓͸ిࢠ΄Ͳٸܹʹ্͕͍͔ͬͯͳ͍ɻ଎౓͕ޫ

଎ʹൺ΂ͯे෼খ͍͞ཅࢠͷՃ଎ʹ͓͍ͯ͸ɺ͜

ͷఆࡏ೾ܕՃ଎؅͕༻͍ΒΕΔɻ

ਤ 1 ΤωϧΪʔͱ଎౓ β ͷؔ܎

௨ৗɺཅࢠͷઢܗՃ଎ثʢຊςΩετͰ͸Ϧχ ΞοΫͱݺͿʣͰ͸ɺෳ਺ͷՃ଎؅ߏ଄͕࠾༻͞

ΕΔɻ͜Ε͸ɺͦͷΤωϧΪʔྖҬʹΑͬͯޮ཰

ͷྑ͍Ճ଎؅ߏ଄͕ҟͳΔ͔ΒͰ͋Δɻ೔ຊͷ

େڧ౓ཅࢠՃ଎ثࢪઃ J-PARC ʢ Japan Proton Accelerator Research Complex ʣͷϦχΞοΫʹ

͓͍ͯ͸ɺ

*1

ಋ೾؅Λ఻ൖ͢Δి࣓೾ͷҐ૬଎౓ v

p

͸ޫ଎ c ΑΓେ

͖͍ʢޙड़ʣɻͦ͜Ͱɺಋ೾؅ʹ݀ͷۭ͍ͨԁ൫Λपظత ʹઃஔ͢Δ͜ͱʹΑΓɺҐ૬଎౓Λిࢠͷ଎౓ v

e

≈ c ʹ߹ΘͤΔɻ

• RFQ ʢ Radio-Frequency Quadrupole ʣ ^*2

• DTL ʢ Drift Tube Linac ʣ ^*3

• ACS ʢ Annular-ring Coupled Structure ʣ ^*4 ͱෳ਺ͷछྨͷՃ଎؅ʹΑΓɺΠΦϯݯͰੜ੒͠

ͨେڧ౓ͷཅࢠϏʔϜ ^*5 Λ 400MeV ·ͰՃ଎͢

ΔɻຊߨٛͰ͸ɺి࣓ؾֶΛجૅͱͯ͠ɺ͜ΕΒ ఆࡏ೾ܕՃ଎؅ͷجૅతࣄ߲Λղઆ͢Δɻ

ɹ

*2

ߴप೾࢛ॏۃܕՃ଎؅ɻຊςΩετͰ͸ղઆ͍ͯ͠ͳ

͍͕ɺେڧ౓ཅࢠϦχΞοΫʹ͓͍ͯඇৗʹॏཁͳՃ଎

؅ɻ

*3

ޙड़͢ΔΞϧόϨܕՃ଎؅ɻ

*4

ޙड़͢Δ π/2 Ϟʔυ݁߹ۭಎܕՃ଎؅ͷҰͭɻ

*5

ਖ਼֬ʹ͸ɺෛਫૉΠΦϯ H

⁻

ϏʔϜɻ

(2)

1 ి࣓೾ͷجຊੑ࣭

1.1 ϚΫε΢Σϧͷํఔࣜ

࣍ͷ 4 ͭͷࣜ

∇ × E ⃗ = − ∂ ⃗ B

∂t (1.1)

∇ × H ⃗ = ⃗i + ∂ ⃗ D

∂t (1.2)

∇ · D ⃗ = ρ (1.3)

∇ · B ⃗ = 0 (1.4)

Λ·ͱΊͯɺϚΫε΢Σϧͷํఔࣜʢ Maxwell’s equations ʣͱ͍͏ɻ ^*6 ֤ه߸͸ɺ

E ⃗ ɿి৔ʢڧ౓ʣʢ Electric field intensity ʣ H ⃗ ɿ࣓৔ʢڧ౓ʣʢ Magnetic field intensity ʣ D ⃗ ɿిଋີ౓ʢ Electric flux density ʣ B ⃗ ɿ࣓ଋີ౓ʢ Magnetic flux density ʣ

⃗i ɿిྲྀີ౓ʢ Electric current density ʣ ρ ɿిՙີ౓ʢ Electric charge density ʣ Λද͢ɻ·ͨ͜ΕΒ͸ɺ

ε ɿ༠ి཰ʢ Permittivity of the dielectric ʣ µ ɿಁ࣓཰ʢ Permeability of the material ʣ σ ɿిؾ఻ಋ཰ʢ Electrical conductivity ʣ ͱͯ͠

D ⃗ = ε ⃗ E (1.5) B ⃗ = µ ⃗ H (1.6)

⃗i = σ ⃗ E (1.7) Λຬͨ͢ɻ

ਅۭதͷޫͷ଎͞ c ͸ɺਅۭͷ༠ి཰ ε 0

ε 0 = 8.854 × 10

⁻

¹² [F/m]

͓Αͼɺਅۭͷಁ࣓཰ µ 0

µ 0 = 4π × 10

⁻

⁷ [H/m]

*6

ࣜ (1.2), (1.3) ͱɺϕΫτϧղੳͷެࣜ ∇· ( ∇× A) = 0 ⃗ Λ༻͍Δ͜ͱʹΑΓɺిՙͷอଘଇ

∇ ·⃗i + ∂ρ

∂t = 0

͕ಘΒΕΔɻ

Λ༻͍ͯɺ c = 1

√ ε 0 µ 0

= 2.99792458 × 10 ⁸ [m/s]

ͱද͢͜ͱ͕Ͱ͖Δɻ

ਅۭதʢిྲྀີ౓ ⃗i = 0 ɺిՙີ౓ ρ = 0 ʣͷ

ి࣓೾Λߟ͑Α͏ɻ͜ͷͱ͖ɺࣜ (1.1) ͓Αͼࣜ

(1.2) ͸

∇ × E ⃗ = − µ ₀ ∂ ⃗ H

∂t (1.8)

∇ × H ⃗ = ε 0

∂ ⃗ E

∂t (1.9)

ͱͳΔɻ͜ΕΒͷࣜΛ t Ͱඍ෼ͯ͠ɺϕΫτϧղ ੳͷެࣜ ∇ × ∇ × A ⃗ = ∇ ( ∇ · A) ⃗ − ∇ ² A ⃗ Λ༻͍Δ ͱɺ࣍ͷ೾ಈํఔࣜ

∇ ² ( E ⃗

H ⃗ )

− ε 0 µ 0

∂ ²

∂t ² ( E ⃗

H ⃗ )

= 0 (1.10)

͕ಘΒΕΔɻ֯प೾਺ ω Λ༻͍ͯɺ E, ⃗ ⃗ H ∝ e ^jωt ͱ͢Δͱɺࣜ (1.10) ͸ɺϔϧϜϗϧπํఔࣜ

∇ ² ( E ⃗

H ⃗ )

+ ω ² ε 0 µ 0

( E ⃗ H ⃗

)

= 0 (1.11) ͱͳΔɻ

ɹ

1.2 ਅۭதͷฏ໘ి࣓೾

௚ߦ࠲ඪܥͷதͰɺਅۭதΛ +z ํ޲ʹਐΉฏ ໘ి࣓೾Λߟ͑Α͏ɻ͜ͷͱ͖ɺࣜ (1.11) ͸

d ² dz ²

( E ⃗ H ⃗

)

+ ω ² ε 0 µ 0

( E ⃗ H ⃗

)

= 0 (1.12) ͱͳΔɻΑͬͯɺ +z ํ޲ʹਐΉฏ໘ి࣓೾͸ɺ೾

਺ k 0 = ω √ ε 0 µ 0 Λ༻͍ͯɺ ( E ⃗

H ⃗ )

= ( E ⃗ 0

H ⃗ ₀ )

e ^j ^(ωt

⁻

^k

⁰

^z) (1.13) ͱॻ͚Δɻ࣍ʹɺ͜ΕΛ௚ަ࠲ඪܥͷ֤੒෼ʹॻ

͖෼͚Δͱɺ

E x = E 0x e ^j(ωt

⁻

^k

⁰

^z) , H x = H 0x e ^j(ωt

⁻

^k

⁰

^z) E y = E 0y e ^j(ωt

⁻

^k

⁰

^z) , H y = H 0y e ^j(ωt

⁻

^k

⁰

^z) E z = E 0z e ^j(ωt

⁻

^k

⁰

^z) , H z = H 0z e ^j(ωt

⁻

^k

⁰

^z)

(1.14)

(3)

1 ి࣓೾ͷجຊੑ࣭

1.1 ϚΫε΢Σϧͷํఔࣜ

࣍ͷ 4 ͭͷࣜ

∇ × E ⃗ = − ∂ ⃗ B

∂t (1.1)

∇ × H ⃗ = ⃗i + ∂ ⃗ D

∂t (1.2)

∇ · D ⃗ = ρ (1.3)

∇ · B ⃗ = 0 (1.4)

Λ·ͱΊͯɺϚΫε΢Σϧͷํఔࣜʢ Maxwell’s equations ʣͱ͍͏ɻ ^*6 ֤ه߸͸ɺ

E ⃗ ɿి৔ʢڧ౓ʣʢ Electric field intensity ʣ H ⃗ ɿ࣓৔ʢڧ౓ʣʢ Magnetic field intensity ʣ D ⃗ ɿిଋີ౓ʢ Electric flux density ʣ B ⃗ ɿ࣓ଋີ౓ʢ Magnetic flux density ʣ

⃗i ɿిྲྀີ౓ʢ Electric current density ʣ ρ ɿిՙີ౓ʢ Electric charge density ʣ Λද͢ɻ·ͨ͜ΕΒ͸ɺ

ε ɿ༠ి཰ʢ Permittivity of the dielectric ʣ µ ɿಁ࣓཰ʢ Permeability of the material ʣ σ ɿిؾ఻ಋ཰ʢ Electrical conductivity ʣ ͱͯ͠

D ⃗ = ε ⃗ E (1.5) B ⃗ = µ ⃗ H (1.6)

⃗i = σ ⃗ E (1.7) Λຬͨ͢ɻ

ਅۭதͷޫͷ଎͞ c ͸ɺਅۭͷ༠ి཰ ε 0

ε 0 = 8.854 × 10

⁻

¹² [F/m]

͓Αͼɺਅۭͷಁ࣓཰ µ 0

µ 0 = 4π × 10

⁻

⁷ [H/m]

*6

ࣜ (1.2), (1.3) ͱɺϕΫτϧղੳͷެࣜ ∇· ( ∇× A) = 0 ⃗ Λ༻͍Δ͜ͱʹΑΓɺిՙͷอଘଇ

∇ ·⃗i + ∂ρ

∂t = 0

͕ಘΒΕΔɻ

Λ༻͍ͯɺ c = 1

√ ε 0 µ 0

= 2.99792458 × 10 ⁸ [m/s]

ͱද͢͜ͱ͕Ͱ͖Δɻ

ਅۭதʢిྲྀີ౓ ⃗i = 0 ɺిՙີ౓ ρ = 0 ʣͷ

ి࣓೾Λߟ͑Α͏ɻ͜ͷͱ͖ɺࣜ (1.1) ͓Αͼࣜ

(1.2) ͸

∇ × E ⃗ = − µ ₀ ∂ ⃗ H

∂t (1.8)

∇ × H ⃗ = ε 0

∂ ⃗ E

∂t (1.9)

ͱͳΔɻ͜ΕΒͷࣜΛ t Ͱඍ෼ͯ͠ɺϕΫτϧղ ੳͷެࣜ ∇ × ∇ × A ⃗ = ∇ ( ∇ · A) ⃗ − ∇ ² A ⃗ Λ༻͍Δ ͱɺ࣍ͷ೾ಈํఔࣜ

∇ ² ( E ⃗

H ⃗ )

− ε 0 µ 0

∂ ²

∂t ² ( E ⃗

H ⃗ )

= 0 (1.10)

͕ಘΒΕΔɻ֯प೾਺ ω Λ༻͍ͯɺ E, ⃗ ⃗ H ∝ e ^jωt ͱ͢Δͱɺࣜ (1.10) ͸ɺϔϧϜϗϧπํఔࣜ

∇ ² ( E ⃗

H ⃗ )

+ ω ² ε 0 µ 0

( E ⃗ H ⃗

)

= 0 (1.11) ͱͳΔɻ

ɹ

1.2 ਅۭதͷฏ໘ి࣓೾

௚ߦ࠲ඪܥͷதͰɺਅۭதΛ +z ํ޲ʹਐΉฏ ໘ి࣓೾Λߟ͑Α͏ɻ͜ͷͱ͖ɺࣜ (1.11) ͸

d ² dz ²

( E ⃗ H ⃗

)

+ ω ² ε 0 µ 0

( E ⃗ H ⃗

)

= 0 (1.12) ͱͳΔɻΑͬͯɺ +z ํ޲ʹਐΉฏ໘ి࣓೾͸ɺ೾

਺ k 0 = ω √ ε 0 µ 0 Λ༻͍ͯɺ ( E ⃗

H ⃗ )

= ( E ⃗ 0

H ⃗ ₀ )

e ^j(ωt

⁻

^k

⁰

^z) (1.13) ͱॻ͚Δɻ࣍ʹɺ͜ΕΛ௚ަ࠲ඪܥͷ֤੒෼ʹॻ

͖෼͚Δͱɺ

E x = E 0x e ^j(ωt

⁻

^k

⁰

^z) , H x = H 0x e ^j(ωt

⁻

^k

⁰

^z) E y = E 0y e ^j(ωt

⁻

^k

⁰

^z) , H y = H 0y e ^j(ωt

⁻

^k

⁰

^z) E z = E 0z e ^j(ωt

⁻

^k

⁰

^z) , H z = H 0z e ^j(ωt

⁻

^k

⁰

^z)

(1.14)

ͱͳΔɻ͜ΕΛࣜ (1.8), (1.9) ʹ୅ೖ͢Δ͜ͱʹΑ Γɺ࣍ͷؔ܎ΛಘΔɻ

√ ε 0 E 0x = √ µ 0 H 0y

− √

ε 0 E 0y = √ µ 0 H 0x

E 0z = 0, H 0z = 0

(1.15)

Αͬͯɺਅۭதͷฏ໘ి࣓೾͕ɺ࣍ͷجຊੑ࣭Λ

΋ͭ͜ͱ͕Θ͔Δɻ

1. ి࣓৔͸ਐߦํ޲ʹରͯ͠ਨ௚Ͱ͋Γɺਐߦ

ํ޲ʹͦͷ੒෼Λ΋ͨͳ͍ɻ͜ͷΑ͏ͳ೾Λɺ TEM ೾ʢ Transverse electromagnetic waves ʣ ͱݺͿɻ

2. ి৔ͱ࣓৔͸ޓ͍ʹਨ௚Ͱ͋Γʢ E ⃗ · H ⃗ = 0 ʣɺ

͜ΕΒ͸࣍ͷؔ܎Λ΋ͭɻ E _x

H y

= − E _y H x

=

√ µ ₀ ε 0

= Z ₀ (1.16)

͜͜Ͱɺ Z 0 ͸ಛੑΠϯϐʔμϯεʢ Charac- teristic impedance ʣͱݺ͹ΕɺਅۭதͰ͸

Z 0 = 376.7 [Ohm] ≈ 120π [Ohm] Ͱ͋Δɻ

·ͨɺి৔ͱ࣓৔͸ಉҐ૬Ͱ͋Δɻ

3. ి࣓೾ͷਐΉํ޲͸ɺϙΠϯςΟϯάϕΫτ ϧʢ Poynting vector ʣ

S ⃗ = E ⃗ × H ⃗ (1.17) ͷ޲͖ͱҰக͢Δɻ·ͨͦͷ଎͞ v p ͸ɺ

v p = ω

k ₀ = 1

√ ε ₀ µ ₀ = c (1.18) ͱͳΓɺਅۭதͷޫͷ଎͞ c ͱҰக͢Δɻ ɹ

1.3 ి࣓৔ͷΤωϧΪʔͱϙΠϯςΟϯάϕΫ τϧ

ిؾΤωϧΪʔີ౓ u e ͸ u e = 1

2 ε ⃗ E · E ⃗ (1.19)

͋Δۭؒʢମੵ V ʣ಺ͷిؾΤωϧΪʔ U e ͸ U _e =

∫∫∫

V

1 2 ε ⃗ E · E dV ⃗ (1.20) ͱهड़͞ΕΔɻ·ͨɺ࣓ؾΤωϧΪʔີ౓ u m ͸

u m = 1

2 µ ⃗ H · H ⃗ (1.21)

͋Δۭؒʢମੵ V ʣ಺ͷ࣓ؾΤωϧΪʔ U m ͸ U m =

∫∫∫

V

1 2 µ ⃗ H · H dV ⃗ (1.22) ͱهड़͞ΕΔɻ

ࣜ (1.17) Ͱද͞ΕΔϙΠϯςΟϯάϕΫτϧͷ

ൃࢄ

∇ · S ⃗ = ∇ · ( E ⃗ × H) ⃗ (1.23) ʹ͍ͭͯߟ͑Α͏ɻ͜Ε͸ɺϕΫτϧղੳͷެࣜ

∇ · ( A ⃗ × B) = ⃗ B ⃗ · ( ∇ × A) ⃗ − A ⃗ · ( ∇ × B) ⃗ ɺ͓Α ͼࣜ (1.1), (1.2) Λ༻͍ͯɺ

∇ · S ⃗ = − µ ⃗ H · ∂ ⃗ H

∂t − E ⃗ · ⃗i − ε ⃗ E · ∂ ⃗ E

∂t

= − ∂

∂t ( 1

2 ε ⃗ E · E ⃗ + 1

2 µ ⃗ H · H ⃗ )

− E ⃗ · ⃗i

= − ∂

∂t (u e + u m ) − E ⃗ · ⃗i (1.24) ͱॻ͚Δɻ͜ͷ྆ลΛดۭؒʢମੵ V ɺද໘ੵ S

^′

ʣ Ͱੵ෼͢Δͱɺࠨล͸Ψ΢εͷੵ෼ఆཧΛ༻͍ͯ

∫∫∫

V ∇ · S dV ⃗ =

∫∫

S

^′

S ⃗ · ⃗n dS

^′

(1.25) ͱද͞Εʢ ⃗n ͸ดۂ໘্Ͱ֎޲͖ͷ୯Ґ๏ઢϕΫ τϧʣɺӈล͸

− ∂

∂t

∫∫∫

V

(u e + u m ) dV −

∫∫∫

V

E ⃗ · ⃗i dV

= − ∂

∂t (U e + U m ) −

∫∫∫

V

E ⃗ · ⃗i dV (1.26) ͱද͞ΕΔɻΑͬͯɺ

− ∂

∂t (U e + U m )

=

∫∫∫

V

E ⃗ · ⃗i dV +

∫∫

S

^′

S ⃗ · ⃗n dS

^′

(1.27) ͱͳΓɺࠨล͸୯Ґ࣌ؒʹݮগ͢Δดۭؒ಺ͷి

࣓৔ΤωϧΪʔΛɺӈลୈҰ߲͸୯Ґ࣌ؒʹดۭ

ؒ಺Ͱδϡʔϧ೤ʢిѹ × ^{ిྲྀʣͱࣦͯ͠ΘΕΔ} ΤωϧΪʔΛද͍ͯ͠ΔɻΤωϧΪʔͷอଘଇʹ ΑΓɺӈลୈೋ߲͸ɺ୯Ґ࣌ؒʹดۂ໘͔Β֎ʹ ग़Δి࣓৔ͷΤωϧΪʔΛද͢ͱղऍ͢Δ͜ͱ͕

Ͱ͖Δɻͭ·ΓɺϙΠϯςΟϯάϕΫτϧ͸ɺ୯

Ґ࣌ؒʹ୯Ґ໘ੵΛ௨ա͢Δి࣓৔ͷΤωϧΪʔ

Λҙຯ͢Δ͜ͱ͕Θ͔Δɻ

(4)

ి࣓৔ͷΤωϧΪʔ͸࣌ؒͱͱ΋ʹมԽ͢Δͷ Ͱɺ͜ΕΛ࣌ؒฏۉͰදͦ͏ɻ֯प೾਺ ω Λ༻͍

ͯɺి৔͓Αͼ࣓৔Λ ( E ⃗

H ⃗ )

= ( E ⃗ 0

H ⃗ 0

)

e ^jωt (1.28) ͱද͢ͱɺిؾΤωϧΪʔີ౓ u e ͷ࣌ؒฏۉ w e

͸ɺ

w e = 1

4 ε ⃗ E · E ⃗

^∗

(1.29)

࣓ؾΤωϧΪʔີ౓ u m ͷ࣌ؒฏۉ w m ͸ɺ w m = 1

4 µ ⃗ H · H ⃗

^∗

(1.30) ͱॻ͚Δɻ ^*7 ͜͜Ͱɺࣜதͷ

^∗

͸ɺෳૉڞ໾Λද

͢ɻͭ·Γɺ೚ҙͷෳૉ਺ z ʹ͍ͭͯɺͦͷ࣮෦ Λ Re { z } = x ɺڏ෦Λ Im { z } = y ͱද͢ͱɺ

z = x + jy

z

^∗

= x − jy (1.31) Ͱ͋Δɻ·ͨɺϙΠϯςΟϯάϕΫτϧ S ⃗ ͷ࣌ؒ

ฏۉ ⟨ S ⃗ ⟩

͸ɺ

⟨ S ⃗ ⟩

= 1

2 Re { E ⃗ × H ⃗

^∗

} (1.32)

*7

E, ⃗ ⃗ H ͷ֤੒෼Λ



 E

x

E

y

E

z



 =



 E

0x

e

^jωt

E

0y

e

^jωt

E

0z

e

^jωt



 =



 | E

0x

| e

^jθ^ex

e

^jωt

| E

0y

| e

^jθ^ey

e

^jωt

| E

0z

| e

^jθ^ez

e

^jωt







 H

x

H

y

H

z



 =



 H

0x

e

^jωt

H

0y

e

^jωt

H

0z

e

^jωt



 =



 |H

0x

|e

^jθ^mx

e

^jωt

| H

0y

| e

^jθ^my

e

^jωt

| H

0z

| e

^jθ^mz

e

^jωt





ͱॻ͘ͱɺ u

e

͸ u

e

= 1

2 ε (

(Re{E

x

})

²

+ (Re{E

y

})

²

+ (Re{E

z

})

²

)

= 1 2 ε (

|E

0x

|

²

cos

²

(ωt + θ

ex

) + |E

0y

|

²

cos

²

(ωt + θ

ey

) + | E

_0z

|

²

cos

²

(ωt + θ

_ez

) ) ͱͳΔɻΑͬͯɺ u

e

ͷ࣌ؒฏۉ w

e

͸ɺ

w

_e

= 1 2 ε

( 1

2 | E

_0x

|

²

+ 1

2 | E

_0y

|

²

+ 1 2 | E

_0z

|

²

)

= 1 4 ε ⃗ E · E ⃗

^∗

ͱදͤΔɻಉ༷ʹɺ u

m

ͷ࣌ؒฏۉ w

m

͸ɺ w

m

= 1

2 µ ( 1

2 | H

0x

|

²

+ 1

2 | H

0y

|

²

+ 1 2 | H

0z

|

²

)

= 1

4 µ ⃗ H · H ⃗

^∗

ͱදͤΔɻ

ͱॻ͚Δɻ ^*8 ·ͨҰൠʹɺ

Re { E ⃗ × H ⃗

^∗

} = 1

2 { ( E ⃗ × H ⃗

^∗

) + ( E ⃗

^∗

× H) ⃗ } (1.33)

͕੒ΓཱͭͷͰɺϙΠϯςΟϯάϕΫτϧͷ࣌ؒ

ฏۉ ⟨ S ⃗ ⟩

ͷൃࢄ͸ɺ

∇ · ⟨ S ⃗ ⟩

= 1

4 {∇ · ( E ⃗ × H ⃗

^∗

) + ∇ · ( E ⃗

^∗

× H) ⃗ }

= − 1 4 ε

(

E ⃗

^∗

· ∂ ⃗ E

∂t + E ⃗ · ∂ ⃗ E

^∗

∂t )

− 1 4 µ

(

H ⃗

^∗

· ∂ ⃗ H

∂t + H ⃗ · ∂ ⃗ H

^∗

∂t )

− 1

4 σ( E ⃗

^∗

· E ⃗ + E ⃗ · E ⃗

^∗

)

= − ∂

∂t ( 1

4 ε ⃗ E · E ⃗

^∗

+ 1

4 µ ⃗ H · H ⃗

^∗

)

− 1

2 σ ⃗ E · E ⃗

^∗

= − ∂

∂t (w _e + w _m ) − ⟨ E ⃗ · ⃗i ⟩

(1.34) ͱͳΔɻ͜ͷ྆ลΛดۭؒʢମੵ V ɺද໘ੵ S

^′

ʣ

*8

ྫ͑͹ɺ S ⃗ ͷ x ੒෼ S

x

͸ɺ

S

x

= Re { E

y

} Re { H

z

} − Re { E

z

} Re { H

y

}

= | E

0y

|| H

0z

| cos(ωt + θ

ey

) cos(ωt + θ

mz

)

− | E

0z

|| H

0y

| cos(ωt + θ

ez

) cos(ωt + θ

my

)

= 1

2 |E

0y

||H

0z

|{cos(2ωt + θ

ey

+ θ

mz

) + cos(θ

ey

− θ

mz

)}

− 1

2 | E

_0z

|| H

_0y

|{ cos(2ωt + θ

ez

+ θ

my

) + cos(θ

ez

− θ

my

)}

ͱͳΔɻΑͬͯɺ S

x

ͷ࣌ؒฏۉ ⟨ S

x

⟩

͸ɺ

⟨ S

x

⟩

= 1

2 {| E

0y

|| H

0z

| cos(θ

ey

− θ

mz

)

− | E

_0z

|| H

_0y

| cos(θ

ez

− θ

my

) }

= 1

2 (Re{E

y

H

_z^∗

} − Re{E

z

H

_y^∗

})

= 1

2 Re{( E ⃗ × H ⃗

^∗

)

x

} ͱॻ͚Δɻͭ·Γɺ S ⃗ ͷ࣌ؒฏۉ ⟨ S ⃗ ⟩

͸ɺ

⟨ S ⃗ ⟩

= 1

2 Re { E ⃗ × H ⃗

^∗

}

ͱදͤΔɻ

(5)

ి࣓৔ͷΤωϧΪʔ͸࣌ؒͱͱ΋ʹมԽ͢Δͷ Ͱɺ͜ΕΛ࣌ؒฏۉͰදͦ͏ɻ֯प೾਺ ω Λ༻͍

ͯɺి৔͓Αͼ࣓৔Λ ( E ⃗

H ⃗ )

= ( E ⃗ 0

H ⃗ 0

)

e ^jωt (1.28) ͱද͢ͱɺిؾΤωϧΪʔີ౓ u e ͷ࣌ؒฏۉ w e

͸ɺ

w e = 1

4 ε ⃗ E · E ⃗

^∗

(1.29)

࣓ؾΤωϧΪʔີ౓ u m ͷ࣌ؒฏۉ w m ͸ɺ w m = 1

4 µ ⃗ H · H ⃗

^∗

(1.30) ͱॻ͚Δɻ ^*7 ͜͜Ͱɺࣜதͷ

^∗

͸ɺෳૉڞ໾Λද

͢ɻͭ·Γɺ೚ҙͷෳૉ਺ z ʹ͍ͭͯɺͦͷ࣮෦ Λ Re { z } = x ɺڏ෦Λ Im { z } = y ͱද͢ͱɺ

z = x + jy

z

^∗

= x − jy (1.31) Ͱ͋Δɻ·ͨɺϙΠϯςΟϯάϕΫτϧ S ⃗ ͷ࣌ؒ

ฏۉ ⟨ S ⃗ ⟩

͸ɺ

⟨ S ⃗ ⟩

= 1

2 Re { E ⃗ × H ⃗

^∗

} (1.32)

*7

E, ⃗ ⃗ H ͷ֤੒෼Λ



 E

x

E

y

E

z



 =



 E

0x

e

^jωt

E

0y

e

^jωt

E

0z

e

^jωt



 =



 | E

0x

| e

^jθ^ex

e

^jωt

| E

0y

| e

^jθ^ey

e

^jωt

| E

0z

| e

^jθ^ez

e

^jωt







 H

x

H

y

H

z



 =



 H

0x

e

^jωt

H

0y

e

^jωt

H

0z

e

^jωt



 =



 |H

0x

|e

^jθ^mx

e

^jωt

| H

0y

| e

^jθ^my

e

^jωt

| H

0z

| e

^jθ^mz

e

^jωt





ͱॻ͘ͱɺ u

e

͸ u

e

= 1

2 ε (

(Re{E

x

})

²

+ (Re{E

y

})

²

+ (Re{E

z

})

²

)

= 1 2 ε (

|E

0x

|

²

cos

²

(ωt + θ

ex

) + |E

0y

|

²

cos

²

(ωt + θ

ey

) + | E

_0z

|

²

cos

²

(ωt + θ

_ez

) ) ͱͳΔɻΑͬͯɺ u

e

ͷ࣌ؒฏۉ w

e

͸ɺ

w

_e

= 1 2 ε

( 1

2 | E

_0x

|

²

+ 1

2 | E

_0y

|

²

+ 1 2 | E

_0z

|

²

)

= 1 4 ε ⃗ E · E ⃗

^∗

ͱදͤΔɻಉ༷ʹɺ u

m

ͷ࣌ؒฏۉ w

m

͸ɺ w

m

= 1

2 µ ( 1

2 | H

0x

|

²

+ 1

2 | H

0y

|

²

+ 1 2 | H

0z

|

²

)

= 1

4 µ ⃗ H · H ⃗

^∗

ͱදͤΔɻ

ͱॻ͚Δɻ ^*8 ·ͨҰൠʹɺ

Re { E ⃗ × H ⃗

^∗

} = 1

2 { ( E ⃗ × H ⃗

^∗

) + ( E ⃗

^∗

× H) ⃗ } (1.33)

͕੒ΓཱͭͷͰɺϙΠϯςΟϯάϕΫτϧͷ࣌ؒ

ฏۉ ⟨ S ⃗ ⟩

ͷൃࢄ͸ɺ

∇ · ⟨ S ⃗ ⟩

= 1

4 {∇ · ( E ⃗ × H ⃗

^∗

) + ∇ · ( E ⃗

^∗

× H) ⃗ }

= − 1 4 ε

(

E ⃗

^∗

· ∂ ⃗ E

∂t + E ⃗ · ∂ ⃗ E

^∗

∂t )

− 1 4 µ

(

H ⃗

^∗

· ∂ ⃗ H

∂t + H ⃗ · ∂ ⃗ H

^∗

∂t )

− 1

4 σ( E ⃗

^∗

· E ⃗ + E ⃗ · E ⃗

^∗

)

= − ∂

∂t ( 1

4 ε ⃗ E · E ⃗

^∗

+ 1

4 µ ⃗ H · H ⃗

^∗

)

− 1

2 σ ⃗ E · E ⃗

^∗

= − ∂

∂t (w _e + w _m ) − ⟨ E ⃗ · ⃗i ⟩

(1.34) ͱͳΔɻ͜ͷ྆ลΛดۭؒʢମੵ V ɺද໘ੵ S

^′

ʣ

*8

ྫ͑͹ɺ S ⃗ ͷ x ੒෼ S

x

͸ɺ

S

x

= Re { E

y

} Re { H

z

} − Re { E

z

} Re { H

y

}

= | E

0y

|| H

0z

| cos(ωt + θ

ey

) cos(ωt + θ

mz

)

− | E

0z

|| H

0y

| cos(ωt + θ

ez

) cos(ωt + θ

my

)

= 1

2 |E

0y

||H

0z

|{cos(2ωt + θ

ey

+ θ

mz

) + cos(θ

ey

− θ

mz

)}

− 1

2 | E

_0z

|| H

_0y

|{ cos(2ωt + θ

ez

+ θ

my

) + cos(θ

ez

− θ

my

)}

ͱͳΔɻΑͬͯɺ S

x

ͷ࣌ؒฏۉ ⟨ S

x

⟩

͸ɺ

⟨ S

x

⟩

= 1

2 {| E

0y

|| H

0z

| cos(θ

ey

− θ

mz

)

− | E

_0z

|| H

_0y

| cos(θ

ez

− θ

my

) }

= 1

2 (Re{E

y

H

_z^∗

} − Re{E

z

H

_y^∗

})

= 1

2 Re{( E ⃗ × H ⃗

^∗

)

x

} ͱॻ͚Δɻͭ·Γɺ S ⃗ ͷ࣌ؒฏۉ ⟨ S ⃗ ⟩

͸ɺ

⟨ S ⃗ ⟩

= 1

2 Re { E ⃗ × H ⃗

^∗

} ͱදͤΔɻ

Ͱੵ෼͠ɺΨ΢εͷੵ෼ఆཧΛ༻͍Δͱɺ

− ∂

∂t (W e + W m )

=

∫∫∫

V

⟨ E ⃗ · ⃗i ⟩ dV +

∫∫

S

^′

⟨ S ⃗ ⟩

· ⃗n dS

^′

(1.35)

͕ಘΒΕΔɻ͜Ε͸ɺ࣌ؒฏۉʹ͍ͭͯ΋ɺࣜ

(1.27) ͷΤωϧΪʔอଘଇ͕੒Γཱͭ͜ͱΛද͠

͍ͯΔɻ

ྫ͑͹ɺਅۭதΛ +z ํ޲ʹਐΉฏ໘ి࣓೾ʹͭ

͍ͯɺͦͷి࣓৔ͷΤωϧΪʔͱϙΠϯςΟϯά ϕΫτϧͷؔ܎Λௐ΂Α͏ɻి࣓৔ͷΤωϧΪʔ

ີ౓Λ u = u e + u m ͱͯ͠ɺϙΠϯςΟϯάϕΫ τϧ S z ͕

S z = cu (1.36)

ి࣓৔ͷΤωϧΪʔີ౓ͷ࣌ؒฏۉΛ w = w e + w m ͱͯ͠ɺϙΠϯςΟϯάϕΫτϧͷ࣌ؒฏۉ

⟨ S _z ⟩

͕

⟨ S z

⟩ = cw (1.37)

ͱද͞ΕΔ͜ͱ͕ɺ؆୯ͳܭࢉʹΑΓ͔֬ΊΒΕ Δɻͭ·Γɺి࣓৔ͷΤωϧΪʔ͕ฏ໘೾ͷਐߦ

ํ޲ʹޫ଎ c Ͱ఻ൖ͍ͯ͠Δ͜ͱΛද͍ͯ͠Δɻ ɹ

1.4 දൽޮՌ

+z ํ޲ʹਐΉฏ໘ి࣓೾ʹ͍ͭͯɺిؾ఻ಋ཰

σ ̸ = 0 ͷ৔߹Λߟ͑Α͏ɻࣜ (1.1) ͓Αͼࣜ (1.2) Λ t Ͱඍ෼ͯ͠ɺϕΫτϧղੳͷެࣜ ∇×∇× A ⃗ =

∇ ( ∇ · A) ⃗ − ∇ ² A ⃗ Λ༻͍Δͱɺ

∇ ( ∇ · E) ⃗ − ∇ ² E ⃗ = (ω ² εµ − jωσµ) E ⃗

∇ ( ∇ · H) ⃗ − ∇ ² H ⃗ = (ω ² εµ − jωσµ) H ⃗ (1.38) ΛಘΔɻ͜͜Ͱɺ E, ⃗ ⃗ H ʹ͍ͭͯɺฏ໘೾Ͱ͋Δ͜

ͱ͔Βɺ ∇ · E ⃗ = ∇ · H ⃗ = 0 ͱͳΔɻͦͯ͠ɺ x ࣠ ͱ y ࣠ΛͦΕͧΕి৔ͷ޲͖ͱ࣓৔ͷ޲͖ʹҰக

͢ΔΑ͏ʹબͿͱɺ্ࣜ͸

d ² dz ²

( E _x H y

)

+ (ω ² εµ − jωσµ) ( E _x

H y

)

= 0 (1.39) ͱͳΔɻ࣍ʹɺෳૉ਺ n = β − jα Λ༻͍ͯ

ω ² εµ − jωσµ = n ² = (β − jα) ² (1.40)

ͱ͢Δͱɺ্ͷඍ෼ํఔࣜͷղ͸

( E x

H _y )

= ( E 0x

H _0y )

e ^j(ωt

⁻

^nz)

= ( E 0x

H 0y

)

e

⁻

^αz · e ^j(ωt

⁻

^βz) (1.41) ͱॻ͚ɺి࣓৔͕ e

⁻

^αz Ͱݮਰ͢Δ͜ͱ͕Θ͔Δɻ

·ͨɺࣜ (1.40) ͷ࣮਺෦ͱڏ਺෦͔Βɺ α ͱ β ͕

ͦΕͧΕ α ² = 1

2 ω ² εµ

(√ σ ²

ω ² ε ² + 1 − 1 )

(1.42)

β ² = 1 2 ω ² εµ

(√ σ ²

ω ² ε ² + 1 + 1 )

(1.43)

ͱॻ͚Δ͜ͱ͕Θ͔Δɻ

ಛੑΠϯϐʔμϯε Z ͸ɺࣜ (1.41) Λࣜ (1.1) ʹ୅ೖ͢Δ͜ͱʹΑΓɺ

Z = E _x H y

= ωµ

α ² + β ² (β + jα) (1.44) ͱಘΒΕΔɻͭ·Γɺి৔ͷҐ૬͕࣓৔ͷҐ૬ʹ ൺ΂ͯ θ = tan

⁻

¹ (α/β) ͚ͩਐΜͰ͍Δ͜ͱΛࣔ

͍ͯ͠Δɻిؾ఻ಋ཰ σ ͱҐ૬ࠩ θ ͷؔ܎Λɺਤ 2 ʹࣔ͢ɻਅۭதͰ͸ σ = 0 Ͱ͋Δ͔Βɺ α = 0, β = ω √ ε 0 µ 0 = k 0 ͱͳΓɺి৔ͱ࣓৔͕ಉҐ૬ Ͱɺݮਰ͸ੜ͡ͳ͍ɻҰํɺ σ ͕े෼େ͖͘ͳΔ ͱɺి৔ͱ࣓৔ͷҐ૬͕ࠩ 45 ˃ͱͳΓɺి࣓೾͕

ਐΉʹͭΕͯݮਰ͢Δɻ

ਤ 2 ిؾ఻ಋ཰ͱҐ૬ࠩ

௨ৗͷۚଐಋମͰ͸ɺ σ ≃ 10 ⁷ [S/m] ɺ ε ≃ ε 0 Ͱ

͋ΔͷͰɺ σ/ωε ≃ 10 ¹⁸ /ω ͱͳΓɺ ω ͕ඇৗʹେ

͖͘ͳΒͳ͍ݶΓɺ σ/ωε ≫ 1 ͷؔ܎͕੒Γཱͭɻ

(6)

Ώ͑ʹಋମதͰ͸ɺࣜ (1.42), (1.43) ͔Β α = β =

√ ωµσ 2 = 1

δ (1.45)

ͱͳΔɻΑͬͯɺಋମதͷి࣓೾ͷి৔ͱ࣓৔͸ɺ ( E x

H y

)

= ( E 0x

H 0y

)

e

⁻

^z/δ · e ^j(ωt

⁻

^z/δ) (1.46) ͱͳΔɻ͜͜Ͱɺ

δ =

√ 2

ωµσ (1.47)

Λදൽͷް͞ʢ Skin depth ʣͱݺͿɻۚଐಋମͷ

৔߹ɺిؾ఻ಋ཰ σ ͕େ͖͍ͨΊɺදൽͷް͞ δ

͸ඇৗʹখ͍͞ɻಋମ಺෦ʹೖࣹͨ͠ి࣓೾͸ɺ e

⁻

^z/δ ͷ߲ʹΑͬͯٸ଎ʹݮਰ͢Δɻ͜ͷΑ͏ͳ ݱ৅ΛදൽޮՌʢ Skin eﬀect ʣͱݺͿɻදൽޮՌʹ Αͬͯɺి࣓৔͸ z = δ ͷҐஔͰɺද໘ʢ z = 0 ʣ ʹൺ΂ͯ e

⁻

¹ ݮগ͢Δɻಔʢ σ = 5.8 × 10 ⁷ ʣͱΞ ϧϛχ΢Ϝʢ σ = 4.0 × 10 ⁷ ʣʹ͍ͭͯɺप೾਺ f ͱදൽް͞ δ ͷؔ܎Λਤ 3 ʹࣔ͢ɻ

ਤ 3 प೾਺ͱදൽͷް͞

ಋମதΛ +z ํ޲ʹਐΉฏ໘ి࣓೾ʢి৔ͷ޲

͖ʹ x ࣠ɺ࣓৔ͷ޲͖ʹ y ࣠ΛબͿʣʹ͍ͭͯɺಛ

ੑΠϯϐʔμϯε Z ͸ɺࣜ (1.45) Λࣜ (1.44) ʹ

୅ೖ͢Δ͜ͱʹΑΓɺ Z = E x

H _y = (1 + j)

√ ωµ

2σ = (1 + j)R s (1.48) ͱಘΒΕΔɻ͜͜Ͱɺ

R s =

√ ωµ 2σ = 1

δσ (1.49)

Λදൽ఍߅ʢ skin resistance ʣͱݺͿɻΑͬͯɺ͜ͷ ͱ͖ͷϙΠϯςΟϯάϕΫτϧͷ࣌ؒฏۉ ⟨

S _z ⟩

͸ɺ

⟨ S z

⟩ = 1

2 Re { E x H _y

^∗

}

= 1

2 Re { (1 + j)R _s H _y H _y

^∗

}

= 1

2 R s H y H _y

^∗

(1.50) ͱͳΔɻ

ɹ

1.5 ڥք৚݅

ి৔ͷ઀ઢ੒෼

ਤ 4 ి࣓৔ͷ઀ઢ੒෼

ਤ 4 ͷΑ͏ʹɺڥք໘ʹฏߦͳ 2 ຊͷઢ෼ AB, CD ΛͱΓɺ໘ʹަࠩ͢Δดۂઢ ABCD Λ࡞Δɻ

ͦͯ͠ɺ͜ͷઢ෼ AB, CD ͷ௕͞Λ ∆l ɺઢ෼ BC, DA ͷ௕͞Λ ∆d ͱ͢Δɻ·ͨɺڥքʹ͓͚Δ୯ Ґ઀ઢϕΫτϧΛ ⃗t ɺ୯Ґ๏ઢϕΫτϧΛ ⃗n ͱ͠

ͯɺ୯ҐϕΫτϧ ⃗u Λɺ ⃗u = ⃗t × ⃗n ͱ͢Δɻ͜ͷด ۂઢ ABCD ্Ͱి৔ E ⃗ ͷઢੵ෼Λߟ͑Α͏ɻ͜

Ε͸ɺ͜ͷดۂઢ ABCD ্ͷ୯Ґ઀ઢϕΫτϧΛ

⃗t

^′

ͱͯ͠ετʔΫεͷఆཧͱࣜ (1.1) Λ༻͍Δ͜

ͱʹΑΓɺ

�

ABCD

E ⃗ · ⃗t

^′

ds =

∫∫

S

( ∇ × E) ⃗ · ⃗u dS

= − ∂

∂t

∫∫

S

B ⃗ · ⃗u dS

= − ∂Φ

∂t (1.51)

ͱදͤΔɻ Φ ͸ۂ໘ ABCD ΛͭΒ͵࣓͘ଋΛද

͢ɻ͜͜Ͱɺ E t1 , E t2 ΛͦΕͧΕഔ࣭ 1, 2 ʹ͓͚

Δి৔ͷ઀ઢ੒෼ͱͯ͠ɺ ∆d ʢઢ෼ BC, DA ʣΛ

े෼খͯ͘͞͠΍Ε͹ɺࣜ (1.51) ͸

− E t1 ∆l + E t2 ∆l = 0 (1.52)

(7)

Ώ͑ʹಋମதͰ͸ɺࣜ (1.42), (1.43) ͔Β α = β =

√ ωµσ 2 = 1

δ (1.45)

ͱͳΔɻΑͬͯɺಋମதͷి࣓೾ͷి৔ͱ࣓৔͸ɺ ( E x

H y

)

= ( E 0x

H 0y

)

e

⁻

^z/δ · e ^j(ωt

⁻

^z/δ) (1.46) ͱͳΔɻ͜͜Ͱɺ

δ =

√ 2

ωµσ (1.47)

Λදൽͷް͞ʢ Skin depth ʣͱݺͿɻۚଐಋମͷ

৔߹ɺిؾ఻ಋ཰ σ ͕େ͖͍ͨΊɺදൽͷް͞ δ

͸ඇৗʹখ͍͞ɻಋମ಺෦ʹೖࣹͨ͠ి࣓೾͸ɺ e

⁻

^z/δ ͷ߲ʹΑͬͯٸ଎ʹݮਰ͢Δɻ͜ͷΑ͏ͳ ݱ৅ΛදൽޮՌʢ Skin eﬀect ʣͱݺͿɻදൽޮՌʹ Αͬͯɺి࣓৔͸ z = δ ͷҐஔͰɺද໘ʢ z = 0 ʣ ʹൺ΂ͯ e

⁻

¹ ݮগ͢Δɻಔʢ σ = 5.8 × 10 ⁷ ʣͱΞ ϧϛχ΢Ϝʢ σ = 4.0 × 10 ⁷ ʣʹ͍ͭͯɺप೾਺ f ͱදൽް͞ δ ͷؔ܎Λਤ 3 ʹࣔ͢ɻ

ਤ 3 प೾਺ͱදൽͷް͞

ಋମதΛ +z ํ޲ʹਐΉฏ໘ి࣓೾ʢి৔ͷ޲

͖ʹ x ࣠ɺ࣓৔ͷ޲͖ʹ y ࣠ΛબͿʣʹ͍ͭͯɺಛ

ੑΠϯϐʔμϯε Z ͸ɺࣜ (1.45) Λࣜ (1.44) ʹ

୅ೖ͢Δ͜ͱʹΑΓɺ Z = E x

H _y = (1 + j)

√ ωµ

2σ = (1 + j)R s (1.48) ͱಘΒΕΔɻ͜͜Ͱɺ

R s =

√ ωµ 2σ = 1

δσ (1.49)

Λදൽ఍߅ʢ skin resistance ʣͱݺͿɻΑͬͯɺ͜ͷ ͱ͖ͷϙΠϯςΟϯάϕΫτϧͷ࣌ؒฏۉ ⟨

S _z ⟩

͸ɺ

⟨ S z

⟩ = 1

2 Re { E x H _y

^∗

}

= 1

2 Re { (1 + j)R _s H _y H _y

^∗

}

= 1

2 R s H y H _y

^∗

(1.50) ͱͳΔɻ

ɹ

1.5 ڥք৚݅

ి৔ͷ઀ઢ੒෼

ਤ 4 ి࣓৔ͷ઀ઢ੒෼

ਤ 4 ͷΑ͏ʹɺڥք໘ʹฏߦͳ 2 ຊͷઢ෼ AB, CD ΛͱΓɺ໘ʹަࠩ͢Δดۂઢ ABCD Λ࡞Δɻ

ͦͯ͠ɺ͜ͷઢ෼ AB, CD ͷ௕͞Λ ∆l ɺઢ෼ BC, DA ͷ௕͞Λ ∆d ͱ͢Δɻ·ͨɺڥքʹ͓͚Δ୯ Ґ઀ઢϕΫτϧΛ ⃗t ɺ୯Ґ๏ઢϕΫτϧΛ ⃗n ͱ͠

ͯɺ୯ҐϕΫτϧ ⃗u Λɺ ⃗u = ⃗t × ⃗n ͱ͢Δɻ͜ͷด ۂઢ ABCD ্Ͱి৔ E ⃗ ͷઢੵ෼Λߟ͑Α͏ɻ͜

Ε͸ɺ͜ͷดۂઢ ABCD ্ͷ୯Ґ઀ઢϕΫτϧΛ

⃗t

^′

ͱͯ͠ετʔΫεͷఆཧͱࣜ (1.1) Λ༻͍Δ͜

ͱʹΑΓɺ

�

ABCD

E ⃗ · ⃗t

^′

ds =

∫∫

S

( ∇ × E) ⃗ · ⃗u dS

= − ∂

∂t

∫∫

S

B ⃗ · ⃗u dS

= − ∂Φ

∂t (1.51)

ͱදͤΔɻ Φ ͸ۂ໘ ABCD ΛͭΒ͵࣓͘ଋΛද

͢ɻ͜͜Ͱɺ E t1 , E t2 ΛͦΕͧΕഔ࣭ 1, 2 ʹ͓͚

Δి৔ͷ઀ઢ੒෼ͱͯ͠ɺ ∆d ʢઢ෼ BC, DA ʣΛ

े෼খͯ͘͞͠΍Ε͹ɺࣜ (1.51) ͸

− E t1 ∆l + E t2 ∆l = 0 (1.52)

ͱͳΔɻ͜ΕʹΑΓɺి৔ͷ઀ઢ੒෼ʹ͍ͭͯڥ ք৚݅

E _t1 = E _t2 (1.53)

͕ಘΒΕΔɻ͢ͳΘͪɺి৔ͷ઀ઢ੒෼͸྆ഔ࣭

ͷڥք໘Ͱ࿈ଓͰ͋Δɻ

࣍ʹɺഔ࣭ 1 ͕ۭؾʢ·ͨ͸ਅۭʣɺഔ࣭ 2 ͕ σ = ∞ ͷ׬શಋମͰ͋Δ৔߹Λߟ͑Α͏ɻ׬શಋ ମͷதʹ͸ి࣓೾͸৵ೖͤͣɺి࣓৔͸ 0 Ͱ͋Δɻ

ͭ·Γɺ E t2 = 0 Ͱ͋Δ͔Βɺ

E t1 = 0 (1.54)

ͱͳΓɺి࣓೾ͷి৔͸ಋମද໘ʹฏߦͳ੒෼Λ

΋ͨͳ͍͜ͱ͕Θ͔Δɻ ɹ

࣓৔ͷ઀ઢ੒෼

ಉ༷ʹɺਤ 4 ͷดۂઢ ABCD ্Ͱ࣓৔ H ⃗ ͷઢ

ੵ෼Λߟ͑Α͏ɻ͜Ε΋ɺετʔΫεͷఆཧͱࣜ

(1.2) Λ༻͍Δ͜ͱʹΑΓɺ

�

ABCD

H ⃗ · ⃗t

^′

ds =

∫∫

S

( ∇ × H) ⃗ · ⃗u dS

=

∫∫

S

⃗i · ⃗u dS + ∂

∂t

∫∫

S

D ⃗ · ⃗u dS (1.55) ͱදͤΔɻ͜͜Ͱɺ H t1 , H t2 ΛͦΕͧΕഔ࣭ 1, 2 ʹ͓͚Δ࣓৔ͷ઀ઢ੒෼ͱͯ͠ɺ ∆d ʢઢ෼ BC, DA ʣΛे෼খͯ͘͞͠΍Ε͹ɺࣜ (1.55) ͸

− H _t1 ∆l + H _t2 ∆l = 0 (1.56) ͱͳΔɻ͜ΕʹΑΓɺ࣓৔ͷ઀ઢ੒෼ʹ͍ͭͯڥ ք৚݅

H t1 = H t2 (1.57)

͕ಘΒΕΔɻ͢ͳΘͪɺ࣓৔ͷ઀ઢ੒෼΋ڥք໘ Ͱ࿈ଓͰ͋Δɻ

࣍ʹɺഔ࣭ 1 ͕ۭؾʢ·ͨ͸ਅۭʣɺഔ࣭ 2 ͕ σ = ∞ ͷ׬શಋମͰ͋Δ৔߹Λߟ͑Α͏ɻ͜ͷͱ

͖ɺࣜ (1.55) ͷӈลୈҰ߲

∫∫

S

⃗i · ⃗u dS = σ ⃗ E · ⃗u ∆l∆d (1.58)

͸ɺ ∆d → 0 Ͱ 0 ͱ͸ͳΒͳ͍ɻ͜͜Ͱɺද໘ి

ྲྀີ౓ ⃗i s = ⃗i∆d Λ༻͍Δͱɺࣜʢ 1.55 ʣ͸

− H t1 ∆l + H t2 ∆l = ⃗i s · ⃗u ∆l (1.59) ͱͳΔɻ·ͨɺ׬શಋମͷதʹ͸ి࣓೾͸৵ೖͤ

ͣɺ࣓৔͸ 0 Ͱ͋Δɻͭ·Γɺ H t2 = 0 Ͱ͋Δ

͔Βɺ

H _t1 = ⃗i s · ( − ⃗u) (1.60) ͱͳΓɺ࣓৔ʹΑͬͯ ⃗n × H ⃗ 1 ͷํ޲ʹେ͖͞ | H t1 | ͷද໘ిྲྀ͕ྲྀΕΔ͜ͱ͕Θ͔Δɻ

ɹ

ిଋີ౓ͷ๏ઢ੒෼

ਤ 5 ి࣓৔ͷ๏ઢ੒෼

ਤ 5 ͷΑ͏ʹɺڥք໘ʹ·͕ͨͬͨ̎ͭͷ໘Ͱ

࡞ΒΕΔԁபܗʹ͍ͭͯߟ͑Α͏ɻ͜ΕΒͷ໘͸

ڥք໘ʹฏߦͰɺ໘ੵΛ ∆S ͱ͠ɺ͜ͷԁபͷߴ

͞Λ ∆d ͱ͢Δɻ͜ͷԁபܗดۂ໘ʢମੵ V ɺද ໘ੵ S

^′

ʣʹ͍ͭͯɺిଋີ౓ D ⃗ ʹΨ΢εͷൃࢄఆ ཧΛ༻͍Δͱ

∫∫∫

V ∇ · D dV ⃗ =

∫∫

S

^′

D ⃗ · ⃗n

^′

dS

^′

(1.61) ͱͳΔʢ ⃗n

^′

͸ดۂ໘্Ͱ֎޲͖ͷ୯Ґ๏ઢϕΫτ ϧʣɻ͜͜Ͱɺࣜ (1.3) Λ༻͍Δͱɺ͜Ε͸

∫∫∫

V

ρ dV =

∫∫

S

^′

D ⃗ · ⃗n

^′

dS

^′

(1.62) ͱͳΔɻԁப͕े෼ബ͍ʢ ∆d ʣͱͯͦ͠ͷଆ໘Α Γग़ΔిଋΛແࢹ͢Δͱɺ্ࣜ͸

ρ∆S∆d = D n1 ∆S − D n2 ∆S (1.63) ͱͳΔɻ͜͜Ͱɺද໘ిՙີ౓ ρ s = ρ∆d Λ༻͍

Δͱɺిଋີ౓ͷ๏ઢ੒෼ʹ͍ͭͯɺڥք৚݅

D n1 − D n2 = ρ s (1.64)

͕ಘΒΕΔɻ

(8)

࣍ʹɺഔ࣭ 1 ͕ۭؾʢ·ͨ͸ਅۭʣɺഔ࣭ 2 ͕ σ = ∞ ͷ׬શಋମͰ͋Δ৔߹Λߟ͑Α͏ɻ׬શಋ ମͷதʹ͸ి࣓೾͸৵ೖͤͣɺి࣓ք͸ 0 Ͱ͋Δɻ

ͭ·Γɺ D n2 = 0 Ͱ͋Δ͔Βɺ

D _n1 = ρ _s (1.65) ͱͳΔɻ

ɹ

࣓ଋີ౓ͷ๏ઢ੒෼

ਤ 5 ͷԁபܗดۂ໘ʢମੵ V ɺද໘ੵ S

^′

ʣʹͭ

͍ͯɺ࣓ଋີ౓ B ⃗ ʹΨ΢εͷൃࢄఆཧΛ༻͍Δͱ

∫∫∫

V ∇ · B dV ⃗ =

∫∫

S

^′

B ⃗ · ⃗n

^′

dS

^′

(1.66) ͱͳΔɻ͜͜Ͱɺࣜ (1.4) Λ༻͍Δͱɺ͜Ε͸

0 =

∫∫

S

^′

B ⃗ · ⃗n

^′

dS

^′

(1.67) ͱͳΔɻԁப͕े෼ബ͍ʢ ∆d ʣͱͯͦ͠ͷଆ໘Α Γग़Δ࣓ଋΛແࢹ͢Δͱɺ্ࣜ͸

0 = B n1 ∆S − B n2 ∆S (1.68) ͱͳΔɻΑͬͯɺ࣓ଋີ౓ͷ๏ઢ੒෼ʹ͍ͭͯɺڥ ք৚݅

B n1 = B n2 (1.69)

͕ಘΒΕΔɻ

࣍ʹɺഔ࣭ 1 ͕ۭؾʢ·ͨ͸ਅۭʣɺഔ࣭ 2 ͕ σ = ∞ ͷ׬શಋମͰ͋Δ৔߹Λߟ͑Α͏ɻ׬શಋ ମͷதʹ͸ి࣓೾͸৵ೖͤͣɺి࣓ք͸ 0 Ͱ͋Δɻ

ͭ·Γɺ B n2 = 0 Ͱ͋Δ͔Βɺ

B n1 = 0 (1.70)

ͱͳΔɻ

2 ಋ೾؅

2.1 TE ೾ͱ TM ೾

ಋ೾؅ͷ؅࣠ํ޲ʢ +z ํ޲ʣʹ

e ^jωt

⁻

^γz = e

⁻

^αz e ^j(ωt

⁻

^βz) (2.1) Ͱ఻ൖ͢Δి࣓೾Λߟ͑Α͏ɻ͜͜Ͱɺ఻ൖఆ਺

ʢ propagation constant ʣ γ ͸ɺݮਰఆ਺ʢ attenu- ation constant ʣ α ͱҐ૬ఆ਺ʢ phase constant ʣ β Λ༻͍ͯ

γ = α + jβ (2.2)

ͱද͞ΕΔɻ

ಋ೾؅ʹ͓͚Δݮਰ͕ͳ͍৔߹ʢ α = 0 ͷ৔

߹ʣɺి৔ E(x, y, z, t) ⃗ ͓Αͼ࣓৔ H(x, y, z, t) ⃗ ͸

࣍ͷΑ͏ʹهड़Ͱ͖Δɻ ( E(x, y, z, t) ⃗

H(x, y, z, t) ⃗ )

=

( E ⃗ ₀ (x, y) H ⃗ 0 (x, y)

)

e ^j(ωt

⁻

^βz) (2.3)

·ͣɺਅۭதͷϚΫε΢Σϧํఔࣜ (1.8), (1.9) Λ֤੒෼ʹ͍ͭͯॻ͖෼͚Δͱɺ

∂E 0z

∂y − ∂E 0y

∂z = − jωµ 0 H 0x (2.4)

∂E 0x

∂z − ∂E 0z

∂x = − jωµ 0 H 0y (2.5)

∂E _0y

∂x − ∂E _0x

∂y = − jωµ ₀ H _0z (2.6)

͓Αͼɺ

∂H _0z

∂y − ∂H _0y

∂z = jωε ₀ E _0x (2.7)

∂H 0x

∂z − ∂H 0z

∂x = jωε 0 E 0y (2.8)

∂H 0y

∂x − ∂H 0x

∂y = jωε 0 E 0z (2.9) ͱͳΔɻ͜͜Ͱɺ ∂/∂z = − jβ Ͱ͋Δ͔Βɺࣜ

(2.4), (2.5), (2.7), (2.8) ΑΓɺਐߦํ޲ʹਨ௚ͳ

੒෼ E 0x , E 0y , H 0x , H 0y ͸ɺ E 0x = − j

k _c ² (

ωµ 0

∂H 0z

∂y + β ∂E 0z

∂x )

(2.10) E 0y = +j

k _c ² (

ωµ 0

∂H 0z

∂x − β ∂E 0z

∂y )

(2.11) H 0x = +j

k _c ² (

ωε 0

∂E 0z

∂y − β ∂H 0z

∂x )

(2.12) H 0y = − j

k _c ² (

ωε 0

∂E 0z

∂x + β ∂H 0z

∂y )

(2.13)

(9)

࣍ʹɺഔ࣭ 1 ͕ۭؾʢ·ͨ͸ਅۭʣɺഔ࣭ 2 ͕ σ = ∞ ͷ׬શಋମͰ͋Δ৔߹Λߟ͑Α͏ɻ׬શಋ ମͷதʹ͸ి࣓೾͸৵ೖͤͣɺి࣓ք͸ 0 Ͱ͋Δɻ

ͭ·Γɺ D n2 = 0 Ͱ͋Δ͔Βɺ

D _n1 = ρ _s (1.65) ͱͳΔɻ

ɹ

࣓ଋີ౓ͷ๏ઢ੒෼

ਤ 5 ͷԁபܗดۂ໘ʢମੵ V ɺද໘ੵ S

^′

ʣʹͭ

͍ͯɺ࣓ଋີ౓ B ⃗ ʹΨ΢εͷൃࢄఆཧΛ༻͍Δͱ

∫∫∫

V ∇ · B dV ⃗ =

∫∫

S

^′

B ⃗ · ⃗n

^′

dS

^′

(1.66) ͱͳΔɻ͜͜Ͱɺࣜ (1.4) Λ༻͍Δͱɺ͜Ε͸

0 =

∫∫

S

^′

B ⃗ · ⃗n

^′

dS

^′

(1.67) ͱͳΔɻԁப͕े෼ബ͍ʢ ∆d ʣͱͯͦ͠ͷଆ໘Α Γग़Δ࣓ଋΛແࢹ͢Δͱɺ্ࣜ͸

0 = B n1 ∆S − B n2 ∆S (1.68) ͱͳΔɻΑͬͯɺ࣓ଋີ౓ͷ๏ઢ੒෼ʹ͍ͭͯɺڥ ք৚݅

B n1 = B n2 (1.69)

͕ಘΒΕΔɻ

࣍ʹɺഔ࣭ 1 ͕ۭؾʢ·ͨ͸ਅۭʣɺഔ࣭ 2 ͕ σ = ∞ ͷ׬શಋମͰ͋Δ৔߹Λߟ͑Α͏ɻ׬શಋ ମͷதʹ͸ి࣓೾͸৵ೖͤͣɺి࣓ք͸ 0 Ͱ͋Δɻ

ͭ·Γɺ B n2 = 0 Ͱ͋Δ͔Βɺ

B n1 = 0 (1.70)

ͱͳΔɻ

2 ಋ೾؅

2.1 TE ೾ͱ TM ೾

ಋ೾؅ͷ؅࣠ํ޲ʢ +z ํ޲ʣʹ

e ^jωt

⁻

^γz = e

⁻

^αz e ^j(ωt

⁻

^βz) (2.1) Ͱ఻ൖ͢Δి࣓೾Λߟ͑Α͏ɻ͜͜Ͱɺ఻ൖఆ਺

ʢ propagation constant ʣ γ ͸ɺݮਰఆ਺ʢ attenu- ation constant ʣ α ͱҐ૬ఆ਺ʢ phase constant ʣ β Λ༻͍ͯ

γ = α + jβ (2.2)

ͱද͞ΕΔɻ

ಋ೾؅ʹ͓͚Δݮਰ͕ͳ͍৔߹ʢ α = 0 ͷ৔

߹ʣɺి৔ E(x, y, z, t) ⃗ ͓Αͼ࣓৔ H(x, y, z, t) ⃗ ͸

࣍ͷΑ͏ʹهड़Ͱ͖Δɻ ( E(x, y, z, t) ⃗

H(x, y, z, t) ⃗ )

=

( E ⃗ ₀ (x, y) H ⃗ 0 (x, y)

)

e ^j(ωt

⁻

^βz) (2.3)

·ͣɺਅۭதͷϚΫε΢Σϧํఔࣜ (1.8), (1.9) Λ֤੒෼ʹ͍ͭͯॻ͖෼͚Δͱɺ

∂E 0z

∂y − ∂E 0y

∂z = − jωµ 0 H 0x (2.4)

∂E 0x

∂z − ∂E 0z

∂x = − jωµ 0 H 0y (2.5)

∂E _0y

∂x − ∂E _0x

∂y = − jωµ ₀ H _0z (2.6)

͓Αͼɺ

∂H _0z

∂y − ∂H _0y

∂z = jωε ₀ E _0x (2.7)

∂H 0x

∂z − ∂H 0z

∂x = jωε 0 E 0y (2.8)

∂H 0y

∂x − ∂H 0x

∂y = jωε 0 E 0z (2.9) ͱͳΔɻ͜͜Ͱɺ ∂/∂z = − jβ Ͱ͋Δ͔Βɺࣜ

(2.4), (2.5), (2.7), (2.8) ΑΓɺਐߦํ޲ʹਨ௚ͳ

੒෼ E 0x , E 0y , H 0x , H 0y ͸ɺ E 0x = − j

k ² _c (

ωµ 0

∂H 0z

∂y + β ∂E 0z

∂x )

(2.10) E 0y = +j

k ² _c (

ωµ 0

∂H 0z

∂x − β ∂E 0z

∂y )

(2.11) H 0x = +j

k ² _c (

ωε 0

∂E 0z

∂y − β ∂H 0z

∂x )

(2.12) H 0y = − j

k ² _c (

ωε 0

∂E 0z

∂x + β ∂H 0z

∂y )

(2.13)

ͱॻ͚Δɻͨͩ͠ɺ

k ² _c = ω ² ε ₀ µ ₀ − β ² (2.14) Ͱ͋Δɻࣜ (2.10) ʙ (2.13) ͔ΒΘ͔ΔΑ͏ʹɺి

৔ͱ࣓৔͕؅࣠ํ޲ʹ੒෼Λ΋ͨͳ͍ʢ E z = H _z = 0 ʣͱ͢Ε͹ɺ E _x , E _y , H _x , H _y ͸ͱ΋ʹҰ ఆͷղΛ΋ͨͳ͍ɻ্͕ࣜҙຯΛ΋ͭͨΊʹ͸ɺ E z , H z ͷ͍ͣΕ͔Ұํ͕ 0 Ͱͳ͍৔߹ͷΈͰ͋

Δɻ E z = 0, H z ̸ = 0 ͷ৔߹Λ TE ೾ʢ Transverse electric waves ʣɺ E z ̸ = 0, H z = 0 ͷ৔߹Λ TM ೾ ʢ Transverse magnetic waves ʣͱݺͿɻ

ɹ

2.2 ํܗಋ೾؅

ਤ 6 ํܗಋ೾؅

ࣜ (2.3) ΛϔϧϜϗϧπํఔࣜ (1.11) ʹ୅ೖ͢

Δͱɺࣜ (2.14) ͷ k c Λ༻͍ͯ

( ∂ ²

∂x ² + ∂ ²

∂y ² ) ( E ⃗ 0

H ⃗ 0

) + k _c ²

( E ⃗ 0

H ⃗ 0

)

= 0 (2.15) ͱͳΔɻ͜ΕΛ׬શಋମͷڥք৚݅Λ༻͍ͯղ

͘͜ͱʹΑΓɺ TE ೾ʢ E z = 0 ʣ͓Αͼ TM ೾ ʢ H z = 0 ʣͷ৔߹ʹ͍ͭͯɺํܗಋ೾؅Λ఻ൖ͢Δ

ి࣓৔ΛٻΊΑ͏ɻ ɹ

ํܗಋ೾؅Λ఻ൖ͢Δ TE ೾

TE ೾ʢ E z = 0 ʣͷ৔߹ɺ H 0z (x, y) Λ

H 0z (x, y) = X(x)Y (y) (2.16) ͷΑ͏ʹม਺෼཭͢Δͱɺࣜ (2.15) ͸࣓৔ʹͭ

͍ͯ

1 X

d ² X dx ² + 1

Y d ² Y

dy ² + k _c ² = 0 (2.17)

ͱॻ͘͜ͱ͕Ͱ͖Δɻ͜ͷղ͸ɺ

k ² _c = k ² _x + k ² _y (2.18) ͱͯ͠ɺ

X(x) = cos(k x x + θ x ) (2.19) Y (y) = cos(k y y + θ y ) (2.20) ͱද͞ΕΔɻΑͬͯɺ H 0z (x, y) ͸

H 0z = cos(k x x + θ x ) cos(k y y + θ y ) (2.21) ͱॻ͚ΔͷͰɺ͜ΕΛࣜ (2.10) ʙ (2.13) ʹ୅ೖ͢

Δ͜ͱʹΑΓɺ E 0x , E 0y , H 0x , H 0y ͕ٻ·Δɻ

ํܗಋ೾؅ͷ E _x , E _y ʹ͍ͭͯͷڥք৚݅

E 0y (x = 0, y) = E 0y (x = a, y) = 0 (2.22) E 0x (x, y = 0) = E 0x (x, y = b) = 0 (2.23)

͔Βɺ

θ x = θ y = 0 (2.24)

k x = mπ

a (m = 0, 1, 2, ...) (2.25) k y = nπ

b (n = 0, 1, 2, ...) (2.26) Ͱͳ͚Ε͹ͳΒͳ͍ɻΑͬͯɺํܗಋ೾؅Λ఻ൖ

͢Δ TE ೾͸ɺ E 0x = jωµ 0 k y

k ² _c cos(k x x) sin(k y y) (2.27) E 0y = − jωµ 0 k x

k _c ² sin(k x x) cos(k y y) (2.28)

E 0z = 0 (2.29)

H _0x = jβk x

k _c ² sin(k _x x) cos(k _y y) (2.30) H 0y = jβk y

k ² _c cos(k x x) sin(k y y) (2.31) H 0z = cos(k x x) cos(k y y) (2.32) ͱද͞ΕΔɻ࠷௿࣍ͷϞʔυ͸ɺ m = 1, n = 0 ͷɺ TE 10 ϞʔυͰ͋Δɻ

଎؅͸ɺਐߦ೾ܕʢ traveling-wave type ʣՃ଎؅

ཅࢠՃ଎ثͷՃ଎؅

ɹ

͸͡Ίʹ

ిࢠ΍ཅࢠ౳ͷՙిཻࢠͷՃ଎ʹ༻͍ΒΕΔՃ