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Gauss’ Law
Electric Field Lines / Electric Field
Vectors
Electric Flux
Gauss’ Law
Use of Gauss’ Law and Gaussian
Surfaces
Electrostatic Equilibrium
#Conductors
#Non Conductors
Electric Field Vectors and Lines
Electric Force and
TheAcceleration
electric force is
 given by

F = qE
The acceleration by
q
a= E
m
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.
Definition
 = EA
  = E  A
N
N
m = m
C
C
2
2
Flux Picture
Flux Picture
Define Area Vector
Area Vector
A = An
Definition of symbols
A = Area (always positive
number)
n = Unit vector. Its direction
corresponds to the orientation of
the area
Forms a right handed system
Dot
product
Definition
of
Electric Flux
Flux
Number of Field lines
through Perpendicular
surface
 = AE
= AECos 
through
a
FluxFlux
through
closed surface

closed
surface from an
 external source
is zero
Closed Surface Picture
Surface Area Element
 =  E  dA
Flux through Curved
Surface
surface
E  d A= EdA Cos  
A = dA
surface
Spherical Surface
Gaussian Surface defined as
Surface
Gaussian Surface
# surrounding charge
# where magnitude of Electric Field is constant
or zero
# the direction of Electric Field is
same as the Area vectors of the surface
# thus same symmetry as charge
distribution
Flux through any closed
surface surrounding a
charge is the same
 Gauss'
=  ELaw
 dI A
Gaussian surface
=

E  r  dA
Gaussian surface
= E r 
 dA
Gaussian surface
= E r 4 p r
2
Using Coulombs
Law
for a point
Gauss'
Law
III charge
Q
= k 4p r
r
2
2
= 4p kQ
Q
=
e
0
Gauss’
Law
Gauss' Law II
=

E  dA
Gaussian surface
Q
=
e
0
Use of Gauss' Law
To Find Electric Field of
Given Charge Distribution
Surface + Charge
Field
Closed Surfaces
Coulombs Law from
Gauss' Law I
Gauss' Law
Coulombs'
Law
Q
Coulombs
Law
from
 = =  Er   dA
Gauss'
Law
I
e 0 sphere
of
radius r
=




E
r
dA
=
E
r
4
p
r

sphere of
radius r
Q
Q
 E r  =
=
k
2
2
4pr e 0
r
2
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
At
Electrostatic
At Electrostatic
Equilibrium
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
Electric Field just outside
conductor
# is perpendicular to conductors
surface
# has magnitude =
surface density / permitivity
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)
Why
is the
charge
on the surface?
Why
is the
Charge
on the
Surface?
Gaussian
Surface 1
E=0
Q
Gaussian
Surface 2
Use Gauss’
Theorem
Answer
Charge must be
between surface 1 and
surface 2
(why?)
Therefore must be on the
surface of object
What is Electric Field
on surface?
Answer
2
3
1
E
•Zero Flux
through 2
•Zero Flux
through 3
•Only Flux
through 1
Q
e
inside
cylinder
= Answer

E
 d2 A
cylinder
0
= E r 
 dA
disk 1
= E r  A
 E r  =
Q
inside
cylinder
Ae
0
s r 
=
e
0
Direction
Answerof3 Field?
Must be orthogonal to
surface
otherwise there will be net
motion on surface
Graph of Field v. Position
magnitude
of electric
field
radius of
conductor
distance from
center
of charged
conductor
Conductor in Electric Field
 In external field conductor
 becomes polarized
 Induced Electric Field
from the surface must
cancel external Electric
Field inside conductor
Induced Field
E
E
+dq
-dq E
induced
-dq
-dq
+dq
+dq
E
E
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
Electric Field inside
Electric Fields
Cavities
inside Cavities of
Conductors Gaussian
Surface
Cavity
Analysis 1
Total charge within
Gaussian surface must be
zero
Otherwise there is an
Electric Field inside the
conductor around the cavity
Analysis 2
 Therefore NO charge on
surface of cavity
 Can enlarge cavity so
that conductor is hollow
 Faraday cage
Thought Question
Radio
reception
over some
bridges
Electric Field inside
Nonconductor
Electric
Field inside non
conductor?
Graph of Field v. Position
magnitude
of electric
field
radius of non
conductor
distance from center
of charged non
conductor
Field above surface of
charged conductor
Field Above Conductor
Q
s
E = Ae = e
0
0
Does not depend on thickness of conductor
Field above surface of
Field
Above
Very Thin
charged
nonconductor
Nonconductor
 = 2 EA
Q
s
E=
=
2 Ae 0 2 e 0