Chapter 23. Gauss

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Chapter 23. Gauss Law

• 23.1. What is Physics?      
• 23.2. Flux      
• 23.3. Flux of an Electric Field      
• 23.4. Gauss' Law      
• 23.5. Gauss' Law and Coulomb's Law      
• 23.6. A Charged Isolated Conductor     
• 23.7. Applying Gauss' Law Cylindrical
  Symmetry      
• 23.8. Applying Gauss' Law Planar Symmetry     
• 23.9. Applying Gauss' Law Spherical Symmetry

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What is Physics?

• Gaussian surface is a hypothetical (any imaginary
  shape) closed surface enclosing the charge
  distribution.
• Gauss' law relates the electric fields at points
  on a (closed) Gaussian surface to the net charge
  enclosed by that surface.

                                                                                                
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Gaussian surface

• Let us divide the surface into small squares
  of area ?A, each square being small enough to
  permit us to neglect any curvature and to
  consider the individual square to be flat. We
  represent each such element of area with an area
  vector
• Magnitude is the area ?A.
• Direstion is perpendicular to the Gaussian
  surface and directed away from the interior of
  the surface.

                                                                                                              
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Flux

• The rate of volume flow through the loop is

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Flux of an Electric Field

• The electric field for a surface is

• The electric field for a gaussian surface is

The electric flux F through a Gaussian surface is
proportional to the net number of electric field
lines passing through that surface.
SI Unit of Electric Flux Nm2/C
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Problem 1

• The drawing shows an edge-on view of two planar
  surfaces that intersect and are mutually
  perpendicular. Surface 1 has an area of 1.7 m2,
  while surface 2 has an area of 3.2 m2. The
  electric field E in the drawing is uniform and
  has a magnitude of 250 N/C. Find the electric
  flux through (a) surface 1 and (b) surface 2.

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Sample Problem 2

• Figure 23-4 shows a Gaussian surface in the
  form of a cylinder of radius R immersed in a
  uniform electric field E , with the cylinder axis
  parallel to the field. What is the flux F of the
  electric field through this closed surface?

                                                                                                                                            

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Sample Problem 3

• A nonuniform electric field given by
  pierces the Gaussian cube shown in Fig. (E
  is in newtons per coulomb and x is in meters.)
  What is the electric flux through the right face,
  the left face, the top face, and the Gaussian
  surface?

                                                                                                                              

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Gauss Law

• For a point charge

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Gauss Law

• For charge distribution Q

The electric flux through a Gaussian surface
times by e0 ( the permittivity of free space) is
equal to the net charge Q enclosed

• The net charge qenc    is the algebraic sum of
  all the enclosed charges.
• Charge outside the surface, no matter how large
  or how close it may be, is not included in the
  term qenc.

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Check Your Understanding

• The drawing shows an arrangement of three
  charges. In parts (a) and (b) different Gaussian
  surfaces are shown. Through which surface, if
  either, does the greater electric flux pass?

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Sample Problem

• Figure 23-7 shows five charged lumps of
  plastic and an electrically neutral coin. The
  cross section of a Gaussian surface S is
  indicated. What is the net electric flux through
  the surface if q1q43.1 nC, q2q5-5.9 nC, and
  q3-3.1 nC?

                                                                                                           

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A Charged Isolated Conductor

• If an excess charge is placed on an isolated
  conductor, that amount of charge will move
  entirely to the surface of the conductor. None of
  the excess charge will be found within the body
  of the conductor.
• For an Isolated Conductor with a Cavity, There is
  no net charge on the cavity walls all the excess
  charge remains on the outer surface of the
  conductor

                                                                                  
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The External Electric Field of a Conductor
If s is the charge per unit area,
                                                                                          

• according to Gauss' law,

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Sample Problem

• Figure 23-11a shows a cross section of a
  spherical metal shell of inner radius R. A point
  charge of q is located at a distance R/2 from
  the center of the shell. If the shell is
  electrically neutral, what are the (induced)
  charges on its inner and outer surfaces? Are
  those charges uniformly distributed? What is the
  field pattern inside and outside the shell?

                                                                                                                                                                          

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Applying Gauss' Law Cylindrical Symmetry
Figure 23-12 shows a section of an infinitely
long cylindrical plastic rod with a uniform
positive linear charge density ?.


                                                                                      

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Applying Gauss' Law Planar Symmetry

• Figure 23-15 shows a portion of a thin,
  infinite, nonconducting sheet with a uniform
  (positive) surface charge density s

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Applying Gauss' Law Spherical Symmetry

• A shell of uniform charge attracts or repels a
  charged particle that is outside the shell as if
  all the shell's charge were concentrated at the
  center of the shell.
• If a charged particle is located inside a shell
  of uniform charge, there is no electrostatic
  force on the particle from the shell.

                                                                                          

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• Any spherically symmetric charge distribution
  with the volume charge density ?

• For rgtR, the charge produces an electric field on
  the Gaussian surface as if the charge were a
  point charge located at the center,
• For rltR, the electric field is

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Checkpoint

• The figure shows two large, parallel,
  nonconducting sheets with identical (positive)
  uniform surface charge densities, and a sphere
  with a uniform (positive) volume charge density.
  Rank the four numbered points according to the
  magnitude of the net electric field there,
  greatest first.

    
                                                                                                                                                      

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Conceptual Questions

1. Two charges, q and q, are inside a Gaussian
   surface. Since the net charge inside the Gaussian
   surface is zero, Gauss law states that the
   electric flux through the surface is also zero
   that is F0. Does the fact that F 0 imply that
   the electric field E at any point on the Gaussian
   surface is also zero? Justify your answer.
2. The drawing shows three charges, labeled q1, q2,
   and q3. A Gaussian surface is drawn around q1 and
   q2. (a) Which charges determine the electric flux
   through the Gaussian surface? (b) Which charges
   produce the electric field at the point P?
   Justify your answers.

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• (3) A charge q is placed inside a spherical
  Gaussian surface. The charge is not located at
  the center of the sphere. (a) Can Gauss law tell
  us exactly where the charge is located inside the
  sphere? Justify your answer. (b) Can Gauss law
  tell us about the magnitude of the electric flux
  through the Gaussian surface? Why?