PHY127 Summer Session II
• Most of information is available at:
http://nngroup.physics.sunysb.edu/~chiaki/PHY127-08
• The website above is the point of contact outside the class for important
messages, so regularly and frequently check the website.
• At the end of class a quiz is given for the previous chapter covered in
the class. Bring a calculator (no wireless connection), a pencil, an eraser,
and a copy of lecture note for the chapter.
• The lab session is an integrated part of the course and make sure that
you will attend all the sessions. See the syllabus for the detailed
information and the information (e.g. lab manuals) at the website above.
• 5 homework problems for each chapter are in general due a week later
at 11:59 pm and are delivered through MasteringPhysics website at:
http://www.masteringphysics.com. You need to open an account.
• In addition to homework problems, there is naturally a reading
requirement of each chapter, which is very important.
Chapter 20: Electric Charge/Force/Field
Electric charge
‰ When
a plastic rod is rubbed with a piece
of fur, the rod is “negatively” charged
‰ When
a glass rod is rubbed with a piece
of silk, the rod is “positively” charged
‰ Two
equally signed charges repel each
other
‰ Two
opposite signed charges attract each
other
‰
Electric charge is conserved
Electric charge (cont’d)
Electric charge (cont’d)
Particle Physics
What is the world made of?
nucleus
Model of Atoms
proton
Old view
electrons equarks
nucleus
Modern view
Semi-modern view
Electric charge (cont’d)
•
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 and neutron,
– 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
Electric charge (cont’d)
• 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.
Charging by contact
Charging by induction (cont’d)
Conductors, insulators, and induced charges
‰ Conductors
‰ Insulators
: material in which charges can freely
move. metal
: material in which charges are not
readily transported. wood
‰ Semiconductors
: material whose electric property is
in between. silicon
‰ Induction
: A process in which a donor material
gives opposite signed charges to
another material without losing any of
donor’s charges
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 =k
q1q2
r2
r
: distance between two charges
q1,q2 : charges
k
: a proportionality constant
- 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.
q1
q2
q1
q2
q1
q2
+
+
-
-
+
-
F2 on 1
r
F1 on 2
F2 on 1
r
F1 on 2
F2 on 1
r
F1 on 2
Coulomb’s law
‰ Coulomb’s
F =k
law and units
q1q2
r2
r
: distance between two charges (m)
(C)
q1,q2 : charges
k
: a proportionality constant (=ke)
k = 8.987551787 ×109 N ⋅ m 2 / C 2
SI units
≅ 8.988 ×109 N ⋅ m 2 / C 2
≅ 9.0 ×109 N ⋅ m 2 / C 2
c = 2.99792458 × 108 m / s
k = (10 −7 N ⋅ s 2 / C 2 )c 2
=
1
4πε 0
Exact by definition
; ε 0 = 8.854 ×10 −12 C 2 /( N ⋅ m 2 )
e = 1.602176462(63) ×10 −19 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 =
4πε 0 r 2
1
m
Fg = G 2
r
2
gravitational force
m = 6.64 × 10 − 27 kg
q
q
+
+
neutron
proton
r
0
+ +
0
α particle
Fe
1 q2
9.0 × 109 N ⋅ m 2 / C 2
(3.2 × 10 −19 C) 2
=
=
2
Fg 4πε 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
F2 on 1
F1 on 2 =
1
-
r = 3.0 cm
F1 on 2
q1q2
4πε 0 r 2
−9
-9
(
25
10
C)(75
10
C)
×
×
= (9.0 ×109 N ⋅ m 2 / C 2 )
(0.030 m) 2
= 0.019 N
r
F1 on 2
= F2 on 1
r
= − F2 on 1
Coulomb’s law
‰ Superposition of forces Principle of superposition of kforces
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
F2 on 3
q3
+
F1 on 3
q2
q1
+
-
2.0 cm
4.0 cm
Coulomb’s law
‰ Example:
Vector addition of electric forces in a plane
q1=2.0 µC
+
0.50 m
0.30 m
0.40 m
Q=4.0 µC
α
+
0.30 m
0.50 m
+
F1 on Q =
α
q2=2.0 µC
r
( F1 on Q ) y
q1Q
4πε 0 r1Q 2
1
= 0.29 N
r
F1 on Q
0.40m
= 0.23 N
0.50m
0.30m
( F1 on Q ) y = ( F1 on Q ) sin α = −(0.29 N)
= −0.17 N
0.50m
force due to q2
( F1 on Q ) x = ( F1 on Q ) cos α = (0.29 N)
(4.0 ×10 −6 C)(2.0 ×10 -6 C)
= (9.0 × 10 N ⋅ m / C )
(0.50 m) 2
9
r
( F1 on Q ) x
2
2
Fx = 0.23N + 0.23N = 0.46 N
Fy = −0.17 N + 0.17 N = 0
Electric field and electric forces
‰ Electric
A
+ ++
+
+
r
− F0 + + +
field and electric forces
B
q0
+
A
r
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 and electric forces
‰ Electric
field and electric forces (cont’d)
A
A
+ ++
+
+
+ ++
P
placing a test charge
+ ++
+
+
r
− F0 + + +
Test charge
q0
r
F0
• To find out experimentally whether there is an electric field at a
particular point, we place a small charged body (test charge) at
r
point.
r F0
• Electric field is defined by E =
(N/C in SI units)
q0
• The force on a charge q:
r
r
F = qE
Electric field and electric forces
‰ Electric
field of arpoint charge
q0 E
q
r
rˆ = r / r
P
r̂
q
+
P
r̂
S
S
F0 =
r
E
q0
1
qq0
r
r F0
E=
q0
+
4πε 0 r 2
q0
P
r̂
r
E=
q
rˆ
2
4πε 0 r
1
q
+
r'
E
r̂ '
S
r
r'
r>r → E < E
'
r
E
P’
Electric field and electric forces
‰ Electric
field by a continuous charge distribution
q
Electric field and electric forces
‰ Electric
field by a continuous charge distribution (cont’d)
These may be considered in 1, 2 or 3 dimensions.
There are some usual conventions for the notation:
Charge per unit length is λ ; units C/m i.e, dq = λ dl
Charge per unit area is σ ; units C/m2 i.e, dq = σ dA
Charge per unit volume is ρ ; units C/m3 i.e, dq = ρdV
Electric field and electric forces
‰ Example
: Electron in a uniform field
-
y
r
1.0 cm E
O
x
- r
r
F = −eE
100 V
+
ƒ Two large parallel conducting plates connected to a battery produce
E = 1.00 ×10 4 N/C
uniform electric field
ƒ Since the electric force is constant, the acceleration is constant too
− eE (−1.60 × 10 −19 C)(1.00 × 10 4 N/C)
15
2
ay =
1
.
76
10
m/s
=
−
×
=
=
m
m
9.11× 10 −31 kg
Fy
ƒ From the constant-acceleration formula: υ y = υ0 y + 2a y ( y − y0 )
υ y = 2a y y = 5.9 ×106 m/s ← υ0y = 0, y0 = 0 when y = −1.0 cm
2
ƒ The electron’s kinetic energy is:
ƒ The time required is:
t=
υ y − υ0 y
ay
2
K = (1 / 2)mυ 2 = 1.6 × 10 −17 J
= 3.4 ×10 −9 s
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.
r
‰ Electric field lines show the direction of E at each point,
and
r their spacing gives a general idea of the magnitude of
E at each point.
r
‰ Where E is strong, electric
r 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.
‰ Field
line examples
Electric field lines (cont’d)
‰ Field
line examples (cont’d)
Electric Dipoles
‰ 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
Electric Dipoles
‰ Force
and torque on an electric dipole
q
r
r
F− = −qE
torque:
electric dipole moment:
φ
r
r
F+ = qE
q
τ = (qE )(d sin φ )
p = qd
r r
τ = p× E
r
work done by a torque τ
during an infinitesimal
displacement dφ
:
dW = τ dφ = − pE sin φ dφ
Electric Dipoles
‰ Force
and torque on an electric dipole (cont’d)
q
r
r
F− = −qE
φ
r
r
F+ = qE
q
r r
U (φ ) ≡ − pE cos φ = − p ⋅ E
φ2
φ2
φ1
φ1
potential energy for a dipole
in an electric field
W = ∫ τ dφ = ∫ (− pE sin φ )dφ = pE cos φ2 − pE cos φ1
= −(U 2 − U1 )
Exercises
‰ Trajectory of a charged particle in a uniform electric field
Exercises
‰ Cathode ray tube
Exercises
‰ Electric field by finite line charge
Exercises
‰ Electric field by a ring charge
Exercises
‰ Electric field by a uniformly charged disk
Exercises
‰ Electric field by infinite plate charge
+
+
+
+
+
+
+
Exercises
‰ Electric field by two oppositely charged parallel planes
Exercises
‰ Far field by an electric dipole
R+ ≈ R −
φ)
q
d
−q
(d / 2) cos θ
φ
(1 + x) ≅ 1 + nx when x << 1
n
E ( P) =
d
cos θ
2
q
4πε 0
(
R− ≈ R +
d
cos θ
2
1
1
−
)
R+2 R−2
⎡
⎤
⎥
1
1
q ⎢
≅
−
⎢
⎥
4πε 0 ⎢ ( R − d cos) 2 ( R + d cos θ ) 2 ⎥
2
2
⎣
⎦
⎡
⎤
⎥
1
1
q ⎢
=
−
⎢
⎥
4πε 0 ⎢ R 2 (1 − d cos θ ) 2 R 2 (1 + d cos θ ) 2 ⎥
2R
2R
⎣
⎦
≅
=
q
1
4πε 0 R 2
q
1
4πε 0 R 2
⎡
⎤
⎢
⎥
1
1
−
⎢ d
⎥
d
⎢1 − cos θ 1 + cos θ ⎥
R
⎣ R
⎦
1
2d
q 1
cos
θ
cos θ =
∝
2πε 0 R 3
R
R3