Physics 212
Lecture 3
Today's Concepts:
Electric Flux and Field Lines
Gauss’s Law
Physics 212 Lecture 3, Slide 1
Music
Who is the Artist?
A)
B)
C)
D)
E)
Professor Longhair
Henry Butler
David Egan
Dr. John
Allen Toussaint
WHY?
Mardi Gras is 4 weeks from today
Physics 212 Lecture 13
Our Comments
• SmartPhysics scores will not be visible in the main gradebook
for a few weeks
• You will need to understand integrals in this course!!!
• Forces and Fields are Vectors
• Always Draw a Picture First… What do the Forces/Fields
Look Like?
Prelecture
E
Lecture
Worked Example
E
E
Infinite line of charge
Finite line of charge
(constant l)
Homework
dq
1 dq
E   k 2 rˆ  
rˆ
2
r
4 0 r
WORKS FOR ALL !!
E
Arc of charge
(constant l)
Finite line of charge
(non-constant l)
The “1/4 shell problem” is not
special !!
It’s the same as all the others!
Physics 212 Lecture 3, Slide 3
Electric Field Lines
Direction & Density of Lines
represent
Direction & Magnitude of E
Point Charge:
Direction is radial
Density  1/R2
07
Physics 212 Lecture 3, Slide 4
Electric Field Lines
Dipole Charge Distribution:
Direction & Density
much more interesting…
08
simulation
Physics 212 Lecture 3, Slide 5
Checkpoint 3.1
Preflight 3
The following three questions pertain to the electric field
lines due to the two charges shown. Compare the
magnitude of the two charges
A. |Q1| > |Q2|
B. |Q1| = |Q2|
C. |Q1| < |Q2|
D. There isn’t enough information to determine the relative
magnitude of the charges
“more field lines emanating from the charge 1, so must be
greater in magnitude.”
09
simulation
Physics 212 Lecture 3, Slide 6
Checkpoint 3.3
What do we know about the signs of the charges from
looking at the picture?
A. Q1 and Q2 have the same sign
B. Q1 and Q2 have the opposite sign
C. There isn’t enough information to determine the relative
signs of the charges
“The field lines leave one and go to the other..”
10
simulation
Physics 212 Lecture 3, Slide 7
Checkpoint 3.5
Preflight 3
Compare the magnitude of the electric field at point A and
B.
A. |EA| > |EB|
B. |EA| = |EB|
C. |EA| < |EB|
D. There isn’t enough information to determine the relative
magnitude of the electric field at points A and B
“Since the electric field lines are denser near
point B the magnitude of the electric field is
greater at point B than at point A.”
12
Physics 212 Lecture 3, Slide 8
Point Charges
“Telling the difference between positive and negative charges while looking at field lines.
Does field line density from a certain charge give information about the sign of the charge?”
-q
+2q
What charges are inside the red circle?
13
-Q
-Q
-2Q
+Q
+Q
+2Q
+Q
A
B
C
D
-Q
E
Physics 212 Lecture 3, Slide 9
Which of the following field line pictures best represents the
electric field from two charges that have the same sign but
different magnitudes?
15
A
B
C
D
simulation
Physics 212 Lecture 3, Slide 10
Electric Flux “Counts Field Lines”
“I still feel a little unclear about flux integrals. Can you go over some more example problems?”
 S   E  dA
S
Flux through
surface S
18
Integral of E  dA
on surface S
Physics 212 Lecture 3, Slide 11
Checkpoint 1
“The flux is directly proportional to
the charge enclosed by the
Gaussian surface, therefore Case
1 will have 2 times the flux of of
Case 2 because it has two charges
enclosed, versus 1 in Case 2. “
An infinitely long charged rod has uniform
charge density l and passes through a
cylinder (gray). The cylinder in Case 2 has
twice the radius and half the length compared
TAKE s TO BE RADIUS !
with the cylinder in Case 1.
L/2
“The flux is the measure of field
lines passing through the surface.
Once calculated, it can be shown
that the surface area of both
cylinders are the same.”
1=22
(A)
23
1=2
(B)
1=1/22
(C)
none
(D)
Physics 212 Lecture 3, Slide 12
Checkpoint 1
An infinitely long charged rod has uniform
charge density l and passes through a cylinder
WHAT IS WRONG WITH THIS REASONING?? (gray). The cylinder in Case 2 has twice the
radius and half the length compared with the
“The flux is the measure of field
TAKE s TO BE RADIUS !
cylinder
in Case 1.
lines passing through the surface.
Once calculated, it can be shown
that the surface area of both
cylinders are the same.”
THE E FIELD AT THE BARREL SURFACE
IS NOT THE SAME IN THE TWO CASES !!
L/2
Definition of Flux:

 
 E  dA
surface
E constant on barrel of cylinder
E perpendicular to barrel surface
(E parallel to dA)

  E  dA  EAbarrel
Case 1
l
E1 
2 0 s
A1  ( 2 s ) L
26
1=22
(A)
barrel
1RESULT:
=2

GAUSS’
1=1/22LAW none
(D) !
(B)
(C) enclosed
 proportional
to charge
Case 2
1 
lL
0
E2 
l
2 0 ( 2 s )
A2  ( 2 ( 2 s )) L / 2  2 sL
2 
l ( L / 2)
0
Physics 212 Lecture 3, Slide 13
Direction Matters:
E
E
E
dA
For a closed surface,
A points outward
dA
E
E
dA
E
dA
dA
E
E
E
 S   E  dA  0
S
29
Physics 212 Lecture 3, Slide 14
Direction Matters:
E
E
E
dA
For a closed surface,
A points outward
dA
E
E
dA
E
dA
dA
E
E
E
 S   E  dA  0
S
30
Physics 212 Lecture 3, Slide 15
Trapezoid in Constant Field
y
Label faces:
1: x = 0
2: z = +a
3: x = +a
4: slanted
E0 E  E0 xˆ
3
1
2
x
Define n = Flux through Face n
dA
 
E  dA  0
z
31
E
A
1 < 0
A
2 < 0
A
3 < 0
A
4 < 0
B
1 = 0
B
2 = 0
B
3 = 0
B
4 = 0
C
1 > 0
C
2 > 0
C
3 > 0
C
4 > 0
Physics 212 Lecture 3, Slide 16
Trapezoid in Constant Field + Q
E  E0 xˆ
y
E0
Label faces:
1: x = 0
2: z = +a
3: x = +a
3
+Q
2
x
Define n = Flux through Face n
 = Flux through Trapezoid
z
36
1
Add a charge +Q at (-a,a/2,a/2)
How does Flux change?
A
1 increases
A
3 increases
A
 increases
B
1 decreases
B
3 decreases
B
 decreases
C
1 remains same
C
3 remains same C
 remains same
Physics 212 Lecture 3, Slide 17
Gauss Law
E
E
Q
E
E
dA
dA
E
dA
E
dA
dA
E
E
E
S 
  Qenclosed
 E  dA 
closed
surface
41
o
Physics 212 Lecture 3, Slide 18
Checkpoint 2.3
(A)
 increases
43
(B)
 decreases
(C)
 stays same
“The flux throughout the entire
surface does not change as the
charge moves, for it is still in the
sphere. However, if it was moved
outside of the sphere, the fluxes
would be different because one
would be zero!“
Physics 212 Lecture 3, Slide 19
Checkpoint 2.1
“charge is closer, field is larger, more
lines through a so increase in flux.“
“as long as the charge is within the
shell, the flux will remain the same”
(A)
dA increases
dB decreases
44
(B)
dA decreases
dB increases
(C)
dA stays same
dB stays same
Physics 212 Lecture 3, Slide 20
Think of it this way:
1
2
The total flux is the same in both cases (just the total number of lines)
The flux through the right (left) hemisphere is smaller (bigger) for case 2.
45
Physics 212 Lecture 3, Slide 21
Things to notice about Gauss’s Law
S 
  Qenclosed
 E  dA 
o
closed
surface
If Qenclosed is the same, the flux has to
be the same, which means that the
integral must yield the same result
for any surface.
47
Physics 212 Lecture 3, Slide 22
Things to notice about Gauss’s Law
“Gausses law and what concepts are most important that we know .”
  Qenclosed
 E  dA 
closed
surface
o
In cases of high symmetry it may be possible to bring E outside
the integral. In these cases we can solve Gauss Law for E
 
Qenclosed
 E  dA  EA 
closed
surface
o
Qenclosed
E
A 0
So - if we can figure out Qenclosed and the area of the
surface A, then we know E !
This is the topic of the next lecture…
48
Physics 212 Lecture 3, Slide 23
Final Thoughts….
• Keep plugging away
– Prelecture 4 and Checkpoint 4 due Thursday
– Homework 2 due next Tuesday … 7 questions mostly Gauss’ law – try
them today!
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Physics 212 Lecture 3, Slide 24