Transcript 20. Electric Charge, Force, & Field

24. Electric Current
1.
2.
3.
4.
5.
Electric Current
Conduction Mechanism
Resistance & Ohm’s Law
Electric Power
Electrical Safety
How does electric current heat this light bulb filament?
Collisions between e & ions.
Where does the energy come from?
e accelerated by E.
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
GOT IT? 24.1
Which of the following representsa nonzero current?
What’s its direction?
I, R to L (a) a beam of electrons moves from left to right,
I, up
(b) a beam of protons moves upward,
I, L
(c) in a solution, positive ions move left & negative ions move right,
no I
(d) blood, carrying positive & negative ions at the same speed, moves up
through a vein,
no I
(e) a metal car with no net charge speeds westward.
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
GOT IT? 24.2
Two wires carry the same current I.
Wire A has a larger diameter,
a higher density of current carrying electrons,
& a lower resistivity than wire B.
Rank order
JA < JB
EA < EB
vA < vB
(a) the current densities,
(b) the electric fields,
(c) the drift speeds in the two wires.
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
Example 24.4. Starting Your Car
A copper wire 0.50 cm in diameter & 70 cm long
connects your car’s battery to the starter motor.
What’s the wire’s resistance?
If the starter motor draws a current of 170A,
what’s the potential difference across the wire?
R
0.70 m
L
 1.68  108   m 
2
A
  0.25  102 m 
V  I R  170 A 0.60 m   0.10 V
 0.60 m 
GOT IT? 24.3
The figure shows 3 pieces of wire.
(1) & (2) are made from the same material, while
(3) is made from material with twice the resistivity.
(1) & (3) have twice the diameter of (2).
(2)
(a) Which has the highest resistance?
(1)
(b) If the same voltage is applied across each,
which will pass the highest current?
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