-
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1
778'943:2$%:")3;-#63'93&?%42@0())&A43"#6,)B'
=35>3&CD#:34'EF63$,)B'GD$'*"3'=35>3&H#6"#62'
943:2$%%-0%-3'
-
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2
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H,5-2()@N'*,$#6'3")'()B343B23-'343:2$"-#63-'/34*8'
' '
943:2$%%-0%-3.''
?3I3B,)B'3")3$'/4D--"B:3"2'$34(;O'@,'3")3$'12+34)0.")*$"%+,")")*-!"./0'("*
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-
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3
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!
FStokes = "6#$aUep
!
Fel = ZeE
!
Ze
!
"
0"2'*3$'94303)2($4(*,)B'"',)*'*3$'T(43)@&U(64'!8'!
Uep
-
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4
!
FStokes = "6#$aUep
!
Fel = ZeE
!
Ze
!
"
9-':%002'@,0'V,-B43"#6'*3$'R3"5,)B-&JH2%:3-N&P$(W',)*'*3$'343:2$"-#63)'P$(W.'
!
Ftot = FStokes +Fel = 0
" Uep =Ze6#$a
E = ionE
!
Uep
X%)3)53I3B4"#6:3"2'
L>1"-#63$'Y3$2.''
!'Z'7M'$'Z'70[(8-M'#'Z'\8])0'"):4,-"O3'(2%0($3'Y(--3$-#6"#62',0'3")':43")3-'X%)''
!
ion " 4# 10$8 m2 Vs
-
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5
X%)3)&?3I3B4"#6:3"2',)*'X%)3)&Q3"SF6"B:3"2'
!
ionE =Uep =" ionZecion
E
# " ion = Zecionion
'
0"2'*3$'X%)3)&?3I3B4"#6:3"2'%&'()*'*3$'X%)3)&Q3"SF6"B:3"2'+&'(*,)*'*3$'X%)3)'P%)@3)2$(;%)',&'(8'
-
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6
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!("D)
#
-
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7
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!
r( ) = 0 + kBT lnc r( )c0
"
# $
%
& ' Ze( r( )
'
0"2'*3$'P%)@3)2$(;%)',\',)*'*30'#630"-#63)'[%23);(4'%\'%6)3'343:2$"-#63)'/34*'
X%)3)O(43)@3)'-")*'B3)(,'B3B3)-F2@4"#6M'5")F$3$'943:2$%4>2'
X0'A43"#6B3I"#62'B"42'"0'B()@3)'H>-230.'
!
r( ) = const. und " r( ) # 0
$ kBT"lnc r( )c0
%
& '
(
) * = !Ze"+ r( )
V))(603).'
!
c "( ) = c0, # "( ) = 0, #surf = $U32(&[%23);(4'
-
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8
!
" c r( ) = c0 exp !ZekBT
# r( )$
% &
'
( )
=(-'GD6$2'@,'3")3$'Q(*,)B-*"#623.'
!
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$ r( )%
& '
(
) *
!"2'*3$'[%"--%)&A43"#6,)B.'
!
"2# r( ) = $ 1%&el r( )
9$B"52'-"#6'*"3'64&114)784%29:+))7;%"&'(
-
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V)(4>;-#63'QK-,)B'GD$'14()($3'^53$+F#63'")'-.&953)3'53"'/'Z'\8'=(*,$#6'I"$*'3-'@,'3")30'7=&[$%5430M';4(+?:+).'
!
" z( ) = 4kBTZe
arctanh tanhZe#4kBT$
% &
'
( ) exp *
z+D
,
- .
/
0 1
,
- .
/
0 1
' 0"2'*3$'=35>3&QF)B3.'
!
"D #$kBT
2 Ze( )2c0
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"D
#
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10
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!
Ze" 4%$&9)2I"#:4,)B.'
!
sinh u( ) " u
!
"2# r( ) = 2 Ze( )2c0
$kBT# r( ) % 1
&D2 # r( )
-
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11
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!(z)
"D
#
[4()($3'^53$+F#63'")'*3$'-0.&953)3'53"'/'Z'\8'943:2$%4>2'53a)*32'-"#6'")'/'b'\8'
!
"z2# z( ) = 1
$D2 # z( )
!
" z( ) = #exp $ z%D
&
' (
)
* + , z > 0
!
c z( ) = c0 1!Ze"kBT
exp #z$D
%
& '
(
) *
%
& '
(
) * , z > 0
!
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(D2 exp #
z(D
)
* +
,
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-
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!(z)
"D
# #
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!
" z( ) = #cosh
z$D
%
& '
(
) *
coshh
2$D
%
& '
(
) *
fr -h2
< z < h2
!
"el = #$%&D
2
coshz&D
'
( )
*
+ ,
coshh
2&D
'
( )
*
+ ,
fr -h2
< z < h2
-
!"#$%&'()*')()%+,"*"#-.'/$%0'1($(-"23-'2%'4(5'%)'#6"1'
13
778]8'943:2$%%-0%-3'
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14
X*3(43$'343:2$%%-0%;-#63$'/4,--'
z
+ $el(z) v = veo
!
Feo = "eleqEext
"D
"D
Q(*,)B-2$3)),)B'()'*3$'Y()*'GD6$2'@,$'V,-5"4*,)B'3")3$'=35>3&H#6"#628'=(-'$3-,4;3$3)*3'/34*'"-2.'
=(*,$#6'3$GF6$2'*"3'=35>3&H#6"#62'3")3'P$(W''
,)*'53B"))2'@,'+"3--3)8'="3'O"-:%-3'R3"5,)B'-%$B2'*(GD$M'*(--'*(-')3,2$(43'
H%4O3)2'0"2'53I3B2'I"$*8'
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' '
!
"V = #ext
!
Eext = "#$ext
!
Feo = "eleqEext
!
"p = 0
-
!"#$%&'()*')()%+,"*"#-.'/$%0'1($(-"23-'2%'4(5'%)'#6"1'
15
943:2$%%-0%;-#63'/4"3--B3-#6I")*"B:3"2'
E(O"3$&H2%:3-&A43"#6,)B.'
V))(603).'
' '
!
" #tv + v$ %( )v( ) = &%pext +'%2v & "eleq%(ext
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5'"-2'6%0%B3)'"""8 /4,--'"-2'-2(;%)F$'"O8 '34'ff'1d]'' '
9g23$)3-'/34*'")'1%-";O3$'R"#62,)B.'
' ' !
E = Eex
!
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=(0"2'3$B"52'-"#6'GD$'*"3'E(O"3$&H2%:3-&A43"#6,)B.'
' '
!
0 = "#$z2vx z( ) " %$z
2&eq z( )[ ]E
-
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5@I.'
' '!
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$E%&eq z( )
'
( ) *
+ , = 0
!"2'*3)'R()*53*")B3).'
' '
!
vx h2
"
# $
%
& ' = 0
( vx z( ) = ) +*eq z( )[ ]
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' '!
"eq z( )
!
vx z( ) = 1 +cosh
z"D
#
$ %
&
' (
coshh
2"D
#
$ %
&
' (
)
*
+ + + +
,
-
.
.
.
.
veo
0"2.'
' '!
veo "#$%E
,)*'*3$'"%"A2.441:431'(")*8"#"$%&'(A"&2.'
' '
C3406%42@&H0%4,#6%I-:"&R34(;%)'
' '
!
eo "veoE
=#$%
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17
L>1"-#63'Y3$23.'
' ' !
" =100mV, eo # 7$ 10%8 m2 Vs, veo #1mm s
V53$'h'
' '!
e"kBT
# 4
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' ' 9-'3$B"52'-"#6'GD$'*3)'"*3(43)'343:2$%%-0%;-#63)'/4,--.'
' '
!
v r( ) " veoex = eoE fr #D
-
!"#$%&'()*')()%+,"*"#-.'/$%0'1($(-"23-'2%'4(5'%)'#6"1'
18
U,-(003)G(--,)B'
!(B)32%16%$3-3' L63$0"-#63$'L$()-G3$'
V:,-";-#63'Y3443)'JjV#%,-2%+,"*"#-kN' 888'
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