10
Current Lecture Density
Ohms Law in Differential Form
Sections: 5.1, 5.2, 5.3
Homework: See homework file
LECTURE 10
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1
Electric Direct Current Review
DC is the flow of charge under Coulomb (electrostatic) forces in
conductors
Georg Simon Ohm was the 1st to observe and explain the lack of
charge acceleration in metals electrons move with uniform averaged
speed (drift velocity)
the electrostatic force is provided by external sources: battery,
charged capacitor
LECTURE 10
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2
Current Density
the current flowing through the cross-section s of a conductor
is the amount of transferred charge Q per unit time
I ( s )
Q
, A=C s 1
t
Q v V v sL v s vt , C I ( s )
V
L
n
I ( s ) J n s where J n v v, A/m2
L vt
s
e
e
e
e
e
e
e
e
Q
dQ
I
dt
Q
v v s, A
t
J
current density,
normal component
the current density is a vector
I
V
J n J an
LECTURE 10
J v v, A/m2
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3
Current and Current Density
the current I is the flux of the current density J
I I ( s ) J n s J s dI J ds I J ds, A
S
Two cylindrical wires are connected in series. Current I = 10
A flows through the junction. The radii of the wires are: r1 =
1 mm, r2 = 2 mm. Find the current densities J1 and J2 in the
two wires.
LECTURE 10
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4
Charge Mobility
charge velocity in a conductor depends on the charge mobility
ve eE, v h hE, vi i E, m/s
metals support electron current
drift electron velocity in metals: vd = eE
semiconductors support both electron and hole currents
most electrolytes support both electron and ion currents
in general plasmas support both electron and ion currents
mobility may in general depend on E (nonlinear conductors)
LECTURE 10
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5
Specific Conductivity 1
J e v e p v p ( e e p p ) E, p h, i
note: e < 0
specific conductivity depends on the free-charge density and its
mobility
e e p p , S/m=( m)1
charge density depends on the number of charge carriers per unit
volume (number density), e.g., e = eNe e 1.6022 1019 , C
semi ( Ne e N h p )e
in pure semiconductors Ne = Nh
metal Ne ee
LECTURE 10
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6
Specific Conductivity 2
typical carrier number densities, mobilities, conductivities (low
frequency, below THz)
e
h
Ne (m3)
Nh (m3)
(S/m)
pure Ge
0.39
0.19
2.4x1019
2.4x1019 2.2
pure Si
0.14
0.05
1.4x1016
1.4x1016 4.4x10 4
Cu
0.0032
1.13x1029
5.8x107
Al
0.0015
1.46x1029
3.5x107
Ag
0.005
7.74x1028
Au
6.2x107
4.5 107 S/m
Homework: What is the drift velocity of electrons in a Cu wire of
length 10 cm if the voltage applied to both ends of the wire is 1 V.
(Ans.: 3.2 cm/s. Wire may melt if too thin!)
LECTURE 10
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7
Ohms Law in Point (Differential) Form
J E
Ohms law in circuits
I GV V / R, A
assume uniform current distribution in the cross-section of the
conductor between points A and B
V
VAB E L AB E , l | L AB |
I J s
l
use Ohms law in point form to arrive at Ohms law for resistors
s
s
1l
I Js sE V
G , R
l
l
s
G R 1
conductance/resistance of a conductor of length l, constant cross-section s, and
constant current density distribution in s
LECTURE 10
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8
General Expression for Resistance
B
B
V
R
I
A E dl ,
s J ds
A E dl ,
R
s E ds
in homogeneous medium
B
A E dl ,
R
E ds
s
s E ds , S
G R 1 B
A E dl
1
LECTURE 10
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9
DC Resistance per Unit Length
L
twin-lead line
1 L
R 2
,
A
1
R 2
, /m
A
I
A
I
A
coaxial line
1 L 1
L
R
,
2
2 b2 )
a (c
11
1
R
2 2 2 , /m
a c b
LECTURE 10
I cb
I
a
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10
Homework: Resistance per Unit Length
Find the resistance per unit length of a coaxial cable whose inner
wire is of radius = a 0.5 mm and whose shield has inner radius b
= 4 mm and outer radius c = 4.5 mm. Both the inner wire and
the shield are made of copper (Cu = 5.8x107 S/m).
ANS: 21 m/m
LECTURE 10
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11
Conservation of Charge/Continuity of Current 1
s
consider the current flowing through a closed surface I
J ds
[v]
total positive flux corresponds to an outflow of charge (charge
inside volume decreases) continuity of current (conservation
dQencl of charge) in integral form
I J ds
NOTE THE NEGATIVE SIGN!
dt
s[ v ]
in circuits we assume that no charge accumulates at nodes
I J ds J ds J ds J ds 0
s[ v ]
s1
s2
I1
s3
I2
Kirchhoffs current law follows
from conservation of charge
In 0
n
I3
s3
I3
I1
LECTURE 10
s1
s2
s
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12
I2
Conservation of Charge/Continuity of Current 2
apply Gauss (divergence) theorem to conservation of charge law
I
J ds Jdv
s[ v ]
v
v
J
t
dQinside
d
v dv
dt
dt v
continuity of current (conservation
of charge) in point form
the equation of charge relaxation
v
v hm
1 v
( E) v E
, also ( E)
E
t
t
D
J
hm
v
v 0
t
(t ) 0e
t
0e t / , /
charge relaxation constant
LECTURE 10
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13
Charge Relaxation
consider an isolated conductor into which some charge Q0 is
injected initially
Coulomb forces push the charge carriers apart until they redistribute and settle on the surface
the process continues until no free charge is left inside the
conductor
the time for this to happen is about 3 where /
this is also the time required to discharge a charged capacitor
through a shorting conductor
LECTURE 10
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14
Charge Relaxation Illustrated
1
3s
0 1
exp(-t/)
0.8
1/e
curve tangent at t = 0,
intersects time axis at t =
0.6
d
0e t /
dt
0.4
t 0
0
0.2
0
0
= 35
10
time (s)
15
20
Example: Calculate the time required to restore charge neutrality
in Cu where = 0 and = 5.8x107 S/m.
3 8.8542 1012
T 3 3 0 /
4.6 1019 , s
5.8 107
LECTURE 10
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15
Joules Law in Differential Form
consider sufficiently small volume v = sL where the E-field
and the charge density v are constant
since charge is moving with uniform drift velocity ud, the E-field
does work on the charge (this work is converted into heat)
We QE L we v, J
F
power is work done per unit time
We
w v QE L
P
pv e
QE u d , W
t
t
t
power density
QE u d
p
v E u d E J , W/m3
v
Joules law in differential form: dissipated power per unit volume
p E J (E E) | E |2 , W/m3
LECTURE 10
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16
Joules Law in Integral Form
power dissipated in conductors
P pdv E Jdv | E |2 dv, W
v
v
v
Joules law in circuit theory
assume that in a piece of conductor, E does not depend on the
cross-section, only J (or ) does, while J (or ) does not
depend on the length
assume that E and J are collinear
P E J dsdL EJdLds
p
v
dv
SL
P EdL Jds V I , W
L
S
LECTURE 10
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17
You have learned:
what current density is and how it relates to the total current
that drift velocity of charge in conductors is proportional to the
strength of E and the coefficient of proportionality is the mobility
what specific conductivity is and how it relates J to E through
Ohms law in differential (point) form
how to compute the resistance/conductance of conducting bodies
that charge is preserved and the rate of change of the charge
density determines the divergence of the current density
(continuity of current in point form)
what charge relaxation is and how it depends on the permittivity
and conductivity of the material
how to find from the E field the dissipated power using Joules law
LECTURE 10
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18