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Physics II:
Electricity & Magnetism
Chapter 22
Friday
(Day 1)
Add 21.9 Figure: E-field in a
conductor.
Fix derivations for uniform charge
density (dr is wrong, etc)
Add EM Field Activities
Warm-Up
Fri, Feb 20
 Write down the steps you give a visually-disabled individual to explain
to how to fill a coffee cup with water.
 Place your homework on my desk:
 “Foundational Mathematics’ Skills of Physics” Packet (Page 9)
 For future assignments - check online at www.plutonium-239.com
Essential Question(s)
 WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE
NECESSARY IN PHYSICS II?
 HOW DO WE DESCRIBE THE NATURE OF ELECTROSTATICS AND
APPLY IT TO VARIOUS SITUATIONS?
 How do we describe and apply the concept of electric field?
 How do we describe and apply the electric field created by uniformly
charged objects?
 How do we describe and apply the relationship between the electric
field and electric flux?
 How do we describe and apply Gauss’s Law?
 How do we describe and apply the nature of electric fields in and
around conductors?
 How do we describe and apply the concept of induced charge and
electrostatic shielding?
Vocabulary
Flux
Electric Flux
Flux Lines
Surface Area
Perpendicular
Orthogonal
Normal
Dot Product
Gauss’s Law
Charge Density
Foundational Mathematics
Skills in Physics Timeline
Day
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WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Agenda
 Review “Foundational Mathematics’ Skills of Physics”
Packet (Page 9) with answer guide.
 Review electric fields, electrons, and conductors
 Discuss the following:
 Electric Flux
 Work Day:
 Chapter 21 Web Assign Problems
 Uniform Charge Distribution Derivations
 Chapter 22 Web Assign Problems 22.1 - 22.4
A Review of Electric Fields and
Conductors
The electric
field is
perpendicular
to the surface
of a
conductor –
again, if it
were not,
charges
would move.
Chapter 22
Gauss’s Law
Main Points of Chapter 22
• Electric flux
• Gauss’ law
• Using Gauss’ law to determine electric fields
• Conductors and electric fields
• Testing Gauss’ and Coulomb’s laws
Section 22.1
 Given a diagram where the electric field is represented by
flux lines, how do we
 determine the direction of the field at a given point?
 identify the locations where the field is strong and where it is
weak?
 identify where the positive and negative charges must be
present?
 How do we use the relationship between the electric field
and electric flux to calculate the flux of a uniform electric
field E through an arbitrary surface?
Section 22.1
 How do we use the relationship between the electric
field and electric flux to calculate the flux of a uniform
electric field E through a curved surface when E is
uniform in magnitude and perpendicular to the
surface?
 How do we use the relationship between the electric
field and electric flux to calculate the flux of a uniform
electric field E through a rectangle when E is
perpendicular to the rectangle and a function of one
coordinate only?
 How do we state and apply the relationship of between
electric flux and lines of force?
22.1 Electric Flux
A
Electric flux:
E  E A cos
E  E A  E A
Electric flux through an
area is proportional to
the total number of field
lines crossing the area.
A
22.1 Electric Flux
Electric flux:
A
E  E A cos
E  E A
When  = 0°, the flux (field lines) passing through the area
is maximized and when  = 90°, the flux (field lines) is
zero. Therefore, for mathematical simplicity, it is important
to note that the area A of a surface will represented by a
vector A whose magnitude is A but whose direction is
perpendicular to it surface.
22.1 Electric Flux
Flux through a closed surface:
EM Field 6
Using EM Field, determine the flux
through a closed surface (represented
by a 2-D cross-section) by counting the
number of field lines entering the area
and subtracting it from the number of
field lines exiting the area.
This is because the area vector A for a
closed surface is always directed outward.
i.e. cos (0°) = 1 and cos (180°) = -1
EM Field 6
How can we draw a surface that has
more field lines entering or exiting the
closed surface?
EM Field 6
The following are examples of the Flux
calculations using EM Field 6.
Besides its relation to the number of
field lines entering and exiting the
closed surface, how does the flux
calculation relate to charge?
Summary
 How does flux relate to the charge enclosed by a closed surface?
 HW (Place in your agenda):
 “Foundational Mathematics’ Skills of Physics” Packet (Page 10)
 The final copies for all Uniformly Charged Objects: 1 derivation with
reasons for each mathematical step and 3 additional derivations: (*Refer
to rubric)
 Web Assign Final Copies: Chapter 21
 Web Assign Problems: Chapter 22.1 & 22.2
How do we apply integration and the Principle of Superposition to uniformly charged objects?
How do we identify and apply the fields of highly symmetric charge distributions?
How do we describe and apply the electric field created by uniformly charged objects?
Monday
(Day 2)
Warm-Up
Mon, Feb 23
 For an electric field raining straight down into an imaginary box . . .
 What is the direction of E?
 What is the direction of A for the (1) bottom, (2) top, and (3) side
surfaces?
 What is the angle, , between E and A for the (1) bottom, (2) top, and (3)
side surfaces?
 What is value of cos  for the (1) bottom, (2) top, and (3) side surfaces?
 Place your homework on my desk:
 “Foundational Mathematics’ Skills of Physics” Packet (Page 10) POSTPONED
 The final copies for all Uniformly Charged Objects: 1 derivation with
reasons for each mathematical step and 3 additional derivations: (*Refer
to rubric)
 Web Assign Final Copies: Chapter 21
 For future assignments - check online at www.plutonium-239.com
Application:
Electric Flux
 Warm-up:
A
E
E
 What is the direction of E?
 What is the direction of A
for the (1) bottom, (2) top,
and (3) side surfaces?
 What is the angle, ,
between E and A for the (1)
bottom, (2) top, and (3) side
surfaces?
 What is value of cos  for
the (1) bottom, (2) top, and
(3) side surfaces?
Application:
Electric Flux
Ebottom  Eout Abottom cos
The bottom square flux is
Atop
 E  EAcos 0 = EA
Atop
Etop  Ein Atop cos
The top square flux is
 E  EAcos 180  = EA
Ein
Etop  Ein Aside cos
The side square flux is
Abottom
Aside
E  EAcos 90  = 0
The net flux, net , is
net = top bottom +  sides
net = EA + EA  0 = 0
Eout
Abottom
Essential Question(s)
 WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE
NECESSARY IN PHYSICS II?
 HOW DO WE DESCRIBE THE NATURE OF ELECTROSTATICS AND
APPLY IT TO VARIOUS SITUATIONS?
 How do we describe and apply the concept of electric field?
 How do we describe and apply the electric field created by uniformly
charged objects?
 How do we describe and apply the relationship between the electric
field and electric flux?
 How do we describe and apply Gauss’s Law?
 How do we describe and apply the nature of electric fields in and
around conductors?
 How do we describe and apply the concept of induced charge and
electrostatic shielding?
Vocabulary
Flux
Electric Flux
Flux Lines
Surface Area
Perpendicular
Orthogonal
Normal
Dot Product
Gauss’s Law
Charge Density
Foundational Mathematics
Skills in Physics Timeline
Day
Pg(s)
Day
Pg(s)
Day
Pg(s)
Day
Pg(s)
1
1
2
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2
13
14
7
4
12
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17
8
3
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18
18
9
4
24
†12
9
6
14
19
19
10
5
15
10
7
15
20
20
11
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Agenda
 Review “Foundational Mathematics’ Skills of Physics”
Packet (Page 10) with answer guide. - POSTPONED
 Review electric flux
 Discuss the following:
 Gauss’s Law
 Applications of Gauss’s Law
 Spherical Conductor, Point Charge, & Line Charges
 Work Time: Web Assign Problems 22.1 - 22.5
Section 22.2
 How do we state Gauss’s Law in integral form and apply it
qualitatively to relate electric flux and electric charge for a
specified surface?
22-2 What Does Gauss’ Law Do?
Imagine field lines emanating from a
positive charge.
Now imagine a sphere of tissue paper
around the charge. How many field
lines penetrate the tissue? It doesn’t
really matter how many we draw in
the first place, as long as we are
consistent; they all go through.
Now imagine the charge being offcenter; all the lines still go through:
22-2 What Does Gauss’ Law Do?
Suppose the tissue is some shape other than
spherical, but still surrounds the charge.
All the field lines still go through:
Now, imagine the paper is crinkled
and overlaps itself; how shall we deal
with the lines that pierce the tissue
three times?
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.
22-2 What Does Gauss’ Law Do?
Now, look at an open (flat) sheet.
If it is perpendicular to the field,
the maximum number of lines
penetrates:
If it is at an angle, fewer lines
penetrate:
22-2 What Does Gauss’ Law Do?
The number of field lines piercing the surface
is proportional to the surface area, the
orientation, and the field strength. If we stop
counting lines and just use the field strength
itself, we can define the electric flux through
an infinitesimal area:
Integrating gives the total flux:
22-2 What Does Gauss’ Law Do?
For a closed surface, we can uniquely define
the direction of the normal to the surface as
pointing outwards and define:
Then:
Note the circle on the integral sign, which means
that the integration is over a closed surface.
22-2 What Does Gauss’ Law Do?
Note that the surface does not have to be
made of real matter – it is a surface that we
can imagine, but that does not have to exist in
reality.
This kind of imaginary surface is called a
Gaussian surface. We can imagine it to be any
shape we want; it is very useful to choose one
that makes the problem you are trying to
solve as easy as possible.
22-2 Gauss’ Law
Without further ado, we can state Gauss’ law:
The electric flux through a closed surface
that encloses a net charge is equal to the
net charge divided by the permittivity of free
space.
22-2 Gauss’ Law
The electric flux through a closed surface
that encloses no net charge is zero.
22.2 Gauss’s Law
Restating Gauss’ Law, the net number of
field lines through the surface is
proportional to the charge enclosed, and
also to the flux:
E 
 EdA 
Qencl
0
Gauss’ Law can easily be used to find the
electric field in situations with a high
degree of symmetry.
E-Field for a Sphere or
Point Charge
Spherical Conductor
At radius r1 r  r0 ,
E
+
+
dAsphere
+
A2
+
r2
r0
+
+
+
r1
+
A1


2

E
4

r

EA
cos


EA
E

dA

1
1

Therefore, Eoutside 
1
Q
4 0 r12
Qencl
0
To calculate the electric field inside of a
conductor, we look at radius r2 r  r0 ,


Qencl
2

EA
cos


EA

E
4

r
E

dA

2
2

Because the enclosed charge inside
of a conductor is zero, E
0
inside
0
22-2 Gauss’ Law for a Point Charge
The electric field through a
Gaussian sphere with a
single point charge at the
center is easily calculated
using:
E   E  dA 


Qencl
 E  E 4 r 2 
 E point charge
0
Qencl
Recall that F  qE.
1 q1q2
 1 q1 
Therefore,
 F
F  q2 
2
4 0 r 2
 4 0 r 
0
1
Q

4 0 r 2
This is Coulomb's Law!
22-2 Gauss’ Law
But the result would be the same if the surface was
not spherical, or if the charge was anywhere inside
it!
Therefore, we can quickly generalize this to any
surface and any charge distribution; all can be
considered as a collection of point or infinitesimal
charges:
Here, Q is the total net charge enclosed by
the surface.
Applications of Gauss’ Law
22-3 Using Gauss’ Law to Determine
Electric Fields
Problem-solving techniques:
1. Make a sketch.
2. Identify any symmetries.
3. Choose a Gaussian surface that matches the
symmetry – that is, the electric field is either
parallel to the surface or constant and
perpendicular to it.
4. The correct choice in 3 should allow you to get
the field outside the integral. Then solve.
Section 22.3
 How do we describe the electric field of a long, uniformly
charged wire?
 How do we describe the electric field of a thin spherical
shell?
 How do we use superposition to determine the electric
fields of coaxial cylinders?
 How do we use superposition to determine the electric
fields of concentric spheres?
Section 22.3
 How do we apply Gauss’s Law, along with symmetry arguments, to
determine the electric field







inside a uniformly charged long cylinder?
outside a uniformly charged long cylinder?
inside a uniformly charged cylindrical shell?
outside a uniformly charged cylindrical shell?
inside a uniformly charged sphere?
outside a uniformly charged sphere?
inside a uniformly charged spherical shell?
 outside a uniformly charged spherical shell?
Section 22.3
 How do we apply Gauss’s Law to determine the charge density on a surface in
terms of the electric field near the surface?
 How do we apply Gauss’s Law to determine the total charge on a surface in
terms of the electric field near the surface?
 How do we prove and apply the relationship between the surface charge
density on a conductor and the electric field strength near its surface?
 How do we qualitatively explain why there can be no electric field in a chargefree region completely surrounded by a single conductor?
 How do we qualitatively explain why the electric field outside of a closed
conducting surface cannot depend on the precise location of charge in the
space enclosed by the conductor?
 What is the significance of why the electric field outside of a closed
conducting surface cannot depend on the precise location of charge in the
space enclosed by the conductor?
E-Field for a Line of
Charge
Long Uniform Line of
Charge
 E  dA  EAtop EAtube  EAbottom

dAtop
Qencl
0
Note: There is no electric field lines
going through the top or bottom of
the cylinder.
E
dAtube
E
dAbottom
Therefore,
E Atop  E Atube  E Abottom  E Atube  E 2 rL 
0
E 2 rL  
0
Qencl
0
 Ewire
1
Q

2 0 rL
Uniformly Charged
Vertical Wire (–∞+∞)

dq
1 x
x
;

dq
dE

dE
cos

 dE
2
x
3
2
2
4 0 h
h 4 0 h
h x y
Q
dE x  dE cos 
Q


 Q  y

y
l
x
dE
dq dq
 dq   dy


dl dy
dE 
dy
y
-
Ex  
Etot
0
dEx 
1
4 0

Qtot
0
1
x
1
dq

h 3  dy 4 0




x
h
x 2 y 2

3  dy

1
4 0
x


1
x
2
 y2

3
dy
2



1 
1 
y
1 
y

1
1





Ex 
x  2 2 2  
 2



2
4

x
2 0 x
4 0
4

x
0
0
 x x  y  
 x  y  
1 y
1 Q
1 Q Note: "r" replaced "x" and "L" replaced "y"
Ex 


2 0 x y
2 0 xy 2 0 rL Do they agree?
1
Summary
 What does the closed integral sign mean in Gauss’ Law?
 HW (Place in your agenda):
 “Foundational Mathematics’ Skills of Physics” Packet (Page 10)
 †“Foundational Mathematics’ Skills of Physics” Packet (Page 11)
 Web Assign 22.1 - 22.5
How do we apply integration and the Principle of Superposition to uniformly charged objects?
How do we identify and apply the fields of highly symmetric charge distributions?
How do we describe and apply the electric field created by uniformly charged objects?
Tuesday
(Day 3)
Warm-Up
Tues, Feb 24
 Calculate the electric flux for a charged object on the following slide.
 Place your homework on my desk:
 “Foundational Mathematics’ Skills of Physics” Packet (Page 10)
 †“Foundational Mathematics’ Skills of Physics” Packet (Page 11) POSTPONED
 Web Assign Problems 22.1 - 22.5
 For future assignments - check online at www.plutonium-239.com
Application: Electric Flux
Two objects, O1 and O2, have charges +1 C and -2.0 C, respectively, and a
third object, O3, is electrically neutral.
(a) What is the electric flux through the surface A1 that encloses all three objects?
(b) What is the electric flux through the surface A2 that encloses the third object
only?
 A1   E  dA 
Qencl
 A2   E  dA 
Qencl
0
0

1.0 C  2.0 C
 1.13 x 105 N  m 2 C
8.85 x 10 12 C2 N  m 2
0 C

2


0
N

m
C
12
2
2
8.85 x 10 C N  m
Essential Question(s)
 WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE
NECESSARY IN PHYSICS II?
 HOW DO WE DESCRIBE THE NATURE OF ELECTROSTATICS AND
APPLY IT TO VARIOUS SITUATIONS?
 How do we describe and apply the concept of electric field?
 How do we describe and apply the electric field created by uniformly
charged objects?
 How do we describe and apply the relationship between the electric
field and electric flux?
 How do we describe and apply Gauss’s Law?
 How do we describe and apply the nature of electric fields in and
around conductors?
 How do we describe and apply the concept of induced charge and
electrostatic shielding?
Vocabulary
Flux
Electric Flux
Flux Lines
Surface Area
Perpendicular
Orthogonal
Normal
Dot Product
Gauss’s Law
Charge Density
Foundational Mathematics
Skills in Physics Timeline
Day
Pg(s)
Day
Pg(s)
Day
Pg(s)
Day
Pg(s)
1
1
2
6
3
11
16
16
21
2
13
14
7
4
12
17
17
8
3
22
23
8
5
13
18
18
9
4
24
†12
9
6
14
19
19
10
5
15
10
7
15
20
20
11
WHAT PRIOR FOUNDATIONAL MATHEMATICS’ SKILLS ARE NECESSARY IN PHYSICS II?
Agenda
 Review “Foundational Mathematics’ Skills of Physics”
Packet (Page 11) with answer guide.
 Discuss the following:
 Applications of Gauss’ Law
 Charged Plate(s), Charged Surfaces, & Charged Spherical Insulators
 Conductors in Electric Fields
 Experimental Basis of Gauss’ and Coulomb’s Law
 Work Time: Web Assign Problems 22.6 - 22.15
Section 22.3
 How do we apply Gauss’s Law, along with symmetry arguments, to
determine the electric field?
 near a large, uniformly charged plane?
Section 22.3
 How do we apply Gauss’s Law to determine the charge density on a surface in
terms of the electric field near the surface?
 How do we apply Gauss’s Law to determine the total charge on a surface in
terms of the electric field near the surface?
 How do we prove and apply the relationship between the surface charge
density on a conductor and the electric field strength near its surface?
 How do we qualitatively explain why there can be no electric field in a chargefree region completely surrounded by a single conductor?
 How do we qualitatively explain why the electric field outside of a closed
conducting surface cannot depend on the precise location of charge in the
space enclosed by the conductor?
 What is the significance of why the electric field outside of a closed
conducting surface cannot depend on the precise location of charge in the
space enclosed by the conductor?
Infinite Plane of Charge
(nonconducting uniform )
 E  dA 
dA
E
A
dAtube
Qencl
0
A

0
Q


Note:


and
Q


A


A
Also Note: The electric field, E,
is in the direction of dA and exists
on both sides of the charged plate.
Therefore,
A net = Atop + Abottom = 2A circle
E
dA

A

E

for each plate
E 2A  
2 0
0



Enet   En  E1  E2


2 0 2 0  0
Infinite Plane of Charge
(nonconducting uniform )
 E  dA 
Qencl
0
A

0
Q


Note:


and
Q


A


A
Also Note: The electric field, E,
is in the direction of dA and exists
on both sides of the charged plate.
Therefore,
dA
E
A
dAtube
E =
 E  dA +  E  dA+ 
top EAtop
tube
0
E  dA
bottom EAbottom
E = 2EAcircle
E
dA
A

2EA 
E
for each side
0
2 0
For two oppositely charged plates:
Enet   En  E1  E2      
2 0 2 0  0
R
Uniformly Charged Disk
(0∞)
z
dq dE  dE cos   dE  1 z dq
dE 
; z
3
2
h
4

h
0
4 0 h
r
2
2
Q
Q
h r z
  2
 Q   r 2
A r

dA
 dA  2 rdr
z
 2 r
dr
dr
dEz  dE cos 
dq
dq
 dq  2 r dr


dA 2 rdr
dE
Qtot z

1
Etot
1
z

dq

Ez   dEz
3

3 2 r dr

0
0
0
4 0
h 2 r dr 4 0
h
1

Ez 
1
4 0
2 z 

0
r
z
2
r
2

3
dr 
2
1


1
1

Ez 
z  2

2
2 0
2 0
z
z  

z 2 r 2
0




1
1
1
1
 z  2 2  
z 2 2 
2 0
2

z r 0
0

 z r 
z  
 z 
2 0
Uniformly Charged
Infinite Plate
dE
-
dEz  dE cos 
z
1 z
dq

dE

dq
dE 
; dEz  dE cos 
3
2
h 4 0 h
4 0 h
Q
Q
 Q   xy
 
A xy

z
2
2
dA  dx dy
h

r

z

dq
dq

 dq   dx dy
 x 2  y2  z 2

dx dy
dA
2
2
y
r x y
Qtot z
1
Etot
dy

dq
x
E

dE
3

z
z
0

dx
4 0
h  dx dy
0
-





1
1
1
z


z
Ez 

dx
dy
3 dx dy


3






4 0
4 0
h
x 2  y2  z 2 2
1
x y z
2
Ez 
1
4 0
 z


2
y2  z 2
dy 
2
1
4 0

2
 2   
 z  2 0
z

No radius?!? What
does that mean?!?
21-7 The Field of a Continuous Distribution
From the electric field due to a nonconducting uniform
sheet of charge, we can calculate what would happen if
we put two oppositely-charged sheets next to each
other:
The
The individual fields:
result:
The superposition:
How do we apply integration and the Principle of Superposition to uniformly charged objects?
How do we identify and apply the fields of highly symmetric charge distributions?
How do we describe and apply the electric field created by uniformly charged objects?
21.8 Field Lines
The electric field between
two closely spaced,
oppositely charged parallel
plates is constant.
E-Field at the Surface of a
Conductor
Surface of a Conductor
 E  dA 
dA
+
+
+
A
+
+
+
E
+
+
+
dAtube
+
+
+
+
+
+
+
Qencl
0

A
0
Q


Note:


and
Q


A


A
Note: Since we are Gaussian surface
is just below the surface of the
conductor, the electric field, E,
is only in the direction of dA and
does not exist within the conductor.
Therefore,
E = EAtop + E Ainside = EA circle
0
EA 

A
E
for the surface
0
0
So why are the E-Fields
different for the plate and
the surface?
Nonconducting: Charge remains
localized (ie. +2Q remains fixed)
Conducting: Like charges repel (ie.
+2Q total -> +Q move to each
side and conduct=1/2 insulator )
Therefore, both E-fields will have
the same magnitude and direction.
EM Field 6
Find the field lines for:
2 Parallel Oppositely Charged Plates
2 Parallel Plates of the Same Charge
Spherical Insulator
(Uniformly distributed charge)
At radius r1 r  r0 ,
+ +
+
+ + + +
+ r2 + r0 +
+ + + +
+
A
+ + 2
r1
dA2




2

E
4

r

EA
cos


EA
E

dA

1
1

E
dA1= dAsphere
Therefore, Eoutside 
At radius r2 r  r0 ,
1
Q
4 0 r12
Qencl
0
Qencl
2

EA
cos


EA

E
4

r
E

dA

2
2

A1
Qtot
Then, E 
Vtot
Note: Since the enclosed charge density is
evenly distributed,   Qtot  dQ  constant
E
Vtot dV
r3
 43  r 3 E 
Qencl  Q  Vencl Q
Q  3 Qtot

encl
tot   4
3

r0
Vtot
 3  r0 E 
Vencl
 E  dA  E 4 r 
2
0
Qencl
0
1 Qtot
r 3 Qtot
Therefore, Einside 
r
 3
3
4 0 r0
r0  0
22-3 Conductors and Electric Fields
In conductors, the charges are
free to move if there is an
external electric field exerting a
force on them.
• Therefore, in equilibrium, there
is no static field inside a
conductor. This also means that
the external electric field is
perpendicular to the conductor at
its surface.
22-3 Conductors and Electric Fields
What if the conductor is charged
– where does the excess charge
go?
• By making a Gaussian surface
very close to, but just under, the
surface of the conductor, we see
that any excess charge must lie
on the outside of the conductor.
22-3 Conductors and Electric Fields
What if there is a cavity inside the conductor, and
that cavity has charges in it?
The field inside the conductor must still be zero, so
charges will be induced on the inner surface of the
cavity and the outer surface of the conductor:
22-3 Conductors and Electric Fields
Electrostatic Fields Near Conductors
Looking at the electrostatic field very near a
conductor, we find:
and therefore:
The electric field is perpendicular to the
surface, and where the charge density is
higher, the field is larger.
23-3 Conductors and Electric Fields
To summarize:
1. The electrostatic field inside a conductor is zero.
2. The electrostatic field immediately outside a conductor
is perpendicular to the surface and has the value σ/ε0
where σ is the local surface charge density.
3. A conductor in electrostatic equilibrium—even one that
contains nonconducting cavities—can have charge only
on its outer surface, as long as the cavities contain no
net charge. If there is a net charge within the cavity,
then an equal and opposite charge will be distributed on
the surface of the conductor that surrounds the cavity.
22-4 Are Gauss’ and Coulomb’s
Laws Correct?
An experiment to validate Gauss’ law (that there
is no charge within a conductor) can be done as
follows:
Need a hollow conducing sphere, a small
conducting ball on an insulating rod, and an
electroscope attached to the surface of the
conductor.
22-4 Are Gauss’ and Coulomb’s
Laws Correct?
Charge the small sphere
and hold it inside the shell
without touching. Induced
charge will be on outside of
shell and on electroscope.
Now touch the inside of the
shell with the small sphere.
Charge will flow onto it until
it is neutral, leaving the
shell with a net positive
charge.
22-4 Are Gauss’ and Coulomb’s
Laws Correct?
Finally, remove the rod. The electroscope
leaves do not move, indicating that the
excess charge resides on the outside of the
shell.
22-4 Are Gauss’ and Coulomb’s
Laws Correct?
This table shows the results of such
experiments looking for a deviation
from an inverse-square law:
22-4 Are Gauss’ and Coulomb’s
Laws Correct?
One problem with the above experiments is that they
have all been done at short range, 1 meter or so.
Other experiments, more sensitive to cosmic-scale
distances, have been done, testing whether
Coulomb’s law has the form:
No evidence for a nonzero μ has been found.
Summary of Chapter 22
• Electric flux due to field intersecting a surface S:
S   E  dA
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
Summary of Chapter 22
• Electric flux:
• Gauss’s law:
 EdA 
Qencl
0
Summary of Chapter 22 (con’t)
• Properties of conductors:
1. Electric field is zero inside
2. Field just outside conductor is perpendicular
to surface
3. Excess charge resides on the outside of a
conductor, unless there is a nonconducting
cavity in it; in that case, there is an induced
charge on both surfaces
• Gauss’ law has been verified to a very high
degree of accuracy
Summary
 In a conductor, what happens to an electron in the presence of an
electric field?
 What happens to the electric field in the presence of an conductor?
 HW (Place in your agenda):
 “Foundational Mathematics’ Skills of Physics” Packet (Page 10)
 †“Foundational Mathematics’ Skills of Physics” Packet (Page 11)
 LAST ONE!!!!!!!!
 Web Assign 22.6 - 22.15
How do we apply integration and the Principle of Superposition to uniformly charged objects?
How do we identify and apply the fields of highly symmetric charge distributions?
How do we describe and apply the electric field created by uniformly charged objects?
Wednesday
(Day 4)
Warm-Up
Wed, Feb 25
 Two thin concentric spherical shells of radii r1 and r2 (r1 < r2) contain a
uniform surface charge densities 1 and 2, respectively. Determine
the electric field for (a) r < r1, (b) r1 < r < r2, and (c) r > r2. (d) Under
what conditions will E = 0 for r > r2? (e) Under what conditions will E
= 0 for r1 < r < r2?
 Place your homework on my desk:
 †“Foundational Mathematics’ Skills of Physics” Packet (Page 11)
 Web Assign Problems 22.6 - 22.15
 For future assignments - check online at www.plutonium-239.com
Essential Question(s)
 HOW DO WE DESCRIBE THE NATURE OF ELECTROSTATICS
AND APPLY IT TO VARIOUS SITUATIONS?
How do we describe and apply the concept of electric field?
How do we describe and apply the electric field created by
uniformly charged objects?
How do we describe and apply the relationship between the
electric field and electric flux?
How do we describe and apply Gauss’s Law?
How do we describe and apply the nature of electric fields in
and around conductors?
How do we describe and apply the concept of induced charge
and electrostatic shielding?
Vocabulary
Flux
Electric Flux
Flux Lines
Surface Area
Perpendicular
Orthogonal
Normal
Dot Product
Gauss’s Law
Charge Density
Agenda
 Review “Foundational Mathematics’ Skills of Physics”
Packet (Page 11) with answer guide.
 Complete the Gauss’ Law lab using EM Field 6
 Complete Web Assign Problems 22.6 - 22.15
Summary
 What did you learn about drawing Gaussian surfaces when using EM Field?
 HW (Place in your agenda):
 Electrostatics Lab #4 Report (Due in 5 Classes)
 Web Assign 22.6 - 22.15
How do we apply integration and the Principle of Superposition to uniformly charged objects?
How do we identify and apply the fields of highly symmetric charge distributions?
How do we describe and apply the electric field created by uniformly charged objects?
Thursday
(Day 5)
Warm-Up
Thurs, Feb 26
 Complete Chapter 22 Graphic Organizers
 Place your homework on your desk:
 Web Assign Problems 22.6 - 22.15
 For future assignments - check online at www.plutonium-239.com
Essential Question(s)
 HOW DO WE DESCRIBE THE NATURE OF ELECTROSTATICS
AND APPLY IT TO VARIOUS SITUATIONS?
How do we describe and apply the concept of electric field?
How do we describe and apply the electric field created by
uniformly charged objects?
How do we describe and apply the relationship between the
electric field and electric flux?
How do we describe and apply Gauss’s Law?
How do we describe and apply the nature of electric fields in
and around conductors?
How do we describe and apply the concept of induced charge
and electrostatic shielding?
Vocabulary
Flux
Electric Flux
Flux Lines
Surface Area
Perpendicular
Orthogonal
Normal
Dot Product
Gauss’s Law
Charge Density
Agenda
 Complete the Gauss’ Law lab using EM Field 6
 Complete Web Assign Problems 22.6 - 22.15
Summary
 Would Gauss’ Law be helpful in determining the electric field due to an
electric dipole?
 HW (Place in your agenda):
 Electrostatics Lab #4 Report (Due in 4 Classes)
 Web Assign 22.6 - 22.15
How do we apply integration and the Principle of Superposition to uniformly charged objects?
How do we identify and apply the fields of highly symmetric charge distributions?
How do we describe and apply the electric field created by uniformly charged objects?