Download L1 Coulomb

Ben Gurion University of the Negev
www.bgu.ac.il/atomchip
Physics 2B for Materials and Structural Engineering
Lecturer: Daniel Rohrlich
Teaching Assistants: Oren Rosenblatt, Irina Segel
Week 1. Charge, E and Coulomb’s law – Introduction • electrical
charges, quantization and conservation • Coulomb’s law • addition
of electric forces • electric field
Sources: Halliday, Resnick and Krane, 4th Edition, Chap. 27;
Halliday, Resnick and Krane, 5th Edition, Chaps. 25-26;
Purcell (Berkeley course 2), Chap. 1, Sects. 1.3 – 1.4 and 1.7.
Introduction
Both electricity and magnetism were known to the ancient
Greeks 2600 years ago. When they rubbed a piece of amber
with fur, the amber would attract straw and hair.
Introduction
Both electricity and magnetism were known to the ancient
Greeks 2600 years ago. When they rubbed a piece of amber
with fur, the amber would attract straw and hair.
Amber
is resin
that
became
a fossil.
Introduction
Both electricity and magnetism were known to the ancient
Greeks 2600 years ago. When they rubbed a piece of amber
with fur, the amber would attract straw and hair.
amber = ήλεκτρον = ‫ענבר = חַ ְׁשמַ ל‬
Introduction
Both electricity and magnetism were known to the ancient
Greeks 2600 years ago. When they rubbed a piece of amber
with fur, the amber would attract straw and hair.
amber = ήλεκτρον = ‫ענבר = חַ ְׁשמַ ל‬
.‫ּומּתֹוכָּה כְׁ עֵין הַ חַ ְׁשמַ ל ִמּתֹוְך הָ אֵ ש‬
ִ ‫ ָענָן גָדֹול וְׁ אֵ ש ִמ ְׁתלַקַ חַ ת וְׁ נֹ גַּה לֹו סָ בִ יב‬:)‫ד‬,‫יחזקאל (א‬
According to the Septuagint, “‫“ = ”חַ ְׁשמַ ל‬amber”.
Introduction
The ancient Greeks also knew of stones that would attract iron.
Magnetite from Utah, U.S.A. © R.Weller/Cochise College.
2600 years ago, and even 300 years ago, electric and magnetic
phenomena appeared to be very unusual. They appeared to
have very little to do with the rest of nature, and nothing to do
with each other.
2600 years ago, and even 300 years ago, electric and magnetic
phenomena appeared to be very unusual. They appeared to
have very little to do with the rest of nature, and nothing to do
with each other.
Today, our view is about as different from this ancient view as
possible: everything in the natural world, including all of
biology and chemistry, depends on electricity and magnetism;
also electricity and magnetism depend on each other.
2600 years ago, and even 300 years ago, electric and magnetic
phenomena appeared to be very unusual. They appeared to
have very little to do with the rest of nature, and nothing to do
with each other.
Today, our view is about as different from this ancient view as
possible: everything in the natural world, including all of
biology and chemistry, depends on electricity and magnetism;
also electricity and magnetism depend on each other.
Why did it take so long to see that electricity and magnetism
are everywhere?
Electrical charges, quantization and conservation
Electrostatics: the study of electric charges at rest.
Electrical charges, quantization and conservation
Electrostatics: the study of electric charges at rest.
How do we know that there are two kinds of charge? (We call
them “positive” and “negative”, but any other names would be
as good.)
Electrical charges, quantization and conservation
Electrostatics: the study of electric charges at rest.
How do we know that there are two kinds of charge? (We call
them “positive” and “negative”, but any other names would be
as good.)
Since electric charges can repel as well as attract, there must be
at least two kinds.
Electrical charges, quantization and conservation
Electrostatics: the study of electric charges at rest.
How do we know that there are two kinds of charge? (We call
them “positive” and “negative”, but any other names would be
as good.)
Since electric charges can repel as well as attract, there must be
at least two kinds.
We have heard that “same charges repel, opposite charges
attract”, but could it be the other way around? Could our world
be a world in which “opposite charges repel, same charges
attract”?
Electrical charges, quantization and conservation
Electric charge is quantized, and nobody knows why!
Not even quantum mechanics explains (so far) why charge is
quantized.
• All electrons have exactly the same charge –e:
e = (1.602 176 487 ± 0.000 000 040) × 10-19 C
Our unit of charge is C and is called a Coulomb.
Electrical charges, quantization and conservation
Electric charge is quantized, and nobody knows why!
Not even quantum mechanics explains (so far) why charge is
quantized.
• All electrons have exactly the same charge –e:
e = (1.602 176 487 ± 0.000 000 040) × 10-19 C
The first person to measure e was R. A. Millikan (around 1910).
Electrical charges, quantization and conservation
Electric charge is quantized, and nobody knows why!
Not even quantum mechanics explains (so far) why charge is
quantized.
• All electrons have exactly the same charge –e:
e = (1.602 176 487 ± 0.000 000 040) × 10-19 C
The first person to measure e was R. A. Millikan. He sprayed
droplets of oil and measured the mass m of each droplet from
its fall. Then he applied an electric field of strength E to
balance the droplet in mid-air, and extracted e from mg = eE.
Electrical charges, quantization and conservation
• The proton and the electron are very different, but the
electron charge and the proton charge are known to be the
same (except for sign) to an accuracy of one part in 1020.
So why did it take so long to see that electricity and magnetism
are everywhere?
Electrical charges, quantization and conservation
Conservation of electric charge: The total electric charge in an
isolated system does not change.
There are processes that change the number of charged
particles in an isolated system, but no process changes the total
electric charge.
Coulomb’s law
Consider two fixed point charges. One is located at r1 and has
charge q1; the other is located at r2 and has charge q2. The
force of the charge located at r1 on the charge at r2 has
magnitude
F12  k
q1q2
r12 2
,
where r12 = | r1 – r2 | and the constant k in the units of this
course is
k
1
4 0
 8.99  10 9 N  m 2 /C2 .
q2
q1
Coulomb’s law (vector formulation)
Consider two fixed point charges. One is located at r1 and has
charge q1; the other is located at r2 and has charge q2. The
force of the charge located at r1 on the charge at r2 is
F12 
1
q1q2
4 0 r12 2
rˆ12 ,
where r̂12 is a unit vector pointing from r1 to r2. By Newton’s
Third Law, the force F21 of the charge located at r2 on the
charge at r1 is equal in magnitude and opposite in sign.
F12
q2
F21
q1
Coulomb’s law (vector formulation)
Consider two fixed point charges. One is located at r1 and has
charge q1; the other is located at r2 and has charge q2. The
force of the charge located at r1 on the charge at r2 is
F12 
1
q1q2
4 0 r  r
2
1
3
r2  r1 
,
writing it a slightly different way. By Newton’s Third Law, the
force F21 of the charge located at r2 on the charge at r1 is equal
in magnitude and opposite in sign.
F12
q2
F21
q1
Addition of electric forces
Electric forces add like vectors! Suppose we have three fixed
point charges. One is located at r1 and has charge q1; the
second is located at r2 and has charge q2; the third is located at
r3 and has charge q3. What is the total force of the charges at
r1 and r2 on the point charge located at r3?
q3
q2
q1
Addition of electric forces
Electric forces add like vectors! Suppose we have three fixed
point charges. One is located at r1 and has charge q1; the
second is located at r2 and has charge q2; the third is located at
r3 and has charge q3. What is the total force of the charges at
r1 and r2 on the point charge located at r3?
F13
F23 q3
q2
q1
Addition of electric forces
Electric forces add like vectors! Suppose we have three fixed
point charges. One is located at r1 and has charge q1; the
second is located at r2 and has charge q2; the third is located at
r3 and has charge q3. What is the total force of the charges at
r1 and r2 on the point charge located at r3? It is
F13  F23 
1
q1q3
4 0 r132
rˆ13 
1
q2 q3
4 0 r232
rˆ23 .
F13
F23 q3
q2
q1
Addition of electric forces
Electric forces add like vectors! Suppose we have three fixed
point charges. One is located at r1 and has charge q1; the
second is located at r2 and has charge q2; the third is located at
r3 and has charge q3. What is the total force of the charges at
r1 and r2 on the point charge located at r3? It is
F13  F23 
F13 + F23
1
q1q3
4 0 r132
rˆ13 
1
q2 q3
4 0 r232
rˆ23 .
F13
F23 q3
q2
q1
Addition of electric forces
Example 1 (electric dipole): Two equal and opposite point
charges lie on the z-axis: charge e is at z = a and charge –e is
at z = –a. What is the force Fq on a point charge q on the z-axis
at arbitrary z?
q
e
0
–e
Addition of electric forces
Example 1 (electric dipole): Two equal and opposite point
charges lie on the z-axis: charge e is at z = a and charge –e is
at z = –a. What is the force Fq on a point charge q on the z-axis
at arbitrary z?
q
e

eq
eq 

zˆ
Answer: Fq 


4 0  z  a 2 z  a 2 
1
4 zaeq

zˆ
4 0 z 2  a 2 2
1


0
–e
1


aeq
 0 z 3 1  a 2 / z 2

2
zˆ
Addition of electric forces
Example 1 (electric dipole): Two equal and opposite point
charges lie on the z-axis: charge e is at z = a and charge –e is
at z = –a. What is the force Fq on a point charge q on the z-axis
at arbitrary z?
q
e

eq
eq 
Answer: Fq 

zˆ


4 0  z  a 2 z  a 2 
1
4 zaeq

zˆ
4 0 z 2  a 2 2
1


0
–e
1 aeq
 0 z 3
1  a

2

2
/ z  ... zˆ
2
Addition of electric forces
Example 1 (electric dipole): Two equal and opposite point
charges lie on the z-axis: charge e is at z = a and charge –e is
at z = –a. What is the force Fq on a point charge q on the z-axis
at arbitrary z?
q
e

eq
eq 
Answer: Fq 

zˆ


4 0  z  a 2 z  a 2 
1
4 zaeq

zˆ
4 0 z 2  a 2 2
1


0
–e
1 aeq
 0 z 3

zˆ
for z >> a.
Addition of electric forces
Let’s continue this example with the charge q on the x-axis at
arbitrary x. What is the force Fq on the charge q?
e
q
0
–e
Addition of electric forces
Let’s continue this example with the charge q on the x-axis at
arbitrary x. What is the force Fq on the charge q?
Answer: Each charge on the z-axis produces a force of
magnitude F = eq/4πε0(x2 + a2) but the x-components of
these forces cancel. The net force is down:
Fq  (zˆ )2 F sin  ,
 (zˆ )2 F
e
θ
θ
0
–e
q

a
x2  a2
eqa
2 0 [ x 2  a 2 ]3 / 2
zˆ .
Addition of electric forces
Example 2: An infinite, stationary straight line carries uniform
charge per unit length λ. Off the wire, a distance L from it
(closest approach) is a fixed point charge of magnitude q.
What is the force of the line on the point charge?
q
L
Addition of electric forces
Example 2: An infinite, stationary straight line carries uniform
charge per unit length λ. Off the wire, a distance L from it
(closest approach) is a fixed point charge of magnitude q.
What is the force of the line on the point charge?
z
Answer: First we choose convenient coordinates.
q
0
L
x
Addition of electric forces
Example 2: An infinite, stationary straight line carries uniform
charge per unit length λ. Off the wire, a distance L from it
(closest approach) is a fixed point charge of magnitude q.
What is the force of the line on the point charge?
z
Answer: First we choose convenient coordinates.
Now consider two infinitesimal line elements of
length dz and charge λdz symmetrically spaced
λdz
above and below the origin on the z-axis.
q
0
x
λdz
L
Addition of electric forces
Example 2: An infinite, stationary straight line carries uniform
charge per unit length λ. Off the wire, a distance L from it
(closest approach) is a fixed point charge of magnitude q.
What is the force of the line on the point charge?
z
Each line element produces a force of magnitude
F = qλdz/4πε0 (z2 + L2) but the z-components of
these forces cancel. The net force is horizontal:
q
0
L
θ
θ
Fnet  2 F cos xˆ
x
 1
qdz 
L

 2
xˆ

2
2
4

z

L
0

 z 2  L2
1
qLdz

xˆ .
2 0 [ z 2  L2 ]3 / 2
Addition of electric forces
Example 2: An infinite, stationary straight line carries uniform
charge per unit length λ. Off the wire, a distance L from it
(closest approach) is a fixed point charge of magnitude q.
What is the force of the line on the point charge?
z
For the total force on the point charge q, we
integrate this expression:
q
0
L
θ
θ
Fq 
xˆ

qL
dz

2
2
3
/
2
2 0 [ z  L ]
0
.
Addition of electric forces
Example 2: An infinite, stationary straight line carries uniform
charge per unit length λ. Off the wire, a distance L from it
(closest approach) is a fixed point charge of magnitude q.
What is the force of the line on the point charge?
Fq 
xˆ

qL
dz

2
2
3
/
2
2 0 [ z  L ]
0
Substitute z = L tan θ, dz = L dθ /cos2θ:
.
Addition of electric forces
Example 2: An infinite, stationary straight line carries uniform
charge per unit length λ. Off the wire, a distance L from it
(closest approach) is a fixed point charge of magnitude q.
What is the force of the line on the point charge?
Fq 
xˆ
2 0

qL
 [ z 2  L2 ]3 / 2
0
Substitute z = L tan θ, dz = L dθ /cos2θ:
 /2

xˆ
2 0
dz

qL
Ld 
2
2
2 3/ 2
2
[
L
tan


L
]
cos

0
Addition of electric forces
Example 2: An infinite, stationary straight line carries uniform
charge per unit length λ. Off the wire, a distance L from it
(closest approach) is a fixed point charge of magnitude q.
What is the force of the line on the point charge?
Fq 
xˆ

qL
dz

2
2
3
/
2
2 0 [ z  L ]
0

xˆ
 /2
x 2
0

xˆ
2 0 L
qL

Ld 
2
2
2 3/ 2
2
[
L
tan


L
]
cos

0
 /2

0
q cosd 
q
2 0 L
xˆ .
Electric field
In general, a set of fixed point charges q1, q2, q3,… located at
r1, r2, r3,… produces an electric force Fq on a fixed point
charge q located at r. If we divide Fq by q, we get the electric
field E(r) at the point r arising from the charges q1, q2, q3,… at
r1, r2, r3,…:
Fq
q
q3
q5
q4
q2
q6
q1
q7
Electric field
In general, a set of fixed point charges q1, q2, q3,… located at
r1, r2, r3,… produces an electric force Fq on a fixed point
charge q located at r. If we divide Fq by q, we get the electric
field E(r) at the point r arising from the charges q1, q2, q3,… at
r1, r2, r3,…:
E(r)
q3
q5
q4
q2
q6
q1
q7
Electric field
Example 1: What is the electric field of an electron?
Electric field
Example 1: What is the electric field of an electron?
Answer: The force of an electron at the origin on a point
charge at r is
Fq  
1
eq
4 0 r 2
rˆ ,
therefore E(r) is
E(r )  
1
e
4 0 r 2
rˆ .
Electric field
Example 2: What is the electric field of a line of charge with
uniform linear charge density λ?
Answer: We found that the force due to the line of charge on a
point charge q at a distance L from the line is
q
Fq  
ρˆ ,
2 0 
where ρ is the radial coordinate with respect to the line,
therefore E(ρ) is

E(  )  
ρˆ .
2 0 
Electric field
What difference does it make whether we talk about point
charges that produce an electric force, or about an electric
field?
If the charges don’t move, it doesn’t make a difference. But if
the charges move, it does make a difference. We will see that
the electric field E(r) takes on a life of its own; it is not simply
a function of where the charges are.
Halliday, Resnick and Krane, 5th Edition, Chap. 25, Prob. 4(a):
Two small balls of mass m and charge q hang from the same
point on threads of length L. At equilibrium, their separation is
x and their angular separation is 2θ. Assuming θ small so that
sin θ ≈ θ ≈ tan θ, what is x?
θ θ
x
Halliday, Resnick and Krane, 5th Edition, Chap. 25, Prob. 4(a):
Two small balls of mass m and charge q hang from the same
point on threads of length L. At equilibrium, their separation is
x and their angular separation is 2θ. Assuming θ small so that
sin θ ≈ θ ≈ tan θ, what is x?
Answer: Let T denote the tension in the threads. The total
force must vanish in the vertical and horizontal directions,
hence
q2
T cos  mg and T sin  
q2
θ θ
x/2
 tan  
2
L
4 0 mgx
T
x
4 0 x 2
 q L
x
 2 0 mg

2
1/ 3




Halliday, Resnick and Krane, 5th Edition, Chap. 25, Prob. 11:
Two point charges q are held on the z-axis at points z = ±a.
Where on the xy-plane – at what distance R from the origin – is
the electric field strongest?
Halliday, Resnick and Krane, 5th Edition, Chap. 25, Prob. 11:
Two point charges q are held on the z-axis at points z = ±a.
Where on the xy-plane – at what distance R from the origin – is
the electric field strongest?
z
Answer: We use our previous calculation, replacing
the two line elements with two point charges. The
electric field at a distance R from the z-axis is
q
θ
θ
0
q
R
E ( R) 
1
qR
2 0 [ R  a ]
2
2 3/ 2
,
Halliday, Resnick and Krane, 5th Edition, Chap. 25, Prob. 11:
Two point charges q are held on the z-axis at points z = ±a.
Where on the xy-plane – at what distance R from the origin – is
the electric field strongest?
E ( R) 
z
1
qR
2 0 [ R 2  a 2 ]3 / 2
,
and we maximize it by requiring dE/dR = 0:
q
θ
θ
0
q
R
d
0
E ( R)
dR

1 
q
3
2qR 2


 2
 ,
2
3
/
2
2
2
5
/
2
2 0  [ R  a ]
2 [ R  a ] 
Halliday, Resnick and Krane, 5th Edition, Chap. 25, Prob. 11:
Two point charges q are held on the z-axis at points z = ±a.
Where on the xy-plane – at what distance R from the origin – is
the electric field strongest?


q
3
2qR 2
0

,
 2

2 0  [ R  a 2 ]3 / 2 2 [ R 2  a 2 ]5 / 2 
z
1
q
θ
θ
0
q
so R  a  3R
2
2
2
R  a/ 2 .
R