Download Lecture 4-2 - Purdue Physics

Lecture 4-1
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 4-2
©2008 by W.H. Freeman and Company
Lecture 4-3
©2008 by W.H. Freeman and Company
Lecture 4-4
©2008 by W.H. Freeman and Company
Lecture 4-5
©2008 by W.H. Freeman and Company
Lecture 4-6
Reading Quiz 1
Which of the following statements contradict Gauss’s law?
a) The electric flux through a closed surface depends on all
the charges outside the closed surface.
b) The electric flux through a closed surface is
proportional to the net charges enclosed in the surface.
c) Any closed surface can be used with Gauss’s law.
d) An electric dipole inside a closed surface does not change
the net flux through the surface.
Lecture 4-7
Electric flux Summary
To state Gauss’s Law in a quantitative form,
we first need to compute the Electric Flux.
# of field lines N
= density of field lines
x “area”
where “area” = A2 x cos θ
N  E A  E An
Sum over
surface
E
θ
An
General definition of electric flux:
 E   E n dA
S
(must specify sense, i.e., which way)
Lecture 4-8
Electric Flux through a Closed Surface
E 

S
E n dA 

S
En dA
• The integral is over a CLOSED surface.
• Since E n is a scalar product, the electric flux is a
SCALAR quantity.
• The integration element
n is a vector normal to
the surface and always points OUTWARD from the
surface. Out is +.
ΦE proportional to # field lines
coming through outward
Lecture 4-9
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 ndA  
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 4-10
A1 A2 cos 
 2 
r1
r22
+Q outside a closed surface
Lecture 4-11
Physics 241 – Warm up Quiz 2
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
Lecture 4-12
Gaussian Surface R
©2008 by W.H. Freeman and Company
Lecture 4-13
Proof of the Shell Theorem
Electric Field Outside
a Shell of Charge
• 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 4-14
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 4-15
Electric Field of a Uniformly Charged Sphere
Apply Gauss’s Law directly or
use superposition of the shell
results
Lecture 4-16
DOCCAM 2
GAUSS’S LAW
5A-12
Lecture 4-17
Cylindrical coordinates
r perpendicular to z axis
r2 = x2 + y2
©2008 by W.H. Freeman and Company
Lecture 4-18
Line charge λ on z axis
©2008 by W.H. Freeman and Company
Lecture 4-19
Physics 241 –Quiz 3 10:30 September 1, 2011
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) The net flux is larger when the charge is at the center.
b) The net flux is the same (and not zero).
c) The net flux is larger when the charge is near the
surface.
d) Not enough information to tell.
e) The flux is zero in both cases.
+Q
+Q
Lecture 4-20
Physics 241 –Quiz 3 11:30 September 1, 2011
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 is larger when the charge is at the center.
The flux is larger when the charge is near the surface.
The magnitude of the flux is the same (and not zero).
Not enough information to tell.
The flux is zero in both cases.
+Q
+Q
Lecture 4-21
Physics 241 –Quiz 3 11:30 September 1, 2011
Two identical point charges are each placed inside and
outside a large spherical surface. One is at the center
while the other is outside the surface. Which statement
about the net electric flux through the surfaces of the
sphere is true?
a) The flux is larger when the charge is outside the surface.
b) The magnitude of the flux is the same.
c) The flux is zero when the charge is outside the surface.
d) The flux is zero in both cases.
+Q
+Q