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Lehrstuhl für Technische Elektrophysik Technische Universität München

~ B

~ H

E3

En

E3 S

E3

v

L

× v L

= =

20 5

km h

Zoll (inch)

✬

✩

✫

✪

Geschwindigkeit Kraft Arbeit Leistung Ladung elektrische Spannung

L¨ ange Zeit

Masse × Beschleunigung Kraft × Weg

Arbeit Zeit

Stromst¨arke × Zeit

Arbeit Ladung

m s

1 1 1 1 1

( ( ( ( (

) = 1 × 1 2 = 1 kgs2m )=1 ×1 =1 ) = 1 /1 = 1 Js )=1 m2 m2 ) = 1 /1 = 1 kg = 1 kg A s3 s2 A s

v

L

t v=

v=

1 1

=

1, 852 1

L t

= 1, 852

= 1, 852 | {z

}

1

1000 3600

= 0, 514

.

Q = 2qel U = 20 U

Q

Wkin = Q · U = 2qel · 20 qel = |e| = 1, 602 × 10−19 = 6, 408 × 10−15

=⇒ Wkin = 6, 408 × 10−15 ·

Wkin =

Q

·

U

·

=

Q

·

U

·

= 2 · 20

= 40

101

10−1

102

10−2

103

10−3

106

10−6

109

10−9

1012

10−12

1015

10−15

1018

10−18

1021

10−21 10n n > 0

µ

10n n < 0

↔ ↔

1

=1

=1

|e| = qel = 1, 602 · 10−19 .

qel qE = ±N · q

q = ±NQ ·

e 3

N ∈ N.

NQ = 1

2,

q1

~r1

~ 1←2 F F~ 2←1

q2

q2 q1

~r2 q2

q1

~ 2←1 = −F~1←2 F ~r2 − ~r1 |q1 · q2 | |F~2←1 | = |F~1←2 | = γe |~r2 − ~r1 |2

I r2 < rI 1

γe =

1 4π · ε0

ε0 = 8, 854 · 10−12

+

ε0

O

q1

sgn (q1 ) sgn (q1 )

sgn (q2 ) − sgn (q2 )

q2

⇔ ⇔

(~r2 − ~r1 )/|~r2 − ~r1 |

F~2←1 = −F~1←2 =

N q

~r

~r1

1 q1 · q2 · (~r2 − ~r1 ) · 4π · ε0 |~r2 − ~r1 |3

qi (i = 1, ..., N ) F~q (~r ) qi q

~ri (i = 1, ..., N )

~r2

F~q (~r ) =

N X i=1

N X qi q · · (~r − ~ri ) 4π · ε0 |~r − ~ri |3 i=1 | {z }

.

+

F~q (~r ) =

..

q · qi 1 · (~r − ~ri ) · 4π · ε0 | ~r − ~ri |3

O

q q

qi

(qi , r~i )i=1...N

q q

~r

~r ~ q (~r ) = q · E(~ ~ r) , F

~ r) E(~ ~ r ) := 1 F~q (~r ) . E(~ q (qi , r~i )i=1...N ~ r) = E(~

N X qi 1 · · (~r − ~ri ) . 4π · ε0 |~r − ~ri |3 i=1

~ = (|E|) 2 3

=

2

·

1

=

~ (~r ) E

N =1

q0 ~ r) = E(~

~r0

q0 1 · · (~r − ~r0 ) 4π · ε0 |~r − ~r0 |3

-

+

q0 q0

q0

N =2

(Q, r~1 )

~ r) = E(~

(−Q, r~2 )

  Q 1 1 · · (~ r − ~ r ) · (~r − ~r1 ) − 2 4π · ε0 |~r − ~r1 |3 |~r − ~r2 |3

G E = Tangentenvektor +

an Feldlinie

G E

G E+ G E −

G E

G E−

~ r) E(~ ~ r) E(~

~r

λ 7→ ~r(λ)

~r0 ~r ~ r (λ)) , ~r(λ0 ) = ~r0 = E(~ λ ~r(λ)

C (P1 , P2 )

E3

P1

~ (~r ) F P2



α

) dr

C (P1 , P2 ) C (P1 , P2 ) s (0, l) ∋ s 7→ ~r (s) ~r(0) = ~r1

~r(l) = ~r2 C (P1 , P2 )    ~r  ~r ~t(s) = ;   = 1 . s s ~r = ~t s

~ (~r (s)) · ~t(s) s . W = |F~ (~r (s))| cos α(s) s = F

W12 =

Zl 0

F~ (~r (s)) · ~t(s) s = |{z} =

~ r s

Zl 0

~r F~ (~r (s)) · s= s

Z

C(P1 ,P2 )

F~ (~r ) · ~r

~ r) E(~

q ~ ~ r) Fq (~r ) = q · E(~ W12 = q

Z

P1

P2

C (P1 , P2 )

~ r ) · ~r E(~

C(P1 ,P2 )

W12 W12

q

~ (~r ) E

P1 U12 =

Z

W12 = q

P2

~ · ~r E

C(P1 ,P2 )

dim(U12 ) = 1

=1

~ E

P1 C(P1 , P2 )

U12 =

ZP2

~ · ~r E

P1

∂Ej ∂Ei = ∂xj ∂xi

(i, j = 1, 2, 3)

C(P1 , P2 )

U12 =

Z

C(P1 ,P2 )

P2

~E · ~r = −

Z

C(P2 ,P1 )

E~ · ~r = −U21

P1

P2

~ E

C Z

~ · ~r = 0 E

C

C

Z

e 2 , P1 ) C = C(P1 , P2 ) + C(P ~ · ~r = E

Z

~ · ~r + E

P2

C

~ · ~r = U12 + U21 = U12 − U12 = 0 E

˜ 2 ,P1 ) C(P

C(P1 ,P2 )

C

Z

P1

~E Φ(~r ) ~ r) = − E(~

Φ(~r ) ~E(~r )

Φ(~r )

Φ(~r ) = Φ(~r0 ) − P0 =

+ ~r0

ZP

~ r ′ ) · ~r ′ E(~

P0

P = Φ(~r0 )

+ ~r

Φ(~r ) − Φ(~r0 ) = −

ZP

P0

~ · ~r ′ = E

ZP0 P

E~ · ~r ′ = UP P0

UP P0 P2 =

P0 + ~r2

P U12

P1 =

+ ~r1

U12 = Φ(~r1 ) − Φ(~r2 ) . P1

P2

C(P1 , P2 )

P0

C(P1 , P2 ) = C(P1 , P0 ) + C(P0 , P2 )

+

U12 =

ZP2

P1

~ · ~r = E

ZP0

P1

|

~E · ~r + {z

}

Φ(P1 )−Φ(P0 )

ZP2

P0

|

~ · ~r E {z

= Φ(P1 ) − Φ(P2 )

}

−Φ(P2 )+Φ(P0 )

Φ0 F (Φ0 ) = {P = Φ0

+ ~r | Φ(~r) = Φ0 } Φ Φ0

E3 ~ = − E

F (Φ0 )

Q

~ r ) = Q · (~r − ~rQ ) E(~ 4πε0 |~r − ~rQ |3

PQ =

~rQ

Φ

P = + ~r PQ P P P0

P0

Φ(~r ) = Φ(~r0 ) +

ZP0

+ +

 P = O+r

~ r ′ ) · ~r ′ = Φ(~r0 ) + E(~

P

P

P C:

ZP0

Q (~r ′ − ~rQ ) · ~r ′ 4πε0 |~r ′ − ~rQ |3

P0

~r ′ (λ) = ~rQ + λ~e; λ1 ≤ λ ≤ λ0 ~e =

~r − ~rQ ; λ1 = |~r − ~rQ |; λ0 = |~r0 − ~rQ | |~r − ~rQ |

~r ′ = ~e λ ~ r ′ (λ)) = Q · λ~e = Q · ~e E(~ 4πε0 λ3 4πε0 λ2

Zλ0 Zλ0 ZP0 ~e 1 Q Q ′ ~ E · ~r = · · ~e λ = 2 4πε0 λ2 λ 4πε0 P

λ1

λ1

Φ(~r ) = Φ(~r0 ) +

Q · 4πε0



  1 Q 1 · − λ= + 4πε0 λ0 λ1

1 1 − |~r − ~rQ | |~r0 − ~rQ |



|r~0 | → ∞ Φ(~r ) =

Φ(~r0 ) = 0

1 Q · 4πε0 |~r − ~rQ | ~rQ

Φ(~r ) = const. = Φ0

⇔

|~r − ~rQ | =

Q 1 · 4πε0 Φ0

(qi , ~ri )i=1,...,N Φ(~r ) =

N X qi 1 · 4πε0 i=1 |~r − ~ri |

~ E

~ q (~r ) = 1 · F~q, F εr

N

(~r ) =

X qi q · (~r − ~ri ) 4π ε0 εr i=1 |~r − ~ri |3 |{z} =ε

εr ≥ 1

ε0 εr εr

ε0 εr εr εr εr εr

= 1, 0005 . . . 1, 0010 = 1, 5 . . . 10 = 81 = 103 . . . 104

F~q (~r ) = q ·

~ r) D(~ (qi , ~ri )i=1,...,N 1 ε

~ r ) = ε0 εr E(~ ~ r) ~ (~r ) = ε E(~ D

N

X qi ~ r) = 1 · (~r − ~ri ). D(~ 4π |~r − ~ri |3 i=1

1 ε

~ r) · D(~

(qi , ~ri )i=1,...,N

V ∂V ~ N

∂V ∂V

Z

~ · ~a = D

∂V

Z   ~·N ~ a D

∂V

Q ~ r ) = 1 · Q · ~r D(~ 4π r 3

r = |~r | K( , R)

R

∂K ( , R) = {

  d a = Nda

 + ~r ∈ E3  |~r| = R},

+

~ = ~er = ~r , N r

~ a = ~r a. ~a = N r

∂K ( , R) Z

∂K( ,R)

~ · ~a = Q · D 4π

Z

∂K( ,R)

~r ~r Q · a= r| 3{z r} 4πR2 1 = 1 r2 R2

Z

∂K( ,R)

a =Q·

4πR2 = Q. 4πR2

Q

P0 =

V

+~r0

∂V

Q

Q ~ r) = 1 · (~r − ~r0 ) D(~ 4π |~r − ~r0 |3 Z

∂V

~ · ~a = D

(

P0 ∈ V \ ∂V P0 ∈ /V

Q 0

(qi , ~ri )i=1...N V

∂V Q(V ) :=

X

qi

~ ri ∈V

∂V

V ~ri

Z

~ · ~a = Q(V ) = D

∂V

V X

qi

~ ri ∈V

! 2

Ladung - Q

L1

Q

L2

H

Q= H

Z

L1

~ · ~a = D

Z

H

L1

L1

−Q

~ · ~a εE

ε C=

Q U12 U12 .

R

~ · ~a εE C = RHL 2 ~ E · ~r L1

A

~ E

~ E C

d

ε ±Q

U12

ZL2 ~ · ~r = Ez · d = E L1

Q=

Z

H

~ · ~a = Dz · A = ε Ez · A D

C=

A Q =ε· d U12

ε

  E = E z ez

a a≤r≤b ε

b>a Q −Q

Q

a

 D b

E~

~ r ) = E(r)~er E(~ -Q

Q=

Z

~ · ~a = ε · E(r)4πr 2 D

a≤r≤b

   Feld: E = E r ⋅ er

|~ r|=r

()

a≤r≤b 1 Q · 2 E(r) = 4πε r Uab

Uab

Zb

Zb Q 1 · = E(r) r = r 4πε r2 a a   Q 1 1 Q b−a = · = − · 4πε b a 4πε ab

C=

a·b Q = 4πε b −a Uab

N

(Ci )i=1...N ±Qi

U U Qi = Ci · U ⇒ Q

=

N X i=1

Qi = U ·

N X

Ci

i=1

N

Q total= ∑ Q i i=1

Cp Cp =

Q

N

U

=

N X

Ci

i=1

(Ci )i=1...N ±Qi Qi = Q

U Qi

U

Ui Ui =

N N N X X X Q 1 Q Ui = =Q ⇒ U= C C Ci i=1 i=1 i i=1 i

Cs N X 1 1 = C Cs i=1 i

d ε1

ε2 A1

U ~ E

~ D

A2

~ 1 | · d = |E~2 | · d U = |E

⇒

~ 1| = |E

U ~ 2| = |E d

σi ~ 2 | = ε2 · U σ2 = | D d

~ 1 | = ε1 · U σ1 = | D d

Q1 = σ1 A1

Q2 = σ2 A2

Q = Q1 + Q2 = σ1 A1 + σ2 A2 =

C=

U (ε1 A1 + ε2 A2 ) d

Q ε1 A1 ε2 A2 = + d U d C = C1 + C2

C1 =

ε1 A1 d

C2 =

ε2 A2 d

d = d1 + d2 , d1 d2 ε1

ε2

A

U ~1 E

~2 E

~1 D

~2 D

~ 1 | = |σ |D

|=

Q = |σ A

  D2 = ε2 E2

~ 2| | = |D

E~ ~ 1 |d 1 + | E ~ 2 |d 2 = U = U 1 + U 2 = |E

~ 2| ~ 1| |D |D d2 d1 + ε2 ε1

  Q d1 d2 + U= ε2 A ε1 C=

C1 =

L2

U (Q)

d1 ε1

ε1 A d1

C=

Q

Q = U



L1 −Q

A +

d2 ε2

C2 = 1 1 + C1 C2

ε2 A d2

−1

L2

L1 Q

∆Q > 0

∆Q ∆Q ·E~

L2

L1 ∆W

∆W = −

ZL1

L2

~ · ~r = ∆Q · ∆Q · E

ZL2

L1

~ · ~r = ∆Q · U (Q) E

Q

Q+ Q

W = U (Q) Q Q=0

W =

ZQ

Q

U (Q′ ) Q′

0

C

W =

ZQ

Q′ C

Q′ =

0

W =

U (Q) = Q/C

1 Q2 · 2 C

1 Q2 1 1 · = · U · Q = · C · U2 2 2 2 C

w (~r )

A V =A·d

d

W =

1 ~ 1 ~ · A = 1 · | E| ~ ·V ~ · |D| · d · |D| · U · Q = · |E| 2 2 2

Q

A G E

d

−Q

w =

1 ~ W ~ = ε · |E| ~ 2 = 1 · |D ~ |2 · |D| = · | E| V 2 2 2ε

w =

1 ~ ~ E·D 2

Q(A) Q(A) t

I(A) := (I) = 1

=1

∆Q A

⊕ ⊕ ⊕ ⊕ A

~j(~r , t)

~j ~j △A I(△A)

△A

|△A| → 0

I(△A) |~j| := lim |△A |→0 |△A| ~ =1 (|j|)

~j S

2



2

|△A|



I(S) ~a ~ ) a t = ~j · ~a t Q = (~j · N

S

G

G

da = N ⋅ d a

G j

a⊥ = cos ϑ a ~ a = ~j · ~a I = |~j| a⊥ = |~j| cos ϑ a = ~j · N t ~j · N ~

a

~a ~a I( ~a) = ~j · ~a S I(S) =

Z

S

~j · ~a

ρ ~j

ρ

n(~r ) q ρ(~r ) = q · n(~r)

  d r = vdt  N

da

 v

dQ

~v (~r) ~a

t = t0 t

~r = ~v t ~r ~a

~a

~r t0

t0 +

t

Q = ρ · V = qn V = qn · ~a · ~r = qn~v · ~a t ~a Q = ~j · ~a t

~j = q · n · ~v = ρ · ~v K

~j =

K X

α=1

qα n α

qα · nα · ~vα

~vα α

m q m·

Zt2 t1

~v m~v · t

Zt2 t1

t=

~v ~ r) = F~el = q · E(~ t ~v (t) Zt2 t1

1 m 2

t

~ · ~v t = q ·E

 2 ~v t=q

Zt2 t1

ZP2

P1

~· q ·E

~ ~r E

[t1 , t2 ]

~r t t

P1

P2

t1

t2

1 m (v22 − v12) = q · U12 2 vi = |~v (ti )|

ti P1

U12 P2 U12

v1 = 0

v(U )

v(U ) =

r

2q √ · U, m

v(t1 ) = 0

U

= v (E)

t1

t2

t3

t4

τ1 , τ2 , τ3 , ... ~ ~v (E) m∗

τ = hτi i h∆~v i ~ h∆~v i ≈ ~v(E)

~ = m∗ qE



∆~v ∆t



= m∗

~v h∆~v i = m∗ τ τ ~ E

~v =

q·τ ~ ~ · E = sgn(q) · µ · E m∗

µ :=

|q|τ >0 m∗

~j = q · n · ~v = q · n · sgn(q) · µ · E, ~ ~j = |q| · n · µ · E. ~ K ~j =

K X

α=1

µα

~ |qα| · nα · µα · E α

~j

~ E

~ ~j = σ · E

σ=

K X

α=1

|qα|nαµα

dim(σ) = 1

=

Ω

l

A

σ

Spannung U12

Φ2 Klemmen potential

G G j = σE

Strom Ι

Φ1

Φ 1 Klemmenpotential

U = Φ1 − Φ2 > 0

Φ2

~ = |E|

U l

I I=

Z

A

~j · ~a =

Z

~ A=σ AU σE~ · ~a = σ |E| l

A

A l 1 l 1 R= = · G σ A G=σ·

I =G·U U =R·I ρ :=

R =ρ·

l A

1 σ

2

(R) = 1

= 1Ω

(ρ) = 1Ω

1Ω

U ~ r) ~j(~r ) = σ E(~ R

I

U =R·I

Φ1 > Φ2

V ~j

V ∂V t

Q(V ) = −

Z

~j · ~a

∂V

Q(V ) ∂V

Z

∂V

~j · ~a = 0.

N

A1 A2 , ... AN Z

Ik =

~j · ~a

Ak

Ak

Ik > 0

Ak

0=

Z

∂V

~j · ~a =

N Z X

~j · ~a =

k=1 A

k

N X

Ik

k=1

A1

I1

A2 I2

A3 I3 AN −1 I N-1

Ik N X

Ik = 0

k=1

V ρ Q(V ) =

Z

ρ(~r , t)

3

r

V

V

t

Q(V ) =

t

Z

ρ(~r , t)

3

r=

Z

∂ρ (~r, t) ∂t

V

V

~j Z

∂V

~j · ~a =

Z

V

~j

3

r

3

r

Z

~j

3

r+

V

Z

∂ρ ∂t

3

r=0

V

Z 

~j + ∂ρ ∂t

V



3

r=0

~j + ∂ρ = 0 ∂t

~ E

q

~ · ~r Wel = F~ · ~r = q · E

~v Pel =

Wel ~ · ~r = q · E ~ · ~v =q ·E t t

K nα

qα ~vα

~j =

K X

α=1

qα · nα · ~vα

α (α) ~ P el = qα · ~vα · E

pel =

K X

(α) nαPel =

α=1

K X

α=1

~ = (nαqα~vα) · E

K X

nαqα~vα

α=1

!

~ ~ = ~j · E ·E

~ pel = ~j · E

~j = σ · E ~ |{z} >0

σ

~ 2 = 1 |~j|2 > 0 pel = ~j · E~ = σ|E| σ

l

A

  jE ~ |E| Pel =

|~j| Z

Pel

pel (~r)

3

pel ~ ·l r = pel · l · A = |~j| · A · |E|

V

I = |~j| · A

~ ·l U = |E|

U =R·I Pel = U · I =

U2 = R · I2 R

R dim(Pel ) = 1

RL UE UV

I

UV = UE − R L · I

PE = UE · I

PV = UV · I η η :=

η=

UE − R L · I R L · I · UE UV PV = =1− = UE UE PE UE2

η =1− η

PV PE

RL · PE U 2E PE UE

η → 1

UE → ∞

~B(~r , t)

~r(t) ~L F

~ r t

~v (t) = ~ B

~v

~ ~FL = q(~v × B)

 B

~ B ~ B ~ = dim(| B|)

~ B

2

=1

~B

~ E

~ + ~v × ...