Chapter 16: Electric Forces and
Fields
•Electric Charge
•Conductors & Insulators
•Coulomb’s Law
•Electric Field
•Motion of a Point Charge in a Uniform E-field
•Conductors in Electrostatic Equilibrium
•Gauss’s Law
1
As reported by the Greek philosopher
Thales of Miletus around 600 BC
Greeks found that if they rub an amber
(or “elektron”) rod with animal fur the rod
attracts small light weight objects.
They “charged” the amber rod.
Conclusion:
There is a force between amber rod and
particles!
It is a fundamental force like gravity and
called the electric force.
Other combinations of materials like glass
and silk are showing the same effect.
2
3
Electric Charge
Amber-Amber
Repulsion
Amber – Glas
Attraction
“Opposites attracts”
Like charges repel each
other (++ or --)
Unlike charges repel
each other (+ -)
1747 Benjamin Franklin
(Besides he helped to draw up the Declaration of Independence)
There are two kinds of electric charge: positive and negative.
A body is electrically neutral if the sum of all the charges in a
body is zero.
4
Hollywood’s Viewpoint …
5
The elementary unit of charge is e = 1.602×10-19 C.
The charge on the electron is -1e.
The charge on the proton is +1e.
The charge on the neutron is 0e.
Particle
Representati
on
Relati
ve
charg
e
Relati
ve
mass
electron
-1
1
1800
proton
+1
1
neutron
o
1
The electron is part of a family of
fundamental particles known as leptons
The proton and the neutron are not
fundamental particles. Each consists of
three quarks.
The value of a charge has to be an integer number (…is
quantized) of
e (+/-e, +/-2e,+/-3e………. )
6
It can never be a decimal like 1.5e
So what is charge actually?
Charge is a property of fundamental particles that causes
them to exert forces on each other.
The unit of electrical charge is the Coulomb [C] = [1A 1s]
(Electric charge is a conserved property of some subatomic
particles, which determines their electromagnetic
interactions. Electrically charged matter is influenced by, and
produces, electromagnetic fields. The interaction between
charge and field is the source of one of the four fundamental
forces, the electromagnetic force.)
7
Law of conservation of net charge
The amount of charge produced in any process is zero
Or in other words:
Charge is just there. You cannot create nor destroy it but you can
separate or “transfer” it.
Mechanism:
Caused by rubbing the amber rod with animal fur it transfers electrons
from the fur to the rod.
Electrons are at the shell of an atom. They can transferred in most
processes rather than protons
An object is……
Positive charged: Lack of electrons
Negative charged: Surplus of electrons
If the object possesses no net charge it is said to be neutral.
An atom is normally neutral, because it possesses an equal number of
electrons and protons. However, if one or more electrons are removed
or added, an ion is formed, which is charged.
8
Conductors and Insulators
A conductor is made of material that allows electric charge to
move through it easily.
An insulator is made of material that does not allow electric
charge to move through it easily.
This is measured by the conductivity, a quantity how
freely charge can move in a material. Unit: [Ω-1 – Ohm-1]
Conductivity high: Conductor (i.e. metals)
Conductivity very low: Insulator (i.e. ceramics, plastics)
Semiconductor with an intermediate conductivity
9
Example:
Amber rod, Insulator
Metal sphere, Conductor
--
10
This body is electrically neutral.
- - +
+ +
+
+ -
An object can become polarized if the charges within it can
be separated.
+
+
Why can we pickup/attract
+
+
+
+
+
+
light weight with a zero net
+
charge?
+
That is what we call
“Polarization” of an object charged
If bring the positive charged
rod close to the surface of our
particles nuclei/protons were
repelled, electrons attracted.
neutral &
polarized
11
How to tell if you are kissing correctly …
12
Coulomb’s Law
The magnitude of the force
between two point charges is:
F=
k q1 q2
r2
where q1 and q2 are the charges and r is the separation
between the two charges.
k = 8.99 ×10 Nm /C
9
where k =
1
4πε 0
2
2
and ε 0 = 8.85 ×10 −12 C 2 /Nm 2
and ε0 is called the permittivity of free space.
13
The electric force is directed between the centers of the two
point charges.
q1
F12
F21
q2
Attractive force
between q1 and q2.
r
Repulsive force
between q1 and q2.
F12
q1
q2
F21
r
If charges at rest we call the situation “electrostatic”
Newton’s 3rd Law F12 = -F21
The electric force is an example of a long-range or field
force, just like the force of gravity.
14
There may be many
charges around
ÖSuperposition to find
the resulting force
acting on a point charge
(Principle of
Superposition)
15
Example: What is the net force on the charge q1 due to the
other two charges? q1 = +1.2 μC, q2 = -0.60 μC, and q3 =
+0.20 μC.
F12
θ
F13
The net force on q1 is Fnet = F12 + F13
16
Example continued:
The magnitudes of the forces are:
F12 =
k q1 q2
F13 =
k q1 q3
(
9 ×10
=
Nm 2 /C 2 (1.2 ×10 −6 C)(0.60 ×10 −6 C)
(1.2 m) 2 + (0.5 m) 2
9
Nm 2 /C 2 (1.2 ×10 −6 C)(0.20 ×10 −6 C)
(1.2 m) 2
r212
= 3.8 ×10 −3 N
(9 ×10
=
r312
= 1.5 ×10 −3 N
)
9
)
17
Example continued:
The components of the net force are:
Fnet , x = F13, x + F12 , x = − F13 + F12 cosθ = 2.1 × 10 N
−3
Fnet , y = F13, y + F12, y = 0 + F12 sin θ = 1.4 ×10 −3 N
1 .2 m
= 0.92
1.3 m
0 .5 m
sin θ =
= 0.38
1.3 m
cos θ =
Where from the figure
18
Example continued:
The magnitude of the net force is:
Fnet = Fnet2 , x + Fnet2 , y = 2.5 × 10−3 N
The direction of the net force is:
Fnet , y
tan φ =
= 0.66
Fnet , x
φ = 33°
19
q1 q 2
Fe = k
2
r12
Electric Force
m1m2
Fg = G 2
r12
Gravitational force
The electric force…
…varies directly with the magnitude of each charge
…varies inversely with the square of the distance between charges
…is directed along the line joining the charges
…can be either attractive or repulsive depending on the signs of the
charges
Gravitational field constant
G ≈ 7 10-11 N m2 kg-2
Electric field constant k ≈ 9 109 N m2 C-2
Fe is much stronger than Fg
Fe can be attractive or repulsive; Fg is always attractive
q must be an integer number of e
20
Example: What is the ratio of the electric force and
gravitational force between a proton and an electron
separated by 5.3×10-11 m (the radius of a Hydrogen atom)?
Fe =
k q1 q2
r2
Gm1m2
Fg =
r2
q1 = q2 = e
m1 = m p = 1.67 ×10 − 27 kg
m2 = me = 9.11× 10 −31 kg
Fe k q1 q2
ke 2
The ratio is:
=
=
= 2.3 × 1039
Fg Gm1m2 Gme m p
21
The Electric Field
Michael Faraday conceived the concept of electric field
Recall :
Fg = mg
Fe = qE
Where g is the strength
of the gravitational field.
Similarly for electric forces
we can define the strength
of the electric field E.
22
E-field applet
In other words:
The force experienced by any charge is then viewed as the
interaction between the charge and the electric field.
Electric field vector is equal to the quotient of force vector a
positive point charge q0 (so called test charge) would feel
at a certain point and the test charge itself.
r
r FA
E=
q0
23
For a point charge of charge Q, the
magnitude of the force per unit charge
at a distance r (the electric field) is:
Fe k Q
E=
= 2
q
r
The electric field at a point in space is found by adding all of
the electric fields present.
E net = ∑ Ei
i
Be careful! The electric
field is a vector!
24
E points outward for a
positive point charge
E points inward for a
negative point charge
25
Example: Find the electric field at the point P.
P
x
q1 = +e
q2 = -2e
x = 0m
x = 1m
x = 2m
E is a vector. What is its direction?
Place a positive test charge at the point of interest. The
direction of the electric field at the location of the test
charge is the same as the direction of the force on the
test charge.
26
Example continued:
q1 = +e
P
x
q2 = -2e
Locate the
positive test
charge here.
P
x
q1 = +e
q2 = -2e
Direction of E due
to charge 2
Direction of E due
to charge 1
27
Example continued:
The net electric field at point P is:
E net = E1 + E 2
The magnitude of the electric field is:
Enet = E1 − E2
28
Example continued:
E1 =
k q1
E2 =
k q2
r2
r2
(
9 ×10
=
(
9 ×10
=
)
9
Nm 2 / C 2 (1.6 ×10 −19 C)
−10
=
3
.
6
×
10
N/C
2
(2 m)
9
Nm 2 / C 2 (2 *1.6 ×10 −19 C)
−9
=
2
.
9
×
10
N/C
2
(1 m)
)
−9
Enet = E1 − E2 = −2.5 ×10 N/C
The net E-field is
directed to the left>
29
Electric field lines
Electric field lines are a useful way to indicate what the
magnitude and direction of an electric field is in space.
Rules:
Field lines…
- point always in the direction of E (E is always tangential)
- start at positive charges or at infinity
- end at negative charges or at infinity
- are more dense when E has a greater magnitude
- the number of field lines entering or leaving a charge is
proportional to the magnitude of the charge
- field lines always end/start perpendicular to the surface
- never cross or touch each other
30
Pictorial representation of the rules on the previous slide:
31
32
Charged Plate
Parallel plate capacitor
Field lines parallel to each other
and perpendicular to the surface
⇒E=
Q
2ε 0 A
Two parallel conducting
plates with opposite charge,
separated by a distance d.
The electric field is uniform
between the plates (except at
the edges).
Q
⇒E=
ε0 A
33
Motion of a Point Charge in a
Uniform E-Field
A region of space with a uniform
electric field containing a particle
of charge q (q>0) and mass m.
34
FBD for the
charge q
y
Fe
x
Apply Newton’s 2nd Law
and solve for the
acceleration.
∑F
x
= Fe = ma
Fe = qE = ma
q
a= E
m
One could now use the kinematic equations to solve for
distance traveled in a time interval, the velocity at the end of
a time interval, etc.
35
Example: What electric field strength is needed to keep an
electron suspended in the air?
y
FBD for the
electron:
Fe
x
w
To get an upward force on the electron, the electric field
must be directed toward the Earth.
36
Example continued:
Apply Newton’s 2nd Law:
∑F
y
= Fe − w = 0
Fe = w
qE = eE = mg
mg
E=
= 5.6 ×10 −11 N/C
e
37
Example: A horizontal beam of electrons moving 4.0×107 m/s
is deflected vertically by the vertical electric field between two
oppositely charged parallel plates. The magnitude of the field
is 2.00×104 N/C.
(a) What is the direction of the field between the plates?
From the top plate to the bottom plate
38
Example continued:
(b) What is the charge per unit area on the plates?
Q
σ
E=
=
ε0 A ε0
This is the electric field
between two charged plates.
Note that E here is independent of the distance from
the plates!
σ = Eε 0 = (2.00 ×10 4 N/C )(8.85 ×10 −12 C 2 /Nm 2 )
= 1.77 ×10 −7 C/m 2
39
Example continued:
(c) What is the vertical deflection d of the electrons as they
leave the plates?
y
FBD for an electron
in the beam:
Fe
x
w
Apply Newton’s 2nd Law and solve for the acceleration:
∑F
y
= Fe − w = ma y
Fe − w Fe
qE
ay =
=
−g =
− g = 3.52 × 1015 − 9.8 m/s
m
m
m
(
)
40
Example continued:
What is the vertical position of the electron after it travels a
horizontal distance of 2.0 cm?
1 02
x = x0 + vox t + a x t
2
t=
x − x0
0.02 m
−10
=
=
5
.
0
×
10
sec
7
4.0 × 10 m/s
v0 x
Time interval to
travel 2.00 cm
horizontally
0
1 2
y = y0 + voy t + a y t
2
y − y0 = d =
1 2
a y t = 4.4 × 10 − 4 m
2
Deflection of an
electron in the
beam
41
Conductors in Electrostatic
Equilibrium
Conductors are easily polarized. These materials have free
electrons that are free to move around inside the material.
Any charges that are placed on a conductor will arrange
themselves in a stable distribution. This stable situation is
called electrostatic equilibrium.
When a conductor is in electrostatic equilibrium, the E-field
inside it is zero. => Electrostatic Shielding
Any net charge must reside on the surface of a conductor
in electrostatic equilibrium.
42
Just outside the surface of a conductor in electrostatic
equilibrium the electric field must be perpendicular to the
surface.
If this were not true, then any surface
charge would have a net force acting
on it, and the conductor would not be
in electrostatic equilibrium.
Fe
43
Any excess charge on the
surface of a conductor will
accumulate where the
surface is highly curved
(i.e. a sharp point).
44
Gauss’s Law
Enclose a point
charge +Q with an
imaginary sphere.
+Q
Here, E-Field lines exit the sphere.
45
Look at a small patch
of the surface of the
imaginary sphere.
With a positive charge
inside the sphere you
would see electric field
lines leaving the surface.
46
number of field lines
Recall that E ∝
A
so that the
number of field lines ∝ EA
It is only the component of the electric field that is
perpendicular to the surface that exits the surface.
E
θ
Surface
47
Define a quantity called flux, which is related to the number
of field lines that cross a surface:
flux = Φ e = E⊥ A = (E cos θ )A
E
This picture defines
the value of θ.
θ
Flux > 0 when field lines exit the surface and flux < 0
when field lines enter the surface.
48
Φ=EA
Φ=0
Φ = E A cosθ
49
Example: Find the electric flux through each side of a cube of
edge length a in a uniform electric field of magnitude E.
A cube has six sides: The field lines enter one face and exit
through another. What is the flux through each of the other
four faces?
50
Example continued:
There is zero electric flux though the other four faces.
The electric field lines never enter/exit any of them.
The flux through the left face is –EA.
The flux through the right face is +EA.
The net flux through the cube is zero.
51
Gauss’s Law
Consider an arbitrary surface (Gaussian surface)
enclosing a total charge q. The electric flux through the
surface is
Flux is positive if field lines leave an enclosed volume.
Flux is negative if field lines enter an enclosed volume.
Φ=
q
ε0
Example 1:
ε 0 = 4πk = 8.85×10 C /N ⋅ m
−12
1
2
2
ÖCharge must be enclosed by the surface, else Φ=0!
q
Φ = EA = k 2 ⋅ 4πr 2 = 4π k q
r
2
1
−12 C
ε0 =
= 8.85 ⋅ 10
Nm 2
4πk
Permitivity of free space
q
⇒ Φ = [Nm 2 / C ] Gauss Law
ε0
52