I2 Gauss Law

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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
• d?eE.dA
• It represents amount of electric intensity E
  which flows perpendicularly through a small
  surface dA, characterized by its outer normal
  vector. Revisit the scalar product.

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The Gauss Law I

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

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The Gauss Law II

• 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 III

• 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 do not deal with point
  charges but rather with charged bodies with
  macroscopic dimensions. Then it is often
  convenient to define the charge density i.e.
  charge per unit volume or surface or length,
  according to the symmetry of the problem. This
  makes sense mainly if the bodies are uniformly
  charged e.g. conductors in equilibrium.

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A Point Charge

• As a Gaussian surface we choose a spherical
  surface centered on the charge. E is
  perpendicular to the surface in every point and
  so parallel to its normal. At the same time E is
  constant on the surface.
• so E4?r2 q/?0 ? E(r) q/4??0r2
• Here the term 1/4??0 appears !

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An Infinite Uniformly Charged Wire I

• Conductive wire (in equilibrium) must be charged
  uniformly so we define the charge density per
  unit length ? Q/L (both can be infinite yet
  reach a finite ratio).
• The wire is axis of the symmetry of the problem.
• Intensity lies in planes perpendicular to the
  wire and it is radial.

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Infinite Wire II

• As a Gaussian surface we choose a cylindrical
  surface (of some length L) centered on the wire.
  E is in every point perpendicular to the surface
  and so parallel to its normal. At the same time E
  is constant on this surface.
• Flux through the flat caps is zero since here the
  intensity is perpendicular to the normal.

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Infinite Wire III

• So E2?rL ?L/?0 ? E(r) ? /2??0r
• By having one dimension infinite we have decrease
  1/r instead of 1/r2 !
• 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

• We define the charge density ?Q/S. Again both
  values 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 half by the plane.
• Nonzero flux will flow only through both flat
  surfaces (with some magnitude S). E is
  perpendicular to them.
• 2ES ?S/?0 ? E(r) ?/2?0
• E doesnt change with the distance from the
  plane. Such a field is called homogeneous or
  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 you will get tomorrow.

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Things to read

• Giancoli Chapter 22

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The scalar or dot product

• Let ca.b
• 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 E is everywhere parallel to the
  surface normal. Then simply

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Infinite Wire by C.L. die hard!

• E has only radial component Er

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

• Quiz What was easier?