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Chapter 25
Current, Resistance, Electromotive Force
•
•
•
•
•
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
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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.
-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 (~2 meters/day)
-Movement of electrons sets up an
electric field that approaches the
speed of light
- Electrons in the conductor start to
move all along the wire at very nearly
the same time.
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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.
I
dQ
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
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Electric Current Density
Current, Drift Velocity, and Current Density
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
dQ  q(nAvd dt )  nqvd Adt
dQ
 nqAvd  I
dt
J
I
dQ
 nqAvd
dt
amperes
I
= 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).




J  nqvd  nqE  E
where

  nq
1
1
E


 nq J
Sub for Vd
conductivity of the material.
Resistivity of the material.
Chapter 25
6
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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.
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
 1 
J  E  E
Resistance Defined

1
J E

Solve for V
therefore
J
I
A
E
V
L
I
1V

A  L
–
+
for a uniform E
V
I
L
 L  (  ) I  RI
A
A
V  RI
Ohm’s Law
where R is the resistance of the material in ohms ()
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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
I
1
V
R
Nonlinear
Slope 
1
R
Nonlinear
Ohm’s law applies
V  RI
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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)
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Ideal diagrams of “open” and “complete” circuits
Symbols
for circuit diagrams
– Shorthand symbols are in use for all wiring components
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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
External resistance
+
Vab
VR EMF

R
R
R
Vab  EMF  Irs  IR
–
b
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A Source with an Open Circuit
Example 25-5
I = 0 amps
Figure 25-16
Vab  EMF  Ir  12V  0r  12V
Chapter 25
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A source in a complete circuit
Example 25-6
Vab    Ir  IR
  IR  Ir  I ( R  r )

12
I

 2A
Rr 42
Vab    Ir  12  2(2)  8V
Figure 25-17
Vab  Va 'b'  IR  2(4)  8V
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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
I

r

12V
 6A
2
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Potential Rises and Drops in a Circuit
Figure 25-21
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Energy and Power
Vab 
P
dWab
dQ
Divide by
dt
P
dQ  Idt
Defined
dQ
dt
dQ  Idt
I
dWab  VabdQ
Substitute for
dWab
dt
dWab  Vab Idt
dWab
 Vab I
dt
watts
Pure Resistance
1 watt = 1 joule/sec
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Power Output of an EMF Source
I
rs
a
–
+
+
+
Vab
EMF
–
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
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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
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Power Input and Output in a Complete Circuit
Example 25-9
Figure 25-25
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Power in a Short Circuit
Example 25-11
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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.
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Q25.1
Two copper wires of different diameter are joined end-to-end,
and a current flows in the wire combination.
When electrons move from the larger-diameter wire into the
smaller-diameter wire,
A. their drift speed increases.
B. their drift speed decreases.
C. their drift speed stays the same.
D. not enough information given to decide
Q25.2
Electrons in an electric circuit pass through a resistor. The wire
has the same diameter on each side of the resistor.
Compared to the drift speed of the electrons before entering the
resistor, the drift speed of the electrons after leaving the resistor is
A. faster.
B. slower.
C. the same.
D. not enough information given to decide
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Q25.3
Electrons in an electric circuit pass through a resistor. The wire
has the same diameter on each side of the resistor.
Compared to the potential energy of an electron before entering
the resistor, the potential energy of an electron after leaving the
resistor is
A. greater.
B. less.
C. the same.
D. not enough information given to decide
Q25.4
Electrons in an electric circuit pass through a source of emf. The
wire has the same diameter on each side of the source of emf.
Compared to the drift speed of the electrons before entering the
source of emf, the drift speed of the electrons after leaving the
source of emf is
A. faster.
B. slower.
C. the same.
D. not enough information given to decide
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Q25.5
Electrons in an electric circuit pass through a source of emf. The
wire has the same diameter on each side of the source of emf.
Compared to the potential energy of an electron before entering
the source of emf, the potential energy of an electron after leaving
the source of emf is
A. greater.
B. less.
C. the same.
D. not enough information given to decide
Q25.6
In the circuit shown, the two bulbs
A and B are identical. Compared to
bulb A,
A. bulb B glows more brightly.
B. bulb B glows less brightly.
C. bulb B glows just as brightly.
D. answer depends on whether the mobile charges in the wires
are positively or negatively charged
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Q25.7
In the circuit shown in (a), the two bulbs A and B are identical. Bulb
B is removed and the circuit is completed as shown in (b). Compared
to the brightness of bulb A in (a), bulb A in (b) is
A. brighter.
B. less bright.
C. just as bright.
D. any of the above, depending on the rated wattage of the bulb.
Q25.8
An ideal voltmeter
A. has zero resistance and should be connected in
parallel with the circuit element being measured.
B. has zero resistance and should be connected in series
with the circuit element being measured.
C. has infinite resistance and should be connected in
parallel with the circuit element being measured.
D. has infinite resistance and should be connected in
series with the circuit element being measured.
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Q25.9
An ideal ammeter
A. has zero resistance and should be connected in
parallel with the circuit element being measured.
B. has zero resistance and should be connected in series
with the circuit element being measured.
C. has infinite resistance and should be connected in
parallel with the circuit element being measured.
D. has infinite resistance and should be connected in
series with the circuit element being measured.
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