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Title: Lesson 7 Gauss’s Law and Electric Fields

1
Lesson 7Gausss Law and Electric Fields
2
Class 18
• Today, we will
• learn the definition of a Gaussian surface
• learn how to count the net number of field lines
passing into a Gaussian surface
• learn Gausss Law of Electricity
• learn about volume, surface, and linear charge
density
• learn Gausss Law of Magnetism
• show by Gausss law and symmetry that the
electric field inside a hollow sphere is zero

3
Section 1Visualizing Gausss Law
4
Gaussian Surface
• A Gaussian surface is
• any closed surface
• surface that encloses a volume
• Gaussian surfaces include
• balloons
• boxes
• tin cans
• Gaussian surfaces do not include
• sheets of paper
• loops

5
Counting Field Lines
• To count field lines passing through Gaussian
surfaces
• Count 1 for every line that passes out of the
surface.
• Count -1 for every line that comes into the
surface.

1
-1
6
Electric Field Lines
We have a 2 charge and a -2 charge.
7
Electric Field Lines
What is the net number of field lines passing
through the Gaussian surface?
8
Electric Field Lines
What is the net number of field lines passing
through the Gaussian surface?
8
9
Electric Field Lines
What is the net number of field lines passing
through the Gaussian surface?
10
Electric Field Lines
What is the net number of field lines passing
through the Gaussian surface?
8
11
Electric Field Lines
What is the net number of field lines passing
through the Gaussian surface?
12
Electric Field Lines
What is the net number of field lines passing
through the Gaussian surface?
-8
13
Electric Field Lines
What is the net number of field lines passing
through the Gaussian surface?
14
Electric Field Lines
What is the net number of field lines passing
through the Gaussian surface?
-8
15
Electric Field Lines
What is the net number of field lines passing
through the Gaussian surface?
16
Electric Field Lines
What is the net number of field lines passing
through the Gaussian surface?
0
17
Electric Field Lines
What is the net number of field lines passing
through the Gaussian surface?
18
Electric Field Lines
What is the net number of field lines passing
through the Gaussian surface?
0
19
Electric Field Lines
From the field lines coming out of this box, what
can you tell about whats inside?
20
Electric Field Lines
The net charge inside must be 1 (if we draw 4
lines per unit of charge).
21
Gausss Law of Electricity
The net number of electric field lines passing
through a Gaussian surface is proportional to the
charge enclosed within the Gaussian surface.
22
Section 2Charge Density
23
Charge Density
Charge Volume Charge Area Charge Length
Volume ? Surface s Linear ?
24
Charge Density
In general, charge density can vary with
position. In this case, we can more carefully
define density in terms of the charge in a very
small volume at each point in space. The density
then looks like a derivative
You need to understand what we mean by this
equation, but we wont usually need to think of
density as a derivative.
25
Section 3Gausss Law of Magnetism
26
Gausss Law and Magnetic Field Lines
If magnetic field lines came out from point
sources like electric field lines, then we would
have a law that said The net number of magnetic
field lines passing through a Gaussian surface is
proportional to the magnetic charge inside.
N
27
Gausss Law and Magnetic Field Lines
But we have never found a magnetic monopole. -
The thread model suggests that there is no reason
we should expect to find a magnetic monopole as
the magnetic field as we know it is only the
result of moving electrical charges. - The field
line model suggests that theres no reason we
shouldnt find a magnetic monopole as the
electric and magnetic fields are both equally
fundamental.
28
Gausss Law and Magnetic Field Lines
What characteristic would a magnetic monopole
field have?
29
Gausss Law and Magnetic Field Lines
What characteristic would a magnetic monopole
field have?
30
Gausss Law and Magnetic Field Lines
All known magnetic fields have field lines that
form closed loops. So what can we conclude
about the number of lines passing through a
Gaussian surface?
31
Gausss Law of Magnetism
The net number of magnetic field lines passing
through any Gaussian surface is zero.
32
Section 4Gausss Law and Spherical Symmetry
33
Spherically Symmetric Charge Distribution
The charge density, ?, can vary with r
only. Below, we assume that the charge density is
greatest near the center of a sphere.
34
Spherically Symmetric Charge Distribution
Outside the distribution, the field lines will go
radially outward and will be uniformly
distributed.
35
Spherically Symmetric Charge Distribution
The field is the same as if all the charge were
located at the center of the sphere!
36
Inside a Hollow Sphere
Now consider a hollow sphere of inside radius r
with a spherically symmetric charge distribution.
37
Inside a Hollow Sphere
There will be electric field lines outside the
sphere and within the charged region. The field
lines will point radially outward because of
38
Inside a Hollow Sphere
Draw a Gaussian surface inside the sphere. What
is the net number of electric field lines that
pass through the Gaussian surface?
39
Inside a Hollow Sphere
The total number of electric field lines from the
hollow sphere that pass through the Gaussian
surface inside the sphere is zero because there
is no charge inside.
40
How can we get zero net field lines?
1. We could have some lines come in and go out
again
but this violates symmetry!
41
How can we get zero net field lines?
2. We could have some radial lines come in and
but this violates symmetry, too!
42
How can we get zero net field lines?
3. Or we could just have no electric field at all
inside the hollow sphere.
43
How can we get zero net field lines?
3. Or we could just have no electric field at all
inside the hollow sphere.
This is the only way it can be done!
44
The Electric Field inside a Hollow Sphere
Conclusion the static electric field inside a
hollow charged sphere with a spherically
symmetric charge distribution must be zero.
45
Class 19
• Today, we will
• learn how to use Gausss law and symmetry to
find the electric field inside a spherical charge
distribution
• show that all the static charge on a conductor
must reside on its outside surface
• learn why cars are safe in lightning but cows
arent

46
Spherically Symmetric Charge Distribution
Electric field lines do not start or end outside
charge distributions, but that can start or end
inside charge distributions.
47
Spherically Symmetric Charge Distribution
What is the electric field inside a spherically
symmetric charge distribution?
48
Spherically Symmetric Charge Distribution
Inside the distribution, it is difficult to draw
field lines, as some field lines die out as we
move inward. We need to draw many, many field
lines to keep the distribution uniform as we move
inward.
49
Spherically Symmetric Charge Distribution
But we do know that if we drew enough lines, the
distribution would be radial and uniform in every
direction, even inside the sphere.
50
Spherically Symmetric Charge Distribution
Lets draw a spherical Gaussian surface at radius
r.
r
51
Spherically Symmetric Charge Distribution
Now we split the sphere into two parts the part
outside the Gaussian surface and the part inside
the Gaussian surface.
r
r
52
Spherically Symmetric Charge Distribution
The total electric field at r will be the sum of
the electric fields from the two parts of the
sphere.
r
r
53
Spherically Symmetric Charge Distribution
Since the electric field at r from the hollow
sphere is zero, the total electric field at r is
that of the core, the part of the sphere within
the Gaussian surface.
r
r
54
Spherically Symmetric Charge Distribution
Outside the core, the electric field is the same
as that of a point charge that has the same
charge as the total charge inside the Gaussian
surface.
r
55
Spherically Symmetric Charge Distribution
Inside a spherically symmetric charge
distribution, the static electric field is
r
56
Example Uniform Distribution
A uniformly charged sphere of radius R has a
total charge Q. What is the electric field at r lt
R ?
57
Example Uniform Distribution
A uniformly charged sphere of radius R has a
total charge Q. What is the electric field at r lt
R ?
Since the charge density is uniform
r
58
Example Uniform Distribution
59
Section 5Gausss Law and Conductors
60
Gausss Law and Conductors
Take an arbitrarily shaped conductor with charges
on the outside.

61
Gausss Law and Conductors
The static electric field inside the conductor
must be zero. Draw a Gaussian surface inside
the conductor.

62
Gausss Law and Conductors
No field lines go through the Gaussian surface
because E0. Hence, the total enclosed charge
must be zero.

63
Gausss Law and Conductors
The same must be true of all Gaussian surfaces
inside the conductor.

64
Surface Charge and Conductors
What if there are no charges on the outside and
the net charge of the conductor is zero? -- The
volume charge density inside the conductor must
be zero and the surface charge density on the
conductor must also be zero.
65
Surface Charge and Conductors
What if there are no charges on the outside and
there is net charge on the surface of a conductor?

66
Surface Charge and Conductors
The charge distributes itself so the field inside
is zero and the surface is at the same electric
potential everywhere.

67
Example Surface Charge on a Spherical Conductor
A spherical conductor of radius R has a voltage
V. What is the total charge? What is surface
charge density?
68
Example Surface Charge on a Spherical Conductor
A spherical conductor of radius R has a voltage
V. What is the total charge? What is surface
charge density?
On the outside, the potential is that of a point
charge.
On the surface, the voltage is V(R).
69
Take Two Conducting Sphereswith the Same Voltage
The smaller sphere has a larger charge density.

70
Now Connect the Two Spheres
The charge density is greater near the pointy
end. The electric field is also greater near the
pointy end.

71
Edges on Conductors
Charge moves to sharp points on
conductors. Electric field is large near sharp
points. Smooth, gently curved surfaces are the
best for holding static charge. Lightning rods
are pointed.
72
A Hollow Conductor
What if theres a hole in the conductor?

73
A Hollow Conductor
Draw a Gaussian surface around the hole.

74
A Hollow Conductor
There is no net charge inside the Gaussian
surface.

75
A Hollow Conductor
Is there surface charge on the surface of the
hole?

76
A Hollow Conductor
There is no field surrounding the charge to hold
the charges fixed, so the charges migrate and
cancel each other out.

77
Charge on a Conductor
Static charge moves to the outside surface of a
conductor.

78
Lightning and Cars
Why is a car a safe place to be when lightning
strikes?
Note Any car will do it doesnt need to be a
Cord.
79
Lightning and Cars
Is it the insulating tires?
80
Lightning and Cars
Is it the insulating tires?
If lightning can travel 1000 ft through the air
to get to your car, it can go another few inches
to go from your car to the ground!
81
Lightning and Cars
A car is essentially a hollow conductor. Charge
goes to the outside. The electric field inside is
small.
82
Lightning and Cars
A car is essentially a hollow conductor. Charge
goes to the outside. The electric field inside is
small.
83
How should a cow stand to avoid injury when
lightning strikes nearby?
84
Physicists Cow
Cow
Earth
d
I
85
Physicists Cow
When d is bigger, the resistance along the ground
between the cows feet is bigger, the voltage
across the cow is bigger, and the current flowing
through the cow is bigger.
Cow
Earth
d
I
86
How should a cow stand to avoid injury when
lightning strikes nearby?
So the cow should keep her feet close together!
87
Class 20
• Today, we will
• learn how integrate over linear, surface, and
volume charge densities to find the total charge
on an object
• learn that flux is the mathematical quantity
that tells us how many field lines pass through a
surface

88
Section 6Integration
89
Gausss Law of Electricity
• The net number of electric field lines passing
through a Gaussian surface is proportional to the
enclosed charge.
• But, how do we find the enclosed charge?

90
Charge and Density
• is valid when?

91
Charge and Density
• when ? is uniform.
• If ? is not uniform over the whole volume, we
find some small volume dV where it is uniform.
Then
• If we add up all the little bits of dq, we get
the entire charge, q.

92
Integration
The best way to review integration is to work
through some practical integration problems.
93
Integration
The best way to review integration is to work
through some practical integration problems. Our
goal is to turn two- and three- dimensional
integrals into one-dimensional integrals.
94
Fundamental Rule of Integration
Identify the spatial variables on which the
integrand depends. You must slice the volume
(length or surface) into slices on which these
variables are constant.
95
Fundamental Rule of Integration
When integrating densities to find the total
charge, the density must be a constant on the
slice or we cannot write
96
Fundamental Rule of IntegrationExamples
Square in x-y plane Cylinder Sphere
97
Fundamental Rule of IntegrationExamples
Consider a very thin slice.
Is constant on this slice?
98
Fundamental Rule of IntegrationExamples
Consider a very thin slice.
Is constant on this slice?
99
Fundamental Rule of IntegrationExamples
Square in x-y plane Cylinder Sphere
100
Fundamental Rule of IntegrationExamples
Square in x-y plane Cylinder Sphere
101
Fundamental Rule of IntegrationExamples
Square in x-y plane Cylinder Sphere
102
Fundamental Rule of IntegrationExamples
Square in x-y plane Cylinder Sphere
103
Fundamental Rule of IntegrationExamples
Square in x-y plane Cylinder Sphere
104
Rules for Areas and Volumes of SlicesMemorize
These!!!
Square in x-y plane Disk Cylinder Sphere
105
Rules for Areas and Volumes of SlicesMemorize
These!!!
Square in x-y plane Disk Cylinder Sphere
106
Rules for Areas and Volumes of SlicesMemorize
These!!!
Square in x-y plane Disk Cylinder Sphere
107
Rules for Areas and Volumes of SlicesMemorize
These!!!
Square in x-y plane Disk Cylinder Sphere
108
Rules for Areas and Volumes of SlicesMemorize
These!!!
Square in x-y plane Disk Cylinder Sphere
109
Rules for Areas and Volumes of SlicesMemorize
These!!!
Square in x-y plane Disk Cylinder Sphere
110
Lets Do Some Integrals
111
Charge on a Cylinder
A cylinder of length L and radius R has a
charge density where is a
constant and z is the distance from one end of
the cylinder. Find the total charge on the
cylinder. How do you slice the cylinder? What is
the volume of each slice?
112
Charge on a Cylinder
113
Charge on a Sphere
A sphere of radius R has a charge density
where is a constant. Find the total
charge on the sphere. How do you slice the
sphere? What is the volume of each slice?
114
Charge on a Sphere
115
Section 7Gausss Law and Flux
116
Field Lines and Electric Field
• This is valid when
• .A is the area of a section of a perpendicular
surface.
• The electric field is constant on A.

117
Field Lines and Electric Field
• This is valid when
• A is the area of a section of a perpendicular
surface.
• The electric field is constant on A.

-- But E is a constant on A only in a few cases
of high symmetry spheres, cylinders, and planes.
118
Electric Flux
Gausss Law states that
EA is called the electric flux. We write it as
or just .
119
Electric Flux
Gausss Law states that
EA is called the electric flux. We write it as
or just . Flux is a mathematical
expression for number of field lines passing
through a surface!
120
Electric Flux and a Point Charge
Lets calculate the electric flux from a point
charge passing through a sphere of radius r.
121
Electric Flux and a Point Charge
Gausss law says this is proportional to the
charge enclosed in the sphere!
122
Electric Flux and Gausss Law
This means that we can write Gausss Law of
Electricity as
123
For our purposes, we will (almost) always
calculate flux through a section of perpendicular
surface where the field is constant. So we will
evaluate flux simply as
124
But we do need to find a more general expression
for flux so youll know what it really means
125
An Area Vector
• We wish to define a vector area. To do this
• we need a flat surface.
• the direction is perpendicular to the plane of
the area.
• (Dont worry about the fact there are two
choices of direction that are both perpendicular
to the area up and down in the figure below.)
• 3) the magnitude of vector is the area.

126
First, Lets consider the flux passing through a
frame oriented perpendicular to the field.
127
If we tip the frame by an angle ?, the angle
between the field and the normal to the frame,
there are fewer field lines passing through the
frame.
128
Or, using the vector area of the loop, we may
write
129
only holds when the frame is flat and
the field is uniform. What if the surface
(frame) isnt flat, or the electric field isnt
uniform?
130
Area Vectors on a Gaussian Surface
1) We must take a small region of the surface dA
that is essentially flat. 2) We choose a unit
vector perpendicular to the plane of dA going in
an outward direction.
131
The flux through this small region is
132
To find the total flux, we simply add up all the
contributions from every little piece of the
surface.
Recall that the normal to each small area is
taken to be in the outward direction.
133
Thus, the most general equation for flux through
a surface is
If we take the flux through a Gaussian surface,
we usually write the integral sign with a circle
through it to emphasize the fact that the
integral is over a closed surface
134
Class 21
• Today, we will
• learn how to use Gausss law to find the
electric fields in cases of high symmetry
• insdide and outside spheres
• inside and outside cylinders
• outside planes

135
Section 7Gausss Laws in Integral Form
136
Gausss Law of ElectricityIntegral Form
The number of electric field lines passing
through a Gaussian surface is proportional to the
charge enclosed by the surface.
We can make this simple expression look much
more impressive by replacing the flux and
enclosed charge with integrals
137
Gausss Law of MagnetismIntegral Form
The number of magnetic field lines passing
through a Gaussian surface is zero
With the integral for magnetic flux, this is
138
Gausss Law of ElectricityTee-Shirt Form
This can be written in many different ways. A
popular form seen on many tee-shirts is
139
Gausss Law of ElectricityTee-Shirt Form
This can be written in many different ways. A
popular form seen on many tee-shirts is
This is a good form of Gausss law to use if you
want to impress someone with how smart you are.
140
Gausss Law of ElectricityPractical Form
This is the form of Gausss law you will use when
you actually work problems.
141
Gausss Law of ElectricityPractical Form
Now lets think about what this equation really
means!
142
Gausss Law of ElectricityPractical Form
Electric field on Gaussian surface -- Must be the
same everywhere on the surface!
143
Gausss Law of ElectricityPractical Form
Electric field on Gaussian surface -- Must be the
same everywhere on the surface!
Area of the entire Gaussian surface Must be a
perpendicular surface (an element of a field
contour)!
144
Gausss Law of ElectricityPractical Form
Integral of the charge density over the
volume enclosed by the Gaussian surface!
Electric field on Gaussian surface -- Must be the
same everywhere on the surface!
Area of the entire Gaussian surface Must be a
perpendicular surface (an element of a field
contour)!
145
Section 9Using Gausss Law to Find Fields
146
Problem 1 Spherical Charge DistributionOutside
• Basic Plan
• Choose a spherical Gaussian surface of radius r
outside the charge distribution.
• 2)
• 3) Integrate the charge over the entire charge
distribution.

147
Problem 1 Spherical Charge DistributionOutside
148
Problem 1 Spherical Charge DistributionOutside
149
Problem 2 Spherical Charge DistributionInside
• Basic Plan
• Choose a spherical Gaussian surface of radius r
inside the charge distribution.
• 2)
• 3) Integrate the charge over the inside of the
Gaussian surface only.

150
Problem 2 Spherical Charge DistributionInside
151
Problem 2 Spherical Charge DistributionInside
152
Problem 3 Cylindrical Charge DistributionOutside
• Basic Plan
• Choose a cylindrical Gaussian surface of radius r
and length L outside the charge distribution.
• 2)
• 3) Integrate the charge over the entire charge
distribution.

153
Problem 3 Cylindrical Charge DistributionOutside
• Basic Plan
• 4) Note that there are no field lines coming out
the ends of the cylinder, so there is no flux
through the ends!

154
Problem 3 Cylindrical Charge DistributionOutside
155
Problem 3 Cylindrical Charge DistributionOutside
156
Problem 4 Cylindrical Charge DistributionInside
• Basic Plan
• Choose a cylindrical Gaussian surface of radius r
and length L inside the charge distribution.
• 2)
• 3) Integrate the charge over the inside of the
Gaussian surface only.

157
Problem 4 Cylindrical Charge DistributionInside
158
Problem 4 Cylindrical Charge DistributionInside
159
Infinite Sheets of Charge
• Basic Plan
• Choose a box with faces parallel to the plane as
a Gaussian surface. Let A be the area of each
face.
• 2) Find the charge inside the box. No integration
is needed.

160
Problem 5 Infinite Sheet of Charge(Insulator
with s given)
Note there is flux out both sides of the box!
161
Problem 6 Infinite Sheet of Charge(Conductor
with s on each surface)
Note there is flux out both sides of the box, and
the total charge density is 2s!
162
Problem 6 A second way
Now there is flux out only one side of the box,
but the total charge density inside is just s!
163
Problem 7 A Capacitor
The area of the plate is and the area of the
box is .
There is flux out only one side of the box!
164
A Word to the Wise!
If you can do these seven examples, you can do
every Gausss law problem I can give you! Know
them well!