Transcript Short Version : 24. Electric Current

Short Version :
24. Electric Current
24.1. Electric Current
Current (I) = Net rate of (+) charge crossing an area.
Biomedics: I ~ A
Electronics: I ~ mA
Steady current:
[ I ] = Ampere
I
Q
t
Instantaneous current:
+ charges moving right
dQ
dt
Zero net
current
Net
current
 charges moving left
I
Both charges moving right
Curent: A Microscopic Look
v of charge carriers in media with E = 0 is
thermal ( random with  v  = 0 ).
For E  0, vd =  v   0.
Charge in this volume is Q = n A L q.
I
n ALq
Q

L / vd
t
drift velocity
n = number of carriers per unit volume
q = charge on each carrier
I  n A q vd
Example 24.1. Copper Wire
A 5.0-A current flows in a copper wire with cross-sectional area 1.0 mm2,
carried by electrons with number density n = 1.11029 m3.
Find the electron’s drift speed.
vd 

I
n Aq
5.0 A
1.11029 m3  1.0 106 m2  1.6 1019 C 
 2.8 104 m / s
 0.28 mm / s
TIP: Big difference between vd ~ mm/s and signal speed ~ c.
Current Density
Current can flow in ill-defined paths ( vd depends on position ),
e.g., in Earth, chemical solutions, ionized gas, …
Better description of such flows is by
current density ( J ) = current per unit area
J  n q vd
  vd
[ J ] = A /m2
  Charge density
Example 24.2. Cell Membrane
Ion channels are narrow pores that allow ions to pass through cell membranes.
A particular channel has a circular cross section 0.15 nm in radius;
it opens for 1 ms and passes 1.1104 singly ionized potassium ions.
Find both the current & the current density in the channel.
1.1104 1.6 1019 C 
Q

I

t
1103 s
 1.8 1012 A
~0.3 nm
Lipid molecules
ion channels
AWG 10 :
J
30 A
 5.7 MA / m2
2
5.26 mm
I
J
A

 1.8 pA
1.8 1012 A
  0.15 109 m 
 25 106 A / m2
2
 25 MA / m2
~ 4 times max. safe current
density in household wirings
24.2. Conduction Mechanism
E  0 in conductor



Collisions
J  E
non-electrostatic equilibrium
charges accelerated
steady state
  conductivity
Ohmic material:  independent of E
J
1

E
  resistivity
[]  Vm/A  m
[]  (m)1
Ohm’s law,
microscopic version
Material
Resistivity (m)
No band gap
Metallic conductors (20C)
Aluminum
2.65108
Copper
1.68108
Gold
2.24108
Iron
9.71108
Mercury
9.84107
Silver
1.59108
Ionic solutions ( in water, 18C)
1-molar CuSO4
3.9104
1-molar HCl
1.7102
1-molar NaCl
1.4104
H2 O
2.6105
Blood, human
0.70
Seawater (typical)
0.22
Semiconductors (pure, 20C)
Small band gap
Germanium
0.47
Silicon
23.0
Insulators
Large band gap
Ceramics
1011

1014
Glass
1010  1014
Polystyrene
1015  1017
Rubber
1013  1016
Wood (dry)
108  1014
Example 24.3. Household Wiring
A 1.8-mm-diameter copper wires carries 15 A to a household appliance.
Find E in the wire.
E J

I
A

15 A 1.68  108   m 
  0.90  10 m 
3
2
 99 mV / m
Conduction in Metals
Metal:  ~ 108  106 m
Atomic structure: polycrystalline.
Carriers: sea of “free” electrons, v ~ 106 m/s
E = 0: equal # of e moving  directions   v  = 0.
E  0: Collisions between e-ph  vd ~ const.
dv m
 v  eE
dt 
dv

0
Steady state:
dt
 = relaxation time
m
vd  
ne2 
J  n  e vd 
E E
m
 T
Cu
eE
m
Ohm’s law
Due to high T Bose statistics of phonons.
c.f., vth   T
Ionic Solutions
Electrolyte: Carriers = e + ions
 ~ 104  105 m
Examples:
Ions through cell membranes.
Electric eels.
Batteries & fuel cells.
Electroplating.
Hydrolysis.
Corrosion of metal.
Plasmas
Plasma: Ionized gas with e & ions as carriers.
Examples:
Fluorescent lamps.
Plasma displays.
Neon signs.
Ionosphere.
Flames.
Lightning.
Stars.
Rarefied plasma (collisionless) can sustain large I with minimal E.
E.g., solar corona.
Thermal motion
dislodges an e ...
Semiconductor
Pure semiC:
T = 0, no mobile charge carriers.
T  0, thermally excited carriers, e & holes.
  increases with T
… leaving a hole
behind.
e & h move
oppositely in E
Doped semiC:
Mobile charge carriers from impurities.
N-type: carriers = e. Impurities = Donors. E.g. P in Si.
P-type: carriers = h. Impurities = Acceptors. E.g. B in Si.
Phosphorous
with 5 e
Bound e jumps left,
h moved to right
P fits into Si lattice, leaving 1 free e
PN Junction
Current flows in only 1 direction
Depletion
region
No battery:
e & h diffuse across junction &
recombine.
Junction depleted of carriers.
Reverse bias:
e & h pulled away from junction.
Depletion region widens.
I ~ 0.
Forward bias:
e & h drawn to junction.
Depletion region vanishes.
I  0.
Application: Transistor
Large current change controlled by small signal (at gate):
Amplifier, or
Digital switch.
Normally channel is closed
( I = 0 ) as one of the
junctions is reverse biased.
+ V applied to gate attracts e to
channel: I  0
Superconductor
Onnes (1911): Hg = 0 below 4.2K.
Muller et al (1986): TC ~ 100K.
Current record: TC ~ 160K.
TC =
Applications:
Electromagnets for strong B:
Labs, MRI, LHC, trains …
SQUIDS for measuring weak B.
YBaCuO
24.3. Resistance & Ohm’s Law
Ohm’s Law
I
V
R
macrscopic version
Open circuit:
R 
 I=0
Short circuit:
R=0
 I  V
J
E



V
I JA 
R
E

A 
V
A
L
L
A
Resistor: piece of conductor made to have specific resistance.
All heating elements are resistors.
So are incandescent lightbulbs.
Table 24.2. Micrscopic & Macroscopic Quantities in
Ohm’s Law
Microscopic
Macroscopic
Relation
E
V
V   E  dr
 EL
J
I
I   J  dA
 J A

R
J
1

E
I
R
V
R
L
A
24.4. Electric Power
Electric Power :
P
d
q V 
dt
IV
V2
PIV I R 
R
V I R
2
Power increase with R
( for fixed I )
for time independent V
Power decrease with R
( for fixed V )
No contradiction
Conceptual Example 24.1. Electric Power Transmission
Long distance power transmission lines operate at very high voltages – often
hundreds of kVs.
Why?
Power
P = I V.
Transmission loss
PW = I 2 RW = P2 RW / V2
 Low I , i.e., high V , lowers PW for same P.
PW = (V  VL )2 / RW
= (V  P RL / V )2 / RW
VL < V because of power loss in wire
see Prob 56
Making the Connection
What is the current in a typical 120 V, 100 W lightbulb?
What’s the bulb’s resistance?
I
P 100 W

V
120 V
R
V
I

120 V
0.833 A
 0.833 A
 144 
24.5. Electrical Safety
TABLE 24.3. Effects of Externally Applied Current on Humans
Current Range
Effect
-------------------------------------------------------------------------------------------------0.5 
2 mA
Threshold of sensation
10  15 mA
Involuntary muscle contractions; can’t let go
15  100 mA
Severe shock; muscle control lost; breathing difficult
100  200 mA
> 200 mA
Fibrillation of heart; death within minutes
Cardiac arrest; breathing stops; severe burns
Typical human resistance ~ 105 .
Fatal current ~ 100 mA = 0.1 A.

V   0.1A 105    10,000V
A wet person can be electrocuted by 120V.
Large current
thru operator
No current
thru operator
Large current
blows fuse
Ground fault interupter