leifi physik

Electric Current
transport of charge per unit time
Teil 3
STROM und SPANNUNG
1 A = 1 Ampere = 1 C/s
dQ
I=
=
dt
!
⃗
⃗j · dS
current density = A/cm2
EX-II 2014
Charge Carriers
Drift Velocity
‘ mean directed speed ‘
electrons and/or positively or negatively charged ions
Paul Drude (1863-1906)
microscopic model :
N = density [m-3]
drift velocity
⃗j = N q ⃗v
Drude model
electrons move in an !
array of positive ions !
with thermal speed
vthermal =
⃗j = N q ⃗v
− −
−
+ +
+ N q ⃗v
+
for example in an electrolyte
p
hv 2 i ⇡ 105 m/s
1
3
mhv 2 i = kB T
2
2
without external field !
there is no directed !
motion
hvi = 0
Diffusion:
⟨v 2 ⟩ =
̸ 0
Drift Velocity
electrical conductivity
!
#
!
⃗
⃗
⃗j = N q⟨⃗v ⟩ = N q 2 τ E/m
= σE
Drude model
$
!
"
⌧ = mean time between collisions with positive ions
external field accelerates electrons over this mean time
U
U
=
L
R
I = σA
!
%
homogeneous conductor
⃗
⃗a = F⃗ /m = q E/m
R=
define resistance
~
mean drift velocity : h~v i = ⌧ q E/m
!
unit of R
⃗
⃗
⃗j = N q⟨⃗v ⟩ = N q 2 τ E/m
= σE
A
σ=
V ·m
[V/A]=[⌦]
conductivity
Ohm’s Law
L
L
= ρs ·
σA
A
"
specific !
resistance
ρs = [Ω m]
potential difference
$
() (! % (*(!
'
a linear potential drop !
across R is assumed :
&
U (x) = φ1 − φx = R · I
R=
U
I
U
R
I=
I
U
R
!
!
#
!
$
%
U =R·I
!
"
%
#
$
&
U1 = U0
x
L
U2 = U0
L−x
L
&
potential differences along a voltage divider
x
L
Joule Heating
Equation of Continuity
current flow through a closed surface
Work required to move a charge Q !
against a potential difference U
!
!
!
⃗=−dQ=−d
⃗j · dS
dt
dt
S
work : W = Q U
power :
P =
W
=U
t
Q
= U I = I 2 R = U 2 /R
t
Unit of power (”Leistung”) : [Watt] = [V A]
!
⃗=
⃗j · dS
S
"
"
ρ dV
V
⃗ · ⃗j = − ∂ρ
∇
∂t
⃗ · ⃗j) dV
(∇
V
conservation of charge
heat generated in a current circuit
is equal to the power dissipated
stationary current
⃗ · ⃗j = 0
∇
Kirchhoff’s Laws
Kirchhoff’s Laws
applicable to circuits with stationary currents
applicable to circuits with stationary currents
2) potential drops are additive
1) branching of conductors
!
!
Ii = 0
⃗j
!
"
!
!
!
#
i
!
!
"
$
Ui = U0
i
!
!
!
%
%
$
!
#
resistors in series
!
!
!
Iges =
!
#
"
Rges =
resistors in parallel
$
!
i
&
Ri
&
!
! U
Ii =
Ri
i
!
"
"
&
"
#
$
!
!
#
#
i
!
! 1
1
=
Rges
Ri
i
%
!
wiring of an ammeter
!
$
%
bridge circuit
current at the ammeter is zero if
#
#
#
!
#
#
R2
R1
=
Rx
R3
%
!
"
'
%
'
!
$
#
!
!
"
!
$
#
%
#
$
"
U1 = U2
"
!
!
$
& '
( '
Ampere-Meter
#
) '
Volt-Meter
Ri ≪ R ≪ Rv
Ohm-Meter
%
&
ammeters
hot-wire
ammeter
%
&
'
!
!
!spring
" # " $
two pieces of magnetized iron repel each other
http://leifi.physik.uni-muenchen.de/web_ph10/versuche/02_dreheiseninst
ammeter
detection of electric charges
, -&. /
0 ! 12 $3 14 ! 1
" !
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/* ' ' '
" !
" !
=
' (
!
: 2 ; <
) * ' ' ' (
# ! $% &&
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* ' +!
5 " 6 2 7 " 6 1
!
9
$
"#= . $ > ? > 9 > "#@ . $ > ? > 9 A
2D-voltage divider
2D voltage divider
of a position sensitive
detector
!
determine charges
at the corners
'
"
undesirable 2D voltage divider
!
$
% ( )*
Blitz
&
, -&. /
0 ! 12 $3 14 ! 1
!
Equi-!
potential!
lines
%
" !
!
/8 ' ' '
/* ' ' '
" !
#
=
' (
: 2 ; <
&
) * ' ' ' (
5 " 6 2 7 " 6 1
!
x
∝
y
∝
(Q1 + Q2 ) − (Q3 + Q4 )
$ Q +Q +Q +Q
1
2
3
4
> 9 A + Q ) − (Q + Q )
" # = . $ > ? > 9 > " # @ . $ > ? (Q
2
3
1
4
9
* ' +!
Q1 + Q2 + Q3 + Q4
electrical currents in liquids
molecules
in liquid
environment
Dissoziationdissociate
von Molekülen
in flüssiger
Umgebung
Faraday’s constant:
CuSO4 → Cu2+ + SO42−
!
#
"
"
!
gel!
electrophoresis :
!
mobility is !
mass dependent
!" # $ # %& " # ' (
F = 96486 C / mol
Debye screening length
+ - + + -- + - +
- + -+ - +
++
+ - -+ - +- - +
-+
- - + -++
+- - - + -- +
- + +- - -+ + - + - - +- +
- + - - - - -+ +
electrode
D=
!
ϵ0 k T
2e2 n∞
n+ = n−
! = n∞
electrolyte is !
quasineutral
screening by a region !
of negative charge density
Potential drop and EMF
DC current sources
electromotive force EMF
( deutsch: EMK)
internal resistance limits !
the maximal current flow
) * + ,,+
Imax
$
&
EM K
<
Ri
! " #
'
# (
$
Klemmenspannung
UKL = EM K
%
Ra
Ra + Ri
Open-circuit voltage!
= Leerlaufspannung
Electrochemial Series
Galvanic Cell
oxydation: loss of electrons!
reduction: gain of electrons!
!
redox pair :
1.11 V
Cu ↔ Cu2+ + 2e−
solid
aqueous
redox potential :!
a measure of the willingness of ions to pick up electrons
Cu/Cu2+ -Paar +0, 35 V
Zn/Zn2+ -Paar −0, 76 V
gradient of the potential
accelerates electrons
downhill
EMF carries
electrons against
the field to the top
}
1, 11 V
noble metals more readily accept electrons
electro-chemistry
Galvanic corrosion
rechargable
batteries
Ladevorgang:
charging the discharged battery:
Galvanic corrosion is an electrochemical action :
two dissimilar metals in the presence of an electrolyte
and an electron-conductive path.
Occurs when dissimilar metals are in contact with liquid.
lead-sulfate + nearly water
P bSO4 + 2 OH
−
→
P bO2 + H2 SO4 + 2e
P bSO4 + 2H
+
→
P b + H2 SO4 − 2e
Entladevorgang:
discharging the battery:
Anode
anode!
cathode
Kathode
lead-oxide + sulfuric acid
P bO2 + 3H + + HSO4− + 2e
→
P bSO4 + 2H2 O
P b + SO4−−
→
P bSO4 + 2e
specs
Anode
anode!
cathode
Kathode
Fuel cells
Battery specifications!
proton-conducting fuel cell
!
Energy/weight
30-40 Wh/kg!
Energy/size
60-75 Wh/L!
Power/weight
180 W/kg!
Charge/discharge efficiency
70%-92%!
Self-discharge rate
3%-20%/month!
Time durability
6 months!
%
Cycle durability
500-800 cycles!
Nominal Cell Voltage
2.0 Volt!
(
&
! "
)
' %
*
'
&
# $
http://en.wikipedia.org/wiki/Fuel_cell
Surface Contact potential
electrons in a potential well
L
!
!
!
!
"
!
!
)
)
n=4
"
"
!
"
# $ %& ''()
energy
!
# $ %& ''("
* + , %& - %
E/
2n + 1
L2
n=3
n=2
Φ = work function
n=1
0.0
Fermi-Dirac distribution f(E)
0.2
0.0
0.0
0.8
kB Têm = 0.
0.03
0.1
0.6
0.4
f (E) =
1.0
Eêm
1.5
1
e(E
µ)/kB T
+1
2.0
0.0
0.0
free space
for electrons
0.6
kB Têm = 0.
0.1
0.4
0.2
0.2
0.5
2.0
0.8
fHEL DHEL
0.4
DHEL
1.0
fHEL DHEL
fHEL
0.6
1.5
1.0
1.2
kB Têm = 0.
0.03
0.1
1.0
x HnmL
density of states D(E)
1.0
0.8
0.5
0.5
1.0
Eêm
1.5
2.0
0.0
0.0
0.5
1.0
Eêm
1.5
2.0
thermo couple
temperature dependence of resistivity
( )
!
!
"
# $
!
!
* + ,- .
% & $
"
#
!
$
' & & $
!
%
/ 0) / 1" 2
"
lattice
!
superconductor
&
"
'(
thermocouple voltage
!
# with
$ % & ' ( impurities
) # * + %, # ) !
pure
( ) $-
!" # !
doped
r0 1 e
$
p. /u
phonon excitation
' (
' )
* +
&,
' (
' -
* +
&,
"
%
conductor
&'
semiconductor
http://de.wikipedia.org/w/index.php?title=Datei%3AThermocouple_voltages.PNG
doped semiconductor
current flow through gases
# $ %%& $ ' (
Si
Si
Si
Si
Si
) *
+ , - ./0 1 ' %2
Si
-) * 5!
( ) * + !
Si
P
Si
Si
In
"
Si
"
!
, -.' '
' "
Si
Si
Si
Si
Si
Si
! "
!
extra electron
missing electron!
“hole”
"
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!
!
!
! "
"
!
2) ! .3 1 4.) ! 3 & 1 ' ' + 0
!
"
donor n-type
!
acceptor p-type
# $ %
# $ $ %
# & %
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q# E