I-2 Gauss

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I-2 Gauss Law
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Main Topics

• The Electric Flux.
• The Gauss Law.
• The Charge Density.
• Use the G. L. to calculate the field of a
• A Point Charge
• An Infinite Uniformly Charged Wire
• An Infinite Uniformly Charged Plane
• Two Infinite Charged Planes

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The Electric Flux

• The electric flux is defined as
• It represents amount of electric intensity
  which flows perpendicularly through a surface,
  characterized by its outer normal vector .
  The surface must be so small that can be
  considered constant there.
• Lets revisit the scalar product.

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The Gauss Law I

• Total electric flux through a closed surface is
  equal to the net charge contained in the volume
  surrounded by the surface divided by the
  permitivity of vacuum .
• It is equivalent to the statement that field
  lines begin in positive charges and end in
  negative charges.

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The Gauss Law II

• Field lines can both begin or end in the
  infinity.
• G. L. is roughly because the decrease of
  intensity the with r2 in the flux is compensated
  by the increase with r2 of surface of the sphere.
• The scalar product takes care of the mutual
  orientation of the surface and the intensity.

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The Gauss Law III

• If there is no charge in the volume each field
  line which enters it must also leave it.
• If there is a positive charge in the volume then
  more lines leave it than enter it.
• If there is a negative charge in the volume then
  more lines enter it than leave it.
• Positive charges are sources and negative are
  sinks of the field.
• Infinity can be either source or sink of the
  field.

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The Gauss Law IV

• Gauss law can be taken as the basis of
  electrostatics as well as Coulombs law. It is
  actually more general!
• Gauss law is useful
• for theoretical purposes
• in cases of a special symmetry

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The Charge Density

• In real situations we often do not deal with
  point charges but rather with charged bodies with
  macroscopic dimensions.
• Then it is usually convenient to define the
  charge density i.e. charge per unit volume or
  surface or length, according to the symmetry of
  the problem.
• Since charge density may depend on the position,
  its use makes sense mainly if the bodies are
  uniformly charged e.g. conductors in equilibrium.

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A Point Charge I

• As a Gaussian surface we choose a spherical
  surface centered on the charge.
• Intensity is perpendicular to the spherical
  surface in every point and so parallel (or
  antiparallel) to its normal.
• At the same time E is constant on the surface, so
  

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A Point Charge II

• So we get the same expression for the intensity
  as from the Coulombs Law
• Here we also see where from the strange term
  appears in the Coulombs Law!

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An Infinite Uniformly Charged Wire I

• Conductive wire (in equilibrium) must be charged
  uniformly so we can define the length charge
  density as charge per unit length
• Both Q and L can be infinite, yet have a finite
  ratio.
• The wire is axis of the symmetry of the problem.

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An Infinite Uniformly Charged Wire II

• Intensity lies in planes perpendicular to the
  wire and it is radial.
• As a Gaussian surface we choose a cylindrical
  surface (of some length L) centered on the wire.
• Intensity is perpendicular to the surface in
  every point and so parallel to its normal.
• At the same time E is constant everywhere on this
  surface.

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Infinite Wire III

• Flux through the flat caps is zero since here the
  intensity is perpendicular to the normal.
• So

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Infinite Wire IV

• By making one dimension infinite the intensity
  decreases 1/r instead of 1/r2 which was the
  case of a point charge!
• Again, we can obtain the same result using the
  Coulombs law and the superposition principle but
  it is a little more difficult!

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An Infinite Charged Conductive Plane I

• If the charging is uniform, we can define the
  surface charge density
• Again both Q and A can be infinite yet reach a
  finite ratio, which is the charge per unit
  surface.
• From the symmetry the intensity must be
  everywhere perpendicular to the surface.

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Infinite Plane II

• As a Gaussian surface we can take e.g. a cylinder
  whose axis is perpendicular to the plane. It
  should be cut in halves by the plane.
• Nonzero flux will flow only through both flat
  cups (with some magnitude A) since is
  perpendicular to them.

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Infinite Plane III

• This time doesnt change with the distance
  from the plane. Such a field is called
  homogeneous or uniform!
• Note that both magnitude and direction of the
  vectors must be the same if the vector field
  should be uniform.

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Quiz Two Parallel Planes

• Two large parallel planes are d apart. One is
  charged with a charge density ?, the other with
  -?. Let Eb be the intensity between and Eo
  outside of the planes. What is true?
• A) Eb 0, Eo?/?0
• B) Eb ?/?0, Eo0
• C) Eb ?/?0, Eo?/2?0

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Homework

• The one from yesterday is due tomorrow!
• The next one will be assigned tomorrow.

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Things to read

• This lecture covers
• Giancoli Chapter 22
• Advance reading
• Giancoli Chapter 23-1, 23-2

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The scalar or dot product

• Let
• Definition I. (components)

Definition II. (projection)
Can you proof their equivalence?

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Gauss Law

• The exact definition

• In cases of a special symmetry we can find
  Gaussian surface on which the magnitude E is
  constant and is everywhere parallel to the
  surface normal. Then simply

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Infinite Wire by C.L. die hard!

• Only radial component Er of is non-zero

• We have to substitute all variables using ? and
  integrate from 0 to ?

• Quiz What was easier?