Download Electric field of a spherical shell Q

Hw: All Chapter 5 problems and exercises
Outline
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Applications of Gauss’s Law
The single Fixed Charge
Field of a sphere of charge
Field of a spherical shell
A Line of Charge
Conductors and Insulators
The electric field of a conductor
The field in the cavity of a conductor;
Faraday’s Cage
Gauss’s Law
The total flux of electric field out of any
closed surface is equal to the charge
contained inside the surface divided by  0 .
  Qenclosed
 E  dS 
S
0
Conductors and insulators
Charges reside at the surface of the conductor
+
+
+ +
+
+
+
+
+
Conductor
E=0
+
+
+
+
+
+
Solid conducting sphere with charge Q
A
E
r A E0
rA E
V
A
r
1
Q
40 r 2
1 Q
rA V 
40 A
1
Q
40 A
r
rA V 
1 Q
40 r
Electric field of a ball of charge
Q
1
Q
rR E
40 r 2
rR
1
rQ
E
40 R 3
Electric field outside of a charged sphere is exactly the
same as the electric field produced by a point charge,
located at the center of the sphere, with charge equal to the
total charge on the sphere.
Electric field of a spherical shell
Q
The field outside the shell is like that of a point charge,
while the field everywhere inside the shell is zero.
Ionization and corona discharge
There is maximum potential to which a conductor in air
can be raised because of ionization.
Em  3106
V
m
Vm
 Em
R
Small potentials applied to sharp points in air produce
sufficiently high fields just outside the point to ionize the
surrounding air.
A lightning rod has a sharp end so that lightning bolts will pass through a
conducting path in the air that leads to the rod; a conducting wire leads from the
lightning rod to the ground.
The metal mast at the top of the Empire State
Building acts as a lightning rod. It is struck by
lightning as many as 500 times each year.
Benjamin Franklin
"For my own part I wish the Bald Eagle had not been chosen the Representative of
our Country. He is a Bird of bad moral Character. He does not get his Living
honestly. You may have seen him perched on some dead Tree near the River,
where, too lazy to fish for himself, he watches the Labour of the Fishing Hawk; and
when that diligent Bird has at length taken a Fish, and is bearing it to his Nest for
the Support of his Mate and young Ones, the Bald Eagle pursues him and takes it
from him.
"With all this Injustice, he is never in good Case but like those among Men who live
by Sharping & Robbing he is generally poor and often very lousy. Besides he is a
rank Coward: The little King Bird not bigger than a Sparrow attacks him boldly and
drives him out of the District. He is therefore by no means a proper Emblem for the
brave and honest Cincinnati of America who have driven all the King birds from our
Country....
"I am on this account not displeased that the Figure is not known as a Bald Eagle,
but looks more like a Turkey. For the Truth the Turkey is in Comparison a much
more respectable Bird, and withal a true original Native of America... He is besides,
though a little vain & silly, a Bird of Courage, and would not hesitate to attack a
Grenadier of the British Guards who should presume to invade his Farm Yard with
a red Coat on."
--Benjamin Franklin, in a letter to his daughter
What have we learned about
conductors?
• There is no electric field inside a conductor
• Net charge can only reside on the surface
of a conductor
• Any external electric field lines are
perpendicular to the surface (there is no
component of electric field that is tangent
to the surface).
• The electric potential within a conductor is
constant
Electric field near a surface of a conductor
l
a
 
 E  dS 
 EdS  Ea
cap
a
Ea 
0

E
0
Two parallel conducting plates


-
+
+
+
l
-
+
a
+
-
+
d
 
 E  dS 
 EdS  Ea
cap
a
Ea 
0

E
0
(the total field at any point
between the plates)
An Apparent Contradiction
+
+
+
+
+
+

E
0
-
-
An Apparent Contradiction


+
2
+

E
2 0

E
0
+
+
+
+
2
E

-
-

?
2 0
Near the surface of any conductor in electrostatics

E
0
1) There is a conducting spherical shell, inner radius A
and outer radius B. If you put a charge Q on it, find the
charge density everywhere.
2) There is a conducting spherical shell, inner radius A and
outer radius B. A charge Q is put at the center. If you put a
charge Q2 on the shell, find the charge density
everywhere.
A sphere of radius A has a charge Q uniformly spread
throughout its volume. Find the difference in the electric
potential, in other words, the voltage difference, between
the center and a point 2A from the center.
There is a conducting spherical shell, inner radius A and
outer radius B. A charge Q1 is put at the center. If you now
put charge -2Q1 on the shell, find the charge density at r=A
and r=B.

r2
 
Vr2  Vr1    E  dr  0

r1
since

E  0 inside the conductor.

For any two points r1
and

r2
inside the conductor
Vr1  Vr2
The conductor’s surface is an equipotential.
Equipotential Surfaces
An equipotential surface is a surface on which
the electric potential V is the same at every
point.
Because potential energy does not change as a test charge moves over
an equipotential surface, the electric field can do no work on such a

charge. So, electric field must be perpendicular
to the surface at every
point so that the electric force qE
is always perpendicular to the
displacement of a charge moving on the surface.
Field lines and equipotential surfaces are always
mutually perpendicular.
Method of images: What is a force on the point charge near a conducting plate?
Equipotential surface
-
--
The force acting on the positive charge is exactly the same as it would
be with the negative image charge instead of the plate.
a
The point charge feels a force towards the plate with a magnitude:
1
2
q
F
40 (2a) 2
Method of images: A point charge near a conducting plane.

E ?
Equipotential surface
-
--
P
r
a
E  
1
aq
40 (a 2  r 2 ) 3 2
1
aq
E  
40 (a 2  r 2 ) 3 2
2
aq
E
40 (a 2  r 2 )3 2
Equilibrium in electrostatic field: Earnshaw’s theorem
There are NO points of stable equilibrium in any electrostatic field!
How to prove it? Gauss’s Law will help!
Imaginary surface
surrounding P
P
If the equilibrium is to be a stable one, we require that if we move the
charge away from P in any direction, there should be a restoring force
directed opposite to the displacement. The electric field at all nearby points
must be pointing inward – toward the point P. But that is in violation of
Gauss’ law if there is no charge at P.
Thomson’s atom
1899
If charges cannot be held stably, there cannot be matter made up of
static point charges (electrons and protons) governed only by the laws
of electrostatics. Such a static configuration would collapse!
Capacitors
Consider two large metal plates which are parallel to each other
and separated by a distance small compared with their width.
y
Area A
L








The field between plates is






 
 
V

E
0

 [V (top)  V (bottom)]   E y dy 
dy   L
0
0
0
0
L
L


 A
QL
 [V (top)  V (bottom)]   L  
L
0
0 A
0 A
QL
V 
A 0
The capacitance is:
A 0
Q
Q
C


QL
V
L
A 0
Cylindrical Capacitor
Spherical Capacitor
A 0
Q
Q
C


V QL
L
A 0
Capacitors in series:
1
1
1
1



 ...
Ctot C1 C2 C3
Capacitors in parallel: Ctot  C1  C2  C3  ...
1
1 2
2
W  CV 
Q
2
2C
[C ]  farad
Quiz
1) If a 4-F capacitor and an 8-F capacitor are connected in
parallel, which has the larger potential difference across it?
Which has the larger charge?
2)
A
B
Two capacitors are connected in series as shown. If they were
initially uncharged, what will be the charge inside the dotted box
after connecting points A and B to a battery of voltage V?
3) If the wire connecting the capacitors is bent so that capacitors
look like
A
B
how do you now call the arrangement?
a
C3
C1
C2
b
C5
C4
C1=C5=8.4 F and C2=C3=C4=4.2 F
The applied potential is Vab=220 V.
a)What is the equivalent capacitance of the network
between points a and b?
b) Calculate the charge on each capacitor and the
potential difference across each capacitor.
 
E

d
r

0
in
electrosta
tics


E
 
E  dr  0
B
C

E2

E1
D
A
 
E  dr  0
  D 
 E1dr   E2dr  E1 ( B  A)  E2 ( D  C )  0
B
A
C
E1 = E 2
Most capacitors have a non-conducting material, or dielectric, between their
conducting plates. When we insert an uncharged sheet of dielectric
between the plates, experiments show that the potential difference
decreases to a smaller value V.
Q
C0 
without dielectric
V0
Q
C
with dielectric
V
C  C0
When the space between plates is completely filled by the dielectric, the ratio
C
K
C0
is called dielectric constant.
Smaller, Denser, Cheaper
Moore’s “Law” (1965): every 2 years the number
of transistors on a chip is doubled
Gordon Moore, co-founder, Intel
Have a great day!