Today

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2
Today

• Current and Current Density
• Devices
• Batteries
• Resistors
• Read Fishbane Chapter 26
• Remember Quiz on Thursday and Friday.
• Covers chapters 24-25-12,
• Potential, Capacitors, Gravity (and everything
  before)

3
Devices and Circuits

• We are now finished with electrostatics the
  study of fields and potentials produced by static
  charge distributions.
• Next topic Devices and Circuits
• We have studied one device so far the capacitor.
  For the next week we will investigate circuits
  composed of the following devices
• Capacitors
• Batteries
• Resistors

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Current is charge in motion

• Electrons exists in conductors with a density, ne
  (ne approx 1029 m-3) and are constantly in
  random motion
• In general only electrons move, the heavy nucleii
  remain fixed in the material lattice
• In the absence of electric fields there is no net
  motion of the charge, electrons bounce around
  like atoms in a gas

• When an electric field is applied
• a small average velocity, ve ,is added to the
  random motion (an electric current)

• NOTE that the current direction is defined as
  the direction of the field BUT the electrons move
  in the opposite direction

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Current is charge in motion

• Current density, J, is given by J qeneve
• unit of J is C/m2sec or A/m2 (A Ampere) and
  1A 1C/s
• Current, I, is J times cross sectional area, I
  J S
• for 10 Amp in 1mm x 1mm area, J107 A/m2,
• ve is about 10-3 m/s
• (Yes, the average velocity is only 1mm/s!)

6
Devices Batteries

• Batteries (Voltage sources, sources of emf)
  Purpose is to provide a constant potential
  difference and source of current between two
  points.
• Cannot calculate the potential difference from
  first principles... chemical electrical energy
  conversion. Non-ideal batteries will be dealt
  with in terms of an "internal resistance".
• Positive terminal has the higher potential
• Current is defined as flowing from the positive
  to the negative terminal
• Inside the battery chemical processes return the
  charge from the negative to positive terminals
• emf is the term for the electrical potential
  provided by the battery

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Devices Resistors

• Resistors
• Resistors limit the current drawn in a circuit.

• Resistance is a natural property of almost all
  materials which opposes the motion of charge
  through the material
• Resistance can be calculated from knowledge of
  the geometry of the resistor AND the
  resistivity of the material out of which it is
  made (often conductors).

8
Ohms Law

• Set up this circuit
• Vary applied voltage V.
• Measure current I
• Ratio remains constant
• Resistance R

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Resistance

• What is happening in the resistance?

• Voltage means Potential Difference -gt E-field

• E-field -gt constant force on electrons

• Constant force on electrons -gt constant
  acceleration

• Constant acceleration -gt very large and
  increasing currents

• This does not happen large increasing currents
  are not observed
• whats wrong with this picture???

• Simple constant acceleration isnt happening.

• Electrons undergo a lot of rapid and random
  scattering
• No constant acceleration (acceleration
  proportional to Voltage)
• Instead velocity of electrons is proportional to
  Voltage

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What gives rise to non-ballistic behavior?

• E-field in conductor (resistor) provided by a
  battery
• Charges are put in motion, but scatter in a very
  short time from things that get in the way
• its crowded inside that metal
• defects, lattice vibrations (phonons), etc

• Typical scattering time t 10-14 sec
• Charges ballistically accelerated for this time
  and then randomly scattered

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What gives rise to non-ballistic behavior?

• Newtons 2nd Law says Fma
• So the acceleration of the electron is eE/m
• Average velocity attained between scatters is
  given by vat or v eEt/m
• Current density is J env so current is
  proportional to E which is proportional to
  Voltage
• OHMs LAW J (e2nt/m)E or J s E s
  conductivity
• Or

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Resistance

• Resistance
• Resistance is defined to be the ratio of the
  applied voltage to the current passing through.

UNIT OHM W

• Is this a good definition?
• i.e., does the resistance belong only to the
  resistor?
• Recall the case of capacitance (CQ/V) depended
  on the geometry, not on Q or V individually
• Does R depend on V or I ?
• It seems as though it should, at first glance...

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Calculating Resistance

• To calculate R, must calculate current I which
  flows when voltage V is applied.
• Applying voltage V sets up an electric field in
  the resistor. What determines the current?

• Current is charge flowing past a point per unit
  time, which depends on the average velocity of
  the charges.
• Field gives rise to force on the charge carriers
  which reach a terminal velocity.
• Resistance calculation splits into two parts
• Part depends on the resistivity ?, a property
  of the material
• Part depends on the geometry (length L and cross
  sectional area A)

14
Resistivity

• Resistivity is a property of bulk matter related
  to the resistance of a sample.
• The resistivity (r) is defined as

• Where E electric field and j current density
  in conductor.

• For the case of a uniform material

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Resistivity



where
So YES, the property belongs to the material and
we can calculate the resistance if we know the
resistivity and the dimensions of the object
e.g., for a copper wire, r 10-8 W-m, 1mm
radius, 1 m long, then R .01W for glass, r
1012 W-m for semiconductors r 1 W-m
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Makes sense?

• Increase the length, flow of electrons impeded
• Increase the cross sectional area, flow
  facilitated
• The structure of this relation is identical to
  heat flow through materials think of a window
  for an intuitive example

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Question 1

• Two cylindrical resistors, R1 and R2, are made of
  identical material. R2 has twice the length of R1
  but half the radius of R1.
• The resistors are then connected to a battery V
  as shown

• What is the relation between the currents I1 and
  I2

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Question 1

• a
• b
• c

19
Question 1

• Two cylindrical resistors, R1 and R2, are made of
  identical material. R2 has twice the length of R1
  but half the radius of R1.
• The resistors are then connected to a battery V
  as shown

• What is the relation between the currents I1 and
  I2

• The resistivity of both resistors is the same
  (?).
• Therefore the resistances are related as

• The resistors have the same voltage across them
  therefore

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Question 2

• A very thin metal wire patterned as shown is
  bonded to some structure.
• As the structure is deformed this stretches the
  wire (slightly).
• When this happens, the resistance of the wire

(c) stays the same
(a) decreases
(b) increases
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Question 2

• a
• b
• c

22
Question 2

• A very thin metal wire patterned as shown is
  bonded to some structure.
• As the structure is deformed this stretches the
  wire (slightly).
• When this happens, the resistance of the wire

(c) stays the same
(a) decreases
(b) increases

• Because the wire is slightly longer,
  is increased.
• Because the volume of the wire is constant,
  increasing the length, decreases the area, which
  increases the resistance.
• By carefully measuring the change in resistance,
  the strain in the structure may be determined

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Is Ohms Law a good law?

• Our derivation of Ohms law ignored the effects
  of temperature.
• At higher temperatures the random motion of
  electrons is faster,
• time between collisions gets smaller
• Resistance gets bigger
• Temperature coefficient of resistivity (?)
• Typical values for metals ??4?10-3

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Is Ohms Law a good law?

• Our derivation of Ohms law ignored quantum
  mechanical effects
• Many materials, only conduct when sufficient
  voltage is applied to move electrons into a
  conduction band in the material
• Examples are semiconductor diodes which have very
  far from linear voltage versus current plots

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Is Ohms Law a good law? Superconductivity

• At low temperatures (cooled to liquid helium
  temperatures, 4.2K)the resistance of some
  metals?0, measured to be less than
  10-16?conductor (i.e., ?lt10-24 Om)!

• Current can flow, even if E0.
• Current in superconducting rings can flow for
  years with no decrease!

• 1957 Bardeen, Cooper, and Schrieffer (BCS)
  publish theoretical explanation, for which they
  get the Nobel prize in 1972.
• It was Bardeens second Nobel prize (1956
  transistor)

26
Is Ohms Law a good law? Superconductivity

• 1986 High temperature superconductors are
  discovered (Tc77K)
• Important because liquid nitrogen (77 K) is much
  cheaper than liquid helium
• Highest critical temperature to date 140K

• Today Superconducting loops are used to produce
  lossless electromagnets (only need to cool
  them, not fight dissipation of current) for
  particle physics.
• Fermilab accelerator, IL
• The Future Smaller motors, lossless power
  transmission lines, magnetic levitation trains,
  quantum computers?? ...

27
Is Ohms Law a good law?

• Answer NO
• Ohms Law is not a fundamental law of physics
• However it is a good approximation for metallic
  conductors at room temperature as used in
  electrical circuits

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Summary

• Ohms Law states
• Ohms Law is not a physical law but an
  approximation which works well enough in normal
  conditions
• Read Chapter 27 for tomorrow
• Remember the Quiz on Thursday and Friday.