Gauss

1
Chapter 21

• Gausss Law

2
Electric Field Lines

• Electric field lines (convenient for visualizing
  electric field patterns) lines pointing in the
  direction of the field vector at any point
• The electric field vector is tangential to the
  electric field lines at each point
• The number of lines per unit area through a
  surface perpendicular to the lines is
  proportional to the strength of the electric
  field in a given region

3
Electric Field Line Patterns

• For a positive point charge the lines will
  radiate outward equally in all directions
• A positive test charge would be repelled away
  from the positive source charge
• For a negative point charge the lines will point
  inward equally in all directions
• A positive test charge would be attracted toward
  the negative source charge

4
Electric Field Line Patterns

• An electric dipole consists of two equal and
  opposite point charges
• The number of field lines leaving the positive
  charge equals the number of lines terminating on
  the negative charge
• For two equal but like point charges, at a great
  distance from the charges, the field would be
  approximately that of a single charge of 2q
  (bulging out of the field lines between the
  charges repulsion)

5
Electric Field Line Patterns

• For these two unequal and unlike point charges,
  at a great distance from the charges, the field
  would be approximately that of a single charge of
  q (two lines leave the 2q charge for each line
  that terminates on -q)

6
Rules for Drawing Electric Field Lines

• For a group of charges, the lines must begin on
  positive charges and end on negative charges
• In the case of an excess of charge, some lines
  will begin or end infinitely far away
• The number of lines drawn leaving a positive
  charge or ending on a negative charge is
  proportional to the magnitude of the charge
• No two field lines can cross each other

7
Electric Flux

• Electric flux is the product of the magnitude of
  the electric field and the surface area, A,
  perpendicular to the field
• FE EA

8
Electric Flux

• The electric flux is proportional to the number
  of electric field lines penetrating some surface
• The field lines may make some angle ? with the
  perpendicular to the surface
• Then
• The flux is a maximum (zero) when the surface is
  perpendicular (parallel) to the field

9
Electric Flux

• If the field varies over the surface, F EA cos
  ? is valid for only a small element of the area
• In the more general case, look at a small area
  element
• In general, this becomes

10
Electric Flux

• The surface integral means the integral must be
  evaluated over the surface in question
• The value of the flux depends both on the field
  pattern and on the surface
• SI units N.m2/C

11
Electric Flux, Closed Surface

• For a closed surface, by convention, the A
  vectors are perpendicular to the surface at each
  point and point outward
• (1) ? lt 90o, F gt 0
• (2) ? 90o, F 0
• (3) 180o gt ? gt 90o, F lt 0

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Electric Flux, Closed Surface

• The net flux through the surface is proportional
  to the number of lines leaving the surface minus
  the number entering the surface

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Electric Flux, Closed Surface

• Example flux through a cube
• The field lines pass perpendicularly through two
  surfaces and are parallel to the other four
  surfaces
• Side 1 F E l2
• Side 2 F E l2
• For the other sides, F 0
• Therefore, Ftotal 0

14
Chapter 21Problem 23

• The electric field on the surface of a
  10-cm-diameter sphere is perpendicular to the
  sphere and has magnitude 47 kN/C. Whats the
  electric flux through the sphere?

15
Chapter 21Problem 42

• Whats the flux through the hemispherical open
  surface of radius
• R in a uniform field of magnitude E shown in the
  figure?

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Gauss Law

• Gauss Law electric flux through any closed
  surface is proportional to the net charge Q
  inside the surface
• eo 8.85 x 10-12 C2/Nm2 permittivity of free
  space
• The area in F is an imaginary Gaussian surface
  (does not have to coincide with the surface of a
  physical object)

17
Gauss Law

• A positive point charge q is located at the
  center of a sphere of radius r
• The magnitude of the electric field everywhere on
  the surface of the sphere is E keq / r2
• Asphere 4pr2

18
Gauss Law

• Gaussian surfaces of various shapes can surround
  the charge (only S1 is spherical)
• The electric flux is proportional to the number
  of electric field lines penetrating these
  surfaces, and this number is the same
• Thus the net flux through any closed surface
  surrounding a point charge q is given by q/eo and
  is independent of the shape of the surface

19
Gauss Law

• If the charge is outside the closed surface of an
  arbitrary shape, then any field line entering the
  surface leaves at another point
• Thus the electric flux through a closed surface
  that surrounds no charge is zero

20
Gauss Law

• Since the electric field due to many charges is
  the vector sum of the electric fields produced by
  the individual charges, the flux through any
  closed surface can be expressed as
• Although Gausss law can, in theory, be solved to
  find for any charge configuration, in
  practice it is limited to symmetric situations
• One should choose a Gaussian surface over which
  the surface integral can be simplified and the
  electric field determined

21
Field Due to a Spherically Symmetric Charge
Distribution

• For r gt a
• For r lt a

22
Field Due to a Spherically Symmetric Charge
Distribution

• Inside the sphere, E varies linearly with r (E ?
  0 as r ? 0)
• The field outside the sphere is equivalent to
  that of a point charge located at the center of
  the sphere

23
Electric Field of a Charged Thin Spherical Shell

• The calculation of the field outside the shell is
  identical to that of a point charge
• The electric field inside the shell is zero

24
Field Due to a Line of Charge

• Select a cylindrical Gaussian surface (of radius
  r and length l)
• Electric field is constant in magnitude and
  perpendicular to the surface at every point on
  the curved part of the surface
• The end view confirms the field is perpendicular
  to the curved surface
• The field through the ends of the cylinder is 0
  since the field is parallel to these surfaces

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Field Due to a Line of Charge
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Field Due to a Plane of Charge

• The uniform field must be perpendicular to the
  sheet and directed either toward or away from the
  sheet
• Use a cylindrical Gaussian surface
• The flux through the ends is EA and there is no
  field through the curved part of the surface
• Surface charge density s Q / A

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Field Distance Dependencesfor Different Charge
Distributions
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Chapter 21Problem 31

• The electric field strength outside a charge
  distribution and 18 cm from its center has
  magnitude 55 kN/C. At 23 cm the field strength is
  43 kN/C. Does the distribution have spherical or
  line symmetry?

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Conductors in Electrostatic Equilibrium

• When no net motion of charge occurs within a
  conductor, the conductor is said to be in
  electrostatic equilibrium
• An isolated conductor has the following
  properties
• Property 1 The electric field is zero everywhere
  inside the conducting material
• If this were not true there were an electric
  field inside the conductor, the free charge there
  would move and there would be a flow of charge
  the conductor would not be in equilibrium

30
Conductors in Electrostatic Equilibrium

• When no net motion of charge occurs within a
  conductor, the conductor is said to be in
  electrostatic equilibrium
• An isolated conductor has the following
  properties
• Property 1 The electric field is zero everywhere
  inside the conducting material

31
Conductors in Electrostatic Equilibrium

• When no net motion of charge occurs within a
  conductor, the conductor is said to be in
  electrostatic equilibrium
• An isolated conductor has the following
  properties
• Property 1 The electric field is zero everywhere
  inside the conducting material

32
Conductors in Electrostatic Equilibrium

• When no net motion of charge occurs within a
  conductor, the conductor is said to be in
  electrostatic equilibrium
• An isolated conductor has the following
  properties
• Property 2 Any excess charge on an isolated
  conductor resides entirely on its surface
• The electric field (and thus the flux) inside is
  zero whereas the Gaussian surface can be as close
  to the actual surface as desired, thus there can
  be no charge inside the surface and any net
  charge must reside on the surface

33
Conductors in Electrostatic Equilibrium

• When no net motion of charge occurs within a
  conductor, the conductor is said to be in
  electrostatic equilibrium
• An isolated conductor has the following
  properties
• Property 3 The electric field just outside a
  charged conductor is perpendicular to the surface
  and has a magnitude of s/eo
• If this was not true, the component along the
  surface would cause the charge to move no
  equilibrium

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Conductors in Electrostatic Equilibrium

• When no net motion of charge occurs within a
  conductor, the conductor is said to be in
  electrostatic equilibrium
• An isolated conductor has the following
  properties
• Property 4 On an irregularly shaped conductor,
  the charge accumulates at locations where the
  radius of curvature of the surface is smallest
• The forces from the charges at the sharp end
  produce a larger resultant force away from the
  surface.

35
Chapter 21Problem 41

• A total charge of is applied to a thin, square
  metal plate 75 cm on a side. Find the electric
  field strength near the plates surface.

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Answers to Even Numbered Problems Chapter 21
Problem 22 490 N m2/C
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Answers to Even Numbered Problems Chapter 21
Problem 24 1.81 103 N m2/C
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Answers to Even Numbered Problems Chapter 21
Problem 32 -2.0 10-4 C/m2
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Answers to Even Numbered Problems Chapter 21
Problem 38 160 kN/C
40
Answers to Even Numbered Problems Chapter 21
Problem 56 58 nC/m2