V. Electric Current
Dr. Bill Pezzaglia
Updated 2014Feb88
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V. Current & Conduction
A. Current
B. Resistance & Ohms Law
C. Electric Power
A. Current
1) Current as Flow of Charge
2) Conduction Model
3) Conservation of Charge
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1. Current is the Flow of Charge
a. Definition: the flux of (positive) charge
Q
I
t
b. SI Units:
Amp=Coulomb/sec
often use mA or A.
•
•
•
•
•
2 mA
10 mA
20 mA
100 mA
100,000 A
threshold of feeling
pain
can’t let go
DEAD
Lightning
NOTE: Electrons flow
in opposite direction
of “conventional
current”
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1b. Units of Current
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• 1820 abampere (biot) defined as current which creates a force of 2
dynes/cm between wires 1 cm apart (based on Ampere’s experiment)
• 1834 (1832) Faraday equates current with amount of mass deposited
in electrolysis. The “Faraday” is the charge of one mole of electrons.
In modern units: 1 Faraday=96,485.3365 Coulombs
• 1865 Loschmidt estimates Avogadro’s number, hence can determine
mass of one atom.
• 1893 international amp defined as depositing 1.118 mg/sec of Ag
from AgNO3
(0.001118
Amp 
(107.8682
gm
sec
gm
mole
of Ag )
Coul
(96,485.3365 Mole
)
of Ag)
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1c. Current Density
1821 Davy shows current flows throughout interior
of wire. Define:
• Current Density “J”, a vector,
units of current per area
 
I JA
• Surface Current Density “j”
units of Amp/meter
I  jL
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2. Drude Model of Conduction
a) 1834 Wheatstone determines electric signals travel
at speed of light.
b) However it is later found that the charge carriers
move at a quite slow “drift velocity” vd, (takes hours
to move one meter) which depends upon the
strength of the electric field E and a constant “”
called the “electron mobility”.
c) The current density is thus given to be:
“-e” is the charge of the electron
“n” is the valence electron density (the number of
free electrons per unit volume which may be a
function of temperature)
v d  E
J  ( e ) nv d
3. Conservation of Current
a)
Review: Continuity Equation of Fluids
This is based upon “conservation of mass”
1v1 A1   2 v2 A2
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3b. Current Continuity Equation
This is based upon “conservation of current”. If no charged stored
anywhere, the current in must equal the current out.
+ + +
+++
I in  J 1 A1  J 2 A2  I out
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3c. Conservation of Charge
•
Fundamental Law of Universe:
Charge is Conserved
•
Current flowing out of a Leyden Jar (capacitor) must match the
loss of charge in jar:
 
Q
I   J  dA  
t
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B. Resistance
1) Conduction
2) Resistivity
3) Non-Ohmic devices
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1. Conduction
a) Electric Field “E” makes current flow:
where “” is “conductance”


J  E
b) Conductance of materials: Units: sC2/(kgm3)
•
•
•
•
Insulator:
Semiconductor
Conductor
Superconductor
 = 0 (very small, 10-15)
1
  108

c) From Drude model of conduction, we can express
the conductance in terms of the electron mobility:
[units of mobiity: m2/(s-volt) ]
  en 
v d  E
J  ( e ) nv d
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2. Resistivity
a) Definition: Resistivity “” is the inverse of
conductance. Units: ohm-meter
•
•
•
•
•
Superconductor
Copper:
Germanium
Silicon
Insulator


EJ
=0
 = 1.72x10-8
 = 0.46
 = 250


b) Ohm’s Law (1872) [1826?]
• Current through a device is proportional to voltage
• Resistance “R” is in units of “ohms”
• Ohm=Volt/Amp=kgm2/(sC2)
V  IR
V
R
1

2c. Resistance of a device
•
Resistivity is an “intensive” quantity, while
“Resistance” is “extensive” (macroscopic)
•
Resistance of a macroscopic object depends upon
its geometric properties.
•
Derivation:
V  E
I  JA
V E  
R 

I JA
A
The historical definition of the “ohm was a 1 meter column of mercury with
cross section area of 1 mm2. The unit system has been changed slightly
since that time, such that the column of mercury would only be 0.96 ohms.
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3. Non-Ohmic Behavior
a) Resistance changes with
temperature (and the temperature
 (T )   0 exp  T  T0 
changes with current).
• Conductors: resistance increases with
temperature, =0.004
• Semiconductors: resistance decreases
with temperature:
=-0.05
b) Light Bulb: More current, gets hotter,
resistance increases. Net result
voltage is approximately proportional
to square of current!
  0 1   T  T0 
V I
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3c. Vacuum Tube
For example, a “diode” tube
Non linear (Child’s law)
I  bV
3
2
35
30
25
20
15
10
5
0
0
2
4
6
8
10
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3d Diodes
Diode (e.g. LED)
• Has low resistance in one direction
• High resistance in other direction
• Behaves like a “one way street”. Current
can only flow in direction of arrow.
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C. Electric Power
1) Source of Power (Batteries)
2) Electric Work
3) Joule Heating
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1. The Electric Battery
(a) First Battery? 400 AD? The
Baghdad Battery is the common name
for a number of artifacts probably
discovered in the village of Khuyut
Rabbou'a (near Baghdad, Iraq) in 1936.
These artifacts came to wider attention
in 1938, when Wilhelm König, the
German director of the National Museum
of Iraq, found the objects in the
museum's collections, and in 1940
(having returned to Berlin due to illness)
published a paper speculating that they
may have been galvanic cells, perhaps
used for electroplating gold onto silver
objects.
-wikipedia
1b. Luigi Galvani (1737-1798)
•1786 first battery cell (two
different metals in contact)
•1791 Animalistic nature of
electricity (frog legs jump
from electric charge)
•(Mary Shelly used this idea
in her “Frankenstein” book)
•Did he also do work on
corrosion? (galvanized
nails?)
http://www.corrosion-doctors.org/Biographies/VoltaBio.htm
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1c. Voltic Pile
•
1786 Galvani creates first cell
•
1793 Volta shows cell creates an
electric current
•
1800 makes 30 volt “Voltic Pile”
from a column of cells
•
1826 Ohm determines voltage is
the driver of current
•
1830 Faraday figures out the
electrochemical reactions of a
battery.
http://www.corrosion-doctors.org/Biographies/VoltaBio.htm
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Alessandro Volta
(1745-1827)
2. Electric Work
(a) Battery is like a “pump” that increases energy of
charge (current) passing through it
•
The “voltage gain” is called EMF (electromotive force),
measured in units of “volts”
•
Change in potential energy of charge q passing through
battery is: U=V q
•
Often “EMF” is given the symbol “” or E
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2b. Electrical Power
•
Power is units of Watts=Joule/Second
•
1841 Joule shows electrical work is equivalent to
mechanical work
•
Electric Power:
•
Hence: Watt=VoltAmp
Often we use Kilowatts or Megawatts
U Vq

 VI
t
t
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2c. Energy Usage
•
Energy or work done is Power x Time
•
Hence: Joule=Wattsecond
•
However, the PGE uses the weird unit of:
Kilowatt-Hour
•
1 kWh=3.6106 Joules
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3. Joule Heating
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In a resistor, the electrical energy is converted into heat.
(a) Power lost in Resistor:
V2
2
P  VI 
 RI
R
(b) Microscopically: Power lost per unit volume over a
cylinder wire of length “L”, cross section area “A”:
P
VI
I V  
2
2


 J E  E   J
Volume AL A L
3c. Power Transmission Loss
•
Consider transmission lines have resistance “R”
and generator has EMF of 
•
Voltage delivered to house is:
•
Power delivered “P” compared to generated P0:
V    IR
P VI (  IR ) I  RI


 1 
P0 I
I


•



Hence, less power is lost if you transmit with high
voltage and low current.
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References
• Wheatstone: http://micro.magnet.fsu.edu/optics/timeline/people/wheatstone.html
• Drude Model
• http://en.wikipedia.org/wiki/Electron_mobility