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Chapter 25
Current, Resistance, Electromotive Force
•
•
•
•
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Consider current and current density
Study the intrinsic property of resistivity
Use Ohm’s Law and study resistance and resistors
Connect circuits and find emf
Examine circuits and determine the energy and power
in them
• Describe the conduction of metals microscopically, on
an atomic scale
1
The direction of current flow
– In the absence of an external field, electrons move randomly in a
conductor. If a field exists near the conductor, its force on the
electron imposes a drift.
106 m/s electron
motion velocity
10-4m/s Drift velocity
Current flowing
– Positive charges would move with the electric field, electrons move in
opposition.
– The motion of electrons in a wire is analogous to water coursing
through a river.
Electric Current
Electrical current (I) in amperes is defined as the rate of electric charge flow in coulombs
per second. 1 ampere (A) of current is a rate of charge flow of 1 coulomb/second.
dQ
I
dt
(25-1)
1 mA (milliampere) = 1 x 10-3 A (ampere)
1 A(microampere) = 1 x 10-6 A (ampere)
Conventional Current Direction
Chapter 25
4
Electric Current Density
Current, Drift Velocity, and Current Density
dQ  q(nAvd dt )  nqvd Adt
where n = charge carriers per unit volume
q = charge per charge carrier in coulombs
vd = average drift velocity of charge carriers
in meters per second
I
J
= current density in amperes/m2
A
Chapter 25
5
Resistivity


vd  E
Drift Velocity
where  = mobility of conducting material
Drift Velocity is 1010 slower than Random Velocity
E

J
Definition of resistivity in ohm-meters (-m).
where




J  nqvd  nqE  E
  nq
conductivity of the material.
1
1
E
 

 nq J
Chapter 25
6
Resistivity is intrinsic to a metal sample
(like density is)
Resistivity and Temperature
• In metals, increasing temperature increases ion
vibration amplitudes, increasing collisions and
reducing current flow. This produces a positive
temperature coefficient.
• In semiconductors, increasing temperature “shakes
loose” more electrons, increasing mobility and
increasing current flow. This produces a negative
temperature coefficient.
• Superconductors, behave like metals until a phase
transition temperature is reached. At lower
temperatures R=0.
Resistance Defined

 1 
J  E  E

1
J E

J
I
A
E
V
L
for a uniform E
–
+
V
Figure 25-7
therefore
I
1V

A  L
I
L
 L  (  ) I  RI
A
A
V  RI
Ohm’s Law
where R is the resistance of the material in ohms ()
9
Ohm’s law an idealized model
• If current density J is nearly proportional to electric field E
ratio E/J = constant and Ohm’s law applies V = I R
• Ohm’s Law is linear, but current flow through other devices
may not be.
Linear
1
I V
R
Nonlinear
1
Slope 
R
Nonlinear
Ohm’s law applies
V  RI
Resistors are color-coded for assembly work
Examples:
Brown-Black-Red-Gold = 1000 ohms +5% to -5%
Yellow-Violet-Orange-Silver = 47000 ohms +10% to -10%
Electromotive
force and circuits
If an electric field is produced in a conductor
without a complete circuit, current flows
for only a very short time.
An external source is needed to produce a
net electric field in a conductor. This
source is an electromotive force, emf ,
“ee-em-eff”, (1V = 1 J/C)
Ideal diagrams of “open” and “complete” circuits
Symbols
for circuit diagrams
– Shorthand symbols are in use for all wiring components
Electromotive Force and Circuits
Electromotive Force (EMF)
Ideal Source
I
Complete path needed for
current (I) to flow
Voltage rise in
current direction
+
+
VR
–
EMF
Ideal source of
electrical energy
–
VR = EMF = R I
rs
+
EMF
–
Real source of
electrical energy
I
I
Real Source
Internal source
resistance
Voltage drop in
current direction
R
a
+
Vab
VR EMF

R
R
External resistance
R
Vab  EMF  Irs  IR
–
b
15
A Source with an Open Circuit
Example 25-5
I = 0 amps
Figure 25-16
Vab  EMF  Ir  12V  0r  12V
Chapter 25
16
A source in a complete circuit
Example 25-6
Vab    Ir  IR
  IR  Ir  I ( R  r )

12
I

 2A
Rr 42
Figure 25-17
Vab    Ir  12  2(2)  8V
Vab  Va 'b '  IR  2(4)  8V
17
A Source with a Short Circuit
Example 25-8
Vab  0
I=6A
Figure 25-19
Vab    Ir  IR  I (0)  0
  Ir  0
  Ir
Chapter 25

12V
I 
 6A
r
2
18
Potential Rises and Drops in a Circuit
Figure 25-21
19
Energy and Power
I 
dQ
dt
dWab
Vab 
dQ
dWab  Vab Idt
Figure 25-21
1 watt = 1 joule/sec
dWab
P
 Vab I
dt
dQ  Idt
dWab  VabdQ
watts
Pure Resistance
20
Power Output of an EMF Source
I
rs
+
EMF
–
+
a
– +
Vab
R
–
b
Vab  EMF  Irs  IR
Pab  Vab I  ( EMF  Irs ) I  ( EMF ) I  I 2 rs  I 2 R
( EMF ) I  I 2 rs  I 2 R
Power output of battery
Power dissipated in R
Power dissipated in battery resistance
Power supplied by the battery
Chapter 25
21
Power Input to a Source
I
rs
+
–
a
+
Vab greater then the EMF of the battery
+
EMF
Vab
–
–
b
Vab  EMF  Irs
Pab  Vab I  ( EMF  Irs ) I  ( EMF ) I  I 2 rs
Vab I  ( EMF ) I  I 2 rs
Power dissipated in battery resistance
Power charging the battery
Total Power input to battery
22
Power Input and Output in a Complete Circuit
Example 25-9
Figure 25-25
23
Power in a Short Circuit
Example 25-11
24
Theory of Metallic Conduction
• Simple, non-quantum-mechanical model
• Each atom in a metal crystal gives up one or more
electrons that are free to move in the crystal.
• The electrons move at a random velocity and collide with
stationary ions. Velocity in the order of 106 m/s (drift
velocity is approximately 10-4 m/s)
• The average time between collisions is the mean free
time, τ.
• As temperature increases the ions vibrate more and
produce more collisions, reducing τ.
Chapter 25
25
A microscopic look at conduction
– Consider Figure 25.27.
– Consider Figure 25.28.
– Follow Example 25.12.