Gauss

1
Gauss Law

• Electric Field Lines / Electric Field Vectors
• Electric Flux
• Gauss Law
• Use of Gauss Law and Gaussian Surfaces
• Electrostatic Equilibrium
• Conductors
• Non Conductors

2
Electric Field Vectors and Lines
3
Electric Force and Acceleration

• The electric force is
• given by
• F qE
• The acceleration by

q

a
E
m
4
Electric Flux

• A measure of the amount of electric field
  through an area perpendicular to the field
• The number of field lines through the area.

5
Definition
6
Flux Picture
7
Flux Picture
8
Area Vector
Define Area Vector
9
Definition of symbols
A Area (always positive number) n Unit
vector. Its direction is given by the orientation
of the area Forms a right handed system
10
Dot product Definition of Flux

• Electric Flux
• Number of Field lines
• through Perpendicular surface

11
Flux through closed surface

• Flux through a
• closed
• surface from an
• external source

is zero

12
Closed Surface Picture
13
Surface Area Element
14
Flux through Curved Surface
ò
F


E
A
d
surface
(
)


q
E
A
d
EdA
Cos
ò

A
dA
surface
15
Spherical Surface
16
Gaussian Surface

• Gaussian Surface defined as
• Surface
• surrounding charge
• where magnitude of Electric Field is constant
  on the surface as much as possible
• the direction of Electric Field is same as the
  Area vectors of the surface
• thus same symmetry as charge distribution

17
Flux through any closed surface surrounding a
charge is the same
18
Gauss' Law I
ò
F


E
A
d
Gaussian surface
ò
(
)

E
r
dA
Gaussian surface
ò

dA
Gaussian surface
(
)

r
p
E
r
4
2
19
Gauss' Law III
Using Coulombs Law for a point charge
Q

p
r
k
4
2
r
2
Q


p
4
kQ
e
0
20
Gauss' Law II
Gauss Law
21
Use of Gauss' Law
To Find Electric Field of Given Charge
Distribution

• Surface Charge

Field
22
Closed Surfaces
23
Coulombs Law from Gauss' Law I
Gauss' Law
Coulombs' Law
24
Coulombs Law from Gauss' Law I
25
Electrostatic Equilibrium
Electrostatic Equilibrium
for objects in an external Electric Field

• Conductors
• No net motion of charge within conductor
• Non Conductors
• in non conductors there is no movement of charge
• therefore always have equilibrium

26
At Electrostatic Equilibrium
At Electrostatic Equilibrium

• Electric Field is zero within conductor
• Any excess charge on an isolated conductor must
  be on its surface
• accumulates at points where radius of curvature
  is greatest

27
Electric Field just outside conductor

• is perpendicular to conductors surface
• has magnitude
• surface density / permitivity

28
Electric Field inside conductor

• Net Electric Field is zero inside,
• otherwise Net Electric Force on charges
• which then accelerate and move charges (on the
  average)

29
Why is the Charge on the Surface?
Why is the charge on the surface?
Gaussian Surface 1
E0
Q
Gaussian Surface 2
Use Gauss Theorem
30
Answer
Charge must be between surface 1 and surface
2 (why?) Therefore must be on the surface of
object
31

• What is Electric Field on surface?

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Answer

• Zero Flux through 2
• Zero Flux through 3
• Only Flux through 1

2
3
1
E
33
Answer 2
Q
inside
ò


E
A
d
cylinder
e
cylinder
0
ò
(
)

E
r
dA
disk
1
(
)

E
r
A
Q
(
)
s
r
inside
(
)
\


E
r
cylinder
e
e
A
0
0
34
Answer 3
Direction of Field?

• Must be orthogonal to surface
• otherwise there will be net motion on surface

35
Graph of Field v. Position
magnitude of electric field
radius of conductor
distance from center of charged conductor
36
Conductor in Electric Field

• In external field conductor
• becomes polarized
• Induced Electric Field on surface must cancel
  external Electric Field inside conductor

37
Charged Conductor

• If the conductor has a net charge
• then it is also a source of an Electric Field
• that combines with the external field
• producing a resultant field
• external to the conductor

38
Electric Field inside Cavities
Electric Fields inside Cavities of Conductors
Gaussian Surface
Cavity
39
Analysis 1

• Total charge within Gaussian surface must be
  zero
• Otherwise there is an Electric Field inside
  conductor

40
Analysis 2

• Therefore NO charge on surface of cavity
• Can enlarge cavity so that conductor is hollow
• Faraday cage

41
Thought Question
Radio reception over some bridges
42
Electric Field inside Nonconductor

• Electric Field inside non conductor?

43
Graph of Field v. Position
magnitude of electric field
radius of non conductor
distance from center of charged non conductor
44
Field Above Conductor
Field above surface of charged conductor
Does not depend on thickness of conductor
45
Field Above Very Thin Nonconductor
Field above surface of very thin charged
nonconductor