Chapter 17 Gauss

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Chapter 17 Gauss Law
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Main Points of Chapter 17

• Electric flux
• Gauss law
• Using Gauss law to determine electric fields

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17-1 What Does Gauss Law Do?
Consider the electric field lines associated with
a point charge Q
All the lines (12).
All the lines (12).
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Suppose the tissue is some shape other than
spherical, but still surrounds the charge. All
the field lines still go through.
Notice that they go out twice and in once if we
subtract the ins from the outs we are left
with one line going out, which is consistent with
the other situations.
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Like how many hairs passing through your scalp.
In the electric field, the number of electric
field lines through a curved surface A is defined
as electric flux.
The number of field lines penetrating a surface
is proportional to the surface area, the
orientation, and the field strength.
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1. Electric flux for uniform electric field
En
We can define the electric flux through an
infinitesimal area
if
then
A finite plane uniform electric field
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2. General definition of electric flux
3. Electric flux through a closed surface
The direction of
unclosed surface choose one side
Outward positive
Closed surface
Inward negative
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The electric flux through a closed surface is the
sum of the electric field lines penetrating the
surface.
If the electric flux is positive, there are net
electric field lines out of the closed
surface.
If the electric flux is negative , there are net
electric field lines into the closed surface.
The closed surface does not have to be made of
real matter. The imaginary surface is a Gaussian
surface
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• Act Imagine a cube of side a positioned in a
  region of constant electric field as shown below.
  Which of the following statements about the net
  electric flux Fe through the surface of this cube
  is true?

a
a
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• Act Consider 2 spheres (of radius R and 2R)
  drawn around a single charge as shown.
• Which of the following statements about the net
  electric flux through the 2 surfaces (F2R and FR)
  is true?

• Look at the lines going out through
• each circle -- each circle has the
• same number of lines.

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Example Calculate the electric flux of a
constant electric field through a hemispherical
surface of radius R whose circular base is
perpendicular to the direction of the field.
Solution
Consider an infinitesimal strip at latitude q
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Alternative Solution
Consider a closed surface that consists of the
hemisphere and the planar circle
A1
R
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• Act A nonuniform electric field given by
  What is the electric flux through the surface
  of the Gaussian cube shown in figure below ?

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• Act a point charge q is placed at the center of
  a sphere of radius of R. Calculate the electric
  flux through a circular plane of radius r as
  shown below.

r
q
S
S
the electric flux of the closed sphere
the electric flux of any closed surface t
surrounding the q?
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17-2 Gauss Law
1. for a point charge q

• Suppose a point charge q is at the center of a
  sphere,the electric flux of a spherical surface

But the result would be the same if the surface
was not spherical, or if the charge was anywhere
inside it!
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• Suppose a point charge q is out of an arbitrary
  closed surface

q
For the field of point charge, the total electric
flux through a closed surface is only related to
the total (net) electric charge inside the
surface.
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2 for multiple point charges
qj
?e of an arbitrary closed surface
qn
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3 for continuous charge distributions
We can quickly generalize this to any surface and
any charge distribution
Gauss law
Gausss law provides a relationship between the
electric field on a closed surface and the charge
distribution within that surface.
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Discussion
Electric field Lines leave () charges
with source
Electric field Lines return to (-) charges
with sink
without source or sink
Electric field Lines never discontinue in empty
space
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• ACT If a closed surface surrounds a dipole, the
  net flux through the surface is zero.

TRUE

• ACT If the net flux through a closed surface is
  zero, then there can be no charge or charges
  within that surface.

FALSE
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• Act A full Gaussian surface encloses two of the
  four
• positively charged particles. a) Which of the
  particles
• contribute to the electric field at point P on
  the surface.
• b) Determine the electric flux through the
  surface.
• c) If I change the position of q4 outside the
  surface, Does Ep change? How about the electric
  flux ?

a) The four particles
P
c) Ep changes
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• Act Consider a point charge q placed in the
  center
• of a cube. Determine the electric flux through
  each face
• of the cube .

q
Put the charge in the middle of a larger cube
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• Example There is an electric field near Earths
  surface of about 100N/C that points vertically
  down. Assume this field is constant around Earth
  and that it is due to charge evenly spread on
  Earths surface. What is the total charge on the
  Earth?

Solution
choose a concentric sphere of radius R for
Gaussian surface
the total charge
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4. Coulombs Law and Gauss Law
Coulombs Law Superposition of electric field
Now we use Gauss Law to get the distribution of
electric field for a point charge.
Draw a concentric spherical Gaussian surface of
radius r
Although E varies radially with distance from q,
it has the same value everywhere on the
spherical surface
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Gauss law and Coulombs law , although
expressed in different forms, are equivalent ways
of describing the relation between electric
charge and electric field in static situation.
Gauss law is more general in that it is always
valid. It is one of the fundamental laws of
electromagnetism.
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17-3 Using Gauss Law to determine Electric Fields
Gausss law is valid for any distribution of
charges and for any closed surface. However, we
can only calculate the field for several highly
symmetric distributions of charge.
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The steps of calculating the magnitude of the
electric field using Gauss law

• Identify the symmetry of the charge distribution
• and the electric field it produces.

(2) Choose a Gaussian surface that is matched to
the symmetry that is, the electric
field is either parallel to the surface or
constant and perpendicular to it.
(3) Calculate the algebraic sum of the charge
enclosed by the Gaussian surface.
(4) Find the electric field using Gausss law.
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• Example Determine the electric field of an
  infinite
• uniformly charged line. The charge density
  per unit
• length?

Solution
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Solution
we use as our Gaussian surface a coaxial
cylinder surface with height l
E
r
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• Could we use Gauss law to find the field of
• a finite line of charge?

• What is the electric field of an infinite
  uniformly
• charged thin cylindrical shell?

• What is the electric field of an infinite
  uniformly
• charged cylinder?

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Solution
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• Example Determine the electric field both inside
  
• and outside a thin uniformly charged spherical
  shell
• of radius R that has a total charge Q.

Solution


take a concentric sphere of radius r for
Gaussian surface
r
r
R
O



From Gauss law

• outside the sphere ( r gt R )

R

• inside the sphere( r lt R )

• on surface ( r R ) E ?

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• ACT A uniform spherical shell of charge of radius
  R surrounds a point charge at its center. The
  point charge has value Q and the shell has total
  charge -Q. The electric field at a distance R/2
  from the center ________
• A) does not depend on the charge of the spherical
  shell.
• B) is directed inward if Qgt0.
• C) is zero.
• D) is half of what it would be if only the point
  charge were present.
• E) cannot be determined

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• Act A nonconducting spherical shell of radius R
  is uniformly charged with total charge of Q. We
  put point charges q1and q2 inside and outside
  spherical shell respectively . Determine the
  electric forces on the point charges.

q1
q2

• inside the sphere( r lt R )

O
r1
r2

• outside the sphere ( r gt R )

right
Two shells law
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• Example Find the electric field outside and
  inside a solid, nonconducting sphere of radius R
  that contains a total charge Q uniformly
  distributed throughout its volume.

Solution
r
take a concentric sphere of radius r for
Gaussian surface



r





R

• outside the sphere ( r gt R )

E

• inside the sphere ( r lt R )

R
O
r
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• ACT Figure below shows four spheres, each with
  charge Q uniformly distributed through its
  volume. (a) Rank the spheres according to their
  volume charge density, greatest first. The figure
  also shows a point P for each sphere, all at the
  same distance from the center of the sphere. (b)
  Rank the spheres according to the magnitude of
  the electric field they produce at point P,
  greatest first.

a, b, c, d
a and b tie, c, d
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Proof
Replace the sphere-with-cavity with two uniform
spheres of equal positive and negative charge
densities.
Consider a point P in the cavity
The e-Field produced by r sphere at p
r
-r
The e-Field produced by -r sphere at p
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• ACT A thin nonconducting uniformly charged
  spherical shell of radius R has a total charge of
  Q . A small circular plug is removed from the
  surface. (a) What is the magnitude and direction
  of the electric field at the center of the hole.
  (b) The plug is put back in the hole. Using the
  result of part (a), calculate the force acting on
  the plug.

Solution
(a) Replace the spherical shell-with-hole with a
perfect spherical shell and a circle with equal
negative surface charge densities.
R
Q
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(b)
R
Q
The electrostatic pressure( force per unit
area) tending to expand the sphere
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• Example Find the electric field outside an
  infinite, nonconducting plane of charge with
  uniform charge density s

Solution
To take advantage of the
symmetry properties, we use as our Gaussian
surface a cylinder with its axis perpendicular to
the sheet of charge, with ends of area S. The
charged sheet passes through the middle of the
cylinders length, so the cylinder ends are
equidistant from the sheet.
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• ACT Nowhere inside (between the planes) an odd
  number of parallel planes of the same nonzero
  charge density is the electric field zero.

TRUE
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• Act Consider two infinite parallel charged
  plates
• with surface charge density of s1and
  s2respectively.
• What is the electric field in the three
  regions.

III
I
II
x
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• Example Find the electric field of a slab of
  nonconducting material forming an infinite plane
  .It has thickness d and carries a uniform
  positive charge density r

The Gaussian surface is shown in
the figure
Solution
Outside the slab
S
Inside the slab
S
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Find the distribution of charge according to the
electric field (Differential form of Gauss
Law )
Gauss law
We now apply Gauss' divergence theorem
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As this is true for any closed surface the result
requires that at every point in space
This is Gauss' law for electric fields in
differential form. It is the first of Maxwell's
equations.
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The formulas for the divergence of a vector
Cartesian coordinates
Spherical coordinates
Cylindrical coordinates
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• Example The electric field of an atom is

Find the distribution of charge of this atom
Solution
Atom is neutral
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Alternative Solution
The charge density should be r (r).
We take a concentric spheres for Gaussian surface
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Summary of Chapter 17

• Electric flux due to field intersecting a
  surface S

• Gauss law relates flux through a closed surface
  to charge enclosed

• Can use Gauss law to find electric field in
  situations with a high degree of symmetry