II

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II2 Microscopic View of Electric Currents
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Main Topics

• The Resistivity and Conductivity.
• Conductors, Semiconductors and Insulators.
• The Speed of Moving Charges.
• The Ohms Law in Differential Form.
• The Classical Theory of Conductivity.
• The Temperature Dependence of Resistivity

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The Resistivity and Conductivity I

• Lets have an ohmic conductor i.e. the one which
  obeys the Ohms law
• V RI
• The resistance R depends both on the geometry and
  the physical properties of the conductors. If we
  have a conductor of the length l and the
  cross-section A we can define the resistivity r
  and its reciprocal the conductivity ? by
• R rl/A l/?A

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The Resistivity and Conductivity II

• The resistivity is the ability of materials to
  defy the electric current. Roughly a stronger
  field is necessary if the resitivity is high to
  reach a certain current. The SI unit is 1 ?m.
• The conductivity is the ability to conduct the
  electric current. The SI unit is 1 ?-1m-1. There
  is a special unit siemens 1 Si ?-1.

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Mobile Charge Carriers I

• Generally, they are charged particles or
  pseudo-particles which can move freely in
  conductors. They can be electrons, holes or
  various ions.
• The conductive properties of materials depend on
  how freely their charge carriers can move and
  this depends on deep intrinsic construction of
  the particular materials.

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Mobile Charge Carriers II

• E.g. in solid conductors each atom shares some of
  its electrons, those least strongly bounded, with
  the other atoms. In zero electric field these
  electrons normally move chaotically at very high
  speeds and undergo frequent collisions with the
  array of atoms of the solid. It resembles thermal
  movement of gas molecules ? electron gas.

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Mobile Charge Carriers III

• In non-zero field the electrons also have some
  relatively very low drift speed in the opposite
  direction then has the field. The collisions are
  the predominant mechanism for the resistivity (of
  metals at normal temperatures) and they are also
  responsible for the power loses in conductors.

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Differential Ohms Law I

• Let us again have a conductor of the length l and
  the cross-section A and consider only one type of
  charged carriers and a uniform current, depends
  on their
• density n i.e. number per unit volume
• charge q
• drift speed vd

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Differential Ohms Law II

• Within some length ?x of the conductor there is a
  charge
• ?Q n q?x A
• The volume which passes some plane in 1 second is
  A?x/?t vd A so the current is
• I ?Q/?t n q vd A j A
• Where j is so called current density. Using Ohms
  law and the definition of the conductivity
• I j A V/R El ? A/l ? j ?E

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Differential Ohms Law III

• j ?E
• This is Ohms law in differential form. It has a
  similar form as the integral law but it contains
  only microscopic and non-geometrical parameters.
  So it is a the starting point of theories which
  try to explain conductivity.
• Generally, it is valid in vector form j ?E

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Differential Ohms Law IV

• Its meaning is that the magnitude of the current
  density is directly proportional to the field and
  that the charge carriers move along the field
  lines.
• For deeper insight it is necessary to have at
  least rough ideas about the magnitudes of the
  parameters involved.

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An Example I

• Let us have a current of 10 A running through a
  copper conductor with the cross-section of 3 10-6
  m2. What is the charge density and drift velocity
  if every atom contributes by one free electron?
• The atomic weight of Cu is 63.5 g/mol.
• The density ? 8.95 g/cm3.

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An Example II

• 1 m3 contains 8.95 106/63.5 1.4 105 mol.
• If each atom contributes by one free electron,
  this corresponds to n 8.48 1028 electrons/m3.
• vd I/Anq
• 10/(8.48 1028 1.6 10-19 3 10-6) 2.46 10-4 m/s

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The Internal Picture

• The drift speed is very low. It would take the
  electron 68 minutes to travel 1 meter! In
  comparison, the average speed of the chaotic
  movement is of the order of 106 m/s.
• So we have currents of the order of 1012 A
  running in random directions and so compensating
  themselves and relatively a very little currents
  caused by the field. It is similar as in the case
  of charging something a very little
  un-equilibrium.

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A Quiz

• The drift speed of the charge carriers is of the
  order of 10-4 m/s. Why it doesnt take hours
  before a bulb lights when we switch on the light?

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The Answer

• By switching on the light we actually connect the
  voltage across the wires and the bulb and thereby
  create the electric field which moves the charge
  carriers. But the electric field spreads with the
  speed of light c 3 108 m/s, so all the charges
  start to move (almost) simultaneously.

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The Classical Model I

• Lets try to explain the drift speed using more
  elementary parameters. We suppose that during
  some average time between the collision ? the
  charge carriers are accelerated by the field.
  Using what we know from electrostatics
• vd qE?/m

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The Classical Model II

• We substitute the magnitude of the drift velocity
  into the formula for the current density
• j n q vd n q2 ? E/m
• So we obtain conductivity and resistivity
• ? n q2 ? /m
• r 1/? m/nq2?

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The Classical Model III

• It may seem that we have just replaced one set of
  parameters by another. But here only the average
  time is unknown and it can be related to mean
  free path and the average thermal speed using
  well established theories similar to those
  studying ideal gas properties.
• This model predicts dependence on the temperature
  but not on the electric field.

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Temperature Dependence of Resistivity I

• In most cases the behavior is close to linear.
• We define a change in resistivity in relation to
  some reference temperature t0 (0 or 20 C)
• ?r r(t) r(t0)
• The relative change of resistivity is directly
  proportional to the change of the temperature
• ?r/r(t0) ?(t t0) ? ?t ?
• r(t) r(t0)(1 ? ?t)

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Temperature Dependence of Resistivity II

• ? K-1 is the linear temperature coefficient. It
  is given the temperature dependence of n and vd.
  It can be negative e.g. in the case of
  semiconductors (but exponential behavior).
• If it doesnt work we have to add a quadratic
  term etc.
• ?r/r(t0) ?(t t0) ? ?t ? (?t)2 ?
• r(t) r(t0)(1 ? ?t ? (?t)2 )

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Homework

• Please, try to prepare as much as you can for the
  midterm exam!

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Things to read

• Chapter 25 8 and 26 2
• See demonstrations
• http//buphy.bu.edu/duffy/semester2/semester2.htm
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