電磁波 Maxwellの方程式

Maxwell
∂B
∇× E =− ∂ t
∂D
∇× H = J + ∂ t
∇ ⋅ D = ρ
∇ ⋅ B =0
2004/10/29 GPR 1
2004/10/29 GPR 2
∂B
∇× E =− ∂ t
∂D
∇× H = J + ∂ t
2004/10/29 GPR 3
2004/10/29 GPR 4
∇⋅ D = ρ
∇ ⋅ B =0
2004/10/29 GPR 5
2004/10/29 GPR 6

D = ε E
B= µ H
−12
ε = ε 0
= 8.85×
10 (F/m)
µ µ π
−7
=
0
= 4 × 10 (H/m)
ε = εε
r
r
0
µ = µµ ≅ µ
0 0
J
= σΕ
2004/10/29 GPR 7
2004/10/29 GPR 8
Maxwell
∂B
rotE =− ∂ t
∂D
rotH
= J + ∂ t
divD = ρ
D = ε E
B= µ H
−12
ε = ε 0
= 8.85×
10 (F/m)
µ µ π
−7
=
0
= 4 × 10 (H/m)
ε = εε
r
r
0
µ = µµ ≅ µ
0 0
div B = 0
J
= σΕ
2004/10/29 GPR 9
2004/10/29 GPR 10
Maxwell
rot =−µ ∂ H
E
0
∂t
= ε ∂ E
H
rot
0
div E = 0
div H = 0
∂t
E,H
rot
=−µ ∂ H
∂t
rot(
rotE) =−µ ∂ rotH
∂t
E
0
0
E
rot
rot ( rotE) = grad divE− E =−E
2
µ ∂ 0
rot µ ∂ 0
( ε ∂ E
0
) µ
0ε
∂ E
− H= − =−
0 2
∂t ∂t ∂t ∂t
µε ∂ E
− =
∂t
2
0 0
0
2
2004/10/29 GPR 11
2004/10/29 GPR 12

2
µε ∂ E
∆E
− =
∂t
0 0 2
0
∂ E ∂ E ∂ E ∂ E
∂x ∂y ∂z ∂t
2
µε ∂ H
∆H
− =
∂t
2 2 2 2
x + x + x − µε
x
2 2 2 0 0
= 0
2
∂ E ∂ E ∂ E ∂ E
2 2 2 2
+ + − µε
0 0
=
y y y y
2 2 2 2
0
∂x ∂y ∂z ∂t
∂ E ∂ E ∂ E ∂ E
∂x ∂y ∂z ∂t
2 2 2 2
z + z + z − µε
z
2 2 2 0 0
= 0
2
0 0 2
0
2004/10/29 GPR 13
(1) : z
(2) : x-y
.
∂ E ∂ E
∂z
∂t
2 2
∂ Ey
∂ Ey
− µε
0 0
=
∂z
∂t
2 2
x
x
− µε
2 0 0
= 0
2
∂ E
∂z
2 2
0
∂ E
∂t
2 2
z
z
− µε
2 0 0
= 0
2
2004/10/29 GPR 14
Maxwell
i i i
x y z
∂
rotE
= 0 0 =−i
∂z
Ex
0 Ez
∂H
∂H
y
=− µ
0
=−i
yµ
0
∂t
∂t
y
∂Ex
∂z
ix iy iz
2 2
∂ E
∂ ∂H
x
∂ Ex
y
rotH
= 0 0 =−i
− µε
0 0
= 0
2 2
x
∂z
∂z
∂z
∂t
0 H
y
0
∂E ∂E
x
= ε0 = i
xε0
∂t
∂t
x
2004/10/29 GPR 15
∂ E ∂ E
∂z
∂t
2 2
∂ Ey
∂ Ey
− µε
0 0
=
∂z
∂t
2 2
− = 0
2 2
x
x
µε
0 0
∂ E
∂z
2 2
0
∂ E
∂t
2 2
z
z
− µε
2 0 0
= 0
2
Maxwell
2004/10/29 GPR 16
2
ω
2
1 ∂ uxyzt ( , , , )
∆ uxyzt ( , , , ) =
∆ U( x) =− U( x)
2 2
2
v ∂t
v
1 U( x)
= e λx
2
2 ω
λ =−
2 2
∂ uxt ( , ) 1 ∂ uxt ( , )
=
2 2 2
∂x v ∂t
+ −
uxt ( , ) = u( x− vt) + u( x+
vt)
2
v
ω
k =± jλ
=±
v
U( x)
= U e + U e
− jkx + jkx
1 2
2004/10/29 GPR 17
2004/10/29 GPR 18

E = f ( z− ct) + f ( z+
ct)
x
1 2
1
H
y
= { f1( z−ct) − f2( z+
ct)
}
η0
1
8
c= = 3×
10 ( m/ s)
µε
η
∂ E
∂z
2 2
x
x
− µε
2 0 0
= 0
2
µ
0 0
0
0
= ≅ ≅
ε0
∂ E
∂t
376.6 120π
2004/10/29 GPR 19
E = f ( z− ct) + f ( z+
ct)
x
1 2
z− ct = constant
z=constant + c
∂ z = c
∂t
2004/10/29 GPR 20
Ex
= f1( z − ct) + f2( z + ct)
1
H
y
= { f1( z−ct) − f2( z+
ct)
}
η
0
η
µ
0
0
= ≅ ≅
ε0
376.6 120π
− jkz + jkz jωt
{( 1 2 ) }
E (,) zt = Re Ae + Ae e
x
= A cos( ωt− kz) + A cos( ωt+
kz)
1 2
ωt− kz = aconstant
2004/10/29 GPR 21
2004/10/29 GPR 22
9.2
E e k
(1)
( x, t) = E0
cos( ⋅x−ωt)
k ⋅x
−ωt
α
k⋅ r= α + ωt
r cosθ
θ
r
k
2004/10/29 GPR 23
t = t 1
k⋅ r = α + ωt (2.42)
1
r
k
2004/10/29
GPR 24

E e k
(1)
( x, t) = E0
cos( ⋅x−ωt)
divEr
(,) t = 0
∂E
∂E
x y ∂Ez
divEr
(,) t = ( + + )
∂x ∂y ∂z
(1) (1) (1)
= ( ex kx + ey ky + ez kz) E0
cos( k⋅r−ωt)
(1)
=−( e ⋅k)sin( k⋅r− ωt) = 0
e
(1)
⋅ k =0
2004/10/29 GPR 25
divBrt e k k r t
(2)
(,) = −( ⋅ )sin( ⋅ − ω ) = 0
(1) (2)
e ⋅ k = e ⋅ k =0
∂B
rotE =− ∂ t
H e = k×
e
0
E
η
(2) (1) 0
2004/10/29 GPR 26
0
Electromagnetic Compatibility
(EMC)
(1) (2)
e ⋅ k = e ⋅ k =0
H e = k×
e
0
E
η
(2) (1) 0
0
x
e (1)
y
k
e ( 2)
z
2004/10/29 GPR 27
2004/10/29 GPR 28
3.5.1
2004/10/29 GPR 29