PH 222-2C Fall 2012 Gauss’ Law Lectures 3-4 Chapter 23 (Halliday/Resnick/Walker, Fundamentals of Physics 8 th edition) 1
PH 222-2C Fall 2012
Gauss’ Law
Lectures 3-4
Chapter 23(Halliday/Resnick/Walker, Fundamentals of Physics 8th edition)
1
Chapter 23 Gauss’ Law
In this chapter we will introduce the following new concepts:
The flux (symbol Φ ) of the electric field Symmetry Gauss’ law
We will then apply Gauss’ law and determine the electric field generated by: An infinite, uniformly charged insulating plane
An infinite, uniformly charged insulating rod A uniformly charged spherical shell A uniform spherical charge distribution
We will also apply Gauss’ law to determine the electric field inside and outside charged conductors. 2
n̂
Consider an airstream of velocity that is aimed at a loopof area . The velocity vector is at angle with respect to the
ˆloop normal . The product cos is know
n as
vA v
n vA
Flux of a Vector.
the . In this example the flux is equal to the volume flow rate through the loop (thus the name flux).
depends on . It is maximum and equal to for 0 ( perpendicularto the loop
vA v
flux
Note 1 :
plane). It is minimum and equal to zero for 90 ( parallelto the loop plane).
cos . The vector is parallel to the loop normal and h asmagnitude equal to .
v
vA v A AA
Note 2 :
3
n̂
n̂n̂
Consider the closed surface shown in the figure. In the vicinity of the surface assume that we have
a known electric field . The flux of the electric field thro h
ugE
Flux of the Electric Field.
the surface is defined as follows: 1. Divide the surface into small "elements" of area .
2. For each element calculate the term cos .
3. Form the sum .4. Take the limit of the sum a
A
E A EA
E A
2Flux SI unit: N m / C
s the area 0.The limit of the sum becomes the integral:
The circle on the integral sign indicates thatthe surface is closed. When we apply Gauss'la
A
E dA
Note 1 :
w the surface is known as "Gaussian." is proportional to the net number of
electric field lines that pass through the surface.Note 2 :
E dA
4
n̂
n̂n̂
0 enc
0 enc
The flux of through any closed surface net charge enclosed by the surface.
Gauss' law can be formulated as follows:
In equ ation form: Equivalently:
E q
q
Gauss' Law
0 encε E dA q
0 encq
0 encε E dA q
Gauss' law holds for closed surface. Usually one particular surface makes the problem ofdetermining the electric field very simple.
When calculating the net charge inside a c
Note 1 : any
Note 2 : losed surface we take into account the algebraic sign of each charge.
When applying Gauss' law for a closed surface we ignore the charges outside the surface no matter how large they are
.
Note 3 :
Examp
1 0 1 2 0 2
3 0 3 4 0 4
1 2 3 4
Surface : , Surface : Surface : 0, Surface : 0
We refer to , , , as "Gaussian surfaces."
S q S qS S q q
S S S S
le :
Note :5
n̂
dA
Gauss' law and Coulomb's law are different waysof describing the relation between electric chargeand electric field in static cases. One can derive Coulomb's law from Gauss
Gauss' Law and Coulomb's Law
' law and vice versa.Here we will derive Coulomb's law from Gauss' law.Consider a point charge . We will use Gauss' law
to determine the electric field generated at a point at a distance from
q
EP r q
. We choose a Gaussian surface that is a sphere of radius and has its center at .r q
Coulomb's law Gauss' law
2
20 enc 0
We divide the Gaussian surface into elements of area . The flux for each element is:
cos0 Total flux 4
From Gauss' law we have: 44
dA
d EdA EdA EdA E dA E r
qq q r E q Er
20
This is the same answer we got in Chapter 22 using Coulomb's law.
6
eE
vF
We shall prove that the electric field inside a conductor vanishes.Consider the conductor shown in the figure to the left. It is an experimental fact that such an
The Electric Field Inside a Conductor
object contains negatively charged electrons, which are free to move inside the conductor. Let's assume for a moment that the electric field is not equal to zero.
In such a case a nonvanishing force is exerted by thefield on each electron. This force would result in a nonzero velocity , and the moving electrons would constitute an electriccurrent. We will see in subsequent chapters th
F eE
v
at electric currents manifest themselves in a variety of ways: (a) They heat the conductor.(b) They generate magnetic fields around the conductor. No such effects have ever been observed, thus the original assumption that there exists a nonzero electric field inside the conductor. We conclude that :
The electrostatic electric field inside a conductor is equal to zero.E
7
Consider the conductor shown in the figure that has a total charge . In this section we will ask the question:Where is this charge located? To answer the question we wil
q
A Charged Isolated Conductor
l apply Gauss' law to the Gaussian surface shown in the figure, which is located just below the conductor
surface. Inside the conductor the electric field 0.
Thus 0 ( ).
From Gaus
E
E A
eq. 1
enc
0
enc
s's law we have: ( ).
If we compare eq. 1 with eq. 2 we get = 0 .
q
q
eq. 2
Thus no charge exists inside the conductor. Yet we know that the conductorhas a nonzero charge . Where is this charge located? There is only one placefor it to be: On the surface of the conductor.
q
electrostatic charges can exist inside a conductor. charges reside on the conductor surface.
NoAll 8
Consider the conductor shown in the figure that has a total charge . This conductor differs from the one shown on the previous page in one aspect: It has a
q
An Isolated Charged Conductor with a Cavity
cavity. We ask the question: Can charges reside on the walls of the cavity? As before, Gauss's law provides the answer. We will apply Gauss' law to the Gaussian surface shownin the figure, which is
enc
0
located just below the conductor
surface. Inside the conductor the electric field 0.
Thus 0 ( ).
From Gauss's law we have: ( ).
If we compare eq. 1 with eq. 2 we ge
E
E A
q
eq. 1
eq. 2
enct = 0.qConclusion :
There is no charge on the cavity walls. All the excess charge remains on the outer surface of the conductor.
q9
1̂n3n̂
2n̂
S1S2S3
The electric field inside a conductor is zero. This isnot the case for the electric field outside. The
electric field vector is perpendicular to the E
The Electric Field Outside a Charged Conductor
||
conductor
surface. If it were not, then would have a componentparallel to the conductor surface.
EE
||Since charges are free to move in the conductor, would cause the free
electrons to move, which is a contradiction to the assumption that we have stationary charges. We will apply Gauss' law using
E
1 2 3 1 2 3
1
the cylindrical closedsurface shown in the figure. The surface is further divided into three sections
, , and as shown in the figure. The net flux . cos 0
S S SEA EA
2
enc3
0
enc
0
enc
0
cos90 0
0 (because the electric field inside the conductor is zero).
1 . The ratio is known as surface charge density .
EAqEA
qEA
EqA
0
E
10
Symmetry. We say that an object is symmetric under a particular mathematical operation (e.g., rotation, translation, …) if to an observer the object looks the same before and after the operation.Note: Symmetry is a primitive notion and as such is very powerful.
Featureless sphere
Rotation axis
ObserverRotational symmetry Example of Spherical Symmetry
Consider a featureless beach ball that can be rotated about a vertical axis that passes through its center. The observer closes his eyes and we rotate the sphere. When the observer opens his eyes, he cannot tell whether the sphere has been rotated or not. We conclude that the sphere has rotational symmetry about the rotation axis.
11
Featureless cylinder
Rotation axis
Observer
Rotational symmetry A Second Example of Rotational Symmetry
Consider a featureless cylinder that can rotate about its central axis as shown in the figure. The observer closes his eyes and we rotate the cylinder. When he opens his eyes, he cannot tell whether the cylinder has been rotated or not. We conclude that the cylinder has rotational symmetry about the rotation axis.
12
Observer
Magic carpet
Infinite featureless plane
Translational symmetry
Example of Translational Symmetry:
Consider an infinite featureless plane. An observer takes a trip on a magic carpet that flies above the plane. The observer closes his eyes and we move the carpet around. When he opens his eyes the observer cannot tell whether he has moved or not. We conclude that the plane has translational symmetry.
13
Recipe for Applying Gauss’ Law
1. Make a sketch of the charge distribution.
2. Identify the symmetry of the distribution and its effect on the electric field.
3. Gauss’ law is true for any closed surface S. Choose one that makes the calculation of the flux as easy as possible.
4. Use Gauss’ law to determine the electric field vector:enc
0
q
14
S11̂n
2n̂S2
S3
3n̂
Consider the long rod shown in the figure. It is uniformly charged with linear charge density . Using symmetry arguments we can show that the
Electric Field Generated by a Long, Uniformly Charged Rod
electric field vector points radially outward and has the same magnitude for points atthe same distance from the rod. We use a Gaussian surface
that has the same symmetry. It is a cylinder of radr
S ius and height whose axis coincides with the charged rod.
rh
1 2
3 1 2 3 1 3
We divide S into three sections: Top flat section , middle curved section ,and bottom flat section . The net flux through is . Fluxes and vanish because the electric field i
S SS S
en2
0
0 0
c
0
s at right angles with the normal to the surface:
2 cos 0 2 2 . From Gauss's law we have: .
If we compare these two equations we get: .2
2
q hrhE rhE rhE
hr EEr
h
02E
r
15
1̂n
2n̂
3n̂ S1
S2S3
We assume that the sheet has a positive charge ofsurface density . From symmetry, the electric field
vector is peE
Electric Field Generated by a Thin, Infinite, Nonconducting Uniformly Charged Sheet
rpendicular to the sheet and has a
constant magnitude. Furthermore, it points away fromthe sheet. We choose a cylindrical Gaussian surface with the caps of area on either side of the sheet asshown
SA
1 2
3
in the figure. We divide into three sections: is the cap to the right of the sheet, is the curved
surface of the cylinder, and is the cap to the left of the sheet. The net flux through
SS S
SS 1 2 3
1 3 2
enc
0 0
0 0
is .cos 0 . 0 ( = 90 )
2 . From Gauss's law we have:
2 .2
EA EAq AEA
AEA E
02E
16
S
S'
A
A'
1 1
The electric field generated by two parallel conducting infinite planes is charged withsurface densities and - . In figs. and we show the two plates isolated so thatone does not influence the
a b charge distribution of the other. The charge spreads out
equally on both faces of each sheet. When the two plates are moved close to eachother as shown in fig. , then the charges on one plate attrc act those on the other. As a result the charges move on the inner faces of each plate. To find the field betweenthe plates we apply Gauss' law for the cylindrical surface , which has caps of area
iES
enc 10
0 0
enc 1
1
0
00
0
.2The net flux To find the field outside
the plates we apply Gauss' law for the cylindrical surface ', which has caps of area .
The net
2 = .
flux
i i
Aq AE A E
S A
E
qE A
1
00
0 0. E
1
0 0
2iE
0 0E
17
1 Consider a Gaussian surface that is a sphere with radius and whose center coincides with that of the ch
Sr R
The Electric Field Generated by a Spherical Shellof Charge and Radius Inside the shell :
q R
2 enc
0
2
arged shell.
The electric field flux 4 0.
Thus Consider a Gaussian surface
that is a sphere with radius and whose center coincides with that of the ch r
0.
a ge
i
i
qr E
Sr R
E
Outside the shell :
2 enc0
0 0
0 20
d shell.
The electric field flux 4 .
Thus
Outside the shell the electric field is the same as if all the charge of the shell were concentrated at the shell center
4
.
.qEr
q qr E
Note :
1̂niE
2n̂0E
0iE
0 204
qEr
18
S2
S1
iE
2n̂
oE
1̂n
1 Consider a Gaussian surface that is a sphere with radius and whose center coincides with that
Sr R
Electric Field Generated by a Uniformly Charged Sphere of Radius and Charge Outside the sphere :
R q
20 enc 0 0
0
2
0 2
of the charged sphere. The electric field flux 4 / /
Thus
Consider a Gaussian surface that is a sphere with radius and whose center coincides
.4
with
r E q q
Sr
qEr
R
Inside the sphere :
2 enc
0
3 32
enc 3
30
33 3 0
that of the charged sphere.
The electric field flux 4 . Since the charge is
uniform 44 43 3
Thus .4
i
enc
i
i
qr E
q q r r qq
qE rR
q r ER Rr R
19