Formelsammlung Felder und Wellen – WS10/11 - ITE

12. Das stationäre Strömungsfeld
Stromdichte: J = κ E
U -∫
E ds
elektrischer Widerstand R = =
I
∫ J df
dw
j
Verlustleistungsdichte: = J ⋅ E
dt
13. Kapazität
allgemein:
⎛Q1⎞ c11 ⋅ ⋅ c1n ⎛Φ1⎞
⎛
⎞
⋅ ⋅ ⋅ ⋅ ⋅ ⋅
=
⎟⋅
⋅⎟ ⋅ ⋅ ⋅ ⋅ ⋅⎟
Q c
n
n1
c
⎜
⋅ ⋅ nn
⎝ ⎟⎠ ⎜⎝
⎠⎟
⎝
⎜Φ⎜
n⎠⎟
speziell:
Q ∫
D df
für n = 2 und Q
1
= -Q
2
= Q: C = =
U
- E ds
Energie allgemein:
⎛c11 ⋅ ⋅ c1n ⎞ ⎛Φ1
⎞
1
⋅ ⋅ ⋅ ⋅ ⋅ W
el
= ( Φ1,....,
Φn
) ⋅ ⎜ ⋅⎜ 2
⋅ ⋅ ⋅ ⋅ ⋅ cn1 c
⎜
⎝ ⋅ ⋅
nn⎠⎟
⎜Φ⎜
⎝ n⎠⎟
Energie speziell:
1
für n = 2 und Q
1
= -Q
2
= Q: W
el
=
2
C ⋅ U
14. Induktivität
⎛L11 ⋅ ⋅ L1n ⎞ ⎛I1
⎞
1
⋅ ⋅ ⋅ ⋅ ⋅
Energie: W
m
= ( I
1,...., In)
⋅
⋅
2
⋅ ⋅ ⋅ ⋅ ⋅ Ln1 L ⎜
nn
I
⎝ ⋅ ⋅ ⎠⎟ ⎝ ⎜ n⎠⎟
1 2
für n = 1: W
m
= L ⋅ I
2
1 2 1
2
für n = 2: W
m
= L11 ⋅I 1
+ L12I1I 2
+ L22 ⋅I2
2 2
Nk ⋅Φm,ik μ ds
i
dsk
Gegeninduktivität: L
ik
= = (für dünne Leiter)
I 4π
∫∫
r - r
(a)
äußere Selbstinduktivität: L =
Magnetischer Fluss:
N ⋅Φ
I
Φ = B dA
m,ik
∫∫
(A k )
m,ik
Induktionsspannung: Uind,ik
=−Nk
dt
i ci
ck
i k
(a)
i
dΦ
∫
2