Transcript Lecture 15: Electric Forces and Electric Fields

PHY121 Summer Session II, 2006
Instructor : Chiaki Yanagisawa
• Most of information is available at:
http://nngroup.physics.sunysb.edu/~chiaki/PHY122-06.
It will be frequently updated.
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• The lab session will start next Monday (June 5), for the first class
go to A-117 at Physics Building. Your TAs will divide each group
into two classes in alphabetic order.
• There will be recitation classes currently planned to be on Fridays at:
9:30 am - 10:15 am, 10:30 am - 11:15 am
3:00 pm - 3:45 pm, 4:00 pm - 4:45 pm.
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be announced.
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the lectures and MOST of THEM (NOT ALL) will be posted on
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• Find information about which you want to know on the web or during
the lectures as much as possible.
Chapter 15: Electric Forces and
Electric Fields
Homework on WebAssign to be set up: 14,22,40,53,64
Properties of electric charges
 When
a plastic rod is rubbed with a piece
of fur, the rod is “positively” charged
 When
a glass rod is rubbed with a piece
of silk, the rod is “negatively” charged
 Two
equally signed charges repel each
other
 Two
opposite signed charges attract each
other

Electric charge is conserved
Electric charge (cont’d)
Particle Physics
What is the world made of?
nucleus
Model of Atoms
proton
Old view
electrons eSemi-modern view
Modern view
quarks
nucleus
Properties of electric charges
 Origin
of electric charge
• Nature’s basic carriers of positive charge are protons, which, along
with neutron, are located in the nuclei of atoms, while the basic
carriers of negative charge are electrons which orbit around the
nucleus of an atoms. Atoms are in general electrically neutral.
• It is easier to take off electron(s) from an atom than proton(s).
By stripping off an electron from the atom, the atom becomes
positively charged, while an atom that the stripped off electron
is relocated to becomes negatively charged.
• In 1909 Millikan discovered that if an object is charged, its charge is
always a multiple of a fundamental unit of charge, designated by the
symbol e : the electric charge is quantized.
The value of e in the SI unit is 1.60219x10-19 coulomb C.
Properties of electric charges
 Electric
charges
•
Electron: Considered a point object with radius less than 10-18 meters with electric
charge e= -1.6 x 10 -19 Coulombs (SI units) and mass me= 9.11 x 10 - 31 kg
•
Proton: It has a finite size with charge +e, mass mp= 1.67 x 10-27 kg and with radius
– 0.805 +/-0.011 x 10-15 m scattering experiment
– 0.890 +/-0.014 x 10-15 m Lamb shift experiment
•
Neutron: Similar size as proton, but with total charge = 0 and mass mn=
– Positive and negative charges exists inside the neutron
•
Pions: Smaller than proton. Three types: + e, - e, 0 charge.
– 0.66 +/- 0.01 x 10-15 m
•
Quarks: Point objects. Confined to the proton, neutron, pions, and so forth.
– Not free
– Proton (uud) charge = 2/3e + 2/3e -1/3e = +e
– Neutron (udd) charge = 2/3e -1/3e -1/3e = 0
– An isolated quark has never been found
Properties of electric charges
• Two kinds of charges: Positive and Negative
• Like charges repel - unlike charges attract
• Charge is conserved and quantized
1. Electric charge is always a multiple of the fundamental unit of
charge, denoted by e.
2. In 1909 Robert Millikan was the first to measure e.
Its value is e = 1.602 x 10−19 C (coulombs).
3. Symbols Q or q are standard for charge.
4. Always Q = Ne where N is an integer
5. Charges: proton, + e ; electron, − e ; neutron, 0 ; omega, − 3e ;
quarks, ± 1/3 e or ± 2/3 e – how come? – quarks always exist
in groups with the N×e rule applying to the group as a whole.
Insulators and Conductors
 Definition
• In conductors, electric charges move freely in response to an
electric force. All other materials are called insulators.
Insulators : glass, rubber, etc.
When an insulator is charged by rubbing, only the rubbed area
becomes charged, and there is no tendency for the charge to
move into other regions of the material.
Conductors : copper, aluminum, silver, etc.
When a small area of a conductor is charged, the charge readily
distributes itself over the entire surface of the material.
Semiconductors : silicon, germanium, etc.
Electrical properties of semiconductor materials are somewhere
between insulators and conductors.
Insulators and Conductors
 Charging
a material
• Charging by contact
Insulators and Conductors
 Charging
a material
• Charging by induction
Induction : A process in which a donor material gives opposite
signed charges to another material without losing any
of donor’s charges
Insulators and Conductors
 Insulator
• Polarization
• Polarization in an insulator by induction
+
+
-
Coulomb’s Law
 Coulomb’s
law
- The magnitude of the electric force between two point charges
is directly proportional to the product of the charges and inversely
proportional to the square of the distance between them
F  ke
r
: distance between two charges
q1,q2 : charges
ke : Coulomb constant 8.9875 x109 Nm/C2
q1q2
r2
- The directions of the forces the two charges exert on each other
are always along the line joining them.
- When two charges have the same sign, the forces are repulsive.
- When two charges have opposite signs, the forces are attractive.
F21
q1
q2
q1
q2
q1
q2
+
+
-
-
+
-
r
F12
F21
r
F12
F21
r
F12
Coulomb’s Law
 Coulomb’s
F  ke
law and units
q1q2
r2
r
: distance between two charges (m)
q1,q2 : charges
(C)
ke : Coulomb’s constant
ke  8.9875 109 N  m 2 / C 2
SI unit
 8.988 109 N  m 2 / C 2
 9.0 109 N  m 2 / C 2
c  2.99792458  108 m / s
ke  (10 7 N  s 2 / C 2 )c 2

1
40
Exact by definition
;  0  8.854 10 12 C 2 /( N  m 2 )
e  1.602176462(63) 1019 C
1 C  106 C, 1 nC  10-9 C
charge of a proton
Coulomb’s Law
 Example
: Electric forces vs. gravitational forces
q  2e  3.2 1019 C
electric force
q2
Fe 
40 r 2
1
2
gravitational force
m
Fg  G 2
r
m  6.64 1027 kg
q
q
+
+
neutron
proton
r
0
+ +
0
a particle
Fe
1 q2
9.0 109 N  m 2 / C 2
(3.2 1019 C)2


2
Fg 40G m
6.67 1011 N  m 2 / kg 2 (6.64 10 27 kg) 2
 3.11035
Gravitational force is tiny compared with electric force!
Coulomb’s law
 Example
: Forces between two charges
q1  25 nC, q2  75 nC
+
r
F21
F12 
1
r  3.0 cm
F12
q1q2
40 r 2
(25 10 9 C)(75 10-9 C)
 (9.0 10 N  m / C )
(0.030 m) 2
9
 0.019 N
 F21


F12   F21
2
2
Coulomb’s law
 Superposition of forces Principle of superposition
When two charges exert forces simultaneously on a third charge,
the total force acting on that charge is the vector sum of the forces
that the two charges would exert individually.
 Example : Vector addition of electric forces on a line
F23
q3
F13
+
q2
q1
+
-
2.0 cm
4.0 cm
Coulomb’s Law
 Example 15.2: May the force be zero
Three charges lie along the x-axis as
in Fig. The positive charge q1=15 C is
at x=2.0 cm, and the positive charge
q2=6.0 C is at the origin. Where must
a negative charge q3 be placed on the
x-axis so that the resultant electric force
on it is zero?
q2
F23
+
q3
-
F13
q1
+
2.0 cm-x
x
2.0 cm
Coulomb’s law
 Example
: Vector addition of electric forces in a plane
q1=2.0 C
+
0.50 m
0.30 m
0.40 m
Q=2.0 C
a
+
0.30 m
0.50 m
+
F1Q 
a
q1=2.0 C

( F1Q ) y
1
q1Q
40 r1Q 2

( F1Q ) x

F1Q
0.40m
 0.23 N
0.50m
0.30m
( F1Q ) y  ( F1Q ) sin a  (0.29 N)
 0.17 N
0.50m
( F1Q ) x  ( F1Q ) cos a  (0.29 N)
(4.0 10 6 C)(2.0 10-6 C)
 (9.0 10 N  m / C )
(0.50 m) 2
9
2
2
 0.29 N
Fx  0.23N  0.23N  0.46 N
Fy  0.17 N  0.17 N  0
Electric Field
 Electric
A
+ ++
+
+
 +
++
 F0
field and electric forces
B
q0
+
A

F0
remove body B
+ ++
+
+
+ ++
P
•Existence of a charged body A modifies property of space and
produces an “electric field”.
•When a charged body B is removed, although the force exerted on
the body B disappeared, the electric field by the body A remains.
•The electric force on a charged body is exerted by the electric field
created by other charged bodies.
Electric Field
 Electric
field and electric forces (cont’d)
A
A
+ ++
+
+
+ ++
P
placing a test charge
+ ++
+
+
 +
++
 F0
Test charge
q0

F0
• To find out experimentally whether there is an electric field at a
particular point, we place a small charged body (test charge) at

the point.
 F0
• Electric field is defined by E 
(N/C in SI units)
q0
• The force on a charge q:


F  qE
Electric Field
 Electric
field of apoint charge
E
q

q0 E
0
unit vector
P
r̂
q

rˆ  r / r
+
r̂
S
S
F0 
q
P
1
qq0
+
40 r 2

 F0
E
q0
q0
P
r̂

E
1
q
rˆ
2
40 r
q
+
'
E
r̂ '
S

'
rr  E  E
'

E
P’
Electric Field Lines
 An
electric field line is an imaginary line or curve drawn
through a region of space so that its tangent at any point
is in the direction of the electric-field vector at that point.

 Electric field lines show the direction of E at each point,
and their spacing gives a general idea of the magnitude of
E at each point.

 Where E is strong, electric
 field lines are drawn bunched
closely together; where E is weaker, they are farther apart.
 At
any particular point, the electric field has a unique
direction so that only one field line can pass through each
point of the field. Field lines never intersect.
Electric Field Lines
 Field
line drawing rules:
• E-field lines begin on + charges and end on - charges. (or infinity)
• They enter or leave charge symmetrically.
• The number of lines entering or leaving a charge is proportional to
the charge.
• The density of lines indicates the strength of E at that point.
• At large distances from a system of charges, the lines become
isotropic and radial as from a single point charge equal to the net
charge of the system.
• No two field lines can cross.
Electric Field Lines
 Field
line examples
Electric Field Lines
 Field
line examples (cont’d)
Electric Field Lines
 An
electric dipole is a pair of point charges with equal
magnitude and opposite sign separated by a distance d.
electric dipole moment
q
qd
d
 Water
molecule and its electric dipole
q
Millikan Oil-Drop Experiment
 Millikan’s
experiment
drag force
D
if v<0
(i.e. Eq<mg)
D
if v>0
(i.e. Eq>mg)
E=0
When the drag force, which is proportional to the velocity of the
drop, becomes equal to mg, the drop reaches the terminal velocity.
E=0
If the oil drop moves downward, the drag force points upward.
When Eq=mg+D, the drop reaches the terminal velocity. Knowing
the terminal velocity, mass of the drop, and the magnitude of the
electric field, the charge of the drop can be measured.
Electric Flux and Guass’s Law

Some definitions
Closed surface : A closed surface has an inside and outside.
Electric flux
: A measure of how much the electric field vectors
penetrate through a given surface.

Electric flux
electric flux:
A (area)
 E  EA

E

A  Anˆ
n̂ : Normal unit vecto r
Unit v ector perpendicu lar to the plane

A  Anˆ
electric flux:
 
 E  EA cos q  E A  E  A
q

E
Calculating Electric Flux

n̂3
Example : Electric flux through a cube
n̂5

E
n̂2
n̂1
n̂4
n̂6
L

 E1  E  nˆ1 A  EL2 cos180   EL2

 E2  E  nˆ2 A  EL2 cos 0   EL2
 E3   E4   E5   E6  EL2 cos 90  0
 E  i 1  Ei  0
i 6
Calculating Electric Flux

Example : Electric flux through a sphere

dA
r=0.20 m
+
+q


E  E , E // nˆ // dA
3.0 10 6 C
E
 (9.0  10 N  m / C )
2
40 r
(0.20m) 2
q
9
2
2
 6.75  105 N/C
 E  EA  (6.75 105 N/C)(4 )(0.20 m) 2
q=3.0 C
A=2r2
 3.4  105 N  m 2 / C
Gauss’s Law

Preview:
The total electric flux through any closed surface (a surface enclosing
a definite volume) is proportional to the total (net) electric charge inside
the surface.

Case 1: Field of a single positive charge q

E
A sphere with r=R
1
r=R
+
q

E

E  surface
q
E
40 R 2
 E  EA 
at r=R
1
q
q
(4R 2 ) 
40 R
0
The flux is independent of the radius R of
the surface.
Gauss’s Law

Case 2: More general case with a charge +q

E
A
 
E  n  E
 E cos 

E A
q
(A) cos q
+
q
+
surface
perpendicular

to E
E  E A  E cosqA
E 
q
0
Gauss’s Law

Case 3: An closed surface without any charge inside
E  0
Electric field lines that go in come out.
Electric field lines can begin or end inside
a region of space only when there is charge
in that region.
+

Gauss’s law
E 
Qinside
0


; Qinside  i qi , E  i Ei
The total electric flux through a closed surface is equal to the total
(net) electric charge inside the surface divided by 0
Applications of Gauss’s Law

Example 15.8: Field of an infinite plane sheet of charge

E  the sheet  E  E
 : charge density

E
+ +
+
Qinside  A

E
+
 E  2( EA) 
+
+ +
+
+ +
+

E
2 0
Gaussian surface
Note:

Ex 
2 0
Ex  

2 0
x0
x0
two end surfaces
A
0
Gauss’s Law

Example : Field between oppositely charged parallel
conducting plane
  plate 1
E1 E 2
+ b
+
a
+
S1 +
+
+
+
S2
+
+

E1

E2

E
plate 2   Solution 1:
- E 2 E1
c
- S
- 4
S3
-
No electric flux
on these surfaces
A

S1 :
EA 
 E  (right surface)
0
0
outward flux
E0
(left surface)
 A

S 4 :  EA 
 E  (left surface)
inward flux
0
E0
0
(right surface)
Solution 2:


   
At Point a : E1   E2  E  E1  E2  0



b : E1  E2  E  2   
2 0  0
c:


   
E1   E2  E  E1  E2  0
Application of Gauss’s Law
 Trajectory of a charged particle in a uniform electric field
Applications of Gauss’s Law

Example : Field of an infinite line of charges

Gaussian surface
E , E  E 
dA
chosen according
to symmetry
line charge
density
Qencl  
E  E on the cylindrica l Gaussian surface
 E  E (2 r) 
E
1

20 r

0
Applications of Gauss’s Law

Example : Field of a uniformly charged sphere
Gaussian surface
4 3
charge density r  R : EA   (  r ) /  0
+
+
3
4 3
2
Q
+
E
(
4

r
)


(
 r ) / 0
+
+ +
+  4
3
+r=R
R 3
3
1 Qr
+
+
E
+
40 R 3
1 Q
r  R: E 
40 R 2
Q
2
R  r : E (4 r ) 
R
0
+
1
Q
E
40 r 2
Applications of Gauss’s Law

Example 15.7 : Field of a charged spherical shell
Gaussian surface
+
+
b
+
+
r  a : EA  E (4 r ) 
+
2
+
a
+
+
Total charge on the shell = Q
b  r : E (4 r ) 
2
Qinside
0
Qinside
0
Q
0E 0
1
Q
 E
0
40 r 2
Applications of Gauss’s Law

Charge distribution and field
• The charge distribution
the field
• The symmetry can simplify the procedure of application

Electric field by a charge distribution on a conductor
• When excess charge is placed on a solid conductor and is at rest,
it resides entirely on the surface, not in the interior of the material
(excess charge = charge other than the ions and free electrons that
make up the material conductor
A Gaussian surface inside conductor
Charges on surface
Conductor
Applications of Gauss’s Law

Electric field by a charge distribution on a conductor (cont’d)
A Gaussian surface inside condactor
Charges on surface
Conductor
E at every point in the interior of a conducting material
is zero in an electrostatic situation (all charges are at rest).
If E were non-zero, then the charges would move
• Draw a Gaussian surface inside of the conductor
• E=0 everywhere on this surface (inside conductor)
Gauss’s law
• The net charge inside the surface is zero
• There can be no excess charge at any point within a solid conductor
• Any excess charge must reside on the conductor’s surface
• E on the surface is perpendicular to the surface
Charges on Conductors

Case 1: charge on a solid conductor resides entirely on
its outer surface in an electrostatic situation
+ + +
+
+
+
+
+
+ ++
+
++
+ +

The electric field at every point within a conductor
is zero and any excess charge on a solid conductor
is located entirely on its surface.
Case 2: charge on a conductor with a cavity
+ + +
+
+
+
+
+
+ ++
+
++
+ +
If there is no charge within the cavity, the net
charge on the surface of the cavity is zero.
Gauss surface
Charges on Conductors

Case 3: charge on a conductor with a cavity and a charge q
inside the cavity
+ + +
+
+
- - +
+
- +
- +
+
- - +
+ ++
+
++
Gauss surface
• The conductor is uncharged and insulated from
charge q.
• The total charge inside the Gauss surface should
be zero from Gauss’ law and E=0 on this surface.
Therefore there must be a charge –q distributed
on the surface of the cavity.
• The similar argument can be used for the case
where the conductor originally had a charge qC.
In this case the total charge on the outer surface
must be q+qC after charge q is inserted in cavity.