Transcript Lec04

Lecture 4-1
At point P on axis of ring:
ds
Q

2 R
At a point P off axis:
At a point P on axis:
Lecture 4-2
Gauss’s Law: Qualitative Statement
 Form any closed surface around charges
 Count the number of electric field lines coming through the
surface, those outward as positive and inward as negative.
 Then the net number of lines is proportional to the net
charges enclosed in the surface.
Lecture 4-3
Electric flux
To state Gauss’s Law in a quantitative form,
we first need to define Electric Flux.
# of field lines N
= density of field lines
x “area”
where “area” = A2 x cos 

N  E A  E  n A
Sum over
surface


E  n A

General definition of electric flux:
(must specify sense, i.e., which way)
 E   E  ndA
Lecture 4-4
Electric Flux through
 E Closed Surface

 E   E  n dA   En dA
s
s
• The integral is over a CLOSED surface.

• Since E  A n is a scalar product, the electric flux is a
SCALAR quantity
• The integration element n is a vector normal to
the surface and points OUTWARD from the
surface. Out is +, In is E proportional to # field lines
coming through outward
Lecture 4-5
Why are we interested in electric flux?
 E is closely related to the charge(s) which cause it.
Consider Point charge Q


r 
 E   E  ndA  kq 2  n dA
r
kQ
Q
2
 2 (4r ) 
r
0
If we now turn to our previous
discussion and use the analogy to
the number of field lines, then the
flux should be the same even when
the surface is deformed. Thus should
only depend on Q enclosed.
Lecture 4-6
Physics 241 –warm-up quiz
One point charge Q is placed inside a closed surface
with complicated shape. Which of the following
statements is correct on the flux through the surface?
a) It’s not possible to calculate the flux since the details
of the surface shape are not given.
b) The flux is Q/ε0
c) The location of the charge Q inside the surface is
need in order to calculate the flux.
d) The flux is not zero and must be negative.
+Q
Lecture 4-7
Gauss’s Law: Quantitative Statement
The net electric flux through any closed surface equals the
net charge enclosed by that surface divided by 0.

E

n
dA



E

Qenclosed
0
How do we use this equation??
The above equation is TRUE always but it doesn’t
look easy to use.
BUT - It is very useful in finding E when the physical situation
exhibits a lot of SYMMETRY.
Lecture 4-8
Uniformly charged thin shell: outside
Electric Field Outside
• By symmetry, the electric field
must only depend on r and is
along a radial line everywhere.
• Apply Gauss’s law to the blue
surface , we get
Q
E  (4 r ) 
2
1 Q
E 
2
4 0 r
0
Lecture 4-9
Uniformly charged thin shell: Inside
E = 0 inside
• By symmetry, the electric field
must only depend on r and is
along a radial line everywhere.
• Apply Gauss’s law to the blue
surface , we get E = 0.
• Equal and opposite contributions
from charges on diagonally opposite
surface elements.
Discontinuity in E
Lecture 4-10
Electric Field of a Uniformly Charged Sphere
Apply Gauss’s Law directly or
use superposition of the shell
results
Lecture 4-11
Physics 241 –Quiz 2a
Two identical point charges are each placed inside a
large cube. One is at the center while the other is close
to the surface. Which statement about the net electric
flux through the surface of the cube is true?
a)
b)
c)
d)
e)
The flux magnitude is larger when the charge is at the center.
The flux magnitude is the same (and not zero).
The flux magnitude is larger when the charge is near the surface.
Not enough information to tell.
The flux is zero in both cases.
+Q
+Q
Lecture 4-12
Physics 241 –Quiz 2b
Two identical point charges are each placed inside a
large sphere. One is at the center while the other is close
to the surface. Which statement about the net electric
flux through the surface of the sphere is true?
a)
b)
c)
d)
e)
The flux magnitude is larger when the charge is at the center.
The flux magnetude is larger when the charge is near the surface.
The flux magnitude is the same (and not zero).
Not enough information to tell.
The flux is zero in both cases.
+Q
+Q
Lecture 4-13
Physics 241 –Quiz 2c
Two identical point charges are placed, at the center of a
large sphere in one case, and outside an identical sphere in
the other case. Which statement about the net electric flux
through the surface of the sphere is true?
a)
b)
c)
d)
e)
The flux is larger when the charge is inside.
The flux is larger when the charge is outside.
The flux is the same (and not zero).
Not enough information to tell.
The flux is zero in both cases.
Q>0
Q>0