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Chapter 20
Electric Charge, Force,
and Field
Properties of Electric Charges
Benjamin Franklin
1706 – 1790
• Two types of charges exist (named by Benjamin
Franklin): positive and negative
• Like charges repel and unlike charges attract one
another
• Nature’s most basic positive charges are the protons
(held firmly in the nucleus and do not move from one
material to another)
• Nature’s most basic negative charges charge are the
electrons – an object becomes charged by gaining or
losing electrons
Properties of Electric Charges
• Electric charge is always conserved (not created,
only exchanged) in an isolated system
• Objects become charged because negative charge
(electrons) is transferred from one object to another
• Charge is quantized (a multiple of a fundamental unit
of charge, e): electrons have a charge of – e and
protons have a charge of + e
• The SI unit of charge is the Coulomb (C)
e = 1.6 x 10-19 C
Charles Coulomb
1736 – 1806
Particle Summary
Coulomb’s Law
kq1q2
rˆ
Electric force is: F12 
2
r
• Along the line joining the two point charges
• Inversely proportional to the square of the
separation distance, r, between the particles
• Proportional to the product of the magnitudes of
the charges, |q1| and |q2| on the two particles
• k = 8.9875 x 109 N m2/C2 : Coulomb constant
• Attractive if the charges are of opposite signs and
repulsive if the charges have the same signs
Electric Forces
• Electric forces are vector quantities
• Electric force on q1 is equal in magnitude and
opposite in direction to the force on q2
• Electric force is exerted by one object on another
object without physical contact between them – field
force
• The superposition principle applies: resultant force
on any one charge equals the vector sum of the
forces exerted by the other individual charges that
are present
Superposition Principle
Chapter 20
Problem 40
A charge 3q is at the origin, and a charge -2q is on the positive x-axis at x
= a. Where would you place a third charge so it would experience no net
electric force?
Electric Field
• Electric field exists in the region of space around a
charged object (source charge)
• When another charged object q0 (test charge) enters
this electric field, the field exerts an electric force F
on the test charge
• Electric field:
• SI units: N / C
F
E
q
Electric Field
• The field is produced by some charge or charge
distribution, separate from the test charge
• The existence of an electric field is a property of the
source charge; the presence of the test charge is not
necessary for the field to exist
• The test charge serves as a detector of the field
F
E
q
Direction of Electric Field
• The direction of the vector of electric field is
defined as the direction of the electric force
that would be exerted on a small positive test
charge placed at that point
• The electric field produced by a negative
charge is directed toward the charge
(attraction)
• The electric field produced by a positive
charge is directed away from the charge
(repulsion)
Relationship Between F and E
F  qE.
• If q is positive, the force and the field are in the same
direction; if q is negative, the force and the field are
in opposite directions
• Coulomb’s law, between the source and test point
qqo
charges, can be expressed as
Fe  ke
• Then
Fe
q
E
 ke 2 rˆ
qo
r
r
2
rˆ
Superposition of Electric Fields
• At any point P, the total electric field due
to a group of source charges equals the
vector sum of the electric fields of all the
charges
kqi
E   Ei   2 rˆi
ri
Continuous Charge Distribution
• Charge ultimately resides on
individual particles, so that the
distances between charges in a
group of charges may be much
smaller than the distance
between the group and a point
of interest
• In this situation, the system of
charges can be modeled as
continuously distributed along
some line, over some surface,
or throughout some volume
Continuous Charge Distribution
• Divide the charge distribution into
small elements, each of which
contains Δq
• Calculate the electric field due to
one of these elements at point P
q
E  ke 2 rˆ
r
Continuous Charge Distribution
• Evaluate the total field by summing
the contributions of all the charge
elements
q
E  ke 2 rˆ
r
qi
dq
E  ke lim  2 rˆi  ke  2 rˆ
qi 0
ri
r
i
Charge Densities
• Volume charge density: when a charge is distributed
throughout a volume:
dq = ρ dV; [ρ] = [Q ] / [V] with units C/m3
• Surface charge density: when a charge is distributed
over a surface area:
dq = σ dA; [ σ ] = [ Q ] / [ A ] with units C/m2
• Linear charge density: when a charge is distributed
along a line:
dq = λ dℓ; [ λ ] = [ Q ] / [ ℓ ] with units C/m
Charge Densities
2k 
E
y
• Linear charge density: when a charge is distributed
along a line:
dq = λ dℓ; [ λ ] = [ Q ] / [ ℓ ] with units C/m
Problem-Solving Strategy
• Categorize (individual charge? group of individual
charges? continuous distribution of charges?) and take
advantage of any symmetry to simplify calculations
• For a group of individual charges: use the superposition
principle, find the fields due to the individual charges at
the point of interest and then add them as vectors to find
the resultant field
• For a continuous charge distribution: a) the vector sums
for evaluating the total electric field at some point must
be replaced with vector integrals; b) divide the charge
distribution into infinitesimal pieces, calculate the vector
sum by integrating over the entire charge distribution
Chapter 20
Problem 48
A 1.0-µC charge and a charge 2.0-µC are 10 cm apart. Find a point where
the electric field is zero.
Electric Field of a Uniform Ring of
Charge (Example 20.6)
dq
dq
dE x  k e 2 cos   k e 2
cos 
2
r
x a
dq
x
dq x
 ke 2
 ke 2
2
2
x a r
x a
x2  a2
ke xdq
ke x
Ex  

dq
3
/
2
3
/
2

x2  a2
x2  a2
ke xQ

2
2 3/ 2
x a






Electric Field of a Uniformly Charged
Disk (Problem 73)
• The ring has a radius R and a
uniform charge density σ
• Choose dq as a ring of
radius r
• The ring has a surface area
2πr dr
Electric Field of a Uniformly Charged
Disk (Problem 73)
dq  dA   (2rdr )
dE x 
R
Ex  
x
ke xdq
2
r

2 3/ 2
2kexrdr

d r 
 k x 
x  r 
0
x
2
r
R
e
x
2
r
2 3/ 2

2 3/ 2
R
 kex 
2
2
0
2 3/ 2

2kexrdr
0
x
 kex
2rdr
2
r

2 3/ 2
R
2
x r
2
2
0


x
 2ke 1 
2
2
x R 

Motion of Charged Particles
• When a charged particle is placed in an electric field,
it experiences an electrical force
• If this is the only force on the particle, it must be the
net force
• The net force will cause the particle to accelerate
according to Newton’s second law
Fe  qE  ma
• If the field is uniform, then the acceleration is
constant
• If the particle has a positive (negative) charge, its
acceleration is in the direction of (opposite) the field
Particle Summary
Electric Dipole
• An electric dipole consists of two
charges of equal magnitude and
opposite signs separated by 2a
• The electric dipole moment p is directed
along the line joining the charges from
–q to +q and has a magnitude of p ≡ 2aq
• Assume the dipole is placed in a uniform
field, external to the dipole (it is not the
field produced by the dipole) and makes
an angle θ with the field
• Each charge has a force of F = Eq acting
on it
Electric Dipole
• The net force on the dipole is zero
• The forces produce a net torque on the
dipole:
t  2Eqa sin θ  pE sin θ
• The torque can also be expressed as the
cross product of the moment and the
field:
t  p E
Electric Dipole
 ke  q   a  ke q  a 
ˆ
ˆ
E
i

i




2
2
r
r  r
r
r
y2  a2

E
y
2ke qa
2
 a2
ˆ
i
3/ 2


2ke qa ˆ
E   3 i ,  y  a 
y
Classification of Substances vs. Their
Ability to Conduct Electric Charge
• Conductors are materials in which the electric
charges move freely in response to an electric force
(e.g., copper, aluminum, silver, etc.)
• When a conductor is charged in a small region, the
charge readily distributes itself over the entire
surface of the material
• Insulators (dielectrics) are materials in which electric
charges do not move freely (e.g., glass, rubber, etc.)
• When insulators are charged (by rubbing), only the
rubbed area becomes charged (no tendency for the
charge to move into other regions of the material)
Classification of Substances vs. Their
Ability to Conduct Electric Charge
• Semiconductors – their characteristics are between
those of insulators and conductors (e.g., silicon,
germanium , etc.)
An Atomic Description of Dielectrics
• Molecules are said to be polarized when a
separation exists between the average
position of the negative charges and the
average position of the positive charges
• Polar molecules are those in which this
condition is always present
• Molecules without a permanent
polarization are called nonpolar molecules
• The average positions of the positive and
negative charges act as point charges,
thus polar molecules can be modeled as
electric dipoles
An Atomic Description of Dielectrics
• A linear symmetric molecule has no permanent
polarization (a)
• Polarization can be induced by placing the molecule in
an electric field (b)
• Induced polarization is the effect that predominates in
most materials used as dielectrics in capacitors
An Atomic Description of Dielectrics
• In the absence of an electric field the molecules that
make up the dielectric (modeled as dipoles) are
randomly oriented
• An external electric field produces a torque on the
molecules partially aligning them with the electric field;
alignment of dipoles reduces the electric field
Answers to Even Numbered Problems
Chapter 20:
Problem 14
1.6 × 1020
Answers to Even Numbered Problems
Chapter 20:
Problem 28
5.2 × 1011 N/C
Answers to Even Numbered Problems
Chapter 20:
Problem 44
(1.6 iˆ - 0.33 jˆ) N
or 1.7 N at an angle of − 11°