Transcript Slide 1

Lecture PowerPoints
Physics for Scientists and
Engineers, 3rd edition
Fishbane
Gasiorowicz
Thornton
© 2005 Pearson Prentice Hall
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Chapter 26
Currents in Materials
Main Points of Chapter 26
• Electric current
• Current density
• Currents in materials: conductors and
insulators
• Conservation of charge
• Resistors, resistance, and conductivity
• Series and parallel combinations of resistors
• Materials and conductivity: semiconductors
and superconductors
• Electric power
26-1 Electric Current
Definition: the total charge that passes
through a given cross-sectional area per
unit time.
(26-2)
Units: amperes
26-1 Electric Current
• Direction: the direction a positive charge
would take, even if the current consists of
negative charges moving in the opposite
direction
Note: current is a
signed scalar,
not a vector
26-1 Electric Current
• Current density: current per unit area,
defined over an infinitesimal area
• Current is surface integral of current
density
(26-4)
26-1 Electric Current
Current density of moving charges:
(26-7)
• nq is number of charges per unit volume
• q is the magnitude of each charge
• v is the velocity of each charge
26-2 Currents in Materials
• Electrons in conductors are always
colliding with molecules
• Average (rms) speed increases with
temperature
• If no electric field, no net speed in any
direction
26-2 Currents in Materials
• Electric field introduces overall “drift”
• Drift velocity is very small compared to thermal
velocity
26-2 Currents in Materials
Writing the current in terms of the motion of
individual charge carriers:
(26-8)
And solving for the drift velocity:
(26-9)
The current density is then:
(26-10)
26-2 Currents in Materials
Current and the conservation of charge:
If the diameter of the conductor changes, the
current density and drift velocity change too
26-3 Resistance
• Resistance is a measure of how easily
current flows in a material
• For the same voltage, more resistance
means less current, and vice versa
• Definition of resistance:
Units of resistance: ohms (Ω)
(26-11)
26-3 Resistance
Ohm’s Law:
An ohmic material is one where the resistance is
nearly constant over a wide range of voltages.
In that case:
(26-12)
26-3 Resistance
Resistors
• Ohmic material
• Specified resistance
• For use in circuits
26-3 Resistance
Resistivity and Conductivity
• Property of a material
• Independent of geometry and
size
• Definition:
(26-13)
26-3 Resistance
Calculating resistance using resistivity:
(26-14)
for a material of length L and crosssectional area A.
The conductivity is the inverse of the
resistivity:
(26-15)
26-3 Resistance
The Temperature Dependence of Resistivity
• Resistivity has its origin in collisions
between electrons and atoms or molecules
• Higher temperature = faster thermal
velocities = more collisions = higher
resistivity
(26-18)
26-4 Resistances in Series and in Parallel
Resistors in series:
have the same current, but
voltage depends on resistance.
Equivalent resistance (same
current and A-B voltage drop):
(26-21)
26-4 Resistances in Series and in Parallel
Resistors in parallel:
have the same voltage, but
current depends on resistance.
Equivalent resistance (same
current and A-B voltage drop):
(26-22)
26-5 Free-Electron Model of Resistivity
Assumptions:
• Conductors contain free electrons, not
attached to particular atoms
• Electrons form a “gas” at temperature T
• Electric field creates drift; collisions
create drag, yielding constant drift velocity
• Yields resistivity that depends only on
mean free path of electrons in that material
(a success!)
26-5 Free-Electron Model of Resistivity
Failures:
• Electrons move much faster than predicted
• Model predicts electrons move faster with
temperature – they don’t
• Mean free path should be independent of
temperature – it isn’t, and is much larger than
predicted
Successful model uses quantum mechanics.
26-6 Materials and Conductivity
• Quantum mechanics tells us that energy
levels of electrons in materials are quantized –
only certain values are possible
• Each level can contain only two electrons
• Results in filled and empty levels
• Insulators: have gap between filled and
empty energy levels
26-6 Materials and Conductivity
Illustration: insulator
has a gap between filled
and empty bands,
conductor does not
26-6 Materials and Conductivity
Semiconductor: also has gap between filled and
empty bands, but gap is small and can be
breached using external field
26-6 Materials and Conductivity
Superconductor
• At some critical
temperature Tc, resistance
goes abruptly to zero.
• Zero resistance means
currents can last indefinitely
26-7 Electric Power
Using the definition of power and of voltage:
(26-26)
And now the definition of current:
(26-27)
This is the power lost in resistive
circuit elements, and is valid for all
materials.
Summary of Chapter 26
• Electric current: I = dQ/dt
• Current density (a vector) is current passing
through a unit area
• Electrical resistance is the ratio of voltage to
current: R = V/I
• In ohmic materials, this ratio is nearly
constant
• Resistivity is a property of a material; to find
resistance: R = ρL/A
• Resistivity depends on temperature
Summary of Chapter 26, cont.
• Resistors in series:
(26-21)
• Resistors in parallel:
(26-22)
• Power P = V/I